Reduced-Dimensional Capture of High-Dynamic Range Images …downloads.hindawi.com/journals/ddns/2020/6703528.pdf · 2020-04-30 · ResearchArticle Reduced-Dimensional Capture of High-Dynamic

Research ArticleReduced-Dimensional Capture of High-Dynamic RangeImages with Compressive Sensing

Shundao Xie 1 Wenfang Wu1 Rongjun Chen 12 and Hong-Zhou Tan 1

1School of Electronics and Information Technology Sun Yat-sen University Guangzhou 510006 China2School of Computer Science Guangdong Polytechnic Normal University Guangzhou 510665 China

Correspondence should be addressed to Hong-Zhou Tan issthzmailsysueducn

Received 20 December 2019 Revised 26 February 2020 Accepted 5 March 2020 Published 27 April 2020

Guest Editor Jianbiao Zhang

Copyright copy 2020 Shundao Xie et al is 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

e range of light illumination in real scenes is very large and ordinary cameras can only record a small part of this range which isfar lower than the range of human eyesrsquo perception of light High-dynamic range (HDR) imaging technology that has appeared inrecent years can record a wider range of illumination than the perceptual range of the human eye However the currentmainstream HDR imaging technology is to capture multiple low-dynamic range (LDR) images of the same scene with differentexposures and then merge them into one HDR image which greatly increases the amount of data captured e advent of single-pixel cameras (compressive imaging system) has proved the feasibility of obtaining and restoring image data based on compressivesensing erefore this paper proposes a method for reduced-dimensional capture of high dynamic range images with com-pressive sensing which includes algorithms for front end (capturing) and back end (processing) At the front end the K-SVDdictionary is used to compressive sensing the input multiple-exposure image sequence thereby reducing the amount of datatransmitted to the back end At the back end the Orthogonal Matching Pursuit (OMP) algorithm is used to reconstruct the inputmultiple-exposure image sequence A low-rank PatchMatch algorithm is proposed to merge the reconstructed image sequence toobtain anHDR image Simulation results show that under the premise of reducing the complexity of the front-end equipment andthe amount of communication data between the front end and the back end the overall system achieves a good balance betweenthe amount of calculation and the quality of the HDR image obtained

1 Introduction

With the development of mobile Internet and Internet ofings (IoT) technology devices with cameras are becomingmore common such as smart phones network surveillancecameras laptop computers autonomous vehicles and trafficmonitoring cameras Furthermore camera is now an es-sential feature for smartphones and laptops Howevercommon cameras on the market can only capture low-dy-namic range (LDR) images ie these cameras can onlycapture a small part of the rang of illuminance in a real scenee dynamic range of the real scene perceptible to thehuman eye is as high as 108 1 but the dynamic range of theLDR images captured by these cameras is only 28 1 or 216 1which makes the LDR images unable to truly represent thereal scene To solve this problem high-dynamic range

(HDR) imaging technique has been proposed and it cancapture a wider range of illumination than that of humaneye ere are two ways to obtain HDR images software andhardware e hardware method directly captures HDRimages by increasing the dynamic range of the sensor butthe range is very limited and it is expensive [1] ereforethe software method is currently the main method iefusing multiple-exposure LDR images (hereinafter calledimage sequence or sequence) to obtain HDR images efusion method can be further divided into two categoriesone is to restore the Camera Response Function (CRF) andthen reconstruct the HDR light radiation pattern [2] theother is to directly fuse the pixels of multiple-exposuresequences at the pixel level Both categories of methods needto consider all the pixels of the multiple-exposure imagesequence increasing the computational complexity and

HindawiDiscrete Dynamics in Nature and SocietyVolume 2020 Article ID 6703528 13 pageshttpsdoiorg10115520206703528

storage space In the process of transmission and storage theimages are further compressed and transformed to removeredundancy to extract the required information ismethod of sampling after compression results in samplingredundancy excessive storage space and increased trans-mission costs

Compressive sensing (CS also called compressedsensing) [3ndash5] can solve the above problem which com-presses signal while sampling CS breaks through the lim-itation of the traditional Shannon sampling theorem and canperform high-probability reconstruction of incompletesignals at a condition far below the Nyquist sampling rateRice University has developed a single-pixel camera basedon the theory of compressive sensing [6] By replacing theCCD or CMOS sensors with a digital micromirror array(DMD) and a single photon detector it only needs to samplethe image fewer times than the number of pixels Its ap-pearance confirms the feasibility of compressive sensingapplying to imaging systems erefore this paper proposesa method for reduced-dimensional capture of high dynamicrange images with compressive sensing which includesalgorithms for front end (capturing) and back end (pro-cessing) At the front end the K-SVD dictionary is used tocompressive sensing the input multiple-exposure imagesequence thereby reducing the amount of data transmittedto the back end At the back end the Orthogonal MatchingPursuit (OMP) algorithm is used to reconstruct the inputmultiple-exposure image sequence A low-rank PatchMatchalgorithm is proposed to merge the reconstructed imagesequence to obtain an HDR image

2 Materials and Methods

Figure 1 is the schematic of the proposed methode wholesystem of this method includes three parts front endcommunication and back end In practical applicationsespecially for IoT applications the front end is generally alow-power device with very limited computing resourcesand its main role is to sense the real world e back end isgenerally a cloud computing center or edge computing nodewhich has powerful computing resources but is far awayfrom the field that needs to be sensed Communicationbetween front end and back end includes Ethernet mobilecommunication networks (including 2G 3G 4G 5G andNB-IoT) wireless local area networks (WLAN) and low-power wide area networks (LPWAN including Lora Sig-fox) Among these communication methods the wirednetwork (such as Ethernet) is rarely used to directly connectfront-end equipment because of high deployment costs andinflexibility Because WLAN has a limited transmissiondistance it is applicable but is relatively limited ecommunication distance of LPWAN is very long but itsbandwidth is also very small which is difficult to use fortransmitting traditional image sequences e most suitabletechnology for transmitting image sequences is the mobilecommunication network (especially 4G and 5G) but thelarger the bandwidth used is the higher the price is and themore energy the front end uses for communicationerefore it is necessary to reduce the computational

complexity of the front-end device and the amount ofcommunication data between the front end and the backend e method proposed in this paper uses compressivesensing technology to reduce the computational complexityand data volume of HDR image capturing front-end devicesthereby reducing the cost of the entire system

21 Reduced-Dimensional Capture and Reconstruction ofMultiple-Exposure Image Sequences Since compressive-sensing cameras are not common now we assume that thefront end uses a common camera to capture a series of LDRimages with different exposures is assumption makes thesystem not only easier to implement but also easier tocompare with other methods Every image in the imagesequence is resampled using compressive sensing Com-pressive sensing includes sparse representation of signalsdesign of measurement matrices and design of signal re-construction algorithms [7]

22 Sparse Representation of Image with OvercompleteDictionaries Signals are not sparse in practical applicationsbut when a suitable basis is used to represent the signals theyare sparse or compressible [8] ie the number of nonzeroelements is small which is conducive to the improvement ofthe sampling rate e use of sparse representation inmultiple fields has become increasingly mature such ascompression regularization in inverse problems and featureextraction [9] Sparseness is the premise of compressivesensing which means that the signal itself is sparse or sparseafter some transformation for example transformingnonsparse signals into sparse ones by Fourier transformdiscrete wavelet transform [7] ie the nonsparse signal isrepresented by a linear combination of several atoms in afixed dictionary (such as a DCT dictionary a wavelet dic-tionary a Haar dictionary and a Gabor dictionary) efixed dictionary has a simple structure and simple calcula-tion but it can only be applied to a limited range of signalsand the sparse representation cannot be guaranteed to beoptimal ie the sparseness of the signal sparse represen-tation cannot be guaranteed To best suit a set of givensignals we can train an overcomplete dictionary with thegiven signals e K-SVD [9] method can continuously it-erate through sparse coding and dictionary update to op-timize the sparse representation of the signal on the premiseof a given training set K-SVD can be regarded as a gen-eralized form of K-means clustering e only difference isthat the number of atoms used for each signal is different

Blocking each image in the multiple-exposure imagesequence can further reduce storage space and the block sizecan usually be 8 times 8 16 times 16 32 times 32 and so on [10] Fig-ure 2 is the flowchart of the K-SVD algorithm e inputimage sequence is I1 I2 IM where M is the number ofimages in the sequence and Imm 1 2 M is an image inthe sequence Here we suppose that all images in the se-quence have the same size of r times c All images in imagesequence are divided into blocks (with block size b times b) andpixels in each block are rearranged into a column vector yii 1 2 N In case that b is not divisible by r or c the

2 Discrete Dynamics in Nature and Society

Capturing multiple-exposure image

sequence

GeneratingK-SVD

dictionary

Compressivesensing the image

sequence

Reconstructing theimage sequence

Fusing the image sequence to obtain

an HDR image

Transmitting theimage sequence

Front end

Communication

Back end

Figure 1 Schematic of the proposed method

Input multiple-exposure image sequenceI1 I2 hellip IM

Initialize dictionary matrixD(0) = (d1 d2 hellip dK) and J = 1

Solve matrix with a pursuit algorithmX = (x1

T x2T hellip XK

T)T

Images blocking to get signal matrixY = (y1 y2 hellip yN)

Define a group of indices pointing to yi that use dkωk = i| 1leileK Xk

T(i) ne 0

Compute representation error matrixEk = Y ndash sum(jnek)djX

jT

Restrict Ek by choosing only columns corresponding to ωk to get Ek

R

Stopping rule

Output dictionary matrix D(J)

J = J + 1

N

Y

kgtK

Initialize column index of D(Jndash1) k = 1

k = k + 1

N

Y

Update dk in D(Jndash1) by SVD decomposition of EkR

Figure 2 e flowchart of the K-SVD algorithm Im (m) 1 2 (M) is an image in image sequence All images in image sequence aredivided into blocks and pixels in each block are rearranged into a column vector yi i 1 2 N (N) is the number of vectors generatedfrom the image sequence and (n) is the length of yi Y isin RntimesN is a matrix of column vectors yi e dictionary D(J) isin RntimesK is made up of atomvector dk where K is the total number of atoms in D(J) and the superscript (J) is the number of iterations X isin RKtimesN is the sparserepresentation of (Y) under dictionary (D) and is made up of row vectors xi

T i 1 2 K where the subscript Tof xiT indicates that xi

T isa row vector and superscript T indicates matrix transpose e matrix Ek is the error for all the input signal when the (k)th atom is removede detail of ER

k and SVD decomposition of ERk can be found in [9]

Discrete Dynamics in Nature and Society 3

image is expanded by 0 N r times c times Mb2 is the number ofvectors generated from the image sequence and n b2 is thelength of yi Y isin RntimesN is a matrix of column vectors yi edictionary D(J) isin RntimesK is made up of atom vector dkwhere K is the total number of atoms in D(J) and the su-perscript (J) is the number of iterations X isin RKtimesN is thesparse representation of Y under dictionary D and is madeup of row vectors xi

T i 1 2 K where the subscript Tof xi

T indicates that xiT is a row vector and superscript T

indicates matrix transpose Equation (1) is the objectfunction of K-SVD where xi is the ith column of matrix Xand T0 is the predetermined number of nonzero elements inxi

minDX

Y minus DX2F1113966 1113967 subject toforalli xi

0 leT0 (1)

ematrix Ek is the error for all the input signal when thekth atom is removed e detail of ER

k and SVD decom-position of ER

k can be found in [9]

23 MeasurementMatrix Design After the signal is sparselyrepresented a suitable measurement matrix Φ isin RKtimesn isneeded to compressive sense the signal e design principleof the measurement matrix is that the sensing matrix Θ

ΦD should meet the Restricted Isometry Property (RIP)[11] [12] to ensure one-to-one mapping from the originalspace to the sparse space e compressive sensing of signalyi is shown in (2) where zi isin RKtimes1 is the compressed sampleof signal yi

zi Φyi ΦDxi Θxi (2)

When Φ is a Gaussian random matrix the sensingmatrix Φ can satisfy the RIP with large probability [13] eadvantage of a Gaussian measurement matrix is that it is notrelated to almost any sparse signal so it requires very fewmeasurements erefore we use the Gaussian randommatrix as measurement matrix

24 Reconstructing Image Sequence e reconstructionmethod is the core step of compressive sensing e qualityof the reconstruction method determines the quality of thereconstructed image Compressive sensing reconstructionmethods mainly include three categories [14] e first isgreedy algorithm (such as orthogonal matching pursuit(OMP) [15] stagewise orthogonal matching pursuit(StOMP) [16] and regular orthogonal matching pursuit)(ROMP) [17]) is method solves the local optimal solutionto approximate the signal in each iteration e second is aconvex optimization algorithm (such as the base trackingalgorithm (BP) [18] the interior point method [19] thegradient projection method [20] and the iterative thresholdalgorithm [21]) Convex optimization can achieve betterreconstruction results with a small number of samples buthas a higher computational complexity e third is com-bination optimization algorithm which uses the grouptesting to accurately reconstruct the signal e recon-struction speed is fast but the scope of application is limitedsuch as HHS Pursuit [22] In this paper we use the OMP

algorithm to reconstruct the image sequence e perfor-mance of the OMP algorithm is stable and the recon-struction accuracy is high which can ensure that the originalsignal is accurately recovered at a lower sampling rate

Given the sensing matrix Θ ΦD (θ1 θ2 θK)

and the compressed sample zi of signal yi the OMP algo-rithm can estimate the sparse representation xi of signal yien the signal yi can be recovered by 3

yi Dxi (3)

e idea behind the OMP is to pick columns in a greedyfashion ie at each iteration t the column θt of Θ that ismost strongly correlated with the remaining part of xi ischosen [15] Figure 3 is the flowchart of the OMP algorithme input is the sensing matrixΘ and one of the compressedsignals z zi i 1 2 N in (2) After running N times ofOMP we can get the matrix X (the sparse representation ofY) and Y can be calculated column by column using (2) Atlast the image sequence can be reconstructed from Y

25 Low-Rank PatchMatch Algorithm During the captureprocess of the multiple-exposure image sequence camerashaking or unpredictable moving objects in the scene areinevitable which will cause artifacts or noise to appear in thefinal fused HDR image Currently block matching fusionmethod is mainly used to eliminate the artifacts and noisee essence of block matching fusion is to find a mappingrelationship between two different images A and B (given theimage block set of A and B as PA and PB respectively)ie by calculating the correlation find the nearest-neighborfield (NNF) of B so that the error of similar image blocks inthe two images is minimized By looking for the block inPA that is closest to block in PB the artifacts in the fusedimage are reduced

If image block matching is performed through a fullsearch the complexity is as high as O(mM2) where m andMare size of the image and the size of the block respectivelyTo reduce the complexity Connelly Barnes et al [23] [24]proposed a fast PatchMatch algorithm with randomizednearest neighbor and successfully reduced the complexity ofthe algorithm to O(mMlog(M)) e main steps of the al-gorithm can be summarized as initialization propagationand random search Due to the high efficiency and betterperformance of the PatchMatch algorithm it has a profoundimpact in the fields of image stitching image completionand image reorganization

In fact there are generally more than three multiple-exposure LDR images of the same scene to fuse into HDRimage so Pradeep Sen et al [25] proposed multisourcebidirectional similarity (MBDS) as shown in (4) S is theoriginal image and T is the target image N is the number ofsource images P and Q are patches in S and T respectivelyωk(P) weighs the source patches when calculating com-pleteness based on how well-exposed they are In order tomeasure the weight of a well-exposed image block the well-exposed image block has a large weight and vice versa d (middot)

is a distance metric which is usually calculated using the l2norm |T| is the total number of image blocks of the target


image is formula mainly includes the integrity of mappingfrom S to T and the correlation from T to S MBDS selectswell-exposed blocks in the image sequence to fill the regis-tration image so it can achieve better registration results

MBDS T|S1 SN( 1113857 1N

1113944

N

k11113944

PisinS1 SN

ωk(P)minQisinT

d(P Q)

+1

|T|1113944QisinT

minPisinS1 SN

d(Q P)

(4)

From the perspective of low-rank matrix recoverycombined with the idea of MBDS this paper proposes animproved algorithm for removing artifacts from HDR im-agese objective function is shown in (5) e input imagesequence has N images Ii i 1 N and Iref is the ref-erence image selected from the sequence Li i 1 N isthe result image of Ii being aligned to the reference imageIref that is the content of Li is aligned with the referenceimage and the exposure parameters remain the same with IiFunction gi(Ij) is the mapping from exposure parameter i toexposure parameter j Function h(middot) maps the grayscaledomain of LDR to the radiance domain of HDR Functionvec(middot) turns a two-dimensional image into a column vector

1113944

N

i1ine ref

MBDS Li|Ii gi Iref( 1113857( 1113857

st rank vec h L1( 1113857 vec h LN( 1113857( 1113857( 1113857( 1113857 1

(5)

Solve the MBDS problem to get Li More details can befound in [25] In addition the addition of a low-rankconstraint enables the aligned images to ensure a sufficientlylow rank ie to maintain a linear correlation in brightnesse solution is to divide into two independent local opti-mization subproblems namely the problem of MBDS andlow-rank matrix recovery At the same time the iterativesolution under multiresolution scale is used to find theoptimal solution of MBDS e low-rank matrix finallyobtained is the target HDR image with high dynamic rangeand linear brightness of the scene e process is shown inFigure 4

3 Results and Discussion

is section will analyze the convergence of the low-rankPatchMatch algorithm simulate the multiexposure imagecompressive sensing and reconstruction algorithm and theantiartifact fusion algorithm and evaluate the algorithms interms of subjective and objective criteria

31 Convergence of the Low-Rank PatchMatch AlgorithmRandomly generate data matrices with rank of r and size of1000 times 500 and add sparse noise with a noise ratio of pValidate the convergence by two sets of experiments In firstset fixmatrix rank r to 1 and observe the convergence underdifferent noise ratios p In the second set fix the noise ratiop which is set to 02 in the experiment and observe theconvergence under different ranks r e results are shownin Figure 5 In both cases the low-rank PatchMatch algo-rithm converges within 5 iterations

32 Image Evaluation Criteria In this paper we will use themean squared error (MSE) peak signal-to-noise ratio(PSNR) information entropy average gradient and runningtime to objectively evaluate the image quality and algorithm

e definition of mean squared error (MSE) is shown in(4) wherem and n are the width and height of the images andf and g are two different images e standard deviationrepresents howmuch the experimental data deviates from themeane higher the standard deviation themore diverse theresult data and the lower the accuracy of the result

MSE 1mn

1113944

mminus1

i01113944

nminus1

j0f(i j) minus g(i j)

2 (6)

PSNR is a commonly used criterion for reconstructedimage quality evaluation According to the definition of thestandard deviation in equation (4) the definition of PSNR isgiven in equation (5) where MAXI is the maximum value ofthe pixel value of an image for example 255 for an 8 bit greyimage e larger the PSNR value the lower the degree ofdistortion of the image

PSNR 10 times log10MAX2

I

MSE1113888 1113889 (7)

Information entropy represents the average informationof an image that is the average information after removing

Input compressed signal z and sensing matrix Θ = θ1 θ2 θK

Find the index λt byλt = arg max(j = 1 2 hellip K)|ltrtndash1 θjgt|

Augment the index set and the matrix of chosen atoms

Λt = Λtndash1λt Θt = (Θtndash1θt)

Initialize the residual r0 = z the index set Λ = Φ chosen atoms matrix Θ0 is emptyand

the iteration counter t = 1

Estimate xt by least squaresxt = arg minx||z ndash Θtx||2

Calculate the new approximation at of zand the newresidualat = Θtxt rt = z ndash at

Output estimation of xx(i) = xt(i) for i in Λt otherwise set x(i) = 0

tltK

t = t + 1

N

Y

Figure 3 e flowchart of the OMP algorithm


redundant information e definition is shown in equation(7) where p(bi) is the probability that the brightness bi

appears in the image and L is the maximum grey value of theimage

Entropy minus 1113944L

i1p bi( 1113857log2 p bi( 1113857 (8)

e average gradient characterizes the relative sharpnessof the image and reflects the rate of change in contrast ofdetails e larger the average gradient is the larger thechanges of grey level are and the richer the levels of theimage are e definition is shown in equation (8) whereMand N are the number of rows and columns of the image frespectively

Select reference image Iref

Initialize parametersj = 1 Li = Iref

Update nearest-neighbor fieldbetween Irefndashj and Lrefndashj between Iref+j and Lref+j

Lrefndashj = grefndashj (Iref)Lref+j = gref+j (Iref)

Maximum resolution scaleN

Solve the MBDS problem to get Lrefndashj Lref+j

Upsampling nearest-neighbor field to the

next scale

Iterate through all images

Find low-rank matrix ofLi|i = ref refndashj ref+j hellip

Y

j++

N

Output HDR image

Y

Maximum number of iterationsN

Y

Input multiexposure image sequence Ii exposure time and camera response function

Multiscale pyramid decompositionIrefndashj Iref Iref+j

Figure 4 Process of low-rank PatchMatch


g 1

(M minus 1)(N minus 1)

times 1113944Mminus1

i11113944

Nminus1

j1

(f(i j) minus f(i + 1 j))2 +(f(i j) minus f(i j + 1))2

2

1113971

(9)

For the fusion of multiple-exposure images of complexscenes with moving objects the evaluation of deghostingneeds to be further explored At present there is no matureobjective criterion to evaluate the deghosting of HDR imagesIn this paper the deghosting evaluation method proposed byKaraduzovic-Hadziabdic and Telaovic et al [26] is used andthe test image set used is a complex real scene

33 Simulation of Compressive Sensing andReconstruction forMultiple-Exposure Images e simulation platform for thisexperiment is MATLAB 2015b the hardware is 32G memoryIntel Core i5-6600K processor (main frequency 35GHz)Airplane and Lena with an image size of 512 times 512 were se-lected for simulation and the simulation results were com-paredwith BPOMP and StOMP algorithms at lower (R 03)medium (R 05) and higher (R 07) sampling rates re-spectively e simulation results are shown in Table 1

Among the three major types of algorithms for com-pressive sensing reconstruction the convex optimizationalgorithm has the best performance but then it has thehighest complexity and the longest reconstruction time As arepresentative of convex optimization algorithm BP hasbetter reconstruction performance than greedy algorithmand combinatorial optimization algorithm At the samplingrate of 03 and 05 compared with BP algorithm the per-formance of our algorithm is better than BP algorithmWith

a sampling rate of 08 although the PSNR of our algorithm isslightly lower than BP it is higher than the other algorithmsIn addition the reconstruction time of our algorithm isshorter than the reconstruction time of the BP algorithmexcept that the sampling rate is low (R 03)

e simulation results are shown in Figure 6 when thesampling rate is 05 From a subjective point of view for theletter area on the fuselage and wings of the airplane imageboth the BP algorithm and our algorithm can recover theclear letters but the letters recovered by the OMP algorithmand the StOMP algorithm are blurred e images recoveredby BP OMP and StOMP algorithms all have obvious noisee particle noise of StOMP algorithm is the most obviousOur algorithm can basically restore the image informationcorrectly

e reconstruction difference of the Lena image is not asobvious as the aircraft image from the subjective point ofview but it can be observed that the images recovered by BPOMP and StOMP all have different degrees of noise and theparticle noise of StOMP is the most obvious

Among the above algorithms the StOMP algorithm hasthe shortest reconstruction time but the reconstructioneffect is also the worst and the particle noise is very obviousBecause of the existence of noise the average gradient of theStOMP algorithm is higher than that of other algorithmse average gradient of our algorithm is similar to the OMPalgorithm which is better than the BP algorithm From theperspective of MSE the MSE of BP algorithm is the smallestand our algorithm is second Information entropy is similarto the case of MSE

34 Simulation Multiple-Exposure Images Fusion AlgorithmIn this section the multiple-exposure image sequences ArchSculpture Garden and Puppet are compressive sensed

0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

04N

orm

aliz

ed M

SE

p = 01p = 02

p = 03p = 04

(a)

r = 1r = 2

r = 3r = 4

0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

Nor

mal

ized

MSE

(b)

Figure 5 Convergence of the low-rank PatchMatch algorithm (a) e convergence under different noise ratios (p) (the matrix rank (r) isfixed to 1) (b) e convergence under different ranks (r) (the noise ratio (p) is set to 02)


Table 1 Comparison of difference compressive sensing and reconstruction algorithm

Image Sampling rate Algorithm PSNR MSE Entropy Average gradient Running time （s）

Airplane

03 BP 23774 44389 7215 7802 28199OMP 23609 46271 7188 8543 15442StOMP 25271 47781 7065 10049 0075Ours 28861 46341 7097 8227 3631

05 BP 30278 45169 6996 6673 42239OMP 28506 46545 7057 7167 30557StOMP 24913 48297 7113 11379 0079Ours 33356 46365 6999 7158 38024

08 BP 4016 46097 6902 6189 81628OMP 34257 46368 6952 6462 60081StOMP 26219 47882 7117 10788 0311Ours 36601 46348 6938 6551 42084

Lena

03 BP 27394 46153 7862 6303 27167OMP 26285 47967 7864 726 15625StOMP 26579 48876 7861 9157 0089Ours 30116 47846 7862 712 28709

05 BP 32867 47262 7865 588 39993OMP 30808 47803 7866 6371 29406StOMP 25885 49249 7859 10208 0313Ours 33442 47743 7863 6308 29429

08 BP 40415 47678 7862 5795 80541OMP 35008 4785 7865 5971 58919StOMP 27261 49012 7861 9779 0326Ours 36171 47799 7863 6051 32847

(a) (b) (c)

(d) (e)

Figure 6 Continued


(f) (g) (h)

(i) (j)

Figure 6 Comparison of difference compressive sensing and reconstruction algorithm at medium sample rate (R 05) (a) Airplane (b)BP (c) OMP (d) StOMP (e) Ours (f ) Lena (g) BP (h) OMP (i) StOMP (j) Ours

(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )

Figure 7 Multiple-exposure image sequences Arch fusing result (a) Multiple-exposure image sequence Arch (b) Arch after compressivesensed and reconstructed (c) Reference [2] (d) RPCA (e) PSSV (f ) OURS

(a)

(b)

(c) (d) (e)

Figure 8 Multiple-exposure image sequences Puppet fusing result (a) Multiple-exposure image sequences Puppet (b) Puppet aftercompressive sensed and reconstructed (c) Reference [31] (c) SEN [25] (d) HU [30] (e) OURS


(a)

(b)

(c) (d)

(e) (f )

(g)

Figure 9 Multiple-exposure image sequences Sculpture Garden fusing result (a) Multiple-exposure image sequence Sculpture Garden (b)Sculpture Garden after compressive sensed and reconstructed (c) Reference [31] (d) Reference [2] (e) SEN [25] (f ) HU [30] (g) OURS


reconstructed and then fused into HDR images e resultsare compared with robust principal component analysis(RPCA) [27] partial sum of singular value (PSSV) [28] [29]and [2] MBDS algorithm (referred to as SEN) proposed byPradeep Sen et al [25] the brightness and texture consis-tency deghosting method (referred to as HU) proposed byJun Hu et al [30] and the low-rank restoration baseddeghosting fusion proposed by Tae-Hyun Oh et al [31]

Figure 7 shows the Arch image sequence ere aremoving people in the picture Artifacts can occur with directfusion Reference [2] and RPCA cannot suppress the ap-pearance of artifacts and our algorithm and PSSV algorithmcan both suppress artifacts well and have better subjectivevisual effects

Figure 8 shows the results of the Puppet sequence Ouralgorithm adds low-rank constraints to minimize the impactof misaligned regions and keep the resulting image as linearas possible It can be seen from the result that our algorithmis better

Figure 9 shows the results of Sculpture Garden sequenceere are many pedestrians in the picture which makes itdifficult to remove artifacts From the results of the fusion[30] is the worst and there are obvious artifacts in [6] andthe SENmethod has a fuzzy phenomenon Both HU and thisalgorithm suppress artifacts well but due to the effect ofimage blocking block effect exists in the fusion result

Figure 10 is the local area details of the fusion result inFigure 9 Because result of [31] is the worst compared toother algorithms the detail of it is not enlarged e liter-ature [6] is less effective in removing artifacts because of theobvious silhouette cross ere is obvious blurring at thepedestrian edges of the SEN method HU and our algorithmachieve better results but our algorithm has noise caused byblock effects

4 Conclusions

Aiming at the problems of traditional cameras with re-dundant sampling large storage space consumption andinability to fully record the radiance in the real scene due tothe limitation of the dynamic range of the sensor this articleuses the K-SVD dictionary to compressive sensing LDRimages of different exposure in the same scene en theLDR images is reconstructed and fused with low-rankPatchMatch algorithm to get an HDR image e simulationresults show that the method in this paper can effectively

reduce the sampling rate and remove the image artifacts andblurring caused by the camera shake and the motion of theobjects in the scene It provides a method for compressivesensing to obtain HDR images

However due to the introduction of block compressivesensing the size of the image block has become a factor thatcannot be ignored Simulation results show that when theimage block is small the block effect is more obvious and theedge details are distorted But when increasing the imageblock it will increase the storage space and computationalcomplexity In addition adding compressive sensing anddictionary learning before fusion increases the computationtime sacrificing time in exchange for reduction in com-plexity and sampling rate erefore the next step is toperform pixel-level fusion in the compressive sensing do-main of the HDR image to further reduce the time requiredfor the algorithm and improve the quality of the fused image

Data Availability

No data were used to support this study

Conflicts of Interest

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

Acknowledgments

is work was funded by the Project for Distinctive Inno-vation of Ordinary Universities of Guangdong Province (no2018KTSCX120) the PhD Start-Up Fund of Natural Sci-ence Foundation of Guangdong Province (no2016A030310335) and Guangdong Science and TechnologyPlan Project (no 76120-42020022)

References

[1] F Banterle A Artusi K Debattista and A Chalmers Ad-vanced High Dynamic Range Imaging Taylor amp Francis CRCPress Boca Raton FL USA Second edition 2018

[2] P E Debevec and J Malik Recovering High Dynamic RangeRadiance Maps from Photographs pp 31ndash10 ACM SIG-GRAPH 2008 Classes New York NY USA 2008

[3] E Candes ldquoCompressive Samplingrdquo Presented at the Pro-ceedings of the International Congress of MathematiciansMadrid Spain 2006

(a) (b) (c) (d)

Figure 10 Details of the fusion result in Figure 9 (a) Reference [2] (b) SEN [25] (c) HU [30] (d) OURS


[4] D L Donoho ldquoCompressed sensingrdquo IEEE Transactions onInformation eory vol 52 no 4 pp 1289ndash1306 Apr 2006

[5] Y Tsaig and D L Donoho ldquoExtensions of compressedsensingrdquo Signal Processing vol 86 no 3 pp 549ndash571 Mar2006

[6] D Takhar J N Laska M BWakin et al ldquoA new compressiveimaging camera architecture using optical-domain com-pressionrdquo Computational Imaging IV vol 6065 p 6065092006

[7] G Shi D Liu D Gao Z Liu J Lin and L Wang ldquoAdvancesin theory and application of compressed sensingrdquo ActaElectronica Sinica vol 37 no 5 pp 1070ndash1081 2009

[8] X Zhang Matrix Analysis and Application Tsinghua Uni-versity Press Co Ltd Beijing China 2004

[9] M Aharon M Elad and A Bruckstein ldquo$rm K$-SVD analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processingvol 54 no 11 pp 4311ndash4322 2006

[10] L Gan ldquoBlock compressed sensing of natural imagesrdquo inProceedings of the 2007 15th International Conference onDigital Signal Processing pp 403ndash406 Wales UK July 2007

[11] E Candes and J Romberg ldquoSparsity and incoherence incompressive samplingrdquo Inverse Problems vol 23 no 3pp 969ndash985 2007

[12] E J Candes J Romberg and T Tao ldquoRobust uncertaintyprinciples exact signal reconstruction from highly incompletefrequency informationrdquo IEEE Transactions on Informationeory vol 52 no 2 pp 489ndash509 2006

[13] E J Candes J K Romberg and T Tao ldquoStable signal recoveryfrom incomplete and inaccurate measurementsrdquo Communi-cations on Pure and Applied Mathematics vol 59 no 8pp 1207ndash1223 2006

[14] D Needell J A Tropp and ldquo CoSaMP ldquoCoSaMP iterativesignal recovery from incomplete and inaccurate samplesrdquoApplied and Computational Harmonic Analysis vol 26 no 3pp 301ndash321 2009

[15] J A Tropp and A C Gilbert ldquoSignal recovery from randommeasurements via orthogonal matching pursuitrdquo IEEETransactions on Information eory vol 53 no 12pp 4655ndash4666 2007

[16] D L Donoho Y Tsaig I Drori and J-L Starck ldquoSparsesolution of underdetermined systems of linear equations bystagewise orthogonal matching pursuitrdquo IEEE Transactions onInformation eory vol 58 no 2 pp 1094ndash1121 2012

[17] D Needell and R Vershynin ldquoUniform uncertainty principleand signal recovery via regularized orthogonal matchingpursuitrdquo Foundations of Computational Mathematics vol 9no 3 pp 317ndash334 2009

[18] S S Chen D L Donoho and M A Saunders ldquoAtomicdecomposition by basis pursuitrdquo SIAM Review vol 43 no 1pp 129ndash159 2001

[19] K S J M Lustig An interior-point method for large-scale l1-regularized least squaresrdquo IEEE Journal of Selected Topics inSignal Processing vol 1 no 4 pp 606ndash617 2007

[20] M A T Figueiredo R D Nowak and S J Wright ldquoGradientprojection for sparse reconstruction application to com-pressed sensing and other inverse problemsrdquo IEEE Journal ofSelected Topics in Signal Processing vol 1 no 4 pp 586ndash5972007

[21] I Daubechies M Defrise and C De Mol ldquoAn iterativethresholding algorithm for linear inverse problems with asparsity constraintrdquo Communications on Pure and AppliedMathematics vol 57 no 11 pp 1413ndash1457 2004

[22] A C Gilbert M J Strauss J A Tropp and R VershyninOneSketch for All Fast Algorithms for Compressed Sensing ACMin Proceedings of the irty-Ninth Annual ACM Symposiumon eory of Computing pp 237ndash246 San Diego CA USAJune 2007

[23] C Barnes E Shechtman A Finkelstein D B Goldman andldquo PatchMatch A Randomized Correspondence Algorithm forStructural Image Editing pp 24ndash11 no 1ndash24 ACM SIG-GRAPH New York NY USA 2009

[24] C Barnes E Shechtman D B Goldman and A Finkelsteine Generalized PatchMatch Correspondence Algorithmpp 29ndash43 Computer VisionndashECCV 2010 Berlin Heidelberg2010

[25] P Sen N K Kalantari M Yaesoubi S DarabiD B Goldman and E Shechtman ldquoRobust patch-based hdrreconstruction of dynamic scenesrdquo ACM Transactions onGraphics vol 31 no 6 p 1 Nov

[26] K KaraduzovicndashHadziabdic and R Mantiuk ldquoExpert evalu-ation of deghosting algorithms for multi-exposure high dy-namic range imagingrdquo in Proccedings of the HDRi2014-SecondInternational Conference and SME Workshop on HDR Im-aging EBANGOR Sarajevo Bosnia April 2014

[27] E J Candes X Li Y Ma and J Wright ldquoRobust principalcomponent analysisrdquo Journal of the ACM vol 58 no 3pp 11ndash37 2011

[28] T-H Oh H Kim Y-W Tai J-C Bazin and I So KweonldquoPartial sum minimization of singular values in rpca for low-level visionrdquo in Proceedings of the IEEE International Con-ference on Computer Vision IEEE Harbour Sydneypp 145ndash152 April 2013

[29] T-H Oh Y-W Tai J-C Bazin H Kim and I S KweonldquoPartial sum minimization of singular values in robust PCAalgorithm and applicationsrdquo IEEE Transactions on PatternAnalysis and Machine Intelligence vol 38 no 4 pp 744ndash758Apr 2016

[30] J Hu O Gallo K Pulli and X Sun ldquoHDR deghosting how todeal with saturationrdquo in Proceedings of the IEEE Conferenceon Computer Vision and Pattern Recognition IEEE HarbourSydney pp 1163ndash1170 April 2013

[31] T-H Oh J-Y Lee Y-W Tai and I S Kweon ldquoRobust highdynamic range imaging by rank minimizationrdquo IEEETransactions on Pattern Analysis and Machine Intelligencevol 37 no 6 pp 1219ndash1232 2015


storage space In the process of transmission and storage theimages are further compressed and transformed to removeredundancy to extract the required information ismethod of sampling after compression results in samplingredundancy excessive storage space and increased trans-mission costs

Compressive sensing (CS also called compressedsensing) [3ndash5] can solve the above problem which com-presses signal while sampling CS breaks through the lim-itation of the traditional Shannon sampling theorem and canperform high-probability reconstruction of incompletesignals at a condition far below the Nyquist sampling rateRice University has developed a single-pixel camera basedon the theory of compressive sensing [6] By replacing theCCD or CMOS sensors with a digital micromirror array(DMD) and a single photon detector it only needs to samplethe image fewer times than the number of pixels Its ap-pearance confirms the feasibility of compressive sensingapplying to imaging systems erefore this paper proposesa method for reduced-dimensional capture of high dynamicrange images with compressive sensing which includesalgorithms for front end (capturing) and back end (pro-cessing) At the front end the K-SVD dictionary is used tocompressive sensing the input multiple-exposure imagesequence thereby reducing the amount of data transmittedto the back end At the back end the Orthogonal MatchingPursuit (OMP) algorithm is used to reconstruct the inputmultiple-exposure image sequence A low-rank PatchMatchalgorithm is proposed to merge the reconstructed imagesequence to obtain an HDR image

2 Materials and Methods

Figure 1 is the schematic of the proposed methode wholesystem of this method includes three parts front endcommunication and back end In practical applicationsespecially for IoT applications the front end is generally alow-power device with very limited computing resourcesand its main role is to sense the real world e back end isgenerally a cloud computing center or edge computing nodewhich has powerful computing resources but is far awayfrom the field that needs to be sensed Communicationbetween front end and back end includes Ethernet mobilecommunication networks (including 2G 3G 4G 5G andNB-IoT) wireless local area networks (WLAN) and low-power wide area networks (LPWAN including Lora Sig-fox) Among these communication methods the wirednetwork (such as Ethernet) is rarely used to directly connectfront-end equipment because of high deployment costs andinflexibility Because WLAN has a limited transmissiondistance it is applicable but is relatively limited ecommunication distance of LPWAN is very long but itsbandwidth is also very small which is difficult to use fortransmitting traditional image sequences e most suitabletechnology for transmitting image sequences is the mobilecommunication network (especially 4G and 5G) but thelarger the bandwidth used is the higher the price is and themore energy the front end uses for communicationerefore it is necessary to reduce the computational

complexity of the front-end device and the amount ofcommunication data between the front end and the backend e method proposed in this paper uses compressivesensing technology to reduce the computational complexityand data volume of HDR image capturing front-end devicesthereby reducing the cost of the entire system

21 Reduced-Dimensional Capture and Reconstruction ofMultiple-Exposure Image Sequences Since compressive-sensing cameras are not common now we assume that thefront end uses a common camera to capture a series of LDRimages with different exposures is assumption makes thesystem not only easier to implement but also easier tocompare with other methods Every image in the imagesequence is resampled using compressive sensing Com-pressive sensing includes sparse representation of signalsdesign of measurement matrices and design of signal re-construction algorithms [7]

22 Sparse Representation of Image with OvercompleteDictionaries Signals are not sparse in practical applicationsbut when a suitable basis is used to represent the signals theyare sparse or compressible [8] ie the number of nonzeroelements is small which is conducive to the improvement ofthe sampling rate e use of sparse representation inmultiple fields has become increasingly mature such ascompression regularization in inverse problems and featureextraction [9] Sparseness is the premise of compressivesensing which means that the signal itself is sparse or sparseafter some transformation for example transformingnonsparse signals into sparse ones by Fourier transformdiscrete wavelet transform [7] ie the nonsparse signal isrepresented by a linear combination of several atoms in afixed dictionary (such as a DCT dictionary a wavelet dic-tionary a Haar dictionary and a Gabor dictionary) efixed dictionary has a simple structure and simple calcula-tion but it can only be applied to a limited range of signalsand the sparse representation cannot be guaranteed to beoptimal ie the sparseness of the signal sparse represen-tation cannot be guaranteed To best suit a set of givensignals we can train an overcomplete dictionary with thegiven signals e K-SVD [9] method can continuously it-erate through sparse coding and dictionary update to op-timize the sparse representation of the signal on the premiseof a given training set K-SVD can be regarded as a gen-eralized form of K-means clustering e only difference isthat the number of atoms used for each signal is different

Blocking each image in the multiple-exposure imagesequence can further reduce storage space and the block sizecan usually be 8 times 8 16 times 16 32 times 32 and so on [10] Fig-ure 2 is the flowchart of the K-SVD algorithm e inputimage sequence is I1 I2 IM where M is the number ofimages in the sequence and Imm 1 2 M is an image inthe sequence Here we suppose that all images in the se-quence have the same size of r times c All images in imagesequence are divided into blocks (with block size b times b) andpixels in each block are rearranged into a column vector yii 1 2 N In case that b is not divisible by r or c the



sequence

GeneratingK-SVD

dictionary


sequence



an HDR image


Front end

Communication

Back end





T x2T hellip XK

T)T



T(i) ne 0


jT


R

Stopping rule


J = J + 1

N

Y

kgtK


k = k + 1

N

Y











minDX


0 leT0 (1)












yi Dxi (3)








MBDS T|S1 SN( 1113857 1N

1113944

N

k11113944

PisinS1 SN

ωk(P)minQisinT

d(P Q)

+1

|T|1113944QisinT

minPisinS1 SN

d(Q P)

(4)


1113944

N

i1ine ref

MBDS Li|Ii gi Iref( 1113857( 1113857


(5)







MSE 1mn

1113944

mminus1

i01113944

nminus1


2 (6)



I

MSE1113888 1113889 (7)











tltK

t = t + 1

N

Y






i1p bi( 1113857log2 p bi( 1113857 (8)









next scale



Y

j++

N

Output HDR image

Y


Y





g 1



i11113944

Nminus1

j1


2

1113971

(9)









0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

04N

orm

aliz

ed M

SE

p = 01p = 02

p = 03p = 04

(a)

r = 1r = 2

r = 3r = 4

0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

Nor

mal

ized

MSE

(b)





Airplane

03 BP 23774 44389 7215 7802 28199OMP 23609 46271 7188 8543 15442StOMP 25271 47781 7065 10049 0075Ours 28861 46341 7097 8227 3631

05 BP 30278 45169 6996 6673 42239OMP 28506 46545 7057 7167 30557StOMP 24913 48297 7113 11379 0079Ours 33356 46365 6999 7158 38024

08 BP 4016 46097 6902 6189 81628OMP 34257 46368 6952 6462 60081StOMP 26219 47882 7117 10788 0311Ours 36601 46348 6938 6551 42084

Lena

03 BP 27394 46153 7862 6303 27167OMP 26285 47967 7864 726 15625StOMP 26579 48876 7861 9157 0089Ours 30116 47846 7862 712 28709

05 BP 32867 47262 7865 588 39993OMP 30808 47803 7866 6371 29406StOMP 25885 49249 7859 10208 0313Ours 33442 47743 7863 6308 29429

08 BP 40415 47678 7862 5795 80541OMP 35008 4785 7865 5971 58919StOMP 27261 49012 7861 9779 0326Ours 36171 47799 7863 6051 32847

(a) (b) (c)

(d) (e)

Figure 6 Continued


(f) (g) (h)

(i) (j)


(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)

































sequence

GeneratingK-SVD

dictionary


sequence



an HDR image


Front end

Communication

Back end





T x2T hellip XK

T)T



T(i) ne 0


jT


R

Stopping rule


J = J + 1

N

Y

kgtK


k = k + 1

N

Y











minDX


0 leT0 (1)












yi Dxi (3)








MBDS T|S1 SN( 1113857 1N

1113944

N

k11113944

PisinS1 SN

ωk(P)minQisinT

d(P Q)

+1

|T|1113944QisinT

minPisinS1 SN

d(Q P)

(4)


1113944

N

i1ine ref

MBDS Li|Ii gi Iref( 1113857( 1113857


(5)







MSE 1mn

1113944

mminus1

i01113944

nminus1


2 (6)



I

MSE1113888 1113889 (7)











tltK

t = t + 1

N

Y






i1p bi( 1113857log2 p bi( 1113857 (8)









next scale



Y

j++

N

Output HDR image

Y


Y





g 1



i11113944

Nminus1

j1


2

1113971

(9)









0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

04N

orm

aliz

ed M

SE

p = 01p = 02

p = 03p = 04

(a)

r = 1r = 2

r = 3r = 4

0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

Nor

mal

ized

MSE

(b)





Airplane

03 BP 23774 44389 7215 7802 28199OMP 23609 46271 7188 8543 15442StOMP 25271 47781 7065 10049 0075Ours 28861 46341 7097 8227 3631

05 BP 30278 45169 6996 6673 42239OMP 28506 46545 7057 7167 30557StOMP 24913 48297 7113 11379 0079Ours 33356 46365 6999 7158 38024

08 BP 4016 46097 6902 6189 81628OMP 34257 46368 6952 6462 60081StOMP 26219 47882 7117 10788 0311Ours 36601 46348 6938 6551 42084

Lena

03 BP 27394 46153 7862 6303 27167OMP 26285 47967 7864 726 15625StOMP 26579 48876 7861 9157 0089Ours 30116 47846 7862 712 28709

05 BP 32867 47262 7865 588 39993OMP 30808 47803 7866 6371 29406StOMP 25885 49249 7859 10208 0313Ours 33442 47743 7863 6308 29429

08 BP 40415 47678 7862 5795 80541OMP 35008 4785 7865 5971 58919StOMP 27261 49012 7861 9779 0326Ours 36171 47799 7863 6051 32847

(a) (b) (c)

(d) (e)

Figure 6 Continued


(f) (g) (h)

(i) (j)


(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)




































minDX


0 leT0 (1)












yi Dxi (3)








MBDS T|S1 SN( 1113857 1N

1113944

N

k11113944

PisinS1 SN

ωk(P)minQisinT

d(P Q)

+1

|T|1113944QisinT

minPisinS1 SN

d(Q P)

(4)


1113944

N

i1ine ref

MBDS Li|Ii gi Iref( 1113857( 1113857


(5)







MSE 1mn

1113944

mminus1

i01113944

nminus1


2 (6)



I

MSE1113888 1113889 (7)











tltK

t = t + 1

N

Y






i1p bi( 1113857log2 p bi( 1113857 (8)









next scale



Y

j++

N

Output HDR image

Y


Y





g 1



i11113944

Nminus1

j1


2

1113971

(9)









0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

04N

orm

aliz

ed M

SE

p = 01p = 02

p = 03p = 04

(a)

r = 1r = 2

r = 3r = 4

0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

Nor

mal

ized

MSE

(b)





Airplane

03 BP 23774 44389 7215 7802 28199OMP 23609 46271 7188 8543 15442StOMP 25271 47781 7065 10049 0075Ours 28861 46341 7097 8227 3631

05 BP 30278 45169 6996 6673 42239OMP 28506 46545 7057 7167 30557StOMP 24913 48297 7113 11379 0079Ours 33356 46365 6999 7158 38024

08 BP 4016 46097 6902 6189 81628OMP 34257 46368 6952 6462 60081StOMP 26219 47882 7117 10788 0311Ours 36601 46348 6938 6551 42084

Lena

03 BP 27394 46153 7862 6303 27167OMP 26285 47967 7864 726 15625StOMP 26579 48876 7861 9157 0089Ours 30116 47846 7862 712 28709

05 BP 32867 47262 7865 588 39993OMP 30808 47803 7866 6371 29406StOMP 25885 49249 7859 10208 0313Ours 33442 47743 7863 6308 29429

08 BP 40415 47678 7862 5795 80541OMP 35008 4785 7865 5971 58919StOMP 27261 49012 7861 9779 0326Ours 36171 47799 7863 6051 32847

(a) (b) (c)

(d) (e)

Figure 6 Continued


(f) (g) (h)

(i) (j)


(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)

































MBDS T|S1 SN( 1113857 1N

1113944

N

k11113944

PisinS1 SN

ωk(P)minQisinT

d(P Q)

+1

|T|1113944QisinT

minPisinS1 SN

d(Q P)

(4)


1113944

N

i1ine ref

MBDS Li|Ii gi Iref( 1113857( 1113857


(5)







MSE 1mn

1113944

mminus1

i01113944

nminus1


2 (6)



I

MSE1113888 1113889 (7)











tltK

t = t + 1

N

Y






i1p bi( 1113857log2 p bi( 1113857 (8)









next scale



Y

j++

N

Output HDR image

Y


Y





g 1



i11113944

Nminus1

j1


2

1113971

(9)









0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

04N

orm

aliz

ed M

SE

p = 01p = 02

p = 03p = 04

(a)

r = 1r = 2

r = 3r = 4

0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

Nor

mal

ized

MSE

(b)





Airplane

03 BP 23774 44389 7215 7802 28199OMP 23609 46271 7188 8543 15442StOMP 25271 47781 7065 10049 0075Ours 28861 46341 7097 8227 3631

05 BP 30278 45169 6996 6673 42239OMP 28506 46545 7057 7167 30557StOMP 24913 48297 7113 11379 0079Ours 33356 46365 6999 7158 38024

08 BP 4016 46097 6902 6189 81628OMP 34257 46368 6952 6462 60081StOMP 26219 47882 7117 10788 0311Ours 36601 46348 6938 6551 42084

Lena

03 BP 27394 46153 7862 6303 27167OMP 26285 47967 7864 726 15625StOMP 26579 48876 7861 9157 0089Ours 30116 47846 7862 712 28709

05 BP 32867 47262 7865 588 39993OMP 30808 47803 7866 6371 29406StOMP 25885 49249 7859 10208 0313Ours 33442 47743 7863 6308 29429

08 BP 40415 47678 7862 5795 80541OMP 35008 4785 7865 5971 58919StOMP 27261 49012 7861 9779 0326Ours 36171 47799 7863 6051 32847

(a) (b) (c)

(d) (e)

Figure 6 Continued


(f) (g) (h)

(i) (j)


(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)



































i1p bi( 1113857log2 p bi( 1113857 (8)









next scale



Y

j++

N

Output HDR image

Y


Y





g 1



i11113944

Nminus1

j1


2

1113971

(9)









0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

04N

orm

aliz

ed M

SE

p = 01p = 02

p = 03p = 04

(a)

r = 1r = 2

r = 3r = 4

0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

Nor

mal

ized

MSE

(b)





Airplane

03 BP 23774 44389 7215 7802 28199OMP 23609 46271 7188 8543 15442StOMP 25271 47781 7065 10049 0075Ours 28861 46341 7097 8227 3631

05 BP 30278 45169 6996 6673 42239OMP 28506 46545 7057 7167 30557StOMP 24913 48297 7113 11379 0079Ours 33356 46365 6999 7158 38024

08 BP 4016 46097 6902 6189 81628OMP 34257 46368 6952 6462 60081StOMP 26219 47882 7117 10788 0311Ours 36601 46348 6938 6551 42084

Lena

03 BP 27394 46153 7862 6303 27167OMP 26285 47967 7864 726 15625StOMP 26579 48876 7861 9157 0089Ours 30116 47846 7862 712 28709

05 BP 32867 47262 7865 588 39993OMP 30808 47803 7866 6371 29406StOMP 25885 49249 7859 10208 0313Ours 33442 47743 7863 6308 29429

08 BP 40415 47678 7862 5795 80541OMP 35008 4785 7865 5971 58919StOMP 27261 49012 7861 9779 0326Ours 36171 47799 7863 6051 32847

(a) (b) (c)

(d) (e)

Figure 6 Continued


(f) (g) (h)

(i) (j)


(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)
































g 1



i11113944

Nminus1

j1


2

1113971

(9)









0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

04N

orm

aliz

ed M

SE

p = 01p = 02

p = 03p = 04

(a)

r = 1r = 2

r = 3r = 4

0 5 10 15 20 25 30Iterations

0

005

01

015

02

025

03

035

Nor

mal

ized

MSE

(b)





Airplane

03 BP 23774 44389 7215 7802 28199OMP 23609 46271 7188 8543 15442StOMP 25271 47781 7065 10049 0075Ours 28861 46341 7097 8227 3631

05 BP 30278 45169 6996 6673 42239OMP 28506 46545 7057 7167 30557StOMP 24913 48297 7113 11379 0079Ours 33356 46365 6999 7158 38024

08 BP 4016 46097 6902 6189 81628OMP 34257 46368 6952 6462 60081StOMP 26219 47882 7117 10788 0311Ours 36601 46348 6938 6551 42084

Lena

03 BP 27394 46153 7862 6303 27167OMP 26285 47967 7864 726 15625StOMP 26579 48876 7861 9157 0089Ours 30116 47846 7862 712 28709

05 BP 32867 47262 7865 588 39993OMP 30808 47803 7866 6371 29406StOMP 25885 49249 7859 10208 0313Ours 33442 47743 7863 6308 29429

08 BP 40415 47678 7862 5795 80541OMP 35008 4785 7865 5971 58919StOMP 27261 49012 7861 9779 0326Ours 36171 47799 7863 6051 32847

(a) (b) (c)

(d) (e)

Figure 6 Continued


(f) (g) (h)

(i) (j)


(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)


































Airplane

03 BP 23774 44389 7215 7802 28199OMP 23609 46271 7188 8543 15442StOMP 25271 47781 7065 10049 0075Ours 28861 46341 7097 8227 3631

05 BP 30278 45169 6996 6673 42239OMP 28506 46545 7057 7167 30557StOMP 24913 48297 7113 11379 0079Ours 33356 46365 6999 7158 38024

08 BP 4016 46097 6902 6189 81628OMP 34257 46368 6952 6462 60081StOMP 26219 47882 7117 10788 0311Ours 36601 46348 6938 6551 42084

Lena

03 BP 27394 46153 7862 6303 27167OMP 26285 47967 7864 726 15625StOMP 26579 48876 7861 9157 0089Ours 30116 47846 7862 712 28709

05 BP 32867 47262 7865 588 39993OMP 30808 47803 7866 6371 29406StOMP 25885 49249 7859 10208 0313Ours 33442 47743 7863 6308 29429

08 BP 40415 47678 7862 5795 80541OMP 35008 4785 7865 5971 58919StOMP 27261 49012 7861 9779 0326Ours 36171 47799 7863 6051 32847

(a) (b) (c)

(d) (e)

Figure 6 Continued


(f) (g) (h)

(i) (j)


(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)
































(f) (g) (h)

(i) (j)


(a)

(b)

Figure 7 Continued


(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)
































(c) (d) (e) (f )


(a)

(b)

(c) (d) (e)



(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)
































(a)

(b)

(c) (d)

(e) (f )

(g)








4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)





































4 Conclusions




Data Availability




Acknowledgments


References




(a) (b) (c) (d)





























































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Reduced-Dimensional Capture of High-Dynamic Range Images …downloads.hindawi.com/journals/ddns/2020/6703528.pdf · 2020-04-30 · ResearchArticle Reduced-Dimensional Capture of High-Dynamic