Research ArticleBlock-Extraction and Haar Transform Based LinearSingularity Representation for Image Enhancement

Yingkun Hou 1 Xiaobo Qu 2 Guanghai Liu 3

Seong-Whan Lee 4 and Dinggang Shen 45

1School of Information Science and Technology Taishan University Taian 271000 China2Department of Electronic Science Fujian Provincial Key Laboratory of Plasma and Magnetic ResonanceState Key Laboratory of Physical Chemistry of Solid Surfaces Xiamen University Xiamen 361005 China3College of Computer Science and Information Technology Guangxi Normal University Guilin 541004 China4Department of Brain and Cognitive Engineering Korea University Seoul 02841 Republic of Korea5Department of Radiology and Biomedical Research Imaging Center University of North Carolina at Chapel HillChapel Hill NC 27599 USA

Correspondence should be addressed to Yingkun Hou ykhoutsueducn and Dinggang Shen dgshenmeduncedu

Received 16 May 2019 Accepted 11 July 2019 Published 6 August 2019

Academic Editor Erik Cuevas

Copyright copy 2019 Yingkun Hou et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we develop a novel linear singularity representation method using spatial K-neighbor block-extraction and Haartransform (BEH) Block-extraction provides a group of image blocks with similar (generally smooth) backgrounds but differentimage edge locations An interblock Haar transform is then used to represent these differences thus achieving a linear singularityrepresentation Next we magnify the weak detailed coefficients of BEH to allow for image enhancement Experimental resultsshow that the proposed method achieves better image enhancement compared to block-matching and 3D filtering (BM3D)nonsubsampled contourlet transform (NSCT) and guided image filtering

1 Introduction

Image enhancement plays an important role in image pro-cessing and pattern recognition Image enhancement tech-niques can be generally divided into two categories (1) spatialmethods and (2) frequency domain methods Because fre-quency domain-based methods can represent image detailsin high-frequency subbands by certain transformations theimage enhancement can be achieved by magnifying the weakdetailed transform coefficients

The contours and textures provide the most importantinformation in most natural images and usually presentlinear singularities Because a classical orthogonal wavelettransform can only effectively represent point singularities [12] to better represent linear singularities a series of beyondwavelets (ie Ridgelet [3 4] Curvelet [5] Contourlet [6] andNSCT [7]) have been developed However either orthogonalwavelets or beyond wavelets will always directly convolute an

image by a certain convolution kernel Due to the convolutionoperation in filtering artifacts are inevitably introduced afterthe inverse transform In addition to better represent linearsingularities more and larger support directional filteringbanks have to be used at the cost of increasing computationalcomplexity

The convolution operation in traditional transforms isoften performed between a kernel and a local neighborhoodof the imageTherefore these methods are called local meth-ods In recent years some nonlocal image processing meth-ods have been developed for image denoising ie nonlocal-means (NL means) [8] and block-matching and 3D filtering(BM3D) [9ndash15] These nonlocal methods attempt to findsome of the most similar image blocks ie by implementingblock-matching and the weighted means on the blocks [8]or by implementing a 3D transform on a 3D array stacked bysimilar blocks for enhanced sparsity representation and thenfor hard-threshold shrinkage on the transformed coefficients

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 6395147 14 pageshttpsdoiorg10115520196395147

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to achieve image denoising [9 10 16ndash19] Because NL-meansand BM3D were both initially developed to achieve imagedenoising the sufficient similar image blocks are crucial fordenoising However overly strong similarities will weakenthe representation of differences among blocks in the 3Dtransformation or equivalently the linear singularity in eachblock This is problematic for applications such as imageenhancement

Some new methods of image enhancement have beenproposed recently For example He et al [20 21] pro-posed a novel guided image filtering method for imageenhancement Wang et al [22] proposed a color face imageenhancement method using adaptive singular value decom-position in Fourier domain for face recognition Gao et al[23] proposed an image enhancement method specificallyfor visual impairments Li et al [24] proposed an adap-tive fractional calculus of small probability strategy-basedmethod for achieving image denoising and enhancementThese methods usually enhance the background or havea halo phenomenon as well when used to enhance imagedetails

In this paper we propose a novel linear singularity repre-sentation method that avoids finding sufficient similar imageblocks The proposed method is based on an observation ofnatural images that depict many image blocks (or patches)with similar smooth backgrounds in a small neighborhoodSuppose that the detail (ie textures or contours) locationsscarcely differ between two isometric image blocks that areboth extracted from the same small local neighborhoodsimply by subtracting one block by one block The imagedetails can then be effectively represented for example thedetails can be preserved only on relatively greater magni-tudes by setting all other magnitudes approximately equalto zero Figure 1 illustrates this situation Inspired by thisfact we propose a novel linear singularity representationmethod called K-neighbor block-extraction and Haar trans-form (BEH) The Haar transform is chosen because ofits low computational complexity but also because of itsability to represent sudden transitional signals Although itlacks continuity and differentiability this property is actuallyadvantageous for analyzing signals with sudden transitions[25]

In the proposed method we first select an image blockusing a sliding window and then extract its spatial K-neighbor image blocks All of the blocks extracted usingthis approach will have similar smooth backgrounds butwith subtle differences in their detail locations Next weimplement a fast Haar transform by calculating either theweighted summation or subtraction among these blocksThus the linear singularity can be effectively representedTo verify the effectiveness of our proposed method weapply it to image enhancement in order to improve thevisibility of images which is crucial for image processing andcomputer vision [26ndash31] Experimental results demonstratethat the proposed method achieves performance in imageenhancement that is superior to that of the existing state-of-the-art linear singularity representation methods includingnonsubsample contourlet transform (NSCT)

Figure 1 An illustration using the proposed linear singularityrepresentation method

2 NSCT and BM3D for Linear SingularityRepresentation in Image Enhancement

21 Linear Singularity Representation Because orthogonalwavelets can only effectively represent point singularitiesa series of beyond wavelets has been developed for linearsingularity representation Among them the NSCT has somefavorable properties such as translation invariability andmultidirectional filtering which makes it one of the state-of-the-art linear singularity representation methods AlthoughNSCT achieves the best linear singularity representationperformance it has drawbacks similar to those of otherlocal transform methods ie it always introduces someartifacts after a certain type of operation on the transformedcoefficients and inverse transform The introduction of arti-facts comes from the convolution operation of the inversetransform

119891 (119909 119910) = ∬1198772119862 (119886 119887) Ψminus1 (119886 119887) 119889119886 119889119887 (1)

where 119862 is the transformed coefficients Ψminus1 is the inversetransform filtering bank and 119891(119909 119910) is the reconstructedimage For image denoising if one implements a hardthresholding operation some of the isolated noisy pointsmayremain after the inverse transform As these noisy points haveinfluences on their surrounding pixels the ringing artifactsfor orthogonal wavelets and the strip artifacts for beyondwavelets will be introduced An illustrated example of thissituation is shown in Figure 2 From this figure we can seethat some strip artifacts are introducedwhen enhancing somecoefficients of a certain directional subband of NSCT

Among recently developed nonlocal methods BM3D isone of the state-of-the-art image denoising methods [32]Specifically exploring other nonlocal approaches to achieveideal image denoising performance resulted in the creation ofBM3D In particular BM3D does not consider linear singu-larity representation whatsoever which may cause problems

Mathematical Problems in Engineering 3

(a) (b)

(c) (d)

Figure 2 An illustration of strip artifacts byNSCT (a) A directional subband ofNSCT and some enhanced coefficients (b)The reconstructedimage by the inverse NSCT (c) A fragment of the original image (d) A reconstructed fragment of (c) by the inverse NSCT

to surface with the use of the block-matching technique Inthe 3D transform only the 2D transform on each imageblock can be seen as a singularity representation Howeverthis singularity representation degenerates the general 2Dwavelet transform method because the third-dimensionaltransform is on considerably similar image blocks whichimplies that singularity cannot be represented by the Haartransform In the extreme case if two image blocks arecompletely alike the subtraction between the two blocks willequal zero and there will be no information in the high-frequency subbands of the Haar transform There are alwayssome differences among image blocks in a group formedby block-matching even though the ideal linear singularityrepresentation cannot be achieved So to achieve betterlinear singularity representation performance some localdifferences must exist among a group of blocks Consideringthis we propose to modify the block-matching in the BM3Dmethod to be block-extraction

In the BM3D method the block-matching is imple-mented by computing the Euclidean distances between agiven reference block and each block in the neighborhood ofthe reference block

119863 = 10038171003817100381710038171198792119863 (119885119877) minus 1198792119863 (119885)10038171003817100381710038172211987321 (2)

where119885119877 is the reference block119885 is the block to be matchedand 1198792119863 is a 2D transform ie either DCT or an orthogonalwavelet transform However this 2D transform does changethe block-matching results corresponding to the case of usingthe original image blocks to calculate distance in block-matching1198731 is the size of each square image block Because1198792119863 does not affect block-matching we simplify (2) to be thefollowing one

119863 = 1003817100381710038171003817119885119877 minus 11988510038171003817100381710038172211987321 (3)

Here we can show a block-matching example by (3) andgive the 3D transform results of the grouped image blocksin Figure 3 From the transformed results in Figure 3 wecan see that there is scarce information in the high-frequencysubbands The main reason is that all of the blocks are toosimilar in a group that is there is not enough singularityamong these blocks

22 Application to Image Enhancement

221 Image Enhancement by NSCT NSCT has excellentlinear singularity representation performance which allowssome relatively weak edges in images to be better representedby NSCT than by BM3D Therefore when using NSCT to

4 Mathematical Problems in Engineering

(b)

(a) (c)

Figure 3 Demonstration of block-matching and 3D filtering (a)-(b) The result of block-matching in a neighborhood of House image (c)The result of 3D transform on the 3D array formed by all of the blocks in (b)

achieve better image enhancement strong edges and weakedges should be processed differently In [7] to achieve betterimage enhancement results the definition below is used todifferentiate among strong edge weak edge and noise119904119905119903119900119899119892 119890119889119892119890 119894119891 119898119890119886119899 ge 119888120590119908119890119886119896 119890119889119892119890 119894119891 119898119890119886119899 lt 119888120590 119898119886119909 ge 119888120590119899119900119894119904119890 119894119891 119898119890119886119899 lt 119888120590 119898119886119909 lt 119888120590 (4)

where 119888 is a parameter ranging from 1 to 5 and 120590 is thestandard deviation of noise in the subbands at a specific pyra-midal level In this case a NSCT-based image enhancementalgorithm can be given as follows

119910 (119909) = 119909 119904119905119903119900119899119892 119890119889119892119890 119901119894119909119890119897119904max(( 119888120590|119909|)119901 1) 119909 119908119890119886119896 119890119889119892119890 1199011198941199091198901198971199040 119899119900119894119904119890 119901119894119909119890119897119904

(5)

where the input 119909 is the original transform coefficient and0 lt 119901 lt 1 is the amplifying gain This function keeps thecoefficients of strong edges amplifies the coefficients of weakedges and zeros the noise coefficients

222 Image Enhancement by BM3D According to the anal-ysis in Section 21 BM3D cannot obtain the linear sin-gularity representation like NSCT Therefore BM3D can-not use the same algorithm as NSCT to achieve imageenhancement In [33] a joint image enhancement anddenoising algorithm by 3D transform-domain collaborativefiltering was proposed In this algorithm block-matchingand 3D transform were implemented Additionally thehard thresholding operation on the 3D transformed coef-ficients was used to remove noise and the alpha-rooting

method was used to amplify coefficients and achieve imageenhancement

Given a transform spectrum 119905 of a signal which containsa DC coefficient termed 119905(0) the alpha-rooting is performedas

119905119904ℎ (119894)=

sign [119905 (119894)] ∙ |119905 (0)| ∙ (10038161003816100381610038161003816100381610038161003816 119905 (119894)119905 (0) 10038161003816100381610038161003816100381610038161003816)1120572 119894119891 119905 (0) = 0119905 (119894) 119900119905ℎ119890119903119908119894119904119890(6)

where 119905119904ℎ is the spectrum of the transformed signal andan exponent 120572 gt 1 leads to the enhancement of imagedetails

In [33] two different approaches are given to per-form image enhancement BM3D-SH3D and BM3D-SH2DThe first one implements the alpha-rooting algorithm onthe 3D transformed spectrum while the second imple-ments the alpha-rooting algorithm only on the 2D trans-formed spectrum after the 1D inverse transform ieon each block-transformed spectrum However actuallyboth methods achieve image enhancement only via oneblock in a certain sense because of extreme similarityamong the blocks in a group the magnitudes of thecoefficients in the high-frequency subbands are all nearlyzero

Accordingly to achieve better enhancement on imagedetails we propose using a K-neighbor block-extractionmethod to replace the block-matching procedure inBM3D and further discard the 2D transform on eachimage block by only implementing an interblock 1D Haartransform

Mathematical Problems in Engineering 5

3 K-Neighbor Block-Extractionand Haar Transform

Considering the effectiveness of the proposed linear sin-gularity representation method the extracted image blocksshould have similar smooth backgrounds as well as somedifferent local details We first select an image block from theinput image and then extract its spatial K-neighbor blocksThus the extracted blocks can follow the above-mentionedcondition ie all the extracted image blocks have similarsmooth backgrounds but different local details After a fastHaar transform on the group of these image blocks theimage details can be effectively represented in the detailssubbands

31 K-Neighbor Block-Extraction Areference image block119861119877with the top-left pixel coordinate 119909119877 isin 119883 (where 119883 sub Z2 isthe coordinate set of the input image 119868) is first selected byan 119899 times 119899 sliding window according to a given sliding stepsize 119899119904119905119890119901 and then its spatial K-neighbor image blocks areextracted to form a vector 119861119864 = 1198611 sdot sdot sdot 119861119894 sdot sdot sdot 119861119870119879 whereevery block 119861119894 can be considered as an element of 119861119864 andtheir top-left pixel coordinates 119909119894 form a set 119878119909119877 = 119909119894 isin 119883 119861119894 is extracted according to 119861119877

Because we want to implement a Haar transform onvector 119861119864 119870 must be a power-of-two integer In addition torepresent more directional details119870 should at least be 8 Forexample we can extract 8 image blocks and then form all oftheir top-left pixel coordinates as the 8-neighborhood top-left pixel coordinates of 119861119877 A block-extraction operation isillustrated in Figure 4 The pink block is an 8times8 referenceblock 119861119877 with the blue solid round as its top-left pixelcoordinate The green solid rounds are the top-left pixelcoordinates of the extracted119870 = 8 image blocks while the redsolid rounds are the top-left pixel coordinates of the extracted119870 = 16 image blocks

32 Haar Transform In this section we give a fast Haartransform method according to the characteristics of block-extraction Because all extracted blocks are isometric we canfully investigate the simplicity of Haar wavelet to construct afast Haar transform on each group of blocks

Forward Transform Here119870 = 2119895 (119895 = 3 4 sdot sdot sdot ) image blocksare denoted as 119861119894 119894 = 1 2 3 sdot sdot sdot 119870 For example if119870 = 8 wecan use the following formulation to realize a complete Haartransform with 3 levels of transform

119861119894 = sum119895=1sdotsdotsdot 119870

Ψ (119894 119895) 119861119895 119894 = 1 sdot sdot sdot 119870 (7)

whereΨ is aHaar transformmatrix119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879is a column vector in which every element denotes an imageblock and 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 is also a column vectorin which every element denotes a transformed subband An8 times 8Ψ of Haar transform can be defined as follows

Ψ =

((((((((((((((((((((

1radic8 1radic8 1radic8 1radic8 1radic8 1radic8 1radic8 1radic81radic8 1radic8 1radic8 1radic8 minus1radic8 minus1radic8 minus1radic8 minus1radic81radic4 1radic4 minus1radic4 minus1radic4 0 0 0 00 0 0 0 1radic4 1radic4 minus1radic4 minus1radic41radic2 minus1radic2 0 0 0 0 0 00 0 1radic2 minus1radic2 0 0 0 00 0 0 0 1radic2 minus1radic2 0 00 0 0 0 0 0 1radic2 minus1radic2

))))))))))))))))))))

(8)

By computing the matrix product 119861119864 can be decomposedinto 8 subbands with 1198611 as the approximated subband andthe rest as detailed subbands Figure 5 shows a way ofdecomposing a group of image blocks with a Haar transformNote that the contours can be effectively represented in thedetailed subbands

Inverse Transform The Haar transform matrix defined in (2)is invertible Thus one can perfectly reconstruct all originalimage blocks using the following inverse transform119861119894 = sum

119895=1sdotsdotsdot 119870

Ψminus1 (119894 119895) 119861119895 119894 = 1 sdot sdot sdot 119870 (9)

where Ψminus1 denotes the inverse matrix of ΨAfter finishing operations on a group of image blocks we

can return them to their original locations in a zero matrixof size 119872 times 119873 by averaging all pixels in the same locationBy finishing operations on all reference blocks we can get theoutput image 119868 by the following aggregation equation

119868 = sum119909119877isin119883sum119909119894isin119878119909119877 119861119894sum119909119877isin119883sum119909119894isin119878119909119877 120594119894 (10)

where120594119894 is the characteristic function of the square support ofa block located at 119909119894 and all of the image blocks are outside-padded by zeros to form an 119872 times 119873 image Because theHaar transform is perfect in reconstruction transform wealways calculate the average of all pixels extracted in the samelocation in the aggregation procedureNamely the pixel valuealways returns to the original value at every location so aperfect reconstructed image can be obtained after conductingall of the above operations Figure 6 shows a Lena image andits reconstructed image with PSNR of 313784 Application to Image Enhancement

We apply the proposed linear singularity representationmethod to image enhancement to verify its effectiveness incapturing image details among a group of blocks Ampli-fying certain transformed coefficients can typically achieveimage enhancement ie amplifying only the transformation

6 Mathematical Problems in Engineering

Figure 4 An illustration of the image block-extraction method

coefficients of image details but suppressing transformationcoefficients of background noise [7 34 35] If image detailscannot be well represented the transformation coefficients ofbackground noises may also be amplified when amplifyingtransformation coefficients In addition when implementingthe traditional convolution-based transform image detailsand their surrounding smooth background can be influencedby each other because the transformation coefficients of thebackground near the image details are usually larger thanthose that are far away Thus the halo phenomenon wouldbe introduced after image enhancement However due to theeffectiveness of our proposed linear singularity representa-tion it can amplify the transformation coefficients of imagedetails and also suppress the transformation coefficients ofbackground and noise simultaneously Most importantlywithout using large filter banks the halo phenomenon canbe effectively alleviated

For the purpose of suppressing background noise we firstestimate the noise deviation 120590 of the input image with therobust median operator [36]

As in the analysis in Section 32 we implement theforward Haar transform on 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879according to (7) to obtain the transformation coefficients119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 Then we can use the followingnonlinear mapping function to amplify the transformationcoefficients of image details

119862119890119899 = 119862119905 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 gt 119862120590120574119862119905 119894119891 119888120590 le 10038161003816100381610038161198621199051003816100381610038161003816 le 1198621205900 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 lt 119888120590

(11)

where 119862119905 is the Haar-transformed coefficients 120574 is the gainfactor to amplify the transformation coefficients of imagedetails 119862 and 119888 are the two constant parameters and 119862119890119899 isthe enhanced coefficient Next we can implement the inverse

Haar transform to get 119861119864 = (1198611 1198612 1198613 sdot sdot sdot 119861119870)119879 by (9)Finally we can use (10) to obtain the enhanced image

The image enhancement algorithm by BEH can be sum-marized as follows

(1) Estimate the noise deviation of the input image(2) Extract 119870 image blocks by the method in Section 21(3) Implement Haar transform according to (7)(4) Amplify the transformation coefficients of image

details and also suppress the transformation coeffi-cients of background noise by (11)

(5) Implement the inverse Haar transform according to(9)

(6) Aggregate all of the blocks to obtain the finalenhanced image

The proposed BEH method has two drawbacks for imagedenoising (1) the extracted image blocks are not similarenough to each other and (2) this algorithm does notimplement the 2D transform on each block These twodrawbacks limit separation of noise from signal Fortunatelybecause the BM3D method can achieve outstanding denois-ing performance we can use BM3D to denoise a noisy imageand then use the proposed method to enhance the detailsof the denoised image This overall method is called BEH-BM3D

It is worth noting that there are two steps in BM3Dmethod for achieving better image denoising The first stepcan basically remove all the noise in the noisy image howeverthe image details are also oversmoothed In order to restorethe oversmoothed image details BM3D uses the second stepie Wiener filtering This two-step strategy is needed forimage denoising problem but for image enhancement theWiener filtering step is not suitable since the enhanced imagewould be restored to the original one if applying Wienerfiltering on the enhanced image We only implement one-step block-extractionHaar transform in ourmethod thus thecomputational complex can also be lowered than the BM3Dmethod

5 Experimental Results

In this section we will provide two groups of experimentsOne is to implement the proposed linear singularity rep-resentation method and the other is to implement imageenhancement

51 Linear Singularity Representation To demonstrate theproposed linear singularity representation method the Bar-bara image with a size of 512 times 512 is decomposed via theproposed KN-BEH The parameter values used are 119899119904119905119890119901 = 3119899 = 8 and 119870 = 8 Figure 7 shows all the transformedsubbands of the proposed method Obviously image detailscan be observed in the detailed subbands This is mainlybecause the KN-BEH can easily capture differences amongthe blocks of the same group where neighboring imageblocks have very similar backgrounds

52 Image Enhancement We have conducted imageenhancement experiments to verify the effectiveness of our

Mathematical Problems in Engineering 7

Figure 5 Example of decomposing a group of 8 image blocks (upper row) with a Haar transform The upper row shows the 8 extractedblocks and the lower row shows Haar-transformed subbands

Figure 6 A perfect reconstruction example of the proposed K-neighbor block-extraction and Haar (KN-BEH) transform The leftis the input image and the right is the reconstructed image (theparameters for KN-BEH are 119899119904119905119890119901 = 3 119899 = 8 119870 = 8)proposed method When 120574 gt 1 the output image details willbe enhanced with the larger 120574 correlated to a more stronglyenhanced image In all experiments the parameters are setas 119862 = 300 and 119888 = 80 for the normalized input image iewith the pixel value ranging from 0 to 1 We just adjust 120574 toget different image enhancement results The enhancementresults on House and Barbara are shown in Figures 8 and 9

To further demonstrate the advantage of the BEHmethod we also compare our image enhancement resultswith results using the NSCT method [7] and the alpha-rooting method [33] Due to the use of multidirectionalfiltering banks and translation invariants NSCT is a powerfullinear singularity representation method Figures 8 and 9show the comparison of image enhancement results by ourproposedmethod NSCT and alpha-rootingWe observe thatthe proposedmethod hardly produces any halo phenomenonaround strong edges but theNSCTmethod results in a stronghalo phenomenon From Figures 8 and 9 we also observethat our proposed method can better preserve weak detailsthan the NSCT method ie for the details on the table legsat the bottom-left part of the Barbara image Because ourmethod can better represent linear singularity as observedfrom Figures 8 and 9 our proposed method has bettersubjective visual sharpness than the alpha-rooting methodAdditionally many small weak details in the image cannotbe sharpened using the alpha-rooting method

We have further conducted other comparison experi-ments For example we used both BEH-BM3D and alpha-rooting BM3D to enhance images with added Gaussianwhite noise Specifically we first added different levels of

noise to the image and then used these two methods toenhance the image and remove noise For BEH-BM3D weused a two-stage method with the first stage of imagedenoising by BM3D and the second stage of enhance-ment by BEH For the alpha-rooting BM3D method byexactly following the procedure in [33] we simultaneouslyenhanced the transformation coefficients of image detailsand removed the transformation coefficients of noise byhard thresholding in BM3D transform domain The exper-imental results are shown in Figures 10 and 11 From thesetwo figures we can see that the BEH-BM3D can betterpreserve image details and achieve better subjective visualresults

To quantitatively validate image enhancement perfor-mance the Local Phase Coherence Sharpness Index (LPC-SI) [37] a novel and effective image sharpness assessmentmethod is used This method considers sharpness as stronglocal phase coherence near the distinctive image featuresevaluated in the complexwavelet transformdomain Further-more this method can assess sharpness without using theoriginal image as reference In our experiments we compareour results with both the NSCT method and the alpha-rooting method In each method we provide its best LPC-SI results by adjusting the respective parameters For thealpha-rooting method we fix the alpha value to 20 and thenadjust the standard deviation of noise to obtain the best LPC-SI values For the NSCT method we use the enhancementalgorithm in [7] by adjusting the gain factor 119888 in (5) to obtainthe best LPC-SI values All of the results are shown in Table 1We can see that our method achieves the best results for mostimages

In addition we further used the traditional backgroundvariation (BV) and detail variation (DV) [35] to evaluateour results The BV and DV values represent the varianceof background and foreground pixels respectively A goodimage enhancement method should increase the DV of theoriginal image but keep or even decrease the BV The BVand DV values of three methods are summarized in Table 2Notice that all of these results were obtained for each imageand each method when the corresponding LPC-SI was thebest From these results we can see that the alpha-rootingmethod achieved the lowest BV and ourmethod achieved thehighest DV However the BV values provided by our methodwere just a little bit higher than those of the alpha-rootingmethod If we use the ratio between DV and BV to assess theobtained results our method achieves the best outcome

8 Mathematical Problems in Engineering

Figure 7 The transformed subbands of the proposed transform To clearly display the details we have increased the brightness of eachhigh-frequency subband

(a) (b) (c)

(d) (e) (f)

Figure 8 The enhancement results (a) is the original image (b) and (c) are the enhanced images by the proposed method by taking 120574 as 3and 5 respectively (d) and (e) are the enhanced images by the NSCTmethod by similarly taking 120574 as 3 and 5 respectively (f) is the enhancedimage by the alpha-rooting method

We also find that NSCT achieves the best LPC-SI valuebut the worst BVDV result on Fingerprint images in Tables1 and 2 respectively This is mainly because the estimatedstandard deviation of noise is too high which results in mostimage edges being removed as noise Therefore the highestLPC-SI value cannot always be trusted This is also the main

reason for us to use both LPC-SI and BVDV to assess theexperimental results

It is worth indicating another significant advantage of ourproposed method ie its lower time complexity comparedto that of the NSCT method For image enhancement on a256times256 grayscale image our method takes only 03 second

Mathematical Problems in Engineering 9

(a) (b)

(c) (d)

Figure 9 Comparison of image enhancement results by the proposed method NSCT and alpha-rooting (a) Input image (b) the enhancedimage by the proposed method (c) the enhanced image by NSCT and (d) the enhanced image by alpha-rooting

while theNSCTmethod takes 126 seconds Ourmethod usesthe same parameter values ie 119899119904119905119890119901 = 3 119899 = 8 and 119870 = 8and the NSCT method uses 2 levels of decomposition ie 2and 4 directional subbands respectively

In the final experiment we also used the proposed BEHmethod to enhance several color images to show superiorenhancement performance and compare the results with theguided image filteringmethod [20 21] with the results shownin Figure 12 Although the image details are enhanced theenhanced images are still visually natural when using theproposed method The guided image filtering method canalso effectively enhance the image details however the imagebackground is also enhanced making the enhanced imageslook rather artificial

6 Conclusions

We have proposed an effective linear singularity representa-tionmethod and have applied thismethod to image enhance-ment in both gray and color images Similar to the BM3Dmethod our method is nonlocal However our methodextracts some spatially adjacent blocks instead of using

block-matching By using our method the image details canbe effectively represented Additionally by using differentparameters to amplify the transformation coefficients ofimage details we can obtain different image enhancementresults Because our method uses no local convolution oper-ation it introduces fewer halo phenomena compared to theNSCT method Furthermore the computational cost of ourmethod is also significantly lower compared to that of theNSCT method

Although our proposed BEHmethod looks like the origi-nal BM3Dmethod the purpose of our proposedmethod is torepresent image details which is different from the originalBM3D Note that edges in the matched image blocks areusually located in the same position thus edge informationcannot be preserved in the high-frequency subbands afterthe third-dimensional transform Therefore our proposedmethod is different from the BM3D method

In summary we have presented a simple but effectiveimage linear singularity representation method that achievesbetter objective and subjective results when applied to imageenhancement than other state-of-the-art methods

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

Mathematical Problems in Engineering 3

(a) (b)

(c) (d)

Figure 2 An illustration of strip artifacts byNSCT (a) A directional subband ofNSCT and some enhanced coefficients (b)The reconstructedimage by the inverse NSCT (c) A fragment of the original image (d) A reconstructed fragment of (c) by the inverse NSCT

to surface with the use of the block-matching technique Inthe 3D transform only the 2D transform on each imageblock can be seen as a singularity representation Howeverthis singularity representation degenerates the general 2Dwavelet transform method because the third-dimensionaltransform is on considerably similar image blocks whichimplies that singularity cannot be represented by the Haartransform In the extreme case if two image blocks arecompletely alike the subtraction between the two blocks willequal zero and there will be no information in the high-frequency subbands of the Haar transform There are alwayssome differences among image blocks in a group formedby block-matching even though the ideal linear singularityrepresentation cannot be achieved So to achieve betterlinear singularity representation performance some localdifferences must exist among a group of blocks Consideringthis we propose to modify the block-matching in the BM3Dmethod to be block-extraction

In the BM3D method the block-matching is imple-mented by computing the Euclidean distances between agiven reference block and each block in the neighborhood ofthe reference block

119863 = 10038171003817100381710038171198792119863 (119885119877) minus 1198792119863 (119885)10038171003817100381710038172211987321 (2)

where119885119877 is the reference block119885 is the block to be matchedand 1198792119863 is a 2D transform ie either DCT or an orthogonalwavelet transform However this 2D transform does changethe block-matching results corresponding to the case of usingthe original image blocks to calculate distance in block-matching1198731 is the size of each square image block Because1198792119863 does not affect block-matching we simplify (2) to be thefollowing one

119863 = 1003817100381710038171003817119885119877 minus 11988510038171003817100381710038172211987321 (3)

Here we can show a block-matching example by (3) andgive the 3D transform results of the grouped image blocksin Figure 3 From the transformed results in Figure 3 wecan see that there is scarce information in the high-frequencysubbands The main reason is that all of the blocks are toosimilar in a group that is there is not enough singularityamong these blocks

22 Application to Image Enhancement

221 Image Enhancement by NSCT NSCT has excellentlinear singularity representation performance which allowssome relatively weak edges in images to be better representedby NSCT than by BM3D Therefore when using NSCT to

4 Mathematical Problems in Engineering

(b)

(a) (c)

Figure 3 Demonstration of block-matching and 3D filtering (a)-(b) The result of block-matching in a neighborhood of House image (c)The result of 3D transform on the 3D array formed by all of the blocks in (b)

achieve better image enhancement strong edges and weakedges should be processed differently In [7] to achieve betterimage enhancement results the definition below is used todifferentiate among strong edge weak edge and noise119904119905119903119900119899119892 119890119889119892119890 119894119891 119898119890119886119899 ge 119888120590119908119890119886119896 119890119889119892119890 119894119891 119898119890119886119899 lt 119888120590 119898119886119909 ge 119888120590119899119900119894119904119890 119894119891 119898119890119886119899 lt 119888120590 119898119886119909 lt 119888120590 (4)

where 119888 is a parameter ranging from 1 to 5 and 120590 is thestandard deviation of noise in the subbands at a specific pyra-midal level In this case a NSCT-based image enhancementalgorithm can be given as follows

119910 (119909) = 119909 119904119905119903119900119899119892 119890119889119892119890 119901119894119909119890119897119904max(( 119888120590|119909|)119901 1) 119909 119908119890119886119896 119890119889119892119890 1199011198941199091198901198971199040 119899119900119894119904119890 119901119894119909119890119897119904

(5)

where the input 119909 is the original transform coefficient and0 lt 119901 lt 1 is the amplifying gain This function keeps thecoefficients of strong edges amplifies the coefficients of weakedges and zeros the noise coefficients

222 Image Enhancement by BM3D According to the anal-ysis in Section 21 BM3D cannot obtain the linear sin-gularity representation like NSCT Therefore BM3D can-not use the same algorithm as NSCT to achieve imageenhancement In [33] a joint image enhancement anddenoising algorithm by 3D transform-domain collaborativefiltering was proposed In this algorithm block-matchingand 3D transform were implemented Additionally thehard thresholding operation on the 3D transformed coef-ficients was used to remove noise and the alpha-rooting

method was used to amplify coefficients and achieve imageenhancement

Given a transform spectrum 119905 of a signal which containsa DC coefficient termed 119905(0) the alpha-rooting is performedas

119905119904ℎ (119894)=

sign [119905 (119894)] ∙ |119905 (0)| ∙ (10038161003816100381610038161003816100381610038161003816 119905 (119894)119905 (0) 10038161003816100381610038161003816100381610038161003816)1120572 119894119891 119905 (0) = 0119905 (119894) 119900119905ℎ119890119903119908119894119904119890(6)

where 119905119904ℎ is the spectrum of the transformed signal andan exponent 120572 gt 1 leads to the enhancement of imagedetails

In [33] two different approaches are given to per-form image enhancement BM3D-SH3D and BM3D-SH2DThe first one implements the alpha-rooting algorithm onthe 3D transformed spectrum while the second imple-ments the alpha-rooting algorithm only on the 2D trans-formed spectrum after the 1D inverse transform ieon each block-transformed spectrum However actuallyboth methods achieve image enhancement only via oneblock in a certain sense because of extreme similarityamong the blocks in a group the magnitudes of thecoefficients in the high-frequency subbands are all nearlyzero

Accordingly to achieve better enhancement on imagedetails we propose using a K-neighbor block-extractionmethod to replace the block-matching procedure inBM3D and further discard the 2D transform on eachimage block by only implementing an interblock 1D Haartransform

Mathematical Problems in Engineering 5

3 K-Neighbor Block-Extractionand Haar Transform

Considering the effectiveness of the proposed linear sin-gularity representation method the extracted image blocksshould have similar smooth backgrounds as well as somedifferent local details We first select an image block from theinput image and then extract its spatial K-neighbor blocksThus the extracted blocks can follow the above-mentionedcondition ie all the extracted image blocks have similarsmooth backgrounds but different local details After a fastHaar transform on the group of these image blocks theimage details can be effectively represented in the detailssubbands

31 K-Neighbor Block-Extraction Areference image block119861119877with the top-left pixel coordinate 119909119877 isin 119883 (where 119883 sub Z2 isthe coordinate set of the input image 119868) is first selected byan 119899 times 119899 sliding window according to a given sliding stepsize 119899119904119905119890119901 and then its spatial K-neighbor image blocks areextracted to form a vector 119861119864 = 1198611 sdot sdot sdot 119861119894 sdot sdot sdot 119861119870119879 whereevery block 119861119894 can be considered as an element of 119861119864 andtheir top-left pixel coordinates 119909119894 form a set 119878119909119877 = 119909119894 isin 119883 119861119894 is extracted according to 119861119877

Because we want to implement a Haar transform onvector 119861119864 119870 must be a power-of-two integer In addition torepresent more directional details119870 should at least be 8 Forexample we can extract 8 image blocks and then form all oftheir top-left pixel coordinates as the 8-neighborhood top-left pixel coordinates of 119861119877 A block-extraction operation isillustrated in Figure 4 The pink block is an 8times8 referenceblock 119861119877 with the blue solid round as its top-left pixelcoordinate The green solid rounds are the top-left pixelcoordinates of the extracted119870 = 8 image blocks while the redsolid rounds are the top-left pixel coordinates of the extracted119870 = 16 image blocks

32 Haar Transform In this section we give a fast Haartransform method according to the characteristics of block-extraction Because all extracted blocks are isometric we canfully investigate the simplicity of Haar wavelet to construct afast Haar transform on each group of blocks

Forward Transform Here119870 = 2119895 (119895 = 3 4 sdot sdot sdot ) image blocksare denoted as 119861119894 119894 = 1 2 3 sdot sdot sdot 119870 For example if119870 = 8 wecan use the following formulation to realize a complete Haartransform with 3 levels of transform

119861119894 = sum119895=1sdotsdotsdot 119870

Ψ (119894 119895) 119861119895 119894 = 1 sdot sdot sdot 119870 (7)

whereΨ is aHaar transformmatrix119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879is a column vector in which every element denotes an imageblock and 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 is also a column vectorin which every element denotes a transformed subband An8 times 8Ψ of Haar transform can be defined as follows

Ψ =

((((((((((((((((((((

1radic8 1radic8 1radic8 1radic8 1radic8 1radic8 1radic8 1radic81radic8 1radic8 1radic8 1radic8 minus1radic8 minus1radic8 minus1radic8 minus1radic81radic4 1radic4 minus1radic4 minus1radic4 0 0 0 00 0 0 0 1radic4 1radic4 minus1radic4 minus1radic41radic2 minus1radic2 0 0 0 0 0 00 0 1radic2 minus1radic2 0 0 0 00 0 0 0 1radic2 minus1radic2 0 00 0 0 0 0 0 1radic2 minus1radic2

))))))))))))))))))))

(8)

By computing the matrix product 119861119864 can be decomposedinto 8 subbands with 1198611 as the approximated subband andthe rest as detailed subbands Figure 5 shows a way ofdecomposing a group of image blocks with a Haar transformNote that the contours can be effectively represented in thedetailed subbands

Inverse Transform The Haar transform matrix defined in (2)is invertible Thus one can perfectly reconstruct all originalimage blocks using the following inverse transform119861119894 = sum

119895=1sdotsdotsdot 119870

Ψminus1 (119894 119895) 119861119895 119894 = 1 sdot sdot sdot 119870 (9)

where Ψminus1 denotes the inverse matrix of ΨAfter finishing operations on a group of image blocks we

can return them to their original locations in a zero matrixof size 119872 times 119873 by averaging all pixels in the same locationBy finishing operations on all reference blocks we can get theoutput image 119868 by the following aggregation equation

119868 = sum119909119877isin119883sum119909119894isin119878119909119877 119861119894sum119909119877isin119883sum119909119894isin119878119909119877 120594119894 (10)

where120594119894 is the characteristic function of the square support ofa block located at 119909119894 and all of the image blocks are outside-padded by zeros to form an 119872 times 119873 image Because theHaar transform is perfect in reconstruction transform wealways calculate the average of all pixels extracted in the samelocation in the aggregation procedureNamely the pixel valuealways returns to the original value at every location so aperfect reconstructed image can be obtained after conductingall of the above operations Figure 6 shows a Lena image andits reconstructed image with PSNR of 313784 Application to Image Enhancement

We apply the proposed linear singularity representationmethod to image enhancement to verify its effectiveness incapturing image details among a group of blocks Ampli-fying certain transformed coefficients can typically achieveimage enhancement ie amplifying only the transformation

6 Mathematical Problems in Engineering

Figure 4 An illustration of the image block-extraction method

coefficients of image details but suppressing transformationcoefficients of background noise [7 34 35] If image detailscannot be well represented the transformation coefficients ofbackground noises may also be amplified when amplifyingtransformation coefficients In addition when implementingthe traditional convolution-based transform image detailsand their surrounding smooth background can be influencedby each other because the transformation coefficients of thebackground near the image details are usually larger thanthose that are far away Thus the halo phenomenon wouldbe introduced after image enhancement However due to theeffectiveness of our proposed linear singularity representa-tion it can amplify the transformation coefficients of imagedetails and also suppress the transformation coefficients ofbackground and noise simultaneously Most importantlywithout using large filter banks the halo phenomenon canbe effectively alleviated

For the purpose of suppressing background noise we firstestimate the noise deviation 120590 of the input image with therobust median operator [36]

As in the analysis in Section 32 we implement theforward Haar transform on 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879according to (7) to obtain the transformation coefficients119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 Then we can use the followingnonlinear mapping function to amplify the transformationcoefficients of image details

119862119890119899 = 119862119905 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 gt 119862120590120574119862119905 119894119891 119888120590 le 10038161003816100381610038161198621199051003816100381610038161003816 le 1198621205900 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 lt 119888120590

(11)

where 119862119905 is the Haar-transformed coefficients 120574 is the gainfactor to amplify the transformation coefficients of imagedetails 119862 and 119888 are the two constant parameters and 119862119890119899 isthe enhanced coefficient Next we can implement the inverse

Haar transform to get 119861119864 = (1198611 1198612 1198613 sdot sdot sdot 119861119870)119879 by (9)Finally we can use (10) to obtain the enhanced image

The image enhancement algorithm by BEH can be sum-marized as follows

(1) Estimate the noise deviation of the input image(2) Extract 119870 image blocks by the method in Section 21(3) Implement Haar transform according to (7)(4) Amplify the transformation coefficients of image

details and also suppress the transformation coeffi-cients of background noise by (11)

(5) Implement the inverse Haar transform according to(9)

(6) Aggregate all of the blocks to obtain the finalenhanced image

The proposed BEH method has two drawbacks for imagedenoising (1) the extracted image blocks are not similarenough to each other and (2) this algorithm does notimplement the 2D transform on each block These twodrawbacks limit separation of noise from signal Fortunatelybecause the BM3D method can achieve outstanding denois-ing performance we can use BM3D to denoise a noisy imageand then use the proposed method to enhance the detailsof the denoised image This overall method is called BEH-BM3D

It is worth noting that there are two steps in BM3Dmethod for achieving better image denoising The first stepcan basically remove all the noise in the noisy image howeverthe image details are also oversmoothed In order to restorethe oversmoothed image details BM3D uses the second stepie Wiener filtering This two-step strategy is needed forimage denoising problem but for image enhancement theWiener filtering step is not suitable since the enhanced imagewould be restored to the original one if applying Wienerfiltering on the enhanced image We only implement one-step block-extractionHaar transform in ourmethod thus thecomputational complex can also be lowered than the BM3Dmethod

5 Experimental Results

In this section we will provide two groups of experimentsOne is to implement the proposed linear singularity rep-resentation method and the other is to implement imageenhancement

51 Linear Singularity Representation To demonstrate theproposed linear singularity representation method the Bar-bara image with a size of 512 times 512 is decomposed via theproposed KN-BEH The parameter values used are 119899119904119905119890119901 = 3119899 = 8 and 119870 = 8 Figure 7 shows all the transformedsubbands of the proposed method Obviously image detailscan be observed in the detailed subbands This is mainlybecause the KN-BEH can easily capture differences amongthe blocks of the same group where neighboring imageblocks have very similar backgrounds

52 Image Enhancement We have conducted imageenhancement experiments to verify the effectiveness of our

Mathematical Problems in Engineering 7

Figure 5 Example of decomposing a group of 8 image blocks (upper row) with a Haar transform The upper row shows the 8 extractedblocks and the lower row shows Haar-transformed subbands

Figure 6 A perfect reconstruction example of the proposed K-neighbor block-extraction and Haar (KN-BEH) transform The leftis the input image and the right is the reconstructed image (theparameters for KN-BEH are 119899119904119905119890119901 = 3 119899 = 8 119870 = 8)proposed method When 120574 gt 1 the output image details willbe enhanced with the larger 120574 correlated to a more stronglyenhanced image In all experiments the parameters are setas 119862 = 300 and 119888 = 80 for the normalized input image iewith the pixel value ranging from 0 to 1 We just adjust 120574 toget different image enhancement results The enhancementresults on House and Barbara are shown in Figures 8 and 9

To further demonstrate the advantage of the BEHmethod we also compare our image enhancement resultswith results using the NSCT method [7] and the alpha-rooting method [33] Due to the use of multidirectionalfiltering banks and translation invariants NSCT is a powerfullinear singularity representation method Figures 8 and 9show the comparison of image enhancement results by ourproposedmethod NSCT and alpha-rootingWe observe thatthe proposedmethod hardly produces any halo phenomenonaround strong edges but theNSCTmethod results in a stronghalo phenomenon From Figures 8 and 9 we also observethat our proposed method can better preserve weak detailsthan the NSCT method ie for the details on the table legsat the bottom-left part of the Barbara image Because ourmethod can better represent linear singularity as observedfrom Figures 8 and 9 our proposed method has bettersubjective visual sharpness than the alpha-rooting methodAdditionally many small weak details in the image cannotbe sharpened using the alpha-rooting method

We have further conducted other comparison experi-ments For example we used both BEH-BM3D and alpha-rooting BM3D to enhance images with added Gaussianwhite noise Specifically we first added different levels of

noise to the image and then used these two methods toenhance the image and remove noise For BEH-BM3D weused a two-stage method with the first stage of imagedenoising by BM3D and the second stage of enhance-ment by BEH For the alpha-rooting BM3D method byexactly following the procedure in [33] we simultaneouslyenhanced the transformation coefficients of image detailsand removed the transformation coefficients of noise byhard thresholding in BM3D transform domain The exper-imental results are shown in Figures 10 and 11 From thesetwo figures we can see that the BEH-BM3D can betterpreserve image details and achieve better subjective visualresults

To quantitatively validate image enhancement perfor-mance the Local Phase Coherence Sharpness Index (LPC-SI) [37] a novel and effective image sharpness assessmentmethod is used This method considers sharpness as stronglocal phase coherence near the distinctive image featuresevaluated in the complexwavelet transformdomain Further-more this method can assess sharpness without using theoriginal image as reference In our experiments we compareour results with both the NSCT method and the alpha-rooting method In each method we provide its best LPC-SI results by adjusting the respective parameters For thealpha-rooting method we fix the alpha value to 20 and thenadjust the standard deviation of noise to obtain the best LPC-SI values For the NSCT method we use the enhancementalgorithm in [7] by adjusting the gain factor 119888 in (5) to obtainthe best LPC-SI values All of the results are shown in Table 1We can see that our method achieves the best results for mostimages

In addition we further used the traditional backgroundvariation (BV) and detail variation (DV) [35] to evaluateour results The BV and DV values represent the varianceof background and foreground pixels respectively A goodimage enhancement method should increase the DV of theoriginal image but keep or even decrease the BV The BVand DV values of three methods are summarized in Table 2Notice that all of these results were obtained for each imageand each method when the corresponding LPC-SI was thebest From these results we can see that the alpha-rootingmethod achieved the lowest BV and ourmethod achieved thehighest DV However the BV values provided by our methodwere just a little bit higher than those of the alpha-rootingmethod If we use the ratio between DV and BV to assess theobtained results our method achieves the best outcome

8 Mathematical Problems in Engineering

Figure 7 The transformed subbands of the proposed transform To clearly display the details we have increased the brightness of eachhigh-frequency subband

(a) (b) (c)

(d) (e) (f)

Figure 8 The enhancement results (a) is the original image (b) and (c) are the enhanced images by the proposed method by taking 120574 as 3and 5 respectively (d) and (e) are the enhanced images by the NSCTmethod by similarly taking 120574 as 3 and 5 respectively (f) is the enhancedimage by the alpha-rooting method

We also find that NSCT achieves the best LPC-SI valuebut the worst BVDV result on Fingerprint images in Tables1 and 2 respectively This is mainly because the estimatedstandard deviation of noise is too high which results in mostimage edges being removed as noise Therefore the highestLPC-SI value cannot always be trusted This is also the main

reason for us to use both LPC-SI and BVDV to assess theexperimental results

It is worth indicating another significant advantage of ourproposed method ie its lower time complexity comparedto that of the NSCT method For image enhancement on a256times256 grayscale image our method takes only 03 second

Mathematical Problems in Engineering 9

(a) (b)

(c) (d)

Figure 9 Comparison of image enhancement results by the proposed method NSCT and alpha-rooting (a) Input image (b) the enhancedimage by the proposed method (c) the enhanced image by NSCT and (d) the enhanced image by alpha-rooting

while theNSCTmethod takes 126 seconds Ourmethod usesthe same parameter values ie 119899119904119905119890119901 = 3 119899 = 8 and 119870 = 8and the NSCT method uses 2 levels of decomposition ie 2and 4 directional subbands respectively

In the final experiment we also used the proposed BEHmethod to enhance several color images to show superiorenhancement performance and compare the results with theguided image filteringmethod [20 21] with the results shownin Figure 12 Although the image details are enhanced theenhanced images are still visually natural when using theproposed method The guided image filtering method canalso effectively enhance the image details however the imagebackground is also enhanced making the enhanced imageslook rather artificial

6 Conclusions

We have proposed an effective linear singularity representa-tionmethod and have applied thismethod to image enhance-ment in both gray and color images Similar to the BM3Dmethod our method is nonlocal However our methodextracts some spatially adjacent blocks instead of using

block-matching By using our method the image details canbe effectively represented Additionally by using differentparameters to amplify the transformation coefficients ofimage details we can obtain different image enhancementresults Because our method uses no local convolution oper-ation it introduces fewer halo phenomena compared to theNSCT method Furthermore the computational cost of ourmethod is also significantly lower compared to that of theNSCT method

Although our proposed BEHmethod looks like the origi-nal BM3Dmethod the purpose of our proposedmethod is torepresent image details which is different from the originalBM3D Note that edges in the matched image blocks areusually located in the same position thus edge informationcannot be preserved in the high-frequency subbands afterthe third-dimensional transform Therefore our proposedmethod is different from the BM3D method

In summary we have presented a simple but effectiveimage linear singularity representation method that achievesbetter objective and subjective results when applied to imageenhancement than other state-of-the-art methods

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

4 Mathematical Problems in Engineering

(b)

(a) (c)

Figure 3 Demonstration of block-matching and 3D filtering (a)-(b) The result of block-matching in a neighborhood of House image (c)The result of 3D transform on the 3D array formed by all of the blocks in (b)

achieve better image enhancement strong edges and weakedges should be processed differently In [7] to achieve betterimage enhancement results the definition below is used todifferentiate among strong edge weak edge and noise119904119905119903119900119899119892 119890119889119892119890 119894119891 119898119890119886119899 ge 119888120590119908119890119886119896 119890119889119892119890 119894119891 119898119890119886119899 lt 119888120590 119898119886119909 ge 119888120590119899119900119894119904119890 119894119891 119898119890119886119899 lt 119888120590 119898119886119909 lt 119888120590 (4)

where 119888 is a parameter ranging from 1 to 5 and 120590 is thestandard deviation of noise in the subbands at a specific pyra-midal level In this case a NSCT-based image enhancementalgorithm can be given as follows

119910 (119909) = 119909 119904119905119903119900119899119892 119890119889119892119890 119901119894119909119890119897119904max(( 119888120590|119909|)119901 1) 119909 119908119890119886119896 119890119889119892119890 1199011198941199091198901198971199040 119899119900119894119904119890 119901119894119909119890119897119904

(5)

where the input 119909 is the original transform coefficient and0 lt 119901 lt 1 is the amplifying gain This function keeps thecoefficients of strong edges amplifies the coefficients of weakedges and zeros the noise coefficients

222 Image Enhancement by BM3D According to the anal-ysis in Section 21 BM3D cannot obtain the linear sin-gularity representation like NSCT Therefore BM3D can-not use the same algorithm as NSCT to achieve imageenhancement In [33] a joint image enhancement anddenoising algorithm by 3D transform-domain collaborativefiltering was proposed In this algorithm block-matchingand 3D transform were implemented Additionally thehard thresholding operation on the 3D transformed coef-ficients was used to remove noise and the alpha-rooting

method was used to amplify coefficients and achieve imageenhancement

Given a transform spectrum 119905 of a signal which containsa DC coefficient termed 119905(0) the alpha-rooting is performedas

119905119904ℎ (119894)=

sign [119905 (119894)] ∙ |119905 (0)| ∙ (10038161003816100381610038161003816100381610038161003816 119905 (119894)119905 (0) 10038161003816100381610038161003816100381610038161003816)1120572 119894119891 119905 (0) = 0119905 (119894) 119900119905ℎ119890119903119908119894119904119890(6)

where 119905119904ℎ is the spectrum of the transformed signal andan exponent 120572 gt 1 leads to the enhancement of imagedetails

In [33] two different approaches are given to per-form image enhancement BM3D-SH3D and BM3D-SH2DThe first one implements the alpha-rooting algorithm onthe 3D transformed spectrum while the second imple-ments the alpha-rooting algorithm only on the 2D trans-formed spectrum after the 1D inverse transform ieon each block-transformed spectrum However actuallyboth methods achieve image enhancement only via oneblock in a certain sense because of extreme similarityamong the blocks in a group the magnitudes of thecoefficients in the high-frequency subbands are all nearlyzero

Accordingly to achieve better enhancement on imagedetails we propose using a K-neighbor block-extractionmethod to replace the block-matching procedure inBM3D and further discard the 2D transform on eachimage block by only implementing an interblock 1D Haartransform

Mathematical Problems in Engineering 5

3 K-Neighbor Block-Extractionand Haar Transform

Considering the effectiveness of the proposed linear sin-gularity representation method the extracted image blocksshould have similar smooth backgrounds as well as somedifferent local details We first select an image block from theinput image and then extract its spatial K-neighbor blocksThus the extracted blocks can follow the above-mentionedcondition ie all the extracted image blocks have similarsmooth backgrounds but different local details After a fastHaar transform on the group of these image blocks theimage details can be effectively represented in the detailssubbands

31 K-Neighbor Block-Extraction Areference image block119861119877with the top-left pixel coordinate 119909119877 isin 119883 (where 119883 sub Z2 isthe coordinate set of the input image 119868) is first selected byan 119899 times 119899 sliding window according to a given sliding stepsize 119899119904119905119890119901 and then its spatial K-neighbor image blocks areextracted to form a vector 119861119864 = 1198611 sdot sdot sdot 119861119894 sdot sdot sdot 119861119870119879 whereevery block 119861119894 can be considered as an element of 119861119864 andtheir top-left pixel coordinates 119909119894 form a set 119878119909119877 = 119909119894 isin 119883 119861119894 is extracted according to 119861119877

Because we want to implement a Haar transform onvector 119861119864 119870 must be a power-of-two integer In addition torepresent more directional details119870 should at least be 8 Forexample we can extract 8 image blocks and then form all oftheir top-left pixel coordinates as the 8-neighborhood top-left pixel coordinates of 119861119877 A block-extraction operation isillustrated in Figure 4 The pink block is an 8times8 referenceblock 119861119877 with the blue solid round as its top-left pixelcoordinate The green solid rounds are the top-left pixelcoordinates of the extracted119870 = 8 image blocks while the redsolid rounds are the top-left pixel coordinates of the extracted119870 = 16 image blocks

32 Haar Transform In this section we give a fast Haartransform method according to the characteristics of block-extraction Because all extracted blocks are isometric we canfully investigate the simplicity of Haar wavelet to construct afast Haar transform on each group of blocks

Forward Transform Here119870 = 2119895 (119895 = 3 4 sdot sdot sdot ) image blocksare denoted as 119861119894 119894 = 1 2 3 sdot sdot sdot 119870 For example if119870 = 8 wecan use the following formulation to realize a complete Haartransform with 3 levels of transform

119861119894 = sum119895=1sdotsdotsdot 119870

Ψ (119894 119895) 119861119895 119894 = 1 sdot sdot sdot 119870 (7)

whereΨ is aHaar transformmatrix119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879is a column vector in which every element denotes an imageblock and 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 is also a column vectorin which every element denotes a transformed subband An8 times 8Ψ of Haar transform can be defined as follows

Ψ =

((((((((((((((((((((

1radic8 1radic8 1radic8 1radic8 1radic8 1radic8 1radic8 1radic81radic8 1radic8 1radic8 1radic8 minus1radic8 minus1radic8 minus1radic8 minus1radic81radic4 1radic4 minus1radic4 minus1radic4 0 0 0 00 0 0 0 1radic4 1radic4 minus1radic4 minus1radic41radic2 minus1radic2 0 0 0 0 0 00 0 1radic2 minus1radic2 0 0 0 00 0 0 0 1radic2 minus1radic2 0 00 0 0 0 0 0 1radic2 minus1radic2

))))))))))))))))))))

(8)

By computing the matrix product 119861119864 can be decomposedinto 8 subbands with 1198611 as the approximated subband andthe rest as detailed subbands Figure 5 shows a way ofdecomposing a group of image blocks with a Haar transformNote that the contours can be effectively represented in thedetailed subbands

Inverse Transform The Haar transform matrix defined in (2)is invertible Thus one can perfectly reconstruct all originalimage blocks using the following inverse transform119861119894 = sum

119895=1sdotsdotsdot 119870

Ψminus1 (119894 119895) 119861119895 119894 = 1 sdot sdot sdot 119870 (9)

where Ψminus1 denotes the inverse matrix of ΨAfter finishing operations on a group of image blocks we

can return them to their original locations in a zero matrixof size 119872 times 119873 by averaging all pixels in the same locationBy finishing operations on all reference blocks we can get theoutput image 119868 by the following aggregation equation

119868 = sum119909119877isin119883sum119909119894isin119878119909119877 119861119894sum119909119877isin119883sum119909119894isin119878119909119877 120594119894 (10)

where120594119894 is the characteristic function of the square support ofa block located at 119909119894 and all of the image blocks are outside-padded by zeros to form an 119872 times 119873 image Because theHaar transform is perfect in reconstruction transform wealways calculate the average of all pixels extracted in the samelocation in the aggregation procedureNamely the pixel valuealways returns to the original value at every location so aperfect reconstructed image can be obtained after conductingall of the above operations Figure 6 shows a Lena image andits reconstructed image with PSNR of 313784 Application to Image Enhancement

We apply the proposed linear singularity representationmethod to image enhancement to verify its effectiveness incapturing image details among a group of blocks Ampli-fying certain transformed coefficients can typically achieveimage enhancement ie amplifying only the transformation

6 Mathematical Problems in Engineering

Figure 4 An illustration of the image block-extraction method

coefficients of image details but suppressing transformationcoefficients of background noise [7 34 35] If image detailscannot be well represented the transformation coefficients ofbackground noises may also be amplified when amplifyingtransformation coefficients In addition when implementingthe traditional convolution-based transform image detailsand their surrounding smooth background can be influencedby each other because the transformation coefficients of thebackground near the image details are usually larger thanthose that are far away Thus the halo phenomenon wouldbe introduced after image enhancement However due to theeffectiveness of our proposed linear singularity representa-tion it can amplify the transformation coefficients of imagedetails and also suppress the transformation coefficients ofbackground and noise simultaneously Most importantlywithout using large filter banks the halo phenomenon canbe effectively alleviated

For the purpose of suppressing background noise we firstestimate the noise deviation 120590 of the input image with therobust median operator [36]

As in the analysis in Section 32 we implement theforward Haar transform on 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879according to (7) to obtain the transformation coefficients119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 Then we can use the followingnonlinear mapping function to amplify the transformationcoefficients of image details

119862119890119899 = 119862119905 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 gt 119862120590120574119862119905 119894119891 119888120590 le 10038161003816100381610038161198621199051003816100381610038161003816 le 1198621205900 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 lt 119888120590

(11)

where 119862119905 is the Haar-transformed coefficients 120574 is the gainfactor to amplify the transformation coefficients of imagedetails 119862 and 119888 are the two constant parameters and 119862119890119899 isthe enhanced coefficient Next we can implement the inverse

Haar transform to get 119861119864 = (1198611 1198612 1198613 sdot sdot sdot 119861119870)119879 by (9)Finally we can use (10) to obtain the enhanced image

The image enhancement algorithm by BEH can be sum-marized as follows

(1) Estimate the noise deviation of the input image(2) Extract 119870 image blocks by the method in Section 21(3) Implement Haar transform according to (7)(4) Amplify the transformation coefficients of image

details and also suppress the transformation coeffi-cients of background noise by (11)

(5) Implement the inverse Haar transform according to(9)

(6) Aggregate all of the blocks to obtain the finalenhanced image

The proposed BEH method has two drawbacks for imagedenoising (1) the extracted image blocks are not similarenough to each other and (2) this algorithm does notimplement the 2D transform on each block These twodrawbacks limit separation of noise from signal Fortunatelybecause the BM3D method can achieve outstanding denois-ing performance we can use BM3D to denoise a noisy imageand then use the proposed method to enhance the detailsof the denoised image This overall method is called BEH-BM3D

It is worth noting that there are two steps in BM3Dmethod for achieving better image denoising The first stepcan basically remove all the noise in the noisy image howeverthe image details are also oversmoothed In order to restorethe oversmoothed image details BM3D uses the second stepie Wiener filtering This two-step strategy is needed forimage denoising problem but for image enhancement theWiener filtering step is not suitable since the enhanced imagewould be restored to the original one if applying Wienerfiltering on the enhanced image We only implement one-step block-extractionHaar transform in ourmethod thus thecomputational complex can also be lowered than the BM3Dmethod

5 Experimental Results

In this section we will provide two groups of experimentsOne is to implement the proposed linear singularity rep-resentation method and the other is to implement imageenhancement

51 Linear Singularity Representation To demonstrate theproposed linear singularity representation method the Bar-bara image with a size of 512 times 512 is decomposed via theproposed KN-BEH The parameter values used are 119899119904119905119890119901 = 3119899 = 8 and 119870 = 8 Figure 7 shows all the transformedsubbands of the proposed method Obviously image detailscan be observed in the detailed subbands This is mainlybecause the KN-BEH can easily capture differences amongthe blocks of the same group where neighboring imageblocks have very similar backgrounds

52 Image Enhancement We have conducted imageenhancement experiments to verify the effectiveness of our

Mathematical Problems in Engineering 7

Figure 5 Example of decomposing a group of 8 image blocks (upper row) with a Haar transform The upper row shows the 8 extractedblocks and the lower row shows Haar-transformed subbands

Figure 6 A perfect reconstruction example of the proposed K-neighbor block-extraction and Haar (KN-BEH) transform The leftis the input image and the right is the reconstructed image (theparameters for KN-BEH are 119899119904119905119890119901 = 3 119899 = 8 119870 = 8)proposed method When 120574 gt 1 the output image details willbe enhanced with the larger 120574 correlated to a more stronglyenhanced image In all experiments the parameters are setas 119862 = 300 and 119888 = 80 for the normalized input image iewith the pixel value ranging from 0 to 1 We just adjust 120574 toget different image enhancement results The enhancementresults on House and Barbara are shown in Figures 8 and 9

To further demonstrate the advantage of the BEHmethod we also compare our image enhancement resultswith results using the NSCT method [7] and the alpha-rooting method [33] Due to the use of multidirectionalfiltering banks and translation invariants NSCT is a powerfullinear singularity representation method Figures 8 and 9show the comparison of image enhancement results by ourproposedmethod NSCT and alpha-rootingWe observe thatthe proposedmethod hardly produces any halo phenomenonaround strong edges but theNSCTmethod results in a stronghalo phenomenon From Figures 8 and 9 we also observethat our proposed method can better preserve weak detailsthan the NSCT method ie for the details on the table legsat the bottom-left part of the Barbara image Because ourmethod can better represent linear singularity as observedfrom Figures 8 and 9 our proposed method has bettersubjective visual sharpness than the alpha-rooting methodAdditionally many small weak details in the image cannotbe sharpened using the alpha-rooting method

We have further conducted other comparison experi-ments For example we used both BEH-BM3D and alpha-rooting BM3D to enhance images with added Gaussianwhite noise Specifically we first added different levels of

noise to the image and then used these two methods toenhance the image and remove noise For BEH-BM3D weused a two-stage method with the first stage of imagedenoising by BM3D and the second stage of enhance-ment by BEH For the alpha-rooting BM3D method byexactly following the procedure in [33] we simultaneouslyenhanced the transformation coefficients of image detailsand removed the transformation coefficients of noise byhard thresholding in BM3D transform domain The exper-imental results are shown in Figures 10 and 11 From thesetwo figures we can see that the BEH-BM3D can betterpreserve image details and achieve better subjective visualresults

To quantitatively validate image enhancement perfor-mance the Local Phase Coherence Sharpness Index (LPC-SI) [37] a novel and effective image sharpness assessmentmethod is used This method considers sharpness as stronglocal phase coherence near the distinctive image featuresevaluated in the complexwavelet transformdomain Further-more this method can assess sharpness without using theoriginal image as reference In our experiments we compareour results with both the NSCT method and the alpha-rooting method In each method we provide its best LPC-SI results by adjusting the respective parameters For thealpha-rooting method we fix the alpha value to 20 and thenadjust the standard deviation of noise to obtain the best LPC-SI values For the NSCT method we use the enhancementalgorithm in [7] by adjusting the gain factor 119888 in (5) to obtainthe best LPC-SI values All of the results are shown in Table 1We can see that our method achieves the best results for mostimages

In addition we further used the traditional backgroundvariation (BV) and detail variation (DV) [35] to evaluateour results The BV and DV values represent the varianceof background and foreground pixels respectively A goodimage enhancement method should increase the DV of theoriginal image but keep or even decrease the BV The BVand DV values of three methods are summarized in Table 2Notice that all of these results were obtained for each imageand each method when the corresponding LPC-SI was thebest From these results we can see that the alpha-rootingmethod achieved the lowest BV and ourmethod achieved thehighest DV However the BV values provided by our methodwere just a little bit higher than those of the alpha-rootingmethod If we use the ratio between DV and BV to assess theobtained results our method achieves the best outcome

8 Mathematical Problems in Engineering

Figure 7 The transformed subbands of the proposed transform To clearly display the details we have increased the brightness of eachhigh-frequency subband

(a) (b) (c)

(d) (e) (f)

Figure 8 The enhancement results (a) is the original image (b) and (c) are the enhanced images by the proposed method by taking 120574 as 3and 5 respectively (d) and (e) are the enhanced images by the NSCTmethod by similarly taking 120574 as 3 and 5 respectively (f) is the enhancedimage by the alpha-rooting method

We also find that NSCT achieves the best LPC-SI valuebut the worst BVDV result on Fingerprint images in Tables1 and 2 respectively This is mainly because the estimatedstandard deviation of noise is too high which results in mostimage edges being removed as noise Therefore the highestLPC-SI value cannot always be trusted This is also the main

reason for us to use both LPC-SI and BVDV to assess theexperimental results

It is worth indicating another significant advantage of ourproposed method ie its lower time complexity comparedto that of the NSCT method For image enhancement on a256times256 grayscale image our method takes only 03 second

Mathematical Problems in Engineering 9

(a) (b)

(c) (d)

Figure 9 Comparison of image enhancement results by the proposed method NSCT and alpha-rooting (a) Input image (b) the enhancedimage by the proposed method (c) the enhanced image by NSCT and (d) the enhanced image by alpha-rooting

while theNSCTmethod takes 126 seconds Ourmethod usesthe same parameter values ie 119899119904119905119890119901 = 3 119899 = 8 and 119870 = 8and the NSCT method uses 2 levels of decomposition ie 2and 4 directional subbands respectively

In the final experiment we also used the proposed BEHmethod to enhance several color images to show superiorenhancement performance and compare the results with theguided image filteringmethod [20 21] with the results shownin Figure 12 Although the image details are enhanced theenhanced images are still visually natural when using theproposed method The guided image filtering method canalso effectively enhance the image details however the imagebackground is also enhanced making the enhanced imageslook rather artificial

6 Conclusions

We have proposed an effective linear singularity representa-tionmethod and have applied thismethod to image enhance-ment in both gray and color images Similar to the BM3Dmethod our method is nonlocal However our methodextracts some spatially adjacent blocks instead of using

block-matching By using our method the image details canbe effectively represented Additionally by using differentparameters to amplify the transformation coefficients ofimage details we can obtain different image enhancementresults Because our method uses no local convolution oper-ation it introduces fewer halo phenomena compared to theNSCT method Furthermore the computational cost of ourmethod is also significantly lower compared to that of theNSCT method

Although our proposed BEHmethod looks like the origi-nal BM3Dmethod the purpose of our proposedmethod is torepresent image details which is different from the originalBM3D Note that edges in the matched image blocks areusually located in the same position thus edge informationcannot be preserved in the high-frequency subbands afterthe third-dimensional transform Therefore our proposedmethod is different from the BM3D method

In summary we have presented a simple but effectiveimage linear singularity representation method that achievesbetter objective and subjective results when applied to imageenhancement than other state-of-the-art methods

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

Mathematical Problems in Engineering 5

3 K-Neighbor Block-Extractionand Haar Transform

Considering the effectiveness of the proposed linear sin-gularity representation method the extracted image blocksshould have similar smooth backgrounds as well as somedifferent local details We first select an image block from theinput image and then extract its spatial K-neighbor blocksThus the extracted blocks can follow the above-mentionedcondition ie all the extracted image blocks have similarsmooth backgrounds but different local details After a fastHaar transform on the group of these image blocks theimage details can be effectively represented in the detailssubbands

31 K-Neighbor Block-Extraction Areference image block119861119877with the top-left pixel coordinate 119909119877 isin 119883 (where 119883 sub Z2 isthe coordinate set of the input image 119868) is first selected byan 119899 times 119899 sliding window according to a given sliding stepsize 119899119904119905119890119901 and then its spatial K-neighbor image blocks areextracted to form a vector 119861119864 = 1198611 sdot sdot sdot 119861119894 sdot sdot sdot 119861119870119879 whereevery block 119861119894 can be considered as an element of 119861119864 andtheir top-left pixel coordinates 119909119894 form a set 119878119909119877 = 119909119894 isin 119883 119861119894 is extracted according to 119861119877

Because we want to implement a Haar transform onvector 119861119864 119870 must be a power-of-two integer In addition torepresent more directional details119870 should at least be 8 Forexample we can extract 8 image blocks and then form all oftheir top-left pixel coordinates as the 8-neighborhood top-left pixel coordinates of 119861119877 A block-extraction operation isillustrated in Figure 4 The pink block is an 8times8 referenceblock 119861119877 with the blue solid round as its top-left pixelcoordinate The green solid rounds are the top-left pixelcoordinates of the extracted119870 = 8 image blocks while the redsolid rounds are the top-left pixel coordinates of the extracted119870 = 16 image blocks

32 Haar Transform In this section we give a fast Haartransform method according to the characteristics of block-extraction Because all extracted blocks are isometric we canfully investigate the simplicity of Haar wavelet to construct afast Haar transform on each group of blocks

Forward Transform Here119870 = 2119895 (119895 = 3 4 sdot sdot sdot ) image blocksare denoted as 119861119894 119894 = 1 2 3 sdot sdot sdot 119870 For example if119870 = 8 wecan use the following formulation to realize a complete Haartransform with 3 levels of transform

119861119894 = sum119895=1sdotsdotsdot 119870

Ψ (119894 119895) 119861119895 119894 = 1 sdot sdot sdot 119870 (7)

whereΨ is aHaar transformmatrix119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879is a column vector in which every element denotes an imageblock and 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 is also a column vectorin which every element denotes a transformed subband An8 times 8Ψ of Haar transform can be defined as follows

Ψ =

((((((((((((((((((((

1radic8 1radic8 1radic8 1radic8 1radic8 1radic8 1radic8 1radic81radic8 1radic8 1radic8 1radic8 minus1radic8 minus1radic8 minus1radic8 minus1radic81radic4 1radic4 minus1radic4 minus1radic4 0 0 0 00 0 0 0 1radic4 1radic4 minus1radic4 minus1radic41radic2 minus1radic2 0 0 0 0 0 00 0 1radic2 minus1radic2 0 0 0 00 0 0 0 1radic2 minus1radic2 0 00 0 0 0 0 0 1radic2 minus1radic2

))))))))))))))))))))

(8)

By computing the matrix product 119861119864 can be decomposedinto 8 subbands with 1198611 as the approximated subband andthe rest as detailed subbands Figure 5 shows a way ofdecomposing a group of image blocks with a Haar transformNote that the contours can be effectively represented in thedetailed subbands

Inverse Transform The Haar transform matrix defined in (2)is invertible Thus one can perfectly reconstruct all originalimage blocks using the following inverse transform119861119894 = sum

119895=1sdotsdotsdot 119870

Ψminus1 (119894 119895) 119861119895 119894 = 1 sdot sdot sdot 119870 (9)

where Ψminus1 denotes the inverse matrix of ΨAfter finishing operations on a group of image blocks we

can return them to their original locations in a zero matrixof size 119872 times 119873 by averaging all pixels in the same locationBy finishing operations on all reference blocks we can get theoutput image 119868 by the following aggregation equation

119868 = sum119909119877isin119883sum119909119894isin119878119909119877 119861119894sum119909119877isin119883sum119909119894isin119878119909119877 120594119894 (10)

where120594119894 is the characteristic function of the square support ofa block located at 119909119894 and all of the image blocks are outside-padded by zeros to form an 119872 times 119873 image Because theHaar transform is perfect in reconstruction transform wealways calculate the average of all pixels extracted in the samelocation in the aggregation procedureNamely the pixel valuealways returns to the original value at every location so aperfect reconstructed image can be obtained after conductingall of the above operations Figure 6 shows a Lena image andits reconstructed image with PSNR of 313784 Application to Image Enhancement

We apply the proposed linear singularity representationmethod to image enhancement to verify its effectiveness incapturing image details among a group of blocks Ampli-fying certain transformed coefficients can typically achieveimage enhancement ie amplifying only the transformation

6 Mathematical Problems in Engineering

Figure 4 An illustration of the image block-extraction method

coefficients of image details but suppressing transformationcoefficients of background noise [7 34 35] If image detailscannot be well represented the transformation coefficients ofbackground noises may also be amplified when amplifyingtransformation coefficients In addition when implementingthe traditional convolution-based transform image detailsand their surrounding smooth background can be influencedby each other because the transformation coefficients of thebackground near the image details are usually larger thanthose that are far away Thus the halo phenomenon wouldbe introduced after image enhancement However due to theeffectiveness of our proposed linear singularity representa-tion it can amplify the transformation coefficients of imagedetails and also suppress the transformation coefficients ofbackground and noise simultaneously Most importantlywithout using large filter banks the halo phenomenon canbe effectively alleviated

For the purpose of suppressing background noise we firstestimate the noise deviation 120590 of the input image with therobust median operator [36]

As in the analysis in Section 32 we implement theforward Haar transform on 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879according to (7) to obtain the transformation coefficients119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 Then we can use the followingnonlinear mapping function to amplify the transformationcoefficients of image details

119862119890119899 = 119862119905 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 gt 119862120590120574119862119905 119894119891 119888120590 le 10038161003816100381610038161198621199051003816100381610038161003816 le 1198621205900 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 lt 119888120590

(11)

where 119862119905 is the Haar-transformed coefficients 120574 is the gainfactor to amplify the transformation coefficients of imagedetails 119862 and 119888 are the two constant parameters and 119862119890119899 isthe enhanced coefficient Next we can implement the inverse

Haar transform to get 119861119864 = (1198611 1198612 1198613 sdot sdot sdot 119861119870)119879 by (9)Finally we can use (10) to obtain the enhanced image

The image enhancement algorithm by BEH can be sum-marized as follows

(1) Estimate the noise deviation of the input image(2) Extract 119870 image blocks by the method in Section 21(3) Implement Haar transform according to (7)(4) Amplify the transformation coefficients of image

details and also suppress the transformation coeffi-cients of background noise by (11)

(5) Implement the inverse Haar transform according to(9)

(6) Aggregate all of the blocks to obtain the finalenhanced image

The proposed BEH method has two drawbacks for imagedenoising (1) the extracted image blocks are not similarenough to each other and (2) this algorithm does notimplement the 2D transform on each block These twodrawbacks limit separation of noise from signal Fortunatelybecause the BM3D method can achieve outstanding denois-ing performance we can use BM3D to denoise a noisy imageand then use the proposed method to enhance the detailsof the denoised image This overall method is called BEH-BM3D

It is worth noting that there are two steps in BM3Dmethod for achieving better image denoising The first stepcan basically remove all the noise in the noisy image howeverthe image details are also oversmoothed In order to restorethe oversmoothed image details BM3D uses the second stepie Wiener filtering This two-step strategy is needed forimage denoising problem but for image enhancement theWiener filtering step is not suitable since the enhanced imagewould be restored to the original one if applying Wienerfiltering on the enhanced image We only implement one-step block-extractionHaar transform in ourmethod thus thecomputational complex can also be lowered than the BM3Dmethod

5 Experimental Results

In this section we will provide two groups of experimentsOne is to implement the proposed linear singularity rep-resentation method and the other is to implement imageenhancement

51 Linear Singularity Representation To demonstrate theproposed linear singularity representation method the Bar-bara image with a size of 512 times 512 is decomposed via theproposed KN-BEH The parameter values used are 119899119904119905119890119901 = 3119899 = 8 and 119870 = 8 Figure 7 shows all the transformedsubbands of the proposed method Obviously image detailscan be observed in the detailed subbands This is mainlybecause the KN-BEH can easily capture differences amongthe blocks of the same group where neighboring imageblocks have very similar backgrounds

52 Image Enhancement We have conducted imageenhancement experiments to verify the effectiveness of our

Mathematical Problems in Engineering 7

Figure 5 Example of decomposing a group of 8 image blocks (upper row) with a Haar transform The upper row shows the 8 extractedblocks and the lower row shows Haar-transformed subbands

Figure 6 A perfect reconstruction example of the proposed K-neighbor block-extraction and Haar (KN-BEH) transform The leftis the input image and the right is the reconstructed image (theparameters for KN-BEH are 119899119904119905119890119901 = 3 119899 = 8 119870 = 8)proposed method When 120574 gt 1 the output image details willbe enhanced with the larger 120574 correlated to a more stronglyenhanced image In all experiments the parameters are setas 119862 = 300 and 119888 = 80 for the normalized input image iewith the pixel value ranging from 0 to 1 We just adjust 120574 toget different image enhancement results The enhancementresults on House and Barbara are shown in Figures 8 and 9

To further demonstrate the advantage of the BEHmethod we also compare our image enhancement resultswith results using the NSCT method [7] and the alpha-rooting method [33] Due to the use of multidirectionalfiltering banks and translation invariants NSCT is a powerfullinear singularity representation method Figures 8 and 9show the comparison of image enhancement results by ourproposedmethod NSCT and alpha-rootingWe observe thatthe proposedmethod hardly produces any halo phenomenonaround strong edges but theNSCTmethod results in a stronghalo phenomenon From Figures 8 and 9 we also observethat our proposed method can better preserve weak detailsthan the NSCT method ie for the details on the table legsat the bottom-left part of the Barbara image Because ourmethod can better represent linear singularity as observedfrom Figures 8 and 9 our proposed method has bettersubjective visual sharpness than the alpha-rooting methodAdditionally many small weak details in the image cannotbe sharpened using the alpha-rooting method

We have further conducted other comparison experi-ments For example we used both BEH-BM3D and alpha-rooting BM3D to enhance images with added Gaussianwhite noise Specifically we first added different levels of

noise to the image and then used these two methods toenhance the image and remove noise For BEH-BM3D weused a two-stage method with the first stage of imagedenoising by BM3D and the second stage of enhance-ment by BEH For the alpha-rooting BM3D method byexactly following the procedure in [33] we simultaneouslyenhanced the transformation coefficients of image detailsand removed the transformation coefficients of noise byhard thresholding in BM3D transform domain The exper-imental results are shown in Figures 10 and 11 From thesetwo figures we can see that the BEH-BM3D can betterpreserve image details and achieve better subjective visualresults

To quantitatively validate image enhancement perfor-mance the Local Phase Coherence Sharpness Index (LPC-SI) [37] a novel and effective image sharpness assessmentmethod is used This method considers sharpness as stronglocal phase coherence near the distinctive image featuresevaluated in the complexwavelet transformdomain Further-more this method can assess sharpness without using theoriginal image as reference In our experiments we compareour results with both the NSCT method and the alpha-rooting method In each method we provide its best LPC-SI results by adjusting the respective parameters For thealpha-rooting method we fix the alpha value to 20 and thenadjust the standard deviation of noise to obtain the best LPC-SI values For the NSCT method we use the enhancementalgorithm in [7] by adjusting the gain factor 119888 in (5) to obtainthe best LPC-SI values All of the results are shown in Table 1We can see that our method achieves the best results for mostimages

In addition we further used the traditional backgroundvariation (BV) and detail variation (DV) [35] to evaluateour results The BV and DV values represent the varianceof background and foreground pixels respectively A goodimage enhancement method should increase the DV of theoriginal image but keep or even decrease the BV The BVand DV values of three methods are summarized in Table 2Notice that all of these results were obtained for each imageand each method when the corresponding LPC-SI was thebest From these results we can see that the alpha-rootingmethod achieved the lowest BV and ourmethod achieved thehighest DV However the BV values provided by our methodwere just a little bit higher than those of the alpha-rootingmethod If we use the ratio between DV and BV to assess theobtained results our method achieves the best outcome

8 Mathematical Problems in Engineering

Figure 7 The transformed subbands of the proposed transform To clearly display the details we have increased the brightness of eachhigh-frequency subband

(a) (b) (c)

(d) (e) (f)

Figure 8 The enhancement results (a) is the original image (b) and (c) are the enhanced images by the proposed method by taking 120574 as 3and 5 respectively (d) and (e) are the enhanced images by the NSCTmethod by similarly taking 120574 as 3 and 5 respectively (f) is the enhancedimage by the alpha-rooting method

We also find that NSCT achieves the best LPC-SI valuebut the worst BVDV result on Fingerprint images in Tables1 and 2 respectively This is mainly because the estimatedstandard deviation of noise is too high which results in mostimage edges being removed as noise Therefore the highestLPC-SI value cannot always be trusted This is also the main

reason for us to use both LPC-SI and BVDV to assess theexperimental results

It is worth indicating another significant advantage of ourproposed method ie its lower time complexity comparedto that of the NSCT method For image enhancement on a256times256 grayscale image our method takes only 03 second

Mathematical Problems in Engineering 9

(a) (b)

(c) (d)

Figure 9 Comparison of image enhancement results by the proposed method NSCT and alpha-rooting (a) Input image (b) the enhancedimage by the proposed method (c) the enhanced image by NSCT and (d) the enhanced image by alpha-rooting

while theNSCTmethod takes 126 seconds Ourmethod usesthe same parameter values ie 119899119904119905119890119901 = 3 119899 = 8 and 119870 = 8and the NSCT method uses 2 levels of decomposition ie 2and 4 directional subbands respectively

In the final experiment we also used the proposed BEHmethod to enhance several color images to show superiorenhancement performance and compare the results with theguided image filteringmethod [20 21] with the results shownin Figure 12 Although the image details are enhanced theenhanced images are still visually natural when using theproposed method The guided image filtering method canalso effectively enhance the image details however the imagebackground is also enhanced making the enhanced imageslook rather artificial

6 Conclusions

We have proposed an effective linear singularity representa-tionmethod and have applied thismethod to image enhance-ment in both gray and color images Similar to the BM3Dmethod our method is nonlocal However our methodextracts some spatially adjacent blocks instead of using

block-matching By using our method the image details canbe effectively represented Additionally by using differentparameters to amplify the transformation coefficients ofimage details we can obtain different image enhancementresults Because our method uses no local convolution oper-ation it introduces fewer halo phenomena compared to theNSCT method Furthermore the computational cost of ourmethod is also significantly lower compared to that of theNSCT method

Although our proposed BEHmethod looks like the origi-nal BM3Dmethod the purpose of our proposedmethod is torepresent image details which is different from the originalBM3D Note that edges in the matched image blocks areusually located in the same position thus edge informationcannot be preserved in the high-frequency subbands afterthe third-dimensional transform Therefore our proposedmethod is different from the BM3D method

In summary we have presented a simple but effectiveimage linear singularity representation method that achievesbetter objective and subjective results when applied to imageenhancement than other state-of-the-art methods

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

6 Mathematical Problems in Engineering

Figure 4 An illustration of the image block-extraction method

coefficients of image details but suppressing transformationcoefficients of background noise [7 34 35] If image detailscannot be well represented the transformation coefficients ofbackground noises may also be amplified when amplifyingtransformation coefficients In addition when implementingthe traditional convolution-based transform image detailsand their surrounding smooth background can be influencedby each other because the transformation coefficients of thebackground near the image details are usually larger thanthose that are far away Thus the halo phenomenon wouldbe introduced after image enhancement However due to theeffectiveness of our proposed linear singularity representa-tion it can amplify the transformation coefficients of imagedetails and also suppress the transformation coefficients ofbackground and noise simultaneously Most importantlywithout using large filter banks the halo phenomenon canbe effectively alleviated

For the purpose of suppressing background noise we firstestimate the noise deviation 120590 of the input image with therobust median operator [36]

As in the analysis in Section 32 we implement theforward Haar transform on 119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879according to (7) to obtain the transformation coefficients119861119864 = 1198611 1198612 1198613 sdot sdot sdot 119861119870119879 Then we can use the followingnonlinear mapping function to amplify the transformationcoefficients of image details

119862119890119899 = 119862119905 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 gt 119862120590120574119862119905 119894119891 119888120590 le 10038161003816100381610038161198621199051003816100381610038161003816 le 1198621205900 119894119891 10038161003816100381610038161198621199051003816100381610038161003816 lt 119888120590

(11)

where 119862119905 is the Haar-transformed coefficients 120574 is the gainfactor to amplify the transformation coefficients of imagedetails 119862 and 119888 are the two constant parameters and 119862119890119899 isthe enhanced coefficient Next we can implement the inverse

Haar transform to get 119861119864 = (1198611 1198612 1198613 sdot sdot sdot 119861119870)119879 by (9)Finally we can use (10) to obtain the enhanced image

The image enhancement algorithm by BEH can be sum-marized as follows

(1) Estimate the noise deviation of the input image(2) Extract 119870 image blocks by the method in Section 21(3) Implement Haar transform according to (7)(4) Amplify the transformation coefficients of image

details and also suppress the transformation coeffi-cients of background noise by (11)

(5) Implement the inverse Haar transform according to(9)

(6) Aggregate all of the blocks to obtain the finalenhanced image

The proposed BEH method has two drawbacks for imagedenoising (1) the extracted image blocks are not similarenough to each other and (2) this algorithm does notimplement the 2D transform on each block These twodrawbacks limit separation of noise from signal Fortunatelybecause the BM3D method can achieve outstanding denois-ing performance we can use BM3D to denoise a noisy imageand then use the proposed method to enhance the detailsof the denoised image This overall method is called BEH-BM3D

It is worth noting that there are two steps in BM3Dmethod for achieving better image denoising The first stepcan basically remove all the noise in the noisy image howeverthe image details are also oversmoothed In order to restorethe oversmoothed image details BM3D uses the second stepie Wiener filtering This two-step strategy is needed forimage denoising problem but for image enhancement theWiener filtering step is not suitable since the enhanced imagewould be restored to the original one if applying Wienerfiltering on the enhanced image We only implement one-step block-extractionHaar transform in ourmethod thus thecomputational complex can also be lowered than the BM3Dmethod

5 Experimental Results

In this section we will provide two groups of experimentsOne is to implement the proposed linear singularity rep-resentation method and the other is to implement imageenhancement

51 Linear Singularity Representation To demonstrate theproposed linear singularity representation method the Bar-bara image with a size of 512 times 512 is decomposed via theproposed KN-BEH The parameter values used are 119899119904119905119890119901 = 3119899 = 8 and 119870 = 8 Figure 7 shows all the transformedsubbands of the proposed method Obviously image detailscan be observed in the detailed subbands This is mainlybecause the KN-BEH can easily capture differences amongthe blocks of the same group where neighboring imageblocks have very similar backgrounds

52 Image Enhancement We have conducted imageenhancement experiments to verify the effectiveness of our

Mathematical Problems in Engineering 7

Figure 5 Example of decomposing a group of 8 image blocks (upper row) with a Haar transform The upper row shows the 8 extractedblocks and the lower row shows Haar-transformed subbands

Figure 6 A perfect reconstruction example of the proposed K-neighbor block-extraction and Haar (KN-BEH) transform The leftis the input image and the right is the reconstructed image (theparameters for KN-BEH are 119899119904119905119890119901 = 3 119899 = 8 119870 = 8)proposed method When 120574 gt 1 the output image details willbe enhanced with the larger 120574 correlated to a more stronglyenhanced image In all experiments the parameters are setas 119862 = 300 and 119888 = 80 for the normalized input image iewith the pixel value ranging from 0 to 1 We just adjust 120574 toget different image enhancement results The enhancementresults on House and Barbara are shown in Figures 8 and 9

To further demonstrate the advantage of the BEHmethod we also compare our image enhancement resultswith results using the NSCT method [7] and the alpha-rooting method [33] Due to the use of multidirectionalfiltering banks and translation invariants NSCT is a powerfullinear singularity representation method Figures 8 and 9show the comparison of image enhancement results by ourproposedmethod NSCT and alpha-rootingWe observe thatthe proposedmethod hardly produces any halo phenomenonaround strong edges but theNSCTmethod results in a stronghalo phenomenon From Figures 8 and 9 we also observethat our proposed method can better preserve weak detailsthan the NSCT method ie for the details on the table legsat the bottom-left part of the Barbara image Because ourmethod can better represent linear singularity as observedfrom Figures 8 and 9 our proposed method has bettersubjective visual sharpness than the alpha-rooting methodAdditionally many small weak details in the image cannotbe sharpened using the alpha-rooting method

We have further conducted other comparison experi-ments For example we used both BEH-BM3D and alpha-rooting BM3D to enhance images with added Gaussianwhite noise Specifically we first added different levels of

noise to the image and then used these two methods toenhance the image and remove noise For BEH-BM3D weused a two-stage method with the first stage of imagedenoising by BM3D and the second stage of enhance-ment by BEH For the alpha-rooting BM3D method byexactly following the procedure in [33] we simultaneouslyenhanced the transformation coefficients of image detailsand removed the transformation coefficients of noise byhard thresholding in BM3D transform domain The exper-imental results are shown in Figures 10 and 11 From thesetwo figures we can see that the BEH-BM3D can betterpreserve image details and achieve better subjective visualresults

To quantitatively validate image enhancement perfor-mance the Local Phase Coherence Sharpness Index (LPC-SI) [37] a novel and effective image sharpness assessmentmethod is used This method considers sharpness as stronglocal phase coherence near the distinctive image featuresevaluated in the complexwavelet transformdomain Further-more this method can assess sharpness without using theoriginal image as reference In our experiments we compareour results with both the NSCT method and the alpha-rooting method In each method we provide its best LPC-SI results by adjusting the respective parameters For thealpha-rooting method we fix the alpha value to 20 and thenadjust the standard deviation of noise to obtain the best LPC-SI values For the NSCT method we use the enhancementalgorithm in [7] by adjusting the gain factor 119888 in (5) to obtainthe best LPC-SI values All of the results are shown in Table 1We can see that our method achieves the best results for mostimages

In addition we further used the traditional backgroundvariation (BV) and detail variation (DV) [35] to evaluateour results The BV and DV values represent the varianceof background and foreground pixels respectively A goodimage enhancement method should increase the DV of theoriginal image but keep or even decrease the BV The BVand DV values of three methods are summarized in Table 2Notice that all of these results were obtained for each imageand each method when the corresponding LPC-SI was thebest From these results we can see that the alpha-rootingmethod achieved the lowest BV and ourmethod achieved thehighest DV However the BV values provided by our methodwere just a little bit higher than those of the alpha-rootingmethod If we use the ratio between DV and BV to assess theobtained results our method achieves the best outcome

8 Mathematical Problems in Engineering

Figure 7 The transformed subbands of the proposed transform To clearly display the details we have increased the brightness of eachhigh-frequency subband

(a) (b) (c)

(d) (e) (f)

Figure 8 The enhancement results (a) is the original image (b) and (c) are the enhanced images by the proposed method by taking 120574 as 3and 5 respectively (d) and (e) are the enhanced images by the NSCTmethod by similarly taking 120574 as 3 and 5 respectively (f) is the enhancedimage by the alpha-rooting method

We also find that NSCT achieves the best LPC-SI valuebut the worst BVDV result on Fingerprint images in Tables1 and 2 respectively This is mainly because the estimatedstandard deviation of noise is too high which results in mostimage edges being removed as noise Therefore the highestLPC-SI value cannot always be trusted This is also the main

reason for us to use both LPC-SI and BVDV to assess theexperimental results

It is worth indicating another significant advantage of ourproposed method ie its lower time complexity comparedto that of the NSCT method For image enhancement on a256times256 grayscale image our method takes only 03 second

Mathematical Problems in Engineering 9

(a) (b)

(c) (d)

Figure 9 Comparison of image enhancement results by the proposed method NSCT and alpha-rooting (a) Input image (b) the enhancedimage by the proposed method (c) the enhanced image by NSCT and (d) the enhanced image by alpha-rooting

while theNSCTmethod takes 126 seconds Ourmethod usesthe same parameter values ie 119899119904119905119890119901 = 3 119899 = 8 and 119870 = 8and the NSCT method uses 2 levels of decomposition ie 2and 4 directional subbands respectively

In the final experiment we also used the proposed BEHmethod to enhance several color images to show superiorenhancement performance and compare the results with theguided image filteringmethod [20 21] with the results shownin Figure 12 Although the image details are enhanced theenhanced images are still visually natural when using theproposed method The guided image filtering method canalso effectively enhance the image details however the imagebackground is also enhanced making the enhanced imageslook rather artificial

6 Conclusions

We have proposed an effective linear singularity representa-tionmethod and have applied thismethod to image enhance-ment in both gray and color images Similar to the BM3Dmethod our method is nonlocal However our methodextracts some spatially adjacent blocks instead of using

block-matching By using our method the image details canbe effectively represented Additionally by using differentparameters to amplify the transformation coefficients ofimage details we can obtain different image enhancementresults Because our method uses no local convolution oper-ation it introduces fewer halo phenomena compared to theNSCT method Furthermore the computational cost of ourmethod is also significantly lower compared to that of theNSCT method

Although our proposed BEHmethod looks like the origi-nal BM3Dmethod the purpose of our proposedmethod is torepresent image details which is different from the originalBM3D Note that edges in the matched image blocks areusually located in the same position thus edge informationcannot be preserved in the high-frequency subbands afterthe third-dimensional transform Therefore our proposedmethod is different from the BM3D method

In summary we have presented a simple but effectiveimage linear singularity representation method that achievesbetter objective and subjective results when applied to imageenhancement than other state-of-the-art methods

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

Mathematical Problems in Engineering 7

Figure 5 Example of decomposing a group of 8 image blocks (upper row) with a Haar transform The upper row shows the 8 extractedblocks and the lower row shows Haar-transformed subbands

Figure 6 A perfect reconstruction example of the proposed K-neighbor block-extraction and Haar (KN-BEH) transform The leftis the input image and the right is the reconstructed image (theparameters for KN-BEH are 119899119904119905119890119901 = 3 119899 = 8 119870 = 8)proposed method When 120574 gt 1 the output image details willbe enhanced with the larger 120574 correlated to a more stronglyenhanced image In all experiments the parameters are setas 119862 = 300 and 119888 = 80 for the normalized input image iewith the pixel value ranging from 0 to 1 We just adjust 120574 toget different image enhancement results The enhancementresults on House and Barbara are shown in Figures 8 and 9

To further demonstrate the advantage of the BEHmethod we also compare our image enhancement resultswith results using the NSCT method [7] and the alpha-rooting method [33] Due to the use of multidirectionalfiltering banks and translation invariants NSCT is a powerfullinear singularity representation method Figures 8 and 9show the comparison of image enhancement results by ourproposedmethod NSCT and alpha-rootingWe observe thatthe proposedmethod hardly produces any halo phenomenonaround strong edges but theNSCTmethod results in a stronghalo phenomenon From Figures 8 and 9 we also observethat our proposed method can better preserve weak detailsthan the NSCT method ie for the details on the table legsat the bottom-left part of the Barbara image Because ourmethod can better represent linear singularity as observedfrom Figures 8 and 9 our proposed method has bettersubjective visual sharpness than the alpha-rooting methodAdditionally many small weak details in the image cannotbe sharpened using the alpha-rooting method

We have further conducted other comparison experi-ments For example we used both BEH-BM3D and alpha-rooting BM3D to enhance images with added Gaussianwhite noise Specifically we first added different levels of

noise to the image and then used these two methods toenhance the image and remove noise For BEH-BM3D weused a two-stage method with the first stage of imagedenoising by BM3D and the second stage of enhance-ment by BEH For the alpha-rooting BM3D method byexactly following the procedure in [33] we simultaneouslyenhanced the transformation coefficients of image detailsand removed the transformation coefficients of noise byhard thresholding in BM3D transform domain The exper-imental results are shown in Figures 10 and 11 From thesetwo figures we can see that the BEH-BM3D can betterpreserve image details and achieve better subjective visualresults

To quantitatively validate image enhancement perfor-mance the Local Phase Coherence Sharpness Index (LPC-SI) [37] a novel and effective image sharpness assessmentmethod is used This method considers sharpness as stronglocal phase coherence near the distinctive image featuresevaluated in the complexwavelet transformdomain Further-more this method can assess sharpness without using theoriginal image as reference In our experiments we compareour results with both the NSCT method and the alpha-rooting method In each method we provide its best LPC-SI results by adjusting the respective parameters For thealpha-rooting method we fix the alpha value to 20 and thenadjust the standard deviation of noise to obtain the best LPC-SI values For the NSCT method we use the enhancementalgorithm in [7] by adjusting the gain factor 119888 in (5) to obtainthe best LPC-SI values All of the results are shown in Table 1We can see that our method achieves the best results for mostimages

In addition we further used the traditional backgroundvariation (BV) and detail variation (DV) [35] to evaluateour results The BV and DV values represent the varianceof background and foreground pixels respectively A goodimage enhancement method should increase the DV of theoriginal image but keep or even decrease the BV The BVand DV values of three methods are summarized in Table 2Notice that all of these results were obtained for each imageand each method when the corresponding LPC-SI was thebest From these results we can see that the alpha-rootingmethod achieved the lowest BV and ourmethod achieved thehighest DV However the BV values provided by our methodwere just a little bit higher than those of the alpha-rootingmethod If we use the ratio between DV and BV to assess theobtained results our method achieves the best outcome

8 Mathematical Problems in Engineering

Figure 7 The transformed subbands of the proposed transform To clearly display the details we have increased the brightness of eachhigh-frequency subband

(a) (b) (c)

(d) (e) (f)

Figure 8 The enhancement results (a) is the original image (b) and (c) are the enhanced images by the proposed method by taking 120574 as 3and 5 respectively (d) and (e) are the enhanced images by the NSCTmethod by similarly taking 120574 as 3 and 5 respectively (f) is the enhancedimage by the alpha-rooting method

We also find that NSCT achieves the best LPC-SI valuebut the worst BVDV result on Fingerprint images in Tables1 and 2 respectively This is mainly because the estimatedstandard deviation of noise is too high which results in mostimage edges being removed as noise Therefore the highestLPC-SI value cannot always be trusted This is also the main

reason for us to use both LPC-SI and BVDV to assess theexperimental results

It is worth indicating another significant advantage of ourproposed method ie its lower time complexity comparedto that of the NSCT method For image enhancement on a256times256 grayscale image our method takes only 03 second

Mathematical Problems in Engineering 9

(a) (b)

(c) (d)

Figure 9 Comparison of image enhancement results by the proposed method NSCT and alpha-rooting (a) Input image (b) the enhancedimage by the proposed method (c) the enhanced image by NSCT and (d) the enhanced image by alpha-rooting

while theNSCTmethod takes 126 seconds Ourmethod usesthe same parameter values ie 119899119904119905119890119901 = 3 119899 = 8 and 119870 = 8and the NSCT method uses 2 levels of decomposition ie 2and 4 directional subbands respectively

In the final experiment we also used the proposed BEHmethod to enhance several color images to show superiorenhancement performance and compare the results with theguided image filteringmethod [20 21] with the results shownin Figure 12 Although the image details are enhanced theenhanced images are still visually natural when using theproposed method The guided image filtering method canalso effectively enhance the image details however the imagebackground is also enhanced making the enhanced imageslook rather artificial

6 Conclusions

We have proposed an effective linear singularity representa-tionmethod and have applied thismethod to image enhance-ment in both gray and color images Similar to the BM3Dmethod our method is nonlocal However our methodextracts some spatially adjacent blocks instead of using

block-matching By using our method the image details canbe effectively represented Additionally by using differentparameters to amplify the transformation coefficients ofimage details we can obtain different image enhancementresults Because our method uses no local convolution oper-ation it introduces fewer halo phenomena compared to theNSCT method Furthermore the computational cost of ourmethod is also significantly lower compared to that of theNSCT method

Although our proposed BEHmethod looks like the origi-nal BM3Dmethod the purpose of our proposedmethod is torepresent image details which is different from the originalBM3D Note that edges in the matched image blocks areusually located in the same position thus edge informationcannot be preserved in the high-frequency subbands afterthe third-dimensional transform Therefore our proposedmethod is different from the BM3D method

In summary we have presented a simple but effectiveimage linear singularity representation method that achievesbetter objective and subjective results when applied to imageenhancement than other state-of-the-art methods

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

8 Mathematical Problems in Engineering

Figure 7 The transformed subbands of the proposed transform To clearly display the details we have increased the brightness of eachhigh-frequency subband

(a) (b) (c)

(d) (e) (f)

Figure 8 The enhancement results (a) is the original image (b) and (c) are the enhanced images by the proposed method by taking 120574 as 3and 5 respectively (d) and (e) are the enhanced images by the NSCTmethod by similarly taking 120574 as 3 and 5 respectively (f) is the enhancedimage by the alpha-rooting method

We also find that NSCT achieves the best LPC-SI valuebut the worst BVDV result on Fingerprint images in Tables1 and 2 respectively This is mainly because the estimatedstandard deviation of noise is too high which results in mostimage edges being removed as noise Therefore the highestLPC-SI value cannot always be trusted This is also the main

reason for us to use both LPC-SI and BVDV to assess theexperimental results

It is worth indicating another significant advantage of ourproposed method ie its lower time complexity comparedto that of the NSCT method For image enhancement on a256times256 grayscale image our method takes only 03 second

Mathematical Problems in Engineering 9

(a) (b)

(c) (d)

Figure 9 Comparison of image enhancement results by the proposed method NSCT and alpha-rooting (a) Input image (b) the enhancedimage by the proposed method (c) the enhanced image by NSCT and (d) the enhanced image by alpha-rooting

while theNSCTmethod takes 126 seconds Ourmethod usesthe same parameter values ie 119899119904119905119890119901 = 3 119899 = 8 and 119870 = 8and the NSCT method uses 2 levels of decomposition ie 2and 4 directional subbands respectively

In the final experiment we also used the proposed BEHmethod to enhance several color images to show superiorenhancement performance and compare the results with theguided image filteringmethod [20 21] with the results shownin Figure 12 Although the image details are enhanced theenhanced images are still visually natural when using theproposed method The guided image filtering method canalso effectively enhance the image details however the imagebackground is also enhanced making the enhanced imageslook rather artificial

6 Conclusions

We have proposed an effective linear singularity representa-tionmethod and have applied thismethod to image enhance-ment in both gray and color images Similar to the BM3Dmethod our method is nonlocal However our methodextracts some spatially adjacent blocks instead of using

block-matching By using our method the image details canbe effectively represented Additionally by using differentparameters to amplify the transformation coefficients ofimage details we can obtain different image enhancementresults Because our method uses no local convolution oper-ation it introduces fewer halo phenomena compared to theNSCT method Furthermore the computational cost of ourmethod is also significantly lower compared to that of theNSCT method

Although our proposed BEHmethod looks like the origi-nal BM3Dmethod the purpose of our proposedmethod is torepresent image details which is different from the originalBM3D Note that edges in the matched image blocks areusually located in the same position thus edge informationcannot be preserved in the high-frequency subbands afterthe third-dimensional transform Therefore our proposedmethod is different from the BM3D method

In summary we have presented a simple but effectiveimage linear singularity representation method that achievesbetter objective and subjective results when applied to imageenhancement than other state-of-the-art methods

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

Mathematical Problems in Engineering 9

(a) (b)

(c) (d)

Figure 9 Comparison of image enhancement results by the proposed method NSCT and alpha-rooting (a) Input image (b) the enhancedimage by the proposed method (c) the enhanced image by NSCT and (d) the enhanced image by alpha-rooting

while theNSCTmethod takes 126 seconds Ourmethod usesthe same parameter values ie 119899119904119905119890119901 = 3 119899 = 8 and 119870 = 8and the NSCT method uses 2 levels of decomposition ie 2and 4 directional subbands respectively

In the final experiment we also used the proposed BEHmethod to enhance several color images to show superiorenhancement performance and compare the results with theguided image filteringmethod [20 21] with the results shownin Figure 12 Although the image details are enhanced theenhanced images are still visually natural when using theproposed method The guided image filtering method canalso effectively enhance the image details however the imagebackground is also enhanced making the enhanced imageslook rather artificial

6 Conclusions

We have proposed an effective linear singularity representa-tionmethod and have applied thismethod to image enhance-ment in both gray and color images Similar to the BM3Dmethod our method is nonlocal However our methodextracts some spatially adjacent blocks instead of using

block-matching By using our method the image details canbe effectively represented Additionally by using differentparameters to amplify the transformation coefficients ofimage details we can obtain different image enhancementresults Because our method uses no local convolution oper-ation it introduces fewer halo phenomena compared to theNSCT method Furthermore the computational cost of ourmethod is also significantly lower compared to that of theNSCT method

Although our proposed BEHmethod looks like the origi-nal BM3Dmethod the purpose of our proposedmethod is torepresent image details which is different from the originalBM3D Note that edges in the matched image blocks areusually located in the same position thus edge informationcannot be preserved in the high-frequency subbands afterthe third-dimensional transform Therefore our proposedmethod is different from the BM3D method

In summary we have presented a simple but effectiveimage linear singularity representation method that achievesbetter objective and subjective results when applied to imageenhancement than other state-of-the-art methods

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

10 Mathematical Problems in Engineering

Figure 10 Image enhancement results by the alpha-rooting BM3D The first row is the original House image noisy image with 120590 = 10 andnoisy image with 120590 = 20 respectively from left to right Row 2 to row 6 give the corresponding simultaneous denoising and enhancementresults by using 120572 = 12 14 16 18 20 respectively

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

Mathematical Problems in Engineering 11

Figure 11 Image enhancement results by BEH-BM3D using different gain factors 120574 = 2 3 5 8 10 from top row to bottom row respectivelyThe left column shows the enhancement results of the original image themiddle column shows the enhancement results of image with addednoise 120590 = 10 and the right column shows the enhancement results of image with added noise 120590 = 20

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

12 Mathematical Problems in Engineering

Figure 12 Enhanced color images Left are the original images middle are the enhanced images by guided image filtering and right are theenhanced images by BEH

Table 1 The comparison of LPC-SI values in the original images and the enhanced images by the NSCT method Alpha-rooting methodand our proposed method All of the values are the best ones for the respective method

Image LPC-SI LPC-SI LPC-SI LPC-SIinput NSCT alpha-rooting our proposed

Barbara 09246 09532 09685 09682House 08760 09302 09395 09603Boat 09341 09503 09529 09590Lena 09060 09411 09549 09629Peppers 09210 09533 09473 09592Couple 09213 09535 09499 09565Hill 08840 09438 09422 09566Man 09104 09508 09499 09594Cameraman 09364 09578 09534 09559Fingerprint 07777 09401 09034 09204

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

Mathematical Problems in Engineering 13

Table 2The BV and DV values in the original image and the enhanced images by the NSCTmethod Alpha-root method and our proposedmethod All of these values are obtained when the LPC-SI is the best

Image Input Alpha-rooting NSCT Our proposedBV DV BV DV BV DV BV DV

Lena 055 660 034 1338 040 1685 036 1838House 062 705 012 2414 026 2077 012 3233Barbara 066 1253 033 3257 048 3148 037 3350Peppers 068 1043 031 1974 047 1304 042 3102Boat 066 947 033 1543 047 2440 034 2307Couple 063 995 034 2420 045 2565 040 3047Hill 058 842 030 1852 041 1878 034 2181Man 053 860 028 1967 048 2166 034 2922Cam 060 1101 026 2836 036 3562 035 4379Fingerprint 053 1895 039 6197 057 1047 052 5054

Data Availability

The images used in this article can be downloaded fromBM3D website

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Science Founda-tion of China [grant numbers 61379015 and 61866005] theNatural Science Foundation of Shandong Province [grantnumber ZR2011FM004] and the Talent Introduction Projectof Taishan University [grant numbers Y-01-2013012 and Y-01-2014018] Dr S-W Lee was partially supported by theInstitute of Information amp Communications TechnologyPlanning amp Evaluation (IITP) grant funded by the Koreagovernment (No 2017-0-00451)

References

[1] S Mallat A Wavelet Tour of Signal Processing Academic Press2nd edition 1999

[2] S Mallat and W L Hwang ldquoSingularity detection and process-ing with waveletsrdquo IEEE Transactions on Information Theoryvol 38 no 2 pp 617ndash643 1992

[3] E J Candes Ridgelets Theory and Applications Department ofStatistics Stanford University 1998

[4] M N Do and M Vetterli ldquoThe finite ridgelet transform forimage representationrdquo IEEE Transactions on Image Processingvol 12 no 1 pp 16ndash28 2003

[5] E J Candes andD LDonoho ldquoCurvelets-a suprisingly effectivenonadaptive representation for objects with edgesrdquo in Curveand Surface Fitting Vanderbilt University Press Saint-MaloFrance 1999

[6] M N Do and M Vetterli ldquoThe contourlet transform an effi-cient directional multiresolution image representationrdquo IEEE

Transactions on Image Processing vol 14 no 12 pp 2091ndash21062005

[7] A L da Cunha J Zhou and M N Do ldquoThe nonsubsampledcontourlet transform theory design and applicationsrdquo IEEETransactions on Image Processing vol 15 no 10 pp 3089ndash31012006

[8] A Buades B Coll and J M Morel ldquoA review of imagedenoising algorithms with a new onerdquoMultiscale Modeling andSimulation vol 4 no 2 pp 490ndash530 2005

[9] K Dabov A Foi V Katkovnik and K Egiazarian ldquoImagedenoising by sparse 3-D transform-domain collaborative filter-ingrdquo IEEE Transactions on Image Processing vol 16 no 8 pp2080ndash2095 2007

[10] Y K Hou C X Zhao D Y Yang and Y Cheng ldquoComments onimage denoising by sparse 3-D transform-domain collaborativefilteringrdquo IEEE Transactions on Image Processing vol 20 no 1pp 268ndash270 2011

[11] X Qu Y Hou F Lam D Guo J Zhong and Z ChenldquoMagnetic resonance image reconstruction fromundersampledmeasurements using a patch-based nonlocal operatorrdquoMedicalImage Analysis vol 18 no 6 pp 843ndash856 2014

[12] X Qu D Guo B Ning et al ldquoUndersampled MRI recon-struction with patch-based directional waveletsrdquoMagnetic Res-onance Imaging vol 30 no 7 pp 964ndash977 2012

[13] YHou SH ParkQWang et al ldquoEnhancement of perivascularspaces in 7 TMR image using haar transformof non-local cubesand block-matching filteringrdquo Scientific Reports vol 7 no 1article no 8569 2017

[14] Y Hou M Liu and D Yang ldquoMulti-stage block-matchingtransform domain filtering for image denoisingrdquo Journal ofComputer-Aided Design and Computer Graphics vol 26 no 2pp 225ndash231 2014

[15] F Zhu Y Hou and J Yang ldquoBlock-matching based multifocusimage fusionrdquoMathematical Problems in Engineering vol 20157 pages 2015

[16] Y Hou and D Shen ldquoImage denoising with morphology-and size-adaptive block-matching transform domain filteringrdquoEurasip Journal on Image and Video Processing vol 59 no 12018

[17] Y Hou J Xu M Liu et al ldquoNLH a blind pixel-level non-localmethod for real-world image denoisingrdquo httpsarxivorgabs190606834 2019

14 Mathematical Problems in Engineering

[18] J Xu L Zhang W Zuo D Zhang and X Feng ldquoPatch groupbased nonlocal self-similarity prior learning for image denois-ingrdquo in Proceedings of the 15th IEEE International Conference onComputer Vision ICCV 2015 pp 244ndash252 Santiago Chile 2015

[19] J Xu D Ren L Zhang and D Zhang ldquoPatch group basedbayesian learning for blind image denoisingrdquo in Proceedings ofthe Asian Conference on Computer VisionWorkshop (ACCVW)Taipei Taiwan 2016

[20] K He J Sun and X Tang ldquoGuided image filteringrdquo inComputer VisionmdashECCV 2010 pp 1ndash14 Springer HeidelbergBerlin Germany 2010

[21] K He J Sun and X Tang ldquoGuided image filteringrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol35 no 6 pp 1397ndash1409 2013

[22] J Wang N T Le J Lee and C Wang ldquoColor face imageenhancement using adaptive singular value decomposition infourier domain for face recognitionrdquo Pattern Recognition vol57 pp 31ndash49 2016

[23] XW Gao andM Loomes ldquoA new approach to image enhance-ment for the visually impairedrdquoElectronic Imaging vol 2016 no20 pp 1ndash7 2016

[24] B Li and W Xie ldquoImage denoising and enhancement basedon adaptive fractional calculus of small probability strategyrdquoNeurocomputing vol 175 pp 704ndash714 2016

[25] B Y Lee and Y S Tarng ldquoApplication of the discrete wavelettransform to the monitoring of tool failure in end millingusing the spindle motor currentrdquo The International Journal ofAdvancedManufacturing Technology vol 15 no 4 pp 238ndash2431999

[26] L Hong YWan and A Jain ldquoFingerprint image enhancementalgorithm and performance evaluationrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 20 no 8 pp 777ndash789 1998

[27] G Liu and J Yang ldquoExploiting color volume and color differ-ence for salient region detectionrdquo IEEE Transactions on ImageProcessing vol 28 no 1 pp 6ndash16 2019

[28] X Jun H Yingkun Y Mengyang et al ldquoSTAR a structure andtexture aware retinex modelrdquo httpsarxivorgabs1906066902019

[29] X Jun Y Huang L Liu et al ldquoNoisy-As-Clean learning unsu-pervised denoising from the corrupted imagerdquo httpsarxivorgabs190606878 2019

[30] J Xu L Zhang and D Zhang ldquoExternal prior guided internalprior learning for real-world noisy image denoisingrdquo IEEETransactions on Image Processing vol 27 no 6 pp 2996ndash30102018

[31] J Xu L Zhang andD Zhang ldquoA trilateral weighted sparse cod-ing scheme for real-world image denoisingrdquo in Proceedings ofthe European Conference on Computer Vision (ECCV) MunichGermany 2018

[32] V Katkovnik A Foi K Egiazarian and J Astola ldquoFromlocal kernel to nonlocal multiple-model image denoisingrdquoInternational Journal of Computer Vision vol 86 no 1 pp 1ndash32 2010

[33] K Dabov A Foi V Katkovnik and K Egiazarian ldquoJointimage sharpening and denoising by 3D transform-domaincollaborative filteringrdquo in Proceedings of the 2007 InternationalTICSP Workshop on Spectral Methods and Multirate SMMSPMoscow Russia 2007

[34] J-L Starck F Murtagh E J Candes and D L DonoholdquoGray and color image contrast enhancement by the curvelet

transformrdquo IEEE Transactions on Image Processing vol 12 no6 pp 706ndash717 2003

[35] G Ramponi N Strobel S K Mitra and T-H Yu ldquoNonlinearunsharp masking methods for image contrast enhancementrdquoJournal of Electronic Imaging vol 5 no 3 pp 353ndash366 1996

[36] S G Chang B Yu and M Vetterli ldquoSpatially adaptive waveletthresholding with context modeling for image denoisingrdquo IEEETransactions on Image Processing vol 9 no 9 pp 1522ndash15312000

[37] R Hassen Z Wang and M M Salama ldquoImage sharpnessassessment based on local phase coherencerdquo IEEE Transactionson Image Processing vol 22 no 7 pp 2798ndash2810 2013

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