Pattern Recognition 43 (2010) 4089–4100

Contents lists available at ScienceDirect

Pattern Recognition

0031-32

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/pr

Homogeneity similarity based image denoising

Qiang Chen n, Quan-sen Sun, De-shen Xia

School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing, China

a r t i c l e i n f o

Article history:

Received 4 March 2009

Received in revised form

11 September 2009

Accepted 4 July 2010

Keywords:

Image denoising

Homogeneity similarity

Patch-based method

Structure similarity

03/$ - see front matter & 2010 Elsevier Ltd. A

016/j.patcog.2010.07.002

esponding author.

ail address: chen2qiang@163.com (Q. Chen).

a b s t r a c t

This paper presents a homogeneity similarity based method, which is a new patch-based image

denoising method. In traditional patch-based methods, such as the NL-means method, block matching

mainly depends on structure similarity. The homogeneity similarity is defined in adaptive weighted

neighborhoods, which can find more similar points than the structure similarity, and so it is more

effective, especially for points with less repetitive patterns, such as corner and end points. Comparative

results on synthetic and real image denoising indicate that our method can effectively remove noise and

preserve effective information, such as edges and contrast, while avoiding artifacts. The application on

medical image denoising also demonstrates that our method is practical.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

The degradation of an image is always unavoidable duringacquisition, and the restoration of degraded images is an importantand widely studied problem in computer vision and imageprocessing. One of the most fundamental image restorationproblems is denoising, whose goal is to recover the original imagefrom a noisy or contaminated observation. There are a number ofgoals we want to meet in designing image denoising algorithms [1].

1.

The perceptually flat regions should be as smooth as possible.Noise should be completely removed from these regions.

2.

Image edges and corners should be well preserved. This meansthat the edge and corner should not be blurred or sharpened.

3.

Texture detail should not be lost. This is one of the hardestcriteria to match. Since image denoising algorithms tend tosmooth the image, it is easy to lose texture detail duringsmoothing.

4.

The global contrast should be preserved (that is, the lowfrequencies of the denoised and input images should be mostlyidentical).

5.

No artifacts, such as staircase and ringing, should appear in thedenoised image.

Summarily, one ideal image denoising algorithm should havethe following properties: noise can be removed completely;effective information (edge, corner, texture and contrast) can bepreserved and artifacts will not appear.

ll rights reserved.

In order to find an ideal image denoising algorithm, researchershave studied for decades and proposed various algorithms. Currently,there exist two popular kinds of image denoising methods: wavelet-based methods [2–7] and partial differential equations (PDE) basedmethods [8–14]. Due to the multi-resolution and directionalitycharacteristics of wavelet transformation, wavelet-based imagedenoising methods can effectively preserve texture details; howeverit is hard to avoid ringing artifacts of wavelet reconstruction. In thispaper, we mainly discuss the image denoising in spatial domain.

PDE based image denoising methods can be categorized intodiffusion-based method and variational-based method. The basicidea behind anisotropic diffusion is to distinguish between edgesand noise by utilizing a gradient operator, then iteratively elimi-nate small variations due to the noise, and meanwhile preservelarge variations due to the edges. Since the introduction ofPerona–Malik (PM) model [8], a considerable amount of researchhas been devoted to the theoretical and practical understandingof this and related methods for image enhancement. Research inthis area has been targeted towards the following four aspects:understanding the mathematical properties of anisotropic diffu-sion and related variational formulation [10,15,16], developingrelated well-posed and stable equations [9,17], extending andmodifying anisotropic diffusion [18–21], and studying the rela-tions between anisotropic diffusion and other image processingmethods [22–24]. A classical variational denoising method is thetotal variation (TV) minimizing process of Rudin–Osher–Fatemi(ROF) [9]. Although these second-order PDEs, such as PM modeland ROF model, have been demonstrated to be able to achieve agood trade-off between noise removal and edge preservation,they tend to cause the processed images to exhibit ‘‘staircaseeffect’’. In order to avoid transforming ramps into stairs, several

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–41004090

higher order PDEs [11,12,25,26] and convex variational penaltyfunctions [27,28] were adopted. Although the higher order PDEscan preserve ramps without penalizing sharp edges; the mathe-matical problem is much more challenging and speckles mayappear.

Traditional spatial denoising methods are based on the intensitysimilarity (such as the bilateral filtering [29]) or gradient informa-tion (such as the PM model) of each pixel, which cannot effectivelypreserve weak edges and texture details. Since Buades et al. [30]presented the non-local means (NL-means) method in 2005, theneighborhood similarity has been widely adopted for imagedenoising. Using a local region can better represent structureinformation than using a single pixel, thus the NL-means methodcan effectively preserve effective information with repetitivepatterns, such as texture details. However, structures with lessrepetitive patterns, such as corners, cannot be effectively pre-served. This paper presents an image denoising method based onhomogeneity similarity, which also utilizes the neighborhoodsimilarity like the NL-means method. The weight of the matchingneighborhood, however, is different from that of the NL-meansmethod, which depends on intensity similarity to preserve effectiveinformation well, even points without repetitive patterns.

The paper is organized as follows: after reviewing relevantwork in Section 2, we introduce the homogeneity similarity basedimage denoising method in Section 3. In Section 4, we analyze anddiscuss our method by comparative results on synthetic and realimage denoising and the application on medical image denoising.Finally, conclusions are drawn in Section 5.

Fig. 1. Neighborhood demonstration.

2. Related work

Since our method also belongs to a patch-based method, webriefly review previous work on patch-based image denoisingmethods in this section.

If both the scene and camera are static, we can simply takemultiple pictures and use the mean to remove the noise. Thismethod is impractical for a single image, but a temporal mean canbe computed from a spatial mean as long as there are enoughsimilar patterns in the single image. The NL-means algorithm [30]computes the denoised image intensity as a weighted average of asample of image intensities, where the weights are derived from theneighborhoods of the pixels in the sample. The region informationcan describe image feature better than a single pixel, so patch-basedmethods work well for texture-like images containing manyrepeated patterns. However, compared with other denoising algo-rithms that have O(n2) complexity, where n is the image width,these algorithms have O(n4) time complexity, which is prohibitivefor real-world applications. Liu et al. [31] presented a fast NL-meansalgorithm based on efficient summed square image scheme and fastFourier transform. Mahmoudi and Sapiro [32] proposed an algo-rithm for accelerating spatial NL-means denoising with filters thateliminate unrelated neighborhoods from weighted average, andBrox et al. [33] presented a cluster tree for the preselection of similarpatches to speedup NL-means filters. Coupe et al. [34] presentedanother preselection based speedup. Awate and Whitaker [35]proposed an unsupervised, information-theoretic, adaptive filter(UINTA) that improves the predictability of pixel intensities fromtheir neighborhoods by decreasing their joint entropy. The NL-means algorithm could be considered a special case of the UINTAalgorithm involving a single iteration and a user-defined Gaussiankernel width. Brox and Cremers [36] formulated a variationaltechnique that leads to an adaptive version of the NL-meansalgorithm, and introduced the idea to replace the neighborhoodweighting by a sorting criterion. By introducing spatial adaptivity, anextended version of the NL-means algorithm was proposed by

Kervrann and Boulanger [37,38], which associates each pixel with aweighted sum of data points within an adaptive neighborhood.A patch-based Tikhonov regularization method was proposed inRef. [39], which is actually quite similar to the NL-means algorithm.Dabov et al. [40] proposed an image denoising strategy based on anenhanced sparse representation in transform domain, namely Block-Matching and 3D filtering (BM3D). Katkovnik et al. [41] presented asingle- and multiple-model transform domain non-local approach,which is a development and extension of the BM3D algorithm. Heand Greenshields [42] presented a non-local maximum likelihoodestimation method for magnetic resonance (MR) image denoising.The patch-based methods are also extended to 3D data [40] orimage sequence [43–45].

Summarily, the improvement or extension for the NL-meansmethod includes fast algorithm [31–33], adaptive neighborhoodor parameters [35–38], collaborative strategy [40–42] and imagesequence [43–45]. The traditional and improved NL-meansmethods are in essence based on structure similarity of neighbor-hoods, because the weights of the neighborhoods with the samestructure are equal. In this paper, we propose a homogeneitysimilarity based image denoising method, where the weightdepends on the homogeneity similarity, instead of the structuresimilarity.

3. Homogeneity similarity based image denoising

3.1. Traditional image filtering framework

The basic idea of image filtering is to update the intensity valueof the current pixel by using its neighborhood information, andthe filtering framework can be represented as follows:

8ðx,yÞAO, uðx,yÞ ¼

ROIðp,qÞwðx,y,p,qÞdpdqR

Owðx,y,p,qÞdpdqð1Þ

with

wðx,y,p,qÞ ¼ cðx,y,p,qÞ � sðx,y,p,qÞ ð2Þ

Fig. 2. Analysis of the neighborhood shape for edges. The left is an edge image, and the right shows the corresponding intensity values (0 or 1) of each neighborhood.

Fig. 3. Analysis of the neighborhood shape for corners marked with a red ‘+’. The

center points of the red region and the dark red regions belong to dark

homogeneous region, while the center points of the blur regions belong to white

homogeneous region.

Fig. 4. Analysis of the neighborhood shape for ends marked with a red ‘+’. The

center points of the red region and the dark red region belong to dark

homogeneous region, while the center points of the blur regions belong to white

homogeneous region.

Fig. 5. Homogeneity similarity weight.

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–4100 4091

where I and u denote an original noisy image and the denoisedimage, respectively. The weight function w(x,y,p,q) depends onboth the spatial similarity between points (x,y) and (p,q) (i.e.c(x,y,p,q)), as well as the range similarity between correspondingpatches centered at (x,y) and (p,q) (i.e. s(x,y,p,q)). In practicalimplementation, the spatial neighborhood is often restricted in asmall search window for computational purpose, instead of theentire image domain O. The main difference of various imagefiltering methods is the construction of the weight function w. Thespatial filtering weight function c( � ) often takes the following twoforms: 1 (i.e. constant) or Gaussian function, and the rangefiltering weight function s( � ) often has the following three forms:1, e�ð9Iðx,yÞ�Iðp,qÞ92

Þ=h2(i.e. point similarity) and e�ð9P

Iðx,yÞ�PI

ðp,qÞ92Þ=h2

(i.e. neighborhood similarity), where h is the range filteringparameter, and the patch PI

ðx,yÞ is defined as the ordered vector ofall image values belonging to a neighborhood of I centered at (x,y).

Fig. 6. Comparison of three neighborhood distances (from the up to down) for the noisy edge, corner and end points, respectively.

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–41004092

By combining the spatial and the range filtering weight functions,we can obtain different image filtering methods.

Patch-based methods utilizing neighborhood similarity canpreserve texture details better than point similarity basedmethods. Homogeneity similarity based image denoising methodproposed in this paper also belongs to patch-based methods,while the construction of the neighborhood (or patch) weight ofour method is different from that of the traditional patch-basedmethods. Currently, for most of patch-based methods theneighborhood weight is constructed based on spatial similarity,such as the Gaussian function. The neighborhood weight of ourmethod is adaptively constructed based on intensity similarity.In the following, we will introduce the origin of the proposedmethod by analyzing the influence of the neighborhood shape onthe weight function.

3.2. Analysis of the neighborhood shape

The common definition of a patch is that PIðx,yÞ located at

(x,y)AO on the image I is the set of all image values belonging to aspatially discretized local d� d (square) neighborhood of I

centered at (x,y). The size d is considered as odd, i.e. d¼2r+1(rAN). Fig. 1 shows a neighborhood of the yellow point (x,y)with a 3�3 square region. The square neighborhood can describethe structure features, such as edges and texture details, andtherefore the denoising methods based on square patch similaritycan preserve edges and texture details well. However, traditionalsquare neighborhood still has some deficiencies; it is not veryeffective for edges, corners and ends.

Here, we analyze the influence of the neighborhood shape foredges in detail, which is similar for corners and ends. Fig. 2 showsa 3�3 square neighborhood of an edge point (x,y) marked with ared ‘+’. In order to omit the influence of the neighborhood filteringparameter h, we define the neighborhood distance based on

structure similarity as

dsðx,y,p,qÞ ¼ PIðx,yÞ�PI

ðp,qÞ

������2¼Xr

i ¼ �r

Xr

j ¼ �r

ðIðxþ i,yþ jÞ�Iðpþ i,qþ jÞÞ2:

ð3Þ

Then, for the edge point (x,y) in Fig. 2, the three neighborhooddistances of (p1,q1), (p2,q2) and (p3,q3) are 0, 3 and 3, respectively.Since the point (p3,q3) belongs to the white region and the otherthree points belong to the black homogeneous region, theneighborhood distances of (p1,q1) and (p2,q2) should be smallerthan that of (p3,q3). Though the edge can be preserved by tradi-tional patch-based methods because the range weight s(x,y,p3,q3)is much smaller than s(x,y,p1,q1), the edge will still be smoothed alittle.

Traditional NL-means filter only has translation invariance. Tofind more pixels with similar structure, several rotation invariantNL-means methods are presented, such as non-local similarityfilter [46]. In the case of additive white Gaussian noise (WGN), themore similar pixels can be correctly found, the more the signal-to-noise ratio (SNR) can be improved [47]. The basic idea of theproposed method is also to find more similar pixels for edges andcorners. If the neighborhood shape was taken as a rectangle, suchas the gray dashed region in Fig. 2, the neighborhood distances,ds(x,y,p1,q1) and ds(x,y,p2,q2), will be equal and both smaller thands(x,y,p3,q3). Thus for the edge point (x,y) homogeneity similaritybased methods can find more similar pixels, and then the edgewill be preserved better than traditional patch-based methods.

For the corner point in Fig. 3 and the end point in Fig. 4, we cansimilarly analyze the neighborhood shape and observe that thegray dashed rectangle regions are better for effective informationpreservation than the traditional red square regions.

According to the analysis of the neighborhood shape above, itcan be seen that traditional square neighborhood is not veryeffective in distinguishing homogenous and inhomogeneous

Fig. 7. Comparison of the effect on a synthetic image. (a) Original image. (b) Noisy image (additive white Gaussian noise). (c) Bilateral filtering. (d) NL-means. (e) UINTA. (f)

Patch-based Tikhonov regularization method. (g) Ours with h2¼0.015.

Fig. 8. Zoomed details from Fig. 7. (a)–(g) are corresponding to Fig. 7(a)–(g).

Table 1PSNR (unit: db) and Mean Structural SIMilarity (MSSIM) of Fig. 7. The MSSIM is

between 0 and 1 with a score of 1 being given only if the denoised image is exactly

equivalent to the original image.

Figs 7(b) 7(c) 7(d) 7(e) 7(f) 7(g)

PSNR 15.63 19.43 20.69 20.38 19.38 20.91MSSIM 0.34 0.63 0.80 0.68 0.79 0.80

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–4100 4093

points, especially for corners and ends, because it is difficult tofind the neighborhoods with similar structures. In order to makethe neighborhood similarities of the points in homogeneous andinhomogeneous regions equal and different, respectively, theneighborhood shape in the following proposed method isadaptively determined by the intensity similarity, which is similarwith the gray dashed regions above.

3.3. Definition of our method

In order to represent the homogeneity similarity well, anadaptive neighborhood is adopted in our method, like the gray

dashed regions in Figs. 2–4. The traditional range similarity ofpatch-based image filtering is weighted by spatial similarity,while the range similarity of our method is weighted by

180

160

140

120

100

80

Inte

nsity

val

ue

60

40

20

0(143,86) (39,176)

Corner position

(10,117) (34,36)

OriginalNoisyBilateral filteringNL-meansUINTATikhonoyOurs

Fig. 9. Intensity values of Fig. 7 in four corner positions.

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–41004094

homogeneity similarity defined as follows:

Hðx,yÞði,jÞ ¼e�kðIðx,yÞ�Iði,jÞÞ2

Pmi ¼ 1

Pnj ¼ 1 e�kðIðx,yÞ�Iði,jÞÞ2

ð4Þ

where m and n are the height and width of the image patch,respectively. The parameter k is a positive constant (the default is2 in this paper), which adjusts the influence of intensity similarity.Eq. (4) represents the weight of each point (i,j) in the image patchcentered at (x,y). When k is close to 0, the influence of intensitysimilarity is small (similar with the spatial weight functionc( � )¼1, or similar with the structure similarity), and vice versa.

Fig. 5 shows the corresponding gray levels mapping of nor-malized homogeneity similarity weight of the edge point (x,y) inFig. 2, the corner point in Fig. 3 and the end point in Fig. 4,respectively, where the gray level value is proportional to theweight. From Fig. 5, we can observe that the weight of the point,which has similar intensity with the points marked with the red‘+’ in Figs. 2–4, is large, while the weight of the point, which haslarge intensity difference with these points, is small. Thus, theneighborhood shapes based on homogeneity similarity, namelythe white regions in Fig. 5, are approximately equal to the graydashed regions in Figs. 2–4. According to the analysis above, it canbe seen that the neighborhood shape of our method is adaptivelydetermined by intensity similarity.

The weight function wHðx,y,p,qÞ of the proposed method isdefined as

wHðx,y,p,qÞ ¼ e� 9Hðx,yÞ3ðPIðx,yÞ�PI

ðp,qÞ Þ92

� �=h2

ð5Þ

where ‘3’ represents the element-wise multiplication operator,namely the multiplications of the corresponding elements of H(x,y)

and ðPIðx,yÞ�PI

ðp,qÞÞ. The homogeneity similarity based imagedenoising is defined by the formula

uðx,yÞ ¼

ROIðp,qÞwHðx,y,p,qÞdpdqR

OwHðx,y,p,qÞdpdq: ð6Þ

The homogeneity similarity based image denoising can be seenas an adaptive patch-based method, because the image patch

similarity is adaptively weighted according to the intensitysimilarity. The weights of the points in homogeneous region willbe larger than those of the points in inhomogeneous region. Thepurpose of our method is to acquire the weighted average of thepoints in homogeneous regions, rather than the points withsimilar structure.

3.4. Comparison of the neighborhood distance

The neighborhood distances based on point similarity isdefined as

dIðx,y,p,qÞ ¼ Iðx,yÞ�Iðp,qÞ�� ��2: ð7Þ

The neighborhood distances based on homogeneity similarityis defined as

dHðx,y,p,qÞ ¼ Hðx,yÞ3ðPIðx,yÞ�PI

ðp,qÞÞ

������2: ð8Þ

In Fig. 6 the neighborhood distances based on structuresimilarity and homogeneity similarity, namely ds and dH, areboth defined on a square neighborhood of 5�5 pixels, andthe neighborhood distance images are all normalized for com-parison.

Fig. 6 shows the comparison of three neighborhood distancesfor the edge, corner and end points marked with the red ‘+’ in thenoisy (WGN) images. For the neighborhood distance images(Fig. 6b–d), the gray values are proportional to the distancevalues. From Fig. 6, it can be seen that homogeneity similarity andpoint similarity can distinguish the black and the white homo-geneous regions more effectively than the structure similaritythat is effective to find the points with similar structure. Patchsimilarity (structure similarity and homogeneity similarity) hasbetter noise-immunity or robustness than point similarity [30].Therefore, the homogeneity similarity can distinguish differenthomogeneous regions better than structure similarity, and hasbetter noise-immunity than point similarity.

Table 2PSNR values when our method with different patch size is applied to noisy (WGN) images (s¼20).

Patch size h2 Lena 512�512 Barbara 512�512 Boat 512�512 House 256�256 Peppers 512�512

3�3 0.009 30.79 28.95 29.01 31.46 30.75

5�5 0.004 31.40 29.97 29.43 32.16 31.33

7�7 0.003 31.64 30.34 29.42 32.46 31.53

9�9 0.003 31.69 30.39 29.24 32.61 31.5411�11 0.003 31.68 30.30 29.06 32.59 31.46

Table 3Performance of our method with a search window of 21�21 pixels and a neighborhood of 7�7 pixels when applied to test noisy (WGN) images.

s/PSNR k h2 Lena 512�512 Barbara 512�512 Boat 512�512 House 256�256 Peppers 512�512

5/34.14 2 0.00025 36.67 35.26 34.42 37.39 35.75

10/28.15 2 0.0008 34.66 33.35 32.41 35.22 34.13

15/24.65 2 0.0015 33.02 31.67 30.80 33.55 32.74

20/22.19 2 0.003 31.64 30.34 29.42 32.46 31.53

25/20.29 2 0.005 30.55 29.12 28.26 31.39 30.39

50/14.69 1 0.02 26.80 24.66 24.61 26.68 26.24

75/11.85 0.5 0.02 24.23 22.18 22.56 23.80 23.36

100/10.18 0.5 0.05 21.98 20.32 21.00 21.59 20.96

Table 4Performance of denoising methods when applied to test noisy (WGN) images.

Image s/PSNR Lena 20/22.15 Barbara 20/22.17 Boat 20/22.18 House 20/22.14 Peppers 20/22.31

Our method 31.64 30.34 29.42 32.46 31.53Bilateral filtering [29] 29.16 26.12 27.11 28.76 29.42

NL-means [30] 31.46 30.07 29.46 32.22 31.39

UINTA [35] 30.86 29.71 29.05 31.52 29.67

Tikhonov regularization [39] 30.82 27.80 28.30 30.92 31.07

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–4100 4095

4. Experimental results and discussion

To verify the proposed image denoising methods, we haveapplied our method to the noise removal of a variety of images. Inthis section, we first analyze the performance of our method bycomparing denoising results of a synthetic image. Then we reportcomparative denoising results with several methods for thestandard noisy images. Finally, our method is used for thedenoising of medical images. The denoising methods areimplemented in Matlab. The processing time of our method isclose to the NL-means method, because the construction ofhomogeneity similarity is similar with that of structure similarity.

1 Publicly available implementation: http://dmi.uib.es/�abuades/software.

html.2 Publicly available implementation: http://www.cs.utah.edu/�suyash/pubs/

uinta.html.

4.1. Experiments with a synthetic image

Fig. 7 shows a comparison of several methods on a 200�200synthetic image, and Fig. 8 shows the corresponding zoomeddetails. The global control parameters of these methods weretuned (we have to try several values) to both eliminate noiseand simultaneously to get the best peak signal-to-noise ratio(PSNR) value. The PSNR is computed according to the standard

formula PSNR¼ 10log10ð2552=MSEÞwhere MSE¼Pðx,yÞAOðu0ðx,yÞ

�

�uðx,yÞÞ2Þ= 9O9 and u0 is the noise-free original image. Fig. 8 is the

zoomed details (i.e. the up angle regions of the five-pointed star)from Fig. 7. Table 1 shows the PSNR and the mean structuralsimilarity (MSSIM) [48], respectively. The MSSIM is defined as

MSSIMðu0,uÞ ¼ ð1=MÞPM

j ¼ 1 SSIMðuj0,ujÞ, where uj

0 and uj are the

image contents at the jth local window, and M is the number oflocal windows of the image. The structural similarity (SSIM) is

defined by

SSIMðu0,uÞ ¼ð2mu0

muþC1Þð2su0uþC2Þ

ðm2u0þm2

uþC1Þðs2u0þs2

uþC2Þ,

where mu0and mu are the mean intensity of u0 and u, respectively;

su0and su are the standard deviation; the correlation coefficient

su0corresponds to the cosine of the angle between the vectors

u0�mu0and u�mu; C1 and C2 are two constants. The detailed

interpretation of SSIM can refer to Ref. [48]. In order to evaluatethe corner preservation, Fig. 9 shows the intensity values in fourcorner positions remarked with red circles as shown in Fig. 7a,where each intensity value is added 1 to make the intensity value‘0’ visible. Fig. 7a is the original image, and Fig. 7b is the noisyimage with additive white Gaussian noise. Here, we compared ourmethod with bilateral filtering [29] and three patch-basedmethods (i.e. NL-means method1 [30], UINTA method2 [35] andTikhonov regularization method [39]). For these patch-basedmethods (Fig. 7d–g and Table 4 in Section 4.2, Figs. 15 and 16 inSection 4.3), we have fixed a search window of 21�21 pixels anda square neighborhood of 7�7 pixels.

From Figs. 7–9 and Table 1, we can observe that (1) the PSNR ofour method is the largest, and the MSSIM of our method isapproximately equal to those of other patch-based methods, suchas the NL-means method. Therefore, our method retains the finecharacteristics of the traditional patch-based methods, such as the

Fig. 10. Comparison of the effect on the Lena image. (a) Original image. (b) Noisy image (s¼20, PSNR¼22.15). (c) Bilateral filtering (PSNR¼29.16). (d) NL-means

(PSNR¼31.46). (e) UINTA (PSNR¼30.86). (f) Patch-based Tikhonov regularization method (PSNR¼30.82). (g) Our method (PSNR¼31.64).

Fig. 11. Zoomed details from Fig. 10. (a)–(g) are corresponding to Fig. 10(a)–(g).

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–41004096

texture detail preservation; (2) our method can preserve edges andcorners better than structure similarity based methods, which canbe seen from Figs. 8 and 9. Therefore, our method is better thanpoint similarity based methods, such as the bilateral filtering, and isa slightly better than structure similarity based methods for pointsthat cannot find neighborhoods with similar structures, such ascorners. For the structure preservation the homogeneity similaritybased method is similar with structure similarity based methods.

It is more difficult for patch-based methods to completelyremove noise in ramp regions or near boundaries than that in flatregions, because it has more obvious structure property. If thereare some points that have the similar intensity value and similar

structure with the noise, it will be probably preserved due to thestructure preservation capability of the patch-based methods.Therefore, there seems to be different kind of artifacts in the rampand boundary regions of Fig. 7d–g. In fact, these ‘‘artifacts’’ are thenoise that cannot be completely removed, not introducedartifacts, which can be removed by increasing the neighborhoodfiltering parameter h. But for the iterated form of the NL-meansfilter, a hallucination of regular patterns in noise will be intro-duced for large iteration numbers [36]. For an increasing numberof iterations, the iterated NL-means filter acts more and morecoherence enhancing reminiscent of curvature motion or coher-ence enhancing anisotropic diffusion [49].

Fig. 12. Method noise (magnified �2) of the denoised results of the Lena image. (a) Bilateral filtering. (b) NL-means. (c) UINTA. (d) Patch-based Tikhonov regularization

method. (e) Our method.

Fig. 13. Comparison between the bilateral filtering, NL-means method and exemplar-based method when applied to the noisy (WGN) Barbara image. (a) Bilateral filtering

(PSNR¼26.12). (b) NL-means method (PSNR¼30.07). (c) Exemplar-based method (PSNR¼30.37). (d) Our method (PSNR¼30.34).

Fig. 14. Illumination of the texture preservation of our method. (a) Texture fragment from the noisy (WGN, s¼20) Barbara image. (b) Denoised result. (c) Method noise

(magnified �2).

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–4100 4097

Fig. 15. Comparison of the effect on a brain MR image. (a) Original image. (b) Noisy image (s¼5, Rician noise, PSNR¼33.23). (c) Bilateral filtering (PSNR¼34.34). (d) NL-

means (PSNR¼35.25). (e) Our method (PSNR¼35.33).

Fig. 16. Denoising results on three medical images. Top row: neural cells (h2¼0.0005); Middle row: cardiac MR image (h2

¼0.0005); Bottom row: brain MR image

(h2¼0.001). (a) Noisy image. (b) Denoised result. (c) Method noise (magnified �2).

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–41004098

4.2. Experiments with standard noisy images

In the section, we show the comparison to the state-of-the-artwith standard noisy images from Portilla et al. [5] available athttp://decsai.ugr.es/� javier/denoise/test_images/. Table 2 reportsthe PSNR values obtained by varying patch size for different testimages. Note the PSNR values are close for every patch sizesexcept for 3�3 pixels. 9�9 patch seems appropriate in mostcases and the PSNR with 7�7 patch is very close to that of 9�9patch. By considering both the denoising result and computing

complexity, we adopt 7�7 patch for our method. Table 3 showsthe PSNR values using our method when applied to this set of testimages for a wide range of noise variances as in Portilla et al. [5]and Kervrann et al. [38]. When noise variance is very large, suchas 100, the parameter k should be small, because the weight ofhomogeneity similarity is unreliable. For this case, our methodis similar with structure similarity based methods, such as theNL-means method.

Table 4 shows the PSNR values of several state-of-the-artmethods for the test images. One point similarity based method

Q. Chen et al. / Pattern Recognition 43 (2010) 4089–4100 4099

(bilateral filtering [29]) and three structure similarity basedmethods (NL-means method [30], UINTA [35] and Tikhonovregularization method [39]) are compared with our method. Inmost cases, our method outperforms any of the tested methodsand patch-based methods are generally better than point-basedmethods. Here, we do not compare our method with some otherimproved patch-based methods, such as the adaptive methods[36–38] and the collaborative method [40], because the purposeof the comparison is to demonstrate the basic performance ofthree kinds of methods, namely point similarity based method,structure similarity based method and homogeneity similaritybased method.

Figs. 10 and 12 show the visual impression and method noiseof the state-of-the-art methods for the Lena image, respectively.Fig. 11 shows the corresponding zoomed details. Compared withthe bilateral filtering (Fig. 12a), the structural information of thepatch-based methods (Fig. 12b–e) is not obvious. Figs. 10–12demonstrate that structure similarity based methods (Fig. 10d–f)are better than point similarity based method (Fig. 10c), andhomogeneity similarity based method (Fig. 10g) is a little betterthan structure similarity based methods.

Fig. 13 shows the comparison between several methods whenapplied to the noisy Barbara image. Fig. 13c is the denoised result ofthe exemplar-based method [38] from Kervrann et al. available athttp://www.irisa.fr/vista/Themes/Demos/Debruitage/ImageDenoising.html. It can be seen from Fig. 13 that patch-based methods (Fig. 13b–d) can preserve details better than bilateral filtering (Fig. 13a); thereexist obvious artifacts in the exemplar-based method; both visuallyand in terms of PSNR, our method outperforms the NL-meansmethod.

Fig. 14 shows the illumination of the texture preservation ofour method. From the visual impression (Fig. 14b) and themethod noise (Fig. 14c), we can see that our homogeneitysimilarity based method can preserve texture details well, likestructure similarity based methods (the main advantage ofstructure similarity based methods is texture preservation).

4.3. Application on medical image denoising

Medical images often need denoising before being subjected tostatistical analysis. Fig. 15 shows the comparison of the effect on abrain magnetic resonance (MR) image obtained from BrainWeb[50]. Fig. 15a is the noise-free image, and Fig. 15b is the noisyimage with Rician noise. Statistically, NL-means will produce anoptimal denoising result if the noise can be modeled as Gaussian[42]. When the standard deviation s is small, such as s¼5, theRician distribution approaches a Gaussian. Thus, the NL-meansmethod and our method are approximately proper for Fig. 15b.Fig. 15c–e are the denoised results of the bilateral filtering, NL-means filtering and our method, respectively. The patch-basedmethods (Fig. 15d and e) are both visually and in terms of PSNRbetter than the bilateral filtering (Fig. 15c), and our method is aslightly better for the edge and corner preservation than the NL-means method.

In additional, we give the denoising results (Fig. 16) of three realmedical images: a confocal image depicts neural cells from Kervrannet al. [38] available at http://www.irisa.fr/vista /Themes/Demos/Debruitage/ImageDenoising.html, a cardiac MR image and a brainMR image. The three real applications demonstrate that our methodcan remove noise and preserve important structures well, and thereis not obvious structure information in the method noise.

According to the experimental analysis above, we can observethat homogeneity similarity based method seems to have theproperties, which one ideal image denoising algorithm should

have, namely good denoising capability, effective information(such as edge, texture and contrast) preservation and no artifacts.

5. Conclusions

In this paper, we present a homogeneity similarity basedimage denoising method, which can effectively distinguish pointsin different homogeneous regions. The homogeneity similarityweight is constructed adaptively based on the intensity similarity,which is more effective for edge and corner points than thestructure similarity weight. Homogeneity similarity based onneighborhoods has a better noise-immunity than point similarity.By comparing with several structure similarity based methods,our method has a good overall performance for the denoising ofsynthetic and real images. The medical image denoising indicatesthat our method is feasible to remove noise and preserveimportant features. The homogeneity similarity of the proposedmethod is based on the pointwise intensity similarity, so somenoise near boundary may be preserved. In the future, we will domore research on the choosing of the homogeneity region that hasbetter noise immunity, in order to make the homogeneitysimilarity based image denoising more effective.

Acknowledgements

The authors sincerely thank the reviewers whose valuablecomments have improved this paper; and the work described inthis paper was supported by two grants from the National NaturalScience Foundations of China under Grant nos. 60805003/60773172 and Special Grade of China Postdoctoral ScienceFoundation under Grant no. 200902519.

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Qiang Chen received B.Sc. degree in computer science and Ph.D. degree in Pattern Recognition and Intelligence System from Nanjing University of Science and Technology,China, in 2002 and 2007, respectively. Currently, he is an associate professor in the School of Computer Science and Technology at the Nanjing University of Science andTechnology. His main research topics are image segmentation, object tracking, image denoising, and image restoration.

Quan-Sen Sun received his Ph.D. degree in pattern recognition and intelligence system from Nanjing University of Science and Technology (NUST), China, in 2006. He is aprofessor in the School of Computer Science at NUST. He visited the Department of Computer Science and Engineering, The Chinese University of Hong Kong in 2004 and2005, respectively. His current interests include pattern recognition, image processing, computer vision and data fusion.

De-Shen Xia received B.Sc. degree in Radiology of Nanjing Institute of Technology, China, in 1963, and Ph.D. degree in the Faculty of Science of Rouen University, France, in1988. Currently, he is a Professor in the School of Computer Science and Technology, Nanjin University of Science and Technology (NJUST), Nanjing, China, HonoraryProfessor in ESIC/ELEC, Rouen, France and Research member in Computer Graphics Lab of CNRS, France. He is also Director of Image Recognition Lab in Nanjing Universityof Science and Technology, China. He is the direction member of National Remote Sensing Federation, China, and the direction member of Micro-Computer ApplicationAssociation of Province Jiangsu, China. His research interests include image processing, analysis, and recognition, remote sensing, medical imaging and mathematics inimaging.