Blind Poissonian Image Deblurring Regularized by a

Research ArticleBlind Poissonian Image Deblurring Regularized by a DenoiserConstraint and Deep Image Prior

Yayuan Feng123 Yu Shi 123 and Dianjun Sun123

1School of Electrical and Information Engineering Wuhan Institute of Technology Wuhan 430205 China2Hubei Key Laboratory of Optical Information and Pattern Recognition Wuhan 430205 China3Laboratory of Hubei Province Video Image and HD Projection Engineering Technology Research Center Wuhan 430205 China

Correspondence should be addressed to Yu Shi shiyu0125163com

Received 4 April 2020 Revised 21 June 2020 Accepted 22 July 2020 Published 24 August 2020

Academic Editor Lotfi Senhadji

Copyright copy 2020 Yayuan Feng 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 denoising and deblurring of Poisson images are opposite inverse problems Single image deblurring methods are sensitive toimage noise A single noise filter can effectively remove noise in advance but it also damages blurred information To si-multaneously solve the denoising and deblurring of Poissonian images better we learn the implicit deep image prior from a singledegraded image and use the denoiser as a regularization term to constrain the latent clear image Combined with the explicit L0regularization prior of the image the denoising and deblurring model of the Poisson image is established+en the split Bregmaniteration strategy is used to optimize the point spread function estimation and latent clear image estimation +e experimentalresults demonstrate that the proposed method achieves good restoration results on a series of simulated and real blurred imageswith Poisson noise

1 Introduction

+e process of image acquisition in electron microscopeimaging astronomical imaging and medical imaging isinevitably affected by environmental factors which oftencause the captured image to be disturbed by Poisson noiseand blur degradation +e image degradation process can bemodeled as follows

y P(Hlowastx) (1)

where y is the observed degraded image P represents theprocess in which the image is corrupted by Poisson noiseH represents the point spread function and lowast is theconvolution operator+e problem of blind Poisson imagedeconvolution is to recover clear images and estimate thepoint spread function from the degraded image +edenoising and deblurring of Poisson images are oppositeinverse problems Single deblurring methods are sensitiveto noise and even a small amount of noise leads to biasedpoint spread function estimation Although the noise filtercan effectively remove noise in advance it also damages

the blurred information and introduces more serious blurwhich leads to amplified and biased point spread functionestimation To suppress Poisson noise and restore theblurred image simultaneously we impose the denoiser as aregularization term to constrain the latent clear imagecombined with an implicit deep image prior and an ex-plicit L0 regularization prior of the image +en the en-ergy functional regularization framework for the latentclear image and point spread function is modeled asfollows

E(x H) minxH

lang1 Hx minus y logHxrang + R(x) + K(H) (2)

where lang1 Hx minus y logHxrang is the data term 1 represents avector with all elements equal to 1 x is the latent clear imagewhich usually ensures that the clear image does not containnegative gray values H is the point spread function R(x) isthe regularization constraint term of the latent clear imageand K(H) is the regularization constraint term of a pointspread function Among them regularization constraintterms R(x) and K(H) play a vital role in the restoration of

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9483521 15 pageshttpsdoiorg10115520209483521

latent clear images and the estimation of point spreadfunctions

11 RelevantWork Poisson image denoising and blurring isan image deconvolution problem Image noise can be di-vided into three categories additive noise multiplicativenoise and Poisson noise +e logarithm of the multiplicativenoise can be processed as additive noise of Gaussian dis-tribution Different from the first two kinds of noise Poissonnoise generally appears in the case of very small illuminanceand amplification with high power electronic circuits andobeys a Poisson distribution Using variance stabilizingtransformation [1ndash5] and Anscombe transformation [6ndash9]the transformed data can be regarded as additive noise ofGaussian distribution However such methods may causesome data loss during the conversion process +us basedon these transformations the denoising and deblurringresults need to be further improved In recent years manyscholars have proposed algorithm models for Poissonianimage deblurring Poisson log-likelihood is inseparable inPoisson image deblurring Many scholars [10ndash15] transformthis problem into an equivalent constrained optimizationproblem through the split Bregmanmethod and then use theaugmented Lagrange method to solve this constrained op-timization problem Setzer et al [12] developed a Poissondenoising model based on minimizing the total variationregularization term (TV) and the I-divergence [16] as asimilarity term to restore blurred images +e TV regula-rization term is sensitive to the regularization parametersetting thus Yan et al [17] proposed an improved spatiallyadaptive total variation regularization algorithm which canautomatically balance the regularization strength betweendifferent regions and retain more edge information in re-stored images to solve this problem Most TV regulariza-tion-based restoration models can effectively reduce thenoise in flat regions but large staircase effects are also in-troduced in flat regions and fine details are not preserved incomplex structural regions which limits the practical ap-plication of the TV regularization term To solve thisproblem Fang et al [13] developed a Poisson imagedeconvolution method based on framelet regularizationwhich aims to adaptively capture multiscale edge structuresin images To make full use of the sparse information andnonlocal information of the image Shi et al [18] proposed alatent image estimation method based on nonlocal totalvariation and framelet regularization constraints whichachieved smooth denoising and deblurring results whilemaintaining details and edges

Generally images corrupted by Poisson noise are ac-companied by blur degradation +erefore the deblurringproblem of Poisson images is not only a denoising problembut also a deblurring problem In the process of imagerestoration the prior information of degraded image plays avery important role [17ndash28] In recent years many scholarshave used regularization priors to denoise or deblur imagessuch as dictionary learning [19] total variation[17 18 21 22] nonlocal mean [26ndash28] and L0 regularizationprior [20] In the field of image denoising a nonlocal mean

regularization term was proposed by Buades et al [26 27] in2005 to remove image noise Almahdi and Hardie [28]developed a new recursive nonlocal mean denoising algo-rithm In 2017 Romano et al [29] provided a new idea forremoving noise they provided an alternative stronger andmore flexible framework for removing additive whiteGaussian noise by proposing an explicit image-adaptiveLaplacian-based regularization function which can use theselected denoiser to define the regularization term calledRegularization by Denoising (RED) +e advantage of REDis that it can flexibly select the denoising engine and useexisting denoising algorithms to define regularization termsWhen using RED to restore degraded images the authorsmainly focus on images contaminated by additive whiteGaussian noise +ey tested and proved that RED canachieve good restoration results as a regularization term toremove additive white Gaussian noise Different from theimage corrupted by Gaussian noise the image corrupted byPoisson noise causes low image contrast and the imagedetails to be covered To make full use of the texture regioninformation in the image and preserve the details of theimage to the greatest extent while removing Poisson noisewe select the nonlocal mean denoiser as the regularizationterm of Poisson image denoising in the proposed method inthis paper In the field of image deblurring Tang et al [21]proposed a nonblind image deblurring method by local andnonlocal total variation models Gradient priors are com-monly used for image deblurring However in practicalapplications if only the gradient prior is used degradedimages cannot be restored well In [20] Pan et al proposedan image deblurring method based on the L0 regularizationintensity and gradient prior However these deblurringmethods all assume that noise is additive white Gaussiannoise or impulse noise To recover clear images from Poissonimages effectively we introduce the intensity prior of L0regularization and the prior of RED as explicit prior reg-ularization terms in the proposed method

With the success of deep networks in image processingan increasing number of scholars have begun to use neuralnetworks to solve the restoration problem of degradedimages [22 30ndash37] Most scholars usually need the model totrain a large number of data sample sets when using the deepCNN network to restore degraded images so that the modelcan learn the prior information of these data thus restoringdegraded images +erefore when using a neural networkfor image restoration a large number of data sample setsneed to be prepared first and then the degraded image canbe restored based on the trained model parameters How-ever Ulyanov et al [33] found in their research that the deepnetwork structure is sufficient for capturing a large numberof low-level image statistics before any learning In otherwords the only prior information when restoring degradedimages is derived from the structure of the network Basedon this discovery they proposed a method of learningimplicit priors from a single degraded image by using anetwork which is called DIP +e authors noted that anuntrained CNN network was used to restore degradedimages while only a randomly initialized network was re-quired Inspired by this idea Mataev et al [32] proposed

2 Mathematical Problems in Engineering

bringing in the concept of Regularization by Denoising(RED) +ey boost DIP by adding an explicit prior whichenriches the overall regularization effect In the same yearLiu et al [22] proposed combining implicit priors in DIPwith a traditional TV regularization prior to improve theimage quality of denoising or deblurring in DIP Howeverthese methods are all aimed at restoring degraded imagescontaminated by additive white Gaussian noise It is still achallenge to denoise and deblur Poisson images using neuralnetworks At present all blind Poisson image deconvolutionmethods have been proposed to use traditional explicitregularization methods to restore degraded images +ere-fore in this paper we propose an algorithm for denoisingand deblurring Poisson images by using neural networksWe combine an implicit regularization prior with two ex-plicit regularization priors that are the prior of RED and theprior of L0 regularization based on intensity and use a neuralnetwork to restore degraded images from Poisson imagesExperimental results demonstrate that compared withtraditional methods Poisson images restored by deep net-works can effectively suppress Poisson noise while pre-serving the real image edge details and detailed textureinformation of degraded images

2 Proposed methodology

21 RED +e regularization term plays an important role inimage deconvolution Compared with other regularizationterms RED [29] has the advantage that existing denoisingalgorithms can be flexibly used to define regularizationterms +e regularization term by denoising which is anexplicit image-adaptive Laplacian-based regularizationfunction uses the selected denoiser to define the regulari-zation term RED as a regularization function is given bythe following formula

ρ(x) 12x

T[x minus f(x)] (3)

where x is a degraded image f(middot) is the selected denoisingengine which is applied to degraded image x and the se-lection of the denoising engine is flexible x minus f(x) is thedenoising residual We found in the experimental researchthat the nonlocal mean denoising device can make full use ofthe texture region information in the degraded image whichcan remove the Poisson noise while maintaining the detailsof the image features to the greatest extent+erefore we usethe nonlocal mean denoiser as the regularization term ofPoisson image denoising in the proposed method in thispaper

22 L0 Regularization Term Based on the observation ofblurred images [20] the pixel intensity histogram of ablurred image is different from that of a clean image In theexperimental observation it is found that the number ofnonzero elements of pixel intensity in blurred images is largeand the distribution is relatively dense According to thepixel intensity property of blurred images for blurred im-ages we have

P(x) x0 (4)

where x0 represents the number of nonzero values inimage x and clear images and blurred images can be dis-tinguished according to the criterion of the pixel value in-tensity distribution Using this property of pixel valueintensity distribution clear images and blurred images canbe distinguished in the process of image restoration thusspeeding up the convergence time of the algorithm+erefore the L0 regularization term based on the pixelintensity prior is adopted as the regularization prior term inthe proposed method in this paper +e result analysis ofconvergence of L0 regularization is described in detail inSection 32

23 DIP In [33] the authors determined that the process ofrecovering degraded images in DIP is different from otherCNN networks It does not require a large amount of data-driven training but only requires the network to learn theimplicit prior of degraded images in a randomly initializedneural network framework +e authors found that theneural network can learn the undamaged part of the imageand then learn the damaged part of the image For exampleif the degraded image of white Gaussian noise is input to thenetwork the network will learn how to copy an imagewithout noise first and then learn to copy the noise+erefore we can remove the network learning before thenetwork learns to copy the noise so that we can obtain animage without noise DIP defines the output of the randomlyinitialized neural network as

x Fθ(z) (5)

where θ represents the parameters of the network z is a fixedrandom vector and Fθ(z) represents the parameterizedform of the network+en the objective function of DIP is asfollows

minθ

HFθ(z) minus x0

2 (6)

where H is a degraded operator and x0 is a degraded image+e network consists of linear convolution upsampling andnonlinear activation functions Based on the given degradedimage and the observation model the model parameters canbe approximated to the maximum likelihood throughiteration

24 Proposed Model In equation (6) to restore degradedimages through a deep network data items for restoringadditive white Gaussian noise images cannot solve theproblem of Poisson image deblurring well To obtain gooddenoising results while preserving the image details weintroduce equation (6) into the objective function 2 forPoissonian image deblurring and use RED as the denoisingregularization term in the Poisson image deblurring modelWe take the L0 norm of the latent clear image x as theregularization constraint in the Poisson image deblurring

Mathematical Problems in Engineering 3

model +en the objective function 2 can be written asfollows

E(x H) minxH

lang1 Hx minus y logHxrang +λ2x

T(x minus f(x))

+β2x0 +

τ2H

22

(7)

where λ is the weight of RED and β is the weight of the L0norm constrained regularization term

To solve the inseparable Poisson log-likelihood problemin equation (7) we use the split Bregman method to in-troduce auxiliary variable d1 Hx and transform equation(7) into a constrained problem In combination withequation (5) if the output x Fθ(z) of the neural network isintroduced into equation (7) as a constraint conditionequation (7) becomes the following equation

mind1 xHlang1 d1 minus y logd1rang +

λ2x

T(x minus f(x)) +

β2x0 +

τ2H

22

st d1 Hx x Fθ(z)

(8)

25OptimalEstimation To avoid the differentiation of theexplicit denoising function two auxiliary variables u1and u2 are introduced by using the split Bregmanmethod +e existence of the L0 norm in the third term ofequation (8) makes the solution somewhat difficult thesplit Bregman method is used to introduce auxiliaryvariable v and equation (8) is changed to the followingequation

mind1 xθHlang1 d1 minus y logd1rang +

μ2

d1 minus HFθ(z) minus u1

22 +

c

2x minus Fθ(z) minus u2

22 +

λ2x

T(x minus f(x)) +

α2

x minus v22 +

β2v0 +

τ2H

22 (9)

According to the split Bregman method seven unknownvariables d1 θ x u1 u2 v and H are iteratively optimizedand updated Variables d1 u1 x and H are fixed andvariable θ can be solved by the following formula

minθ

μ2


22 +

c


22 (10)

To solve equation (10) we use backpropagation to op-timize update variable θ where variables u1 and u2 areinfinitely close to d1 minus HFθ(z) and x minus Fθ(z) respectively

Given fixed variables d1 u1 and θ the point spreadfunction H can be solved by the following formula

minH

μ2


22 +

τ2H

22 (11)

+e update of variable Hk+1 is obtained by the followingequation

Hk+1

μ d1

k minus u1k( 1113857

TFθ(z)k+1

μFθ(z)k+1 Fθ(z)k+11113872 1113873

T+ τ

(12)

Given fixed variables θ u1 and H variable d1 can besolved by the following formula

mind1

lang1 d1 minus y log d1rang +μ2


22 (13)

+e update of variable dk+11 which can be solved by

equation (13) is obtained by the following equation

dk+11

12μ

μHk+1

Fθ(z)k+1

+ μuk1 minus 11113872 1113873

+

μHk+1Fθ(z)k+1 + μuk1 minus 11113872 1113873

2+ 4μy

1113970

(14)

where dk+11 is the value of d1 in the k + 1th iteration and

Fθ(z)k+1 is the output of the neural network in the k + 1thiteration

Given fixed variables θ u2 and v variable x can besolved by the following formula

minx

c


22 +

λ2x

T(x minus f(x)) +

α2

x minus v22

(15)

We use gradient descent to update xk+1

xk+1

xk

minus c c xk

minus Fθ(z)k+1

minus u21113872 1113873 + λ xk

minus f xk

1113872 11138731113872 1113873 + α xk

minus v1113872 11138731113960 1113961

(16)

where c should be selected to ensure a decreaseGiven fixed variables d1 H and θ auxiliary variable

u1k+1 can be updated by the following formula

uk+11 u

k1 minus d

k+11 + H

k+1Fθ(z)

k+1 (17)

Given fixed variables x and θ auxiliary variable uk+12 can

be updated by the following formula

uk+12 u

k2 minus x

k+1+ Fθ(z)

k+1 (18)

Given fixed variable x auxiliary variable vk+1 can beupdated by the following formula

vk+1

xk+1 xk+1

111386811138681113868111386811138681113868111386811138682 ge

βα

0 otherwise

⎧⎪⎪⎨

⎪⎪⎩(19)

Algorithm 1 proposed in this paper is as follows

3 Experimental Results

In this section we carry out experiments and image qualityevaluation on simulated blurred images and real blurredimages that are corrupted with Poisson noise and comparethe experimental results with those of four methods PID-


Input degraded image y

Initialization k 0 H0 25 u01 0 u0

2 0 v0 0 d01 0 x0 y randomly θ0

While θk+1 is not converged doUpdate θk+1 by equation (10)Update Hk+1 by equation (12)Update dk+1

1 by equation (14)Update xk+1 by equation (16)Update uk+1

1 by equation (17)Update uk+1

2 by equation (18)Update vk+1 by equation (19)

EndOutput recovered clear image x

ALGORITHM 1 Proposed method

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

Figure 1 Original images (a) Lena (size 256times 256) (b) Cameraman (size 256times 256) (c) Butterfly (size 256times 256) (d) Zebra(size 584times 387) (e) House (size 256times 256)

Table 1 PSNR (dB) values of the five methods

Image Gaussian blur Imax Degraded image PID-Split PIDSB-FA PIDSB-NLFA DeepRED Our method

Lena

Size 5 σ 163000 2484 2506 2522 2557 2580 26914000 2487 2507 2525 2562 2582 26945000 2490 2511 2529 2568 2585 2711

Size 15 σ 163000 2436 2501 2512 2526 2581 26334000 2438 2503 2516 2528 2592 26475000 2440 2507 2520 2529 2609 2657

Size 25 σ 23000 2299 2370 2444 2455 2457 24614000 2300 2372 2449 2457 2459 24645000 2302 2375 2451 2459 2461 2466

Cameraman

Size 5 σ 163000 2363 2538 2668 2673 2712 27464000 2366 2551 2673 2675 2720 28055000 2367 2569 2685 2692 2740 2814

Size 15 σ 23000 2173 2339 2402 2409 2424 24594000 2175 2343 2415 2417 2438 24645000 2176 2348 2420 2423 2447 2471

Size 25 σ 163000 2283 2439 2529 2537 2551 26294000 2284 2449 2532 2540 2574 26305000 2288 2466 2535 2542 2592 2650

Butterfly

Size 5 σ 163000 2288 2450 2523 2552 2556 25954000 2290 2463 2526 2555 2562 26015000 2291 2478 2527 2558 2566 2612

Size 15 σ 22000 2057 2196 2299 2329 2347 23983000 2060 2204 2303 2347 2359 24054000 2063 2222 2307 2357 2369 2413

Size 25 σ 163000 2181 2317 2396 2433 2446 24814000 2183 2333 2399 2436 2452 24895000 2184 2348 2400 2439 2459 2498


Table 1 Continued


Zebra

Size 5 σ 163000 2358 2481 2589 2650 2687 27174000 2361 2486 2596 2656 2696 27245000 2364 2492 2602 2662 2705 2730

Size 15 σ 23000 2169 2395 2505 2532 2638 26764000 2171 2403 2512 2543 2657 26855000 2173 2411 2520 2554 2669 2692

Size 25 σ 163000 2268 2441 2551 2604 2678 27224000 2271 2447 2557 2617 2690 27355000 2273 2454 2565 2629 2703 2742

House

Size 5 σ 163000 2881 3008 3071 3043 3244 32714000 2893 3017 3091 3048 3254 32885000 2899 3028 3110 3060 3261 3318

Size 15 σ 23000 2611 2712 2769 2752 2836 28474000 2614 2720 2770 2757 2837 28545000 2618 2733 2773 2762 2839 2862

Size 25 σ 163000 2771 2870 2961 2941 3029 30874000 2784 2878 2973 2949 3047 31085000 2785 2887 2981 2956 3056 3115

Table 2 VIF values of the five methods


Lena

Size 5 σ 163000 04482 04310 04513 05147 05182 057794000 04553 04327 04559 05122 05131 059255000 04603 04369 04614 05147 05136 05984

Size 15 σ 163000 04026 04307 04298 04938 05178 055854000 04074 04338 04346 04942 05282 056835000 04116 04393 04404 04946 05410 05745

Size 25 σ 23000 03484 03946 04761 04949 05130 052904000 03531 03973 04812 04962 05116 052975000 03566 04003 04842 04969 05290 05373

Cameraman

Size 5 σ 163000 03086 04045 04332 04336 04513 045334000 03107 04133 04436 04395 04563 046755000 03140 04233 04450 04458 04573 04723

Size 15 σ 23000 02255 03132 03260 03860 03743 039814000 02280 03166 03347 03867 03890 040175000 02301 03195 03372 03861 03950 04105

Size 25 σ 163000 02705 03730 03854 03860 03996 042954000 02733 03805 03894 03876 04040 044075000 02760 03910 03903 03861 04185 04447

Butterfly

Size 5 σ 163000 03940 04709 05535 05597 05768 058874000 03967 04815 05577 05614 05909 060285000 03982 04933 05602 05635 06074 06102

Size 15 σ 22000 02925 03818 04622 04717 05018 051243000 02957 03877 04661 04849 05228 052844000 02991 04002 04713 04967 05335 05390

Size 25 σ 163000 03471 04119 05117 05292 05478 056134000 03492 04189 05142 05325 05488 056305000 03507 04263 05163 05358 05660 05800

Zebra

Size 5 σ 163000 03913 04007 04905 05192 04913 052064000 03979 04038 04957 05278 05106 053025000 04020 04070 04992 05109 05179 05355

Size 15 σ 23000 02832 03348 03982 04094 04295 044904000 02878 03387 04026 04155 04394 045905000 02911 03429 04071 04219 04448 04622

Size 25 σ 163000 03402 03697 04066 04712 04763 049334000 03463 03731 04100 04789 04850 050165000 03504 03768 04136 04857 04959 05082


Split [12] PIDSB-FA [13] PIDSB-NLFA [18] andDeepRED [32] +e experiment of the proposed methodand the DeepRED method is implemented on a GTX 1080iGPU computer using the Python language in the Linuxsystem PID-Split PIDSB-FA and PIDSB-NLFA algo-rithms are tested by MATLAB 8a on a computer with anIntel(R) Core(TM) i5-7400 CPU and 8GB RAM In thesection of parameter setting we set the parameters of thePID-Split PIDSB-FA and PIDSB-NLFA methods

according to the methods suggested in the original paperFor different Poisson degraded images we set differentparameters to obtain the best recovery results +e pa-rameter settings in our proposed method are as followsμ 001 c 004 λ 002 α 001 β 0002 τ 004 Toevaluate the image quality the Peak Signal-to-Noise Ratio(PSNR) and Visual Information Fidelity (VIF) are used inthis section to evaluate the Poisson image recovered byeach algorithm Finally the PSNR value of each algorithm

Table 2 Continued


House

Size 5 σ 163000 03842 04028 04472 05040 05062 051564000 03898 04081 04563 05049 05103 051725000 03924 04127 04659 05047 05133 05320

Size 15 σ 23000 02914 03620 04378 04357 04334 045594000 02929 03688 04380 04373 04370 046805000 02946 03753 04404 04375 04389 04749

Size 25 σ 163000 03429 03743 04653 04719 04522 049464000 03496 03783 04741 04724 04686 050375000 03495 03823 04752 04663 04703 05091

(a) (b) (c)

(g)(f)(e)(d)

(h)

Figure 2 Restoration of the House image (a) Ground truth image (b) Degraded image blurred by Gaussian kernel of size 15times15 σ 2 andcorrupted by Poisson noise with Imax 3000 (c) Restored image by the PID-Split method (d) Restored image by the PIDSB-FAmethod (e)Restored image by the PIDSB-NLFAmethod (f ) Restored image by the DeepREDmethod (g) Restored image by the proposed method (h)Close-up views (close-up views of (a)ndash(g) image regions extracted from this example respectively)


is evaluated by using the ldquocompare_psnrrdquo function in theldquoskimagemeasurerdquo module in the Python language and theVIF value of each algorithm is evaluated by using theldquovifp_mscalerdquo function in MATLAB +e higher the valueof PSNR the better the image recovery effect the higher thevalue of VIF and the higher the visual information fidelityof the image

31 Simulated Poisson Image Denoising and Blurring In thissection we compare the proposed method with the otherfour algorithms (PID-Split PIDSB-FA PIDSB-NLFA andDeepRED) on five images with simulated Poisson imagedenoising and blurring experiments +e five images areLena (size 256times 256) Cameraman (size 256times 256)Butterfly (size 256times 256) Zebra (size 584times 387) andHouse (size 256times 256) as shown in Figure 1

In the simulated Poisson image experiment we ap-plied different levels of blur and noise to different images+e Gaussian blur kernel parameters we set for the Lenaimage of Figure 1(a) were (size 5 σ 16) (size 15σ 16) and (size 25 σ 2) +ree different Gaussianblur kernels were used to convolve the remaining fourimages and the Gaussian blur kernels were set to (size 5σ 16) (size 15 σ 2) and (size 25 σ 16) +en theldquoskimageutilrandom_noiserdquo function of the image pro-cessing algorithm set in Python was used to add threedifferent Poisson noise levels to the five images and thenoise levels ldquoImaxrdquo were set to 3000 4000 and 5000Particularly since the Butterfly image with an image size

of 256 times 256 in Figure 1(c) was insensitive to noise levels of5000 when the Gaussian blur kernel was (size 15 σ 2)we applied noise levels of 2000 3000 and 4000 to thisimage

In the experiment of deblurring simulated Poissonimages with different noise levels and different blurkernel sizes the image quality evaluation data recoveredby the PID-Split PIDSB-FA PIDSB-NLFA andDeepRED algorithms and the proposed method in thispaper are shown in Tables 1 and 2 According to theimage quality evaluation data in Tables 1 and 2 the PSNRvalue and VIF value of the proposed method in this paperwere higher than those of the other four algorithms Weconducted a set of comparison experiments on the Houseimage in Figure 1(e) and the results of the comparisonexperiments are shown in Figure 2 +e House imageshave a large size of flat regions and a small amount ofdetailed information In this group of comparative ex-periments as shown in Figure 2(b) the House image wasdegraded by a Gaussian blur kernel of size 15 times15 andstandard deviation σ 2 and polluted by Poisson noise(Imax 3000) As shown in Figure 2(c) although theimage restored by the PID-Split algorithm suppressednoise well it also caused staircase effects +e imagesrecovered by the PIDSB-NLFA and PIDSB-FA algo-rithms are shown in Figures 2(d) and 2(e) Comparedwith Figure 2(c) the staircase effects were suppressedand the noise in the flat regions was suppressed betterFigure 2(f ) shows the result of restoration of theDeepRED which retains more details but cannot remove

(a) (b) (c)

(g)(f)(e)(d)

(h)

Figure 3 Restoration of the Zebra image (a) Ground truth image (b) Degraded image blurred by Gaussian kernel of size 25times 25 σ 16and corrupted by Poisson noise with Imax 3000 (c) Restored image by the PID-Split method (d) Restored image by the PIDSB-FAmethod (e) Restored image by the PIDSB-NLFAmethod (f ) Restored image by the DeepREDmethod (g) Restored image by the proposedmethod (h) Close-up views (close-up views of (a)ndash(g) image regions extracted from this example respectively)


0 1000 2000 3000 4000

325

300

275

250

225

200

175

PSNR without L0 normPSNR with the L0 norm

(a)

0 100 200 300 400 500

300

275

250

225

200

175


(b)

Figure 5 Continued

(a) (b) (c)

(g)(f)(e)(d)

(h)

Figure 4 Restoration of the Butterfly image (a) Ground truth image (b) Degraded image blurred by Gaussian kernel of size 5times 5 σ 16and corrupted by Poisson noise with Imax 5000 (c) Restored image by the PID-Split method (d) Restored image by the PIDSB-FAmethod (e) Restored image by the PIDSB-NLFAmethod (f ) Restored image by the DeepREDmethod (g) Restored image by the proposedmethod (h) Close-up views (close-up views of (a)ndash(g) image regions extracted from this example respectively)


Table 3 BRISQUE values of the five methods

Image PID-Split PIDSB-FA PIDSB-NLFA DeepRED Our methodSaturn 4513 4331 4896 4142 3922Moon 4981 4334 4775 3928 3856Docking 4865 4914 4905 4195 3985Phoebe 3661 3294 3126 3044 2425

0 50 100 150 2000000

0002

0004

0006

Loss without L0 normLoss with the L0 norm

(c)

50 100 150 20000000

00001

00002

00003

00004

00005


(d)

Figure 5 PSNR values and loss comparison (a) PSNR (b) PSNE (zoomed in) (c) Loss (d) Loss (zoomed in)

(c)(b)(a)

(f)(e)(d)

(g)

Figure 6 Restoration of real blurred image (a) Saturn truth image (b) Restored image by the PID-Split method (c) Restored image by thePIDSB-FA method (d) Restored image by the PIDSB-NLFA method (e) Restored image by the DeepRED method (f ) Restored image bythe proposed method (g) Close-up views (close-up views of (a)ndash(f) image regions extracted from this example respectively)


Poisson noise well +e image restored by the proposedmethod in this paper is shown in Figure 2(g) Comparedwith the other four methods it not only suppressed noiseto the greatest extent but also had clearer details and thebest visual perception

Figure 3 shows a set of comparative results for Zebraimages in Figure 1(d) +e size of Zebra image is 584times 387Figure 3(b) is an image degraded by a Gaussian blur kernelwith a size of 25times 25 and a standard deviation of 16 andPoisson noise with a size of 3000 As can be seen fromFigure 3(a) the zebra has a large size of flat regions and thegrassland contains considerable detailed information Asseen in Figure 3(g) the proposed method in this paper notonly suppressed noise but also preserved considerable de-tailed information Figure 3(c) shows that the image restoredby the PID-Split algorithm was too smooth and had staircaseeffects Figures 3(d)ndash3(f ) show the images recovered by thePIDSB-NLFA PIDSB-FA and DeepRED algorithms re-spectively Compared with Figure 3(c) they suppressed thestaircase effects while retaining some details Compared withthe other four method the proposed method in this papernot only suppressed the staircase effects but also preservedthe details of the image to the greatest extent and was clearer

In addition the Butterfly image in Figure 1(c) was degradedby the Gaussian blur kernel (size 5times 5 σ 16) and thencorrupted by Poisson noise (Imax 5000) As shown in Fig-ure 4 compared with the images restored by the other fouralgorithms the proposed method in this paper suppressednoise and restored clear images to the greatest extentFigure 4(g) restored by ourmethod looks natural and preservedthe most detailed information of the images

32 Convergence Analysis +e explicit regularizationprior term adopts the combination of the intensity-basedL0 regularization term and RED+e purpose is to be ableto use the pixel intensity distribution of the blurredimage and the clear image to distinguish the restoredimage pixels while effectively filtering the image pixelsand removing the Poisson noise of the degraded imageTo verify the effectiveness of the L0 regularization priorwe used the method with the L0 norm regularization termand the method without the L0 norm to carry out sim-ulation and comparison experiments on the House im-ages of Figure 1(c) in Section 31 +e experimentalresults are shown in Figure 5 Figure 5(a) shows the

(c)(b)(a)

(f)(e)(d)

(g)

Figure 7 Restoration of real blurred image (a) Moon truth image (b) Restored image by the PID-Split method (c) Restored image by thePIDSB-FA method (d) Restored image by the PIDSB-NLFA method (e) Restored image by the DeepRED method (f ) Restored image bythe proposed method (g) Close-up views (close-up views of (a)ndash(f) image regions extracted from this example respectively)


PSNR values of the image restored by the two algorithmsIt can be clearly seen that the method with the L0 reg-ularization term has the characteristic of fast conver-gence With the increase in iteration times the restoredimage reached a higher PSNR Figure 5(b) shows theenlarged graph of the abscissa range (0sim500) inFigure 5(a) Figure 5(c) shows the loss of the objectivefunction and Figure 5(d) shows an enlarged graph withabscissa (0sim200) in Figure 5(c) It can be seen that themethod with the L0 regularization term also achieved asmaller loss value

33 Denoising and Deblurring of Real Poisson Images+is section is an experimental comparison of denoising anddeblurring of real Poisson images Tests were carried out onfour real astronomically degraded images To evaluate thereal Poisson image quality we used the BlindReferenceless

Image Spatial Quality Evaluator (BRISQUE) to evaluate thereal data +e smaller the value of BRISQUE the better theimage quality +e image quality evaluation data recoveredby the PID-Split PIDSB-FA PIDSB-NLFA and DeepREDalgorithms and the proposedmethod in this paper are shownin Table 3

Figure 6(a) shows a satellite image and Figure 6(b)shows an image restored by the PID-Split algorithm therestored image is too smooth Figures 6(c)ndash6(e) show theimages recovered by the PIDSB-FA PIDSB-NLFA andDeepRED algorithms respectively Although the staircaseeffects were suppressed to a large extent ring effects exist atthe edge +e image restored by the proposed method in thispaper is shown in Figure 6(f ) +e restored image not onlyrestored a large amount of detailed information and textureinformation but also suppressed the ring effects at the edgewhile suppressing the step effect +e image restored by theproposed method looks more natural

(c)(b)(a)

(f)(e)(d)

(g)

Figure 8 Restoration of real blurred image (a) +e docking truth image of Shenzhou-9 and Tiangong-1 (b) Restored image by the PID-Split method (c) Restored image by the PIDSB-FA method (d) Restored image by the PIDSB-NLFA method (e) Restored image by theDeepRED method (f ) Restored image by the proposed method (g) Close-up views (close-up views of (a)ndash(f) image regions extracted fromthis example respectively)


+e second real image is a moon image as shown inFigure 7(a) Figures 7(b)ndash7(e) show the images restored by thePID-Split PIDSB-FA PIDSB-NLFA and DeepRED algo-rithms respectively It can be seen from these three images thatthe images recovered by thesemethods had ringing effects of theedge and the flat areas were too smooth Figure 7(f) shows theimage restored by the proposed method in this paper It can beseen that the image restored by the proposed method in thispaper retained more detailed information while suppressingnoise and ringing effects

Figure 8 shows a set of comparative results for the dockingimages of Shenzhou-9 and Tiangong-1 Restored images byRLTV PID-Split PIDSB-FA DeepRED and the proposedmethod are shown in Figures 8(b)ndash8(f) respectively +e

image restored by the proposed method suppressed the ringeffects at the edge and retained more detailed informationFigure 9(a) shows the Phoebe image Comparing the restoredimages by the other four methods shown in Figures 9(b)ndash9(e)respectively the image restored by the proposed method inthis paper looks clear with fine details

4 Conclusions

+is paper proposed an algorithm for denoising and deblurringPoisson images by using neural networks We combine animplicit regularization prior with two explicit regularizationpriors that are the prior of RED and the prior of L0 regulari-zation based on intensity and use a neural network to restore

(c)(b)(a)

(f)(e)(d)

(g)

Figure 9 Restoration of real blurred image (a) Phoebe truth image (b) Restored image by the PID-Split method (c) Restored image by thePIDSB-FA method (d) Restored image by the PIDSB-NLFA method (e) Restored image by the DeepRED method (f ) Restored image bythe proposed method (g) Close-up views (close-up views of (a)ndash(f) image regions extracted from this example respectively)


degraded images fromPoisson images In the proposedmethodit has been proved that Poisson images restored by deep net-works can effectively suppress Poisson noise Meanwhile themethod with the L0 regularization term has the characteristic offast convergence

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

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

Acknowledgments

+is work was supported by a project of the National ScienceFoundation of China (61701353 61801337 and 61671337)

References

[1] J Boulanger C Kervrann P Bouthemy P ElbauJ-B Sibarita and J Salamero ldquoPatch-based nonlocal func-tional for denoising fluorescence microscopy image se-quencesrdquo IEEE Transactions on Medical Imaging vol 29no 2 pp 442ndash454 2010

[2] B Zhang M J Fadili and J L Starck ldquoMulti-scale variancestabilizing transform for multi-dimensional Poisson countimage denoisingrdquo in Proceedings of the 2006 IEEE Interna-tional Conference on Acoustics Speech and Signal Processingpp 1329ndash1332 Toulouse France May 2006

[3] B Zhang M J Fadili J L Starck et al ldquoMultiscale variance-stabilizing transform for mixed-Poisson-Gaussian processesand its applications in bioimagingrdquo in Proceedings of the 2007IEEE International Conference on Image Processing pp 233ndash236 San Antonio TX USA October 2007

[4] M Makitalo and A Foi ldquoA closed-form approximation of theexact unbiased inverse of the Anscombe variance-stabilizingtransformationrdquo IEEE Transactions on Image Processingvol 20 no 9 pp 2697-2698 2011

[5] B Zhang J M Fadili and J L Starck ldquoWavelets ridgeletsand curvelets for Poisson noise removalrdquo IEEE Transactionson Image Processing A Publication of the IEEE Signal Pro-cessing Society vol 17 no 7 pp 1093ndash1108 2008

[6] M Makitalo and A Foi ldquoOptimal inversion of the generalizedAnscombe transformation for Poisson-Gaussian noiserdquo IEEETransactions on Image Processing vol 22 no 1 pp 91ndash1032013

[7] F J Anscombe ldquo+e transformation of Poisson binomial andnegative- binomial datardquo Biometrika vol 35 no 3-4pp 246ndash254 1948

[8] R Liu Z Lin D L T Fernando and Z Su ldquoFixed-rankrepresentation for unsupervised visual learningrdquo in Pro-ceedings of the 2012 IEEE Conference on Computer Vision andPattern Recognition pp 598ndash605 Providence RI USA June2012

[9] M Makitalo and A Foi ldquoOptimal inversion of the Anscombetransformation in low-count Poisson image denoisingrdquo IEEETransactions on Image Processing vol 20 no 1 pp 99ndash1092011

[10] M A T Figueiredo and J M Bioucas-Dias ldquoDeconvolutionof Poissonian images using variable splitting and augmentedLagrangian optimizationrdquo in Proceedings of the 2009 IEEEWorkshop on Statistical Signal Processing pp 733ndash736Cardiff UK August 2009

[11] M A T Figueiredo J M Bioucas-Dias and M V AfonsoldquoFast frame-based image deconvolution using variable split-ting and constrained optimizationrdquo in Proceedings of the 2009IEEE Workshop on Statistical Signal Processing pp 109ndash112Cardiff UK August 2009

[12] S Setzer G Steidl and T Teuber ldquoDeblurring Poissonianimages by split Bregman techniquesrdquo Journal of VisualCommunication and Image Representation vol 21 no 3pp 193ndash199 2010

[13] H Fang H Yan Y Liu and Y Chang ldquoBlind Poissonianimages deconvolution with framelet regularizationrdquo OpticsLetters vol 38 no 4 pp 389ndash391 2013

[14] D Q Chen and L Z Cheng ldquoSpatially adapted regularizationparameter selection based on the local discrepancy functionfor Poissonian image deblurringrdquo Inverse Problems vol 28no 1 Article ID 015004-1 2012

[15] M A T Figueiredo and J M Bioucas-Dias ldquoRestoration ofpoissonian images using alternating direction optimizationrdquoIEEE Transactions on Image Processing vol 19 no 12pp 3133ndash3145 2010

[16] L A Shepp and Y Vardi ldquoMaximum likelihood recon-struction for emission tomographyrdquo IEEE Transactions onMedical Imaging vol 1 no 2 pp 113ndash122 2007

[17] L Yan H Fang and S Zhong ldquoBlind image deconvolutionwith spatially adaptive total variation regularizationrdquo OpticsLetters vol 37 no 14 pp 2778ndash2780 2012

[18] Y Shi J Song and X Hua ldquoPoissonian image deblurringmethod by non-local total variation and framelet regulari-zation constraintrdquo Computers amp Electrical Engineeringvol 62 pp 319ndash329 2017

[19] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash3745 2006

[20] J Pan Z Hu Z Su and M H Yang ldquoDeblurring text imagesvia L0 -regularized intensity and gradient priorrdquo in Pro-ceedings of the 2014 International Conference on ComputerVision and Pattern Recogintion pp 2901ndash2908 ColumbusOH USA June 2014

[21] S Tang W Gong W Li and W Wang ldquoNon-blind imagedeblurring method by local and nonlocal total variationmodelsrdquo Signal Processing vol 94 pp 339ndash349 2014

[22] J Liu Y Sun X Xu and U S Kamilov ldquoImage restorationusing total variation regularized deep image priorrdquo in Pro-ceedings of the 2019 International Conference on AcousticsSpeech and Signal Processing pp 7715ndash7719 Brighton UKMay 2019

[23] H Hong X Hua X Zhang and Y Shi ldquoMulti-frame realimage restoration based on double loops with alternativemaximum likelihood estimationrdquo Signal Image and VideoProcessing vol 10 no 8 pp 1489ndash1495 2016

[24] H Hong L Li I k Park and T Zhang ldquoUniversal deblurringmethod for real images using transition regionrdquo OpticalEngineering vol 51 no 4 2012

[25] H Hong Restoration Method and Application for Multi-spectralImage in Object Detection National Defense IndustryPress Beijing China 2017

[26] A Buades B Coll and J M Morel ldquoA non-local algorithmfor image denoisingrdquo in Proceedings of the 2005 IEEE


Computer Society Conference on Computer Vision and PatternRecognition vol 2 pp 60ndash65 San Diego CA USA June 2005

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

[28] R Almahdi and R C Hardie ldquoRecursive non-local meansfilter for video denoisingrdquo Eurasip Journal on Image andVideo Processing vol 2017 no 1 p 29 2017

[29] Y Romano M Elad and P Milanfar ldquo+e little engine thatcould regularization by denoising (RED)rdquo SIAM Journal onImaging Sciences vol 10 no 4 pp 1804ndash1844 2017

[30] K Zhang W Zuo S Gu and L Zhang ldquoLearning deep CNNdenoiser prior for image restorationrdquo in Proceedings of the2017 IEEE Conference on Computer Vision and Pattern Rec-ognition pp 3929ndash3938 Honolulu HI USA July 2017

[31] X Xu J Pan Y-J Zhang and M-H Yang ldquoMotion blurkernel estimation via deep learningrdquo IEEE Transactions onImage Processing vol 27 no 1 pp 194ndash205 2018

[32] G Mataev M Elad and P Milanfar ldquoDeepRED deep imageprior powered by REDrdquo 2019 httpsarxivorgabs190310176

[33] D Ulyanov A Vedaldi and V Lempitsky ldquoDeep imagepriorrdquo in Proceedings of the 2018 IEEE Computer SocietyConference on Computer Vision and Pattern Recognition SaltLake City UT USA June 2018

[34] K Zhang W Zuo Y Chen D Meng and L Zhang ldquoBeyonda Gaussian denoiser residual learning of deep CNN for imagedenoisingrdquo IEEE Transactions on Image Processing vol 26no 7 pp 3142ndash3155 2017

[35] K Zhang W Zuo and L Zhang ldquoDeep plug-and-play super-resolution for arbitrary blur kernelsrdquo in Proceedings of the2019 IEEE Computer Society Conference on Computer Visionand Pattern Recognition pp 1671ndash1681 Long Beach CAUSA June 2019

[36] Z Huang Y Zhang Q Li et al ldquoJoint analysis and weightedsynthesis sparsity priors for simultaneous denoising anddestriping optical remote sensing imagesrdquo IEEE Transactionson Geoscience and Remote Sensing pp 1ndash25 2020

[37] Z Huang Y Zhang Q Li et al ldquoUnidirectional variation anddeep CNN denoiser priors for simultaneously destriping anddenoising optical remote sensing imagesrdquo InternationalJournal of Remote Sensing vol 40 no 15 pp 5737ndash57482019


latent clear images and the estimation of point spreadfunctions

11 RelevantWork Poisson image denoising and blurring isan image deconvolution problem Image noise can be di-vided into three categories additive noise multiplicativenoise and Poisson noise +e logarithm of the multiplicativenoise can be processed as additive noise of Gaussian dis-tribution Different from the first two kinds of noise Poissonnoise generally appears in the case of very small illuminanceand amplification with high power electronic circuits andobeys a Poisson distribution Using variance stabilizingtransformation [1ndash5] and Anscombe transformation [6ndash9]the transformed data can be regarded as additive noise ofGaussian distribution However such methods may causesome data loss during the conversion process +us basedon these transformations the denoising and deblurringresults need to be further improved In recent years manyscholars have proposed algorithm models for Poissonianimage deblurring Poisson log-likelihood is inseparable inPoisson image deblurring Many scholars [10ndash15] transformthis problem into an equivalent constrained optimizationproblem through the split Bregmanmethod and then use theaugmented Lagrange method to solve this constrained op-timization problem Setzer et al [12] developed a Poissondenoising model based on minimizing the total variationregularization term (TV) and the I-divergence [16] as asimilarity term to restore blurred images +e TV regula-rization term is sensitive to the regularization parametersetting thus Yan et al [17] proposed an improved spatiallyadaptive total variation regularization algorithm which canautomatically balance the regularization strength betweendifferent regions and retain more edge information in re-stored images to solve this problem Most TV regulariza-tion-based restoration models can effectively reduce thenoise in flat regions but large staircase effects are also in-troduced in flat regions and fine details are not preserved incomplex structural regions which limits the practical ap-plication of the TV regularization term To solve thisproblem Fang et al [13] developed a Poisson imagedeconvolution method based on framelet regularizationwhich aims to adaptively capture multiscale edge structuresin images To make full use of the sparse information andnonlocal information of the image Shi et al [18] proposed alatent image estimation method based on nonlocal totalvariation and framelet regularization constraints whichachieved smooth denoising and deblurring results whilemaintaining details and edges

Generally images corrupted by Poisson noise are ac-companied by blur degradation +erefore the deblurringproblem of Poisson images is not only a denoising problembut also a deblurring problem In the process of imagerestoration the prior information of degraded image plays avery important role [17ndash28] In recent years many scholarshave used regularization priors to denoise or deblur imagessuch as dictionary learning [19] total variation[17 18 21 22] nonlocal mean [26ndash28] and L0 regularizationprior [20] In the field of image denoising a nonlocal mean

regularization term was proposed by Buades et al [26 27] in2005 to remove image noise Almahdi and Hardie [28]developed a new recursive nonlocal mean denoising algo-rithm In 2017 Romano et al [29] provided a new idea forremoving noise they provided an alternative stronger andmore flexible framework for removing additive whiteGaussian noise by proposing an explicit image-adaptiveLaplacian-based regularization function which can use theselected denoiser to define the regularization term calledRegularization by Denoising (RED) +e advantage of REDis that it can flexibly select the denoising engine and useexisting denoising algorithms to define regularization termsWhen using RED to restore degraded images the authorsmainly focus on images contaminated by additive whiteGaussian noise +ey tested and proved that RED canachieve good restoration results as a regularization term toremove additive white Gaussian noise Different from theimage corrupted by Gaussian noise the image corrupted byPoisson noise causes low image contrast and the imagedetails to be covered To make full use of the texture regioninformation in the image and preserve the details of theimage to the greatest extent while removing Poisson noisewe select the nonlocal mean denoiser as the regularizationterm of Poisson image denoising in the proposed method inthis paper In the field of image deblurring Tang et al [21]proposed a nonblind image deblurring method by local andnonlocal total variation models Gradient priors are com-monly used for image deblurring However in practicalapplications if only the gradient prior is used degradedimages cannot be restored well In [20] Pan et al proposedan image deblurring method based on the L0 regularizationintensity and gradient prior However these deblurringmethods all assume that noise is additive white Gaussiannoise or impulse noise To recover clear images from Poissonimages effectively we introduce the intensity prior of L0regularization and the prior of RED as explicit prior reg-ularization terms in the proposed method

With the success of deep networks in image processingan increasing number of scholars have begun to use neuralnetworks to solve the restoration problem of degradedimages [22 30ndash37] Most scholars usually need the model totrain a large number of data sample sets when using the deepCNN network to restore degraded images so that the modelcan learn the prior information of these data thus restoringdegraded images +erefore when using a neural networkfor image restoration a large number of data sample setsneed to be prepared first and then the degraded image canbe restored based on the trained model parameters How-ever Ulyanov et al [33] found in their research that the deepnetwork structure is sufficient for capturing a large numberof low-level image statistics before any learning In otherwords the only prior information when restoring degradedimages is derived from the structure of the network Basedon this discovery they proposed a method of learningimplicit priors from a single degraded image by using anetwork which is called DIP +e authors noted that anuntrained CNN network was used to restore degradedimages while only a randomly initialized network was re-quired Inspired by this idea Mataev et al [32] proposed





ρ(x) 12x

T[x minus f(x)] (3)



P(x) x0 (4)



x Fθ(z) (5)


minθ

HFθ(z) minus x0

2 (6)





E(x H) minxH


T(x minus f(x))

+β2x0 +

τ2H

22

(7)




λ2x

T(x minus f(x)) +

β2x0 +

τ2H

22

st d1 Hx x Fθ(z)

(8)



μ2


22 +

c


22 +

λ2x

T(x minus f(x)) +

α2

x minus v22 +

β2v0 +

τ2H

22 (9)


minθ

μ2


22 +

c


22 (10)



minH

μ2


22 +

τ2H

22 (11)


Hk+1

μ d1


TFθ(z)k+1

μFθ(z)k+1 Fθ(z)k+11113872 1113873

T+ τ

(12)


mind1



22 (13)



dk+11

12μ

μHk+1

Fθ(z)k+1

+ μuk1 minus 11113872 1113873

+


2+ 4μy

1113970

(14)




minx

c


22 +

λ2x

T(x minus f(x)) +

α2

x minus v22

(15)


xk+1

xk

minus c c xk

minus Fθ(z)k+1

minus u21113872 1113873 + λ xk

minus f xk

1113872 11138731113872 1113873 + α xk

minus v1113872 11138731113960 1113961

(16)



uk+11 u

k1 minus d

k+11 + H

k+1Fθ(z)

k+1 (17)



uk+12 u

k2 minus x

k+1+ Fθ(z)

k+1 (18)


vk+1

xk+1 xk+1

111386811138681113868111386811138681113868111386811138682 ge

βα

0 otherwise

⎧⎪⎪⎨

⎪⎪⎩(19)














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




Lena

Size 5 σ 163000 2484 2506 2522 2557 2580 26914000 2487 2507 2525 2562 2582 26945000 2490 2511 2529 2568 2585 2711

Size 15 σ 163000 2436 2501 2512 2526 2581 26334000 2438 2503 2516 2528 2592 26475000 2440 2507 2520 2529 2609 2657

Size 25 σ 23000 2299 2370 2444 2455 2457 24614000 2300 2372 2449 2457 2459 24645000 2302 2375 2451 2459 2461 2466

Cameraman

Size 5 σ 163000 2363 2538 2668 2673 2712 27464000 2366 2551 2673 2675 2720 28055000 2367 2569 2685 2692 2740 2814

Size 15 σ 23000 2173 2339 2402 2409 2424 24594000 2175 2343 2415 2417 2438 24645000 2176 2348 2420 2423 2447 2471

Size 25 σ 163000 2283 2439 2529 2537 2551 26294000 2284 2449 2532 2540 2574 26305000 2288 2466 2535 2542 2592 2650

Butterfly

Size 5 σ 163000 2288 2450 2523 2552 2556 25954000 2290 2463 2526 2555 2562 26015000 2291 2478 2527 2558 2566 2612

Size 15 σ 22000 2057 2196 2299 2329 2347 23983000 2060 2204 2303 2347 2359 24054000 2063 2222 2307 2357 2369 2413

Size 25 σ 163000 2181 2317 2396 2433 2446 24814000 2183 2333 2399 2436 2452 24895000 2184 2348 2400 2439 2459 2498


Table 1 Continued


Zebra

Size 5 σ 163000 2358 2481 2589 2650 2687 27174000 2361 2486 2596 2656 2696 27245000 2364 2492 2602 2662 2705 2730

Size 15 σ 23000 2169 2395 2505 2532 2638 26764000 2171 2403 2512 2543 2657 26855000 2173 2411 2520 2554 2669 2692

Size 25 σ 163000 2268 2441 2551 2604 2678 27224000 2271 2447 2557 2617 2690 27355000 2273 2454 2565 2629 2703 2742

House

Size 5 σ 163000 2881 3008 3071 3043 3244 32714000 2893 3017 3091 3048 3254 32885000 2899 3028 3110 3060 3261 3318

Size 15 σ 23000 2611 2712 2769 2752 2836 28474000 2614 2720 2770 2757 2837 28545000 2618 2733 2773 2762 2839 2862

Size 25 σ 163000 2771 2870 2961 2941 3029 30874000 2784 2878 2973 2949 3047 31085000 2785 2887 2981 2956 3056 3115



Lena

Size 5 σ 163000 04482 04310 04513 05147 05182 057794000 04553 04327 04559 05122 05131 059255000 04603 04369 04614 05147 05136 05984

Size 15 σ 163000 04026 04307 04298 04938 05178 055854000 04074 04338 04346 04942 05282 056835000 04116 04393 04404 04946 05410 05745

Size 25 σ 23000 03484 03946 04761 04949 05130 052904000 03531 03973 04812 04962 05116 052975000 03566 04003 04842 04969 05290 05373

Cameraman

Size 5 σ 163000 03086 04045 04332 04336 04513 045334000 03107 04133 04436 04395 04563 046755000 03140 04233 04450 04458 04573 04723

Size 15 σ 23000 02255 03132 03260 03860 03743 039814000 02280 03166 03347 03867 03890 040175000 02301 03195 03372 03861 03950 04105

Size 25 σ 163000 02705 03730 03854 03860 03996 042954000 02733 03805 03894 03876 04040 044075000 02760 03910 03903 03861 04185 04447

Butterfly

Size 5 σ 163000 03940 04709 05535 05597 05768 058874000 03967 04815 05577 05614 05909 060285000 03982 04933 05602 05635 06074 06102

Size 15 σ 22000 02925 03818 04622 04717 05018 051243000 02957 03877 04661 04849 05228 052844000 02991 04002 04713 04967 05335 05390

Size 25 σ 163000 03471 04119 05117 05292 05478 056134000 03492 04189 05142 05325 05488 056305000 03507 04263 05163 05358 05660 05800

Zebra

Size 5 σ 163000 03913 04007 04905 05192 04913 052064000 03979 04038 04957 05278 05106 053025000 04020 04070 04992 05109 05179 05355

Size 15 σ 23000 02832 03348 03982 04094 04295 044904000 02878 03387 04026 04155 04394 045905000 02911 03429 04071 04219 04448 04622

Size 25 σ 163000 03402 03697 04066 04712 04763 049334000 03463 03731 04100 04789 04850 050165000 03504 03768 04136 04857 04959 05082




Table 2 Continued


House

Size 5 σ 163000 03842 04028 04472 05040 05062 051564000 03898 04081 04563 05049 05103 051725000 03924 04127 04659 05047 05133 05320

Size 15 σ 23000 02914 03620 04378 04357 04334 045594000 02929 03688 04380 04373 04370 046805000 02946 03753 04404 04375 04389 04749

Size 25 σ 163000 03429 03743 04653 04719 04522 049464000 03496 03783 04741 04724 04686 050375000 03495 03823 04752 04663 04703 05091

(a) (b) (c)

(g)(f)(e)(d)

(h)








(a) (b) (c)

(g)(f)(e)(d)

(h)



0 1000 2000 3000 4000

325

300

275

250

225

200

175


(a)

0 100 200 300 400 500

300

275

250

225

200

175


(b)

Figure 5 Continued

(a) (b) (c)

(g)(f)(e)(d)

(h)





0 50 100 150 2000000

0002

0004

0006


(c)

50 100 150 20000000

00001

00002

00003

00004

00005


(d)


(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)






4 Conclusions


(c)(b)(a)

(f)(e)(d)

(g)




Data Availability




Acknowledgments


References












































ρ(x) 12x

T[x minus f(x)] (3)



P(x) x0 (4)



x Fθ(z) (5)


minθ

HFθ(z) minus x0

2 (6)





E(x H) minxH


T(x minus f(x))

+β2x0 +

τ2H

22

(7)




λ2x

T(x minus f(x)) +

β2x0 +

τ2H

22

st d1 Hx x Fθ(z)

(8)



μ2


22 +

c


22 +

λ2x

T(x minus f(x)) +

α2

x minus v22 +

β2v0 +

τ2H

22 (9)


minθ

μ2


22 +

c


22 (10)



minH

μ2


22 +

τ2H

22 (11)


Hk+1

μ d1


TFθ(z)k+1

μFθ(z)k+1 Fθ(z)k+11113872 1113873

T+ τ

(12)


mind1



22 (13)



dk+11

12μ

μHk+1

Fθ(z)k+1

+ μuk1 minus 11113872 1113873

+


2+ 4μy

1113970

(14)




minx

c


22 +

λ2x

T(x minus f(x)) +

α2

x minus v22

(15)


xk+1

xk

minus c c xk

minus Fθ(z)k+1

minus u21113872 1113873 + λ xk

minus f xk

1113872 11138731113872 1113873 + α xk

minus v1113872 11138731113960 1113961

(16)



uk+11 u

k1 minus d

k+11 + H

k+1Fθ(z)

k+1 (17)



uk+12 u

k2 minus x

k+1+ Fθ(z)

k+1 (18)


vk+1

xk+1 xk+1

111386811138681113868111386811138681113868111386811138682 ge

βα

0 otherwise

⎧⎪⎪⎨

⎪⎪⎩(19)














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




Lena

Size 5 σ 163000 2484 2506 2522 2557 2580 26914000 2487 2507 2525 2562 2582 26945000 2490 2511 2529 2568 2585 2711

Size 15 σ 163000 2436 2501 2512 2526 2581 26334000 2438 2503 2516 2528 2592 26475000 2440 2507 2520 2529 2609 2657

Size 25 σ 23000 2299 2370 2444 2455 2457 24614000 2300 2372 2449 2457 2459 24645000 2302 2375 2451 2459 2461 2466

Cameraman

Size 5 σ 163000 2363 2538 2668 2673 2712 27464000 2366 2551 2673 2675 2720 28055000 2367 2569 2685 2692 2740 2814

Size 15 σ 23000 2173 2339 2402 2409 2424 24594000 2175 2343 2415 2417 2438 24645000 2176 2348 2420 2423 2447 2471

Size 25 σ 163000 2283 2439 2529 2537 2551 26294000 2284 2449 2532 2540 2574 26305000 2288 2466 2535 2542 2592 2650

Butterfly

Size 5 σ 163000 2288 2450 2523 2552 2556 25954000 2290 2463 2526 2555 2562 26015000 2291 2478 2527 2558 2566 2612

Size 15 σ 22000 2057 2196 2299 2329 2347 23983000 2060 2204 2303 2347 2359 24054000 2063 2222 2307 2357 2369 2413

Size 25 σ 163000 2181 2317 2396 2433 2446 24814000 2183 2333 2399 2436 2452 24895000 2184 2348 2400 2439 2459 2498


Table 1 Continued


Zebra

Size 5 σ 163000 2358 2481 2589 2650 2687 27174000 2361 2486 2596 2656 2696 27245000 2364 2492 2602 2662 2705 2730

Size 15 σ 23000 2169 2395 2505 2532 2638 26764000 2171 2403 2512 2543 2657 26855000 2173 2411 2520 2554 2669 2692

Size 25 σ 163000 2268 2441 2551 2604 2678 27224000 2271 2447 2557 2617 2690 27355000 2273 2454 2565 2629 2703 2742

House

Size 5 σ 163000 2881 3008 3071 3043 3244 32714000 2893 3017 3091 3048 3254 32885000 2899 3028 3110 3060 3261 3318

Size 15 σ 23000 2611 2712 2769 2752 2836 28474000 2614 2720 2770 2757 2837 28545000 2618 2733 2773 2762 2839 2862

Size 25 σ 163000 2771 2870 2961 2941 3029 30874000 2784 2878 2973 2949 3047 31085000 2785 2887 2981 2956 3056 3115



Lena

Size 5 σ 163000 04482 04310 04513 05147 05182 057794000 04553 04327 04559 05122 05131 059255000 04603 04369 04614 05147 05136 05984

Size 15 σ 163000 04026 04307 04298 04938 05178 055854000 04074 04338 04346 04942 05282 056835000 04116 04393 04404 04946 05410 05745

Size 25 σ 23000 03484 03946 04761 04949 05130 052904000 03531 03973 04812 04962 05116 052975000 03566 04003 04842 04969 05290 05373

Cameraman

Size 5 σ 163000 03086 04045 04332 04336 04513 045334000 03107 04133 04436 04395 04563 046755000 03140 04233 04450 04458 04573 04723

Size 15 σ 23000 02255 03132 03260 03860 03743 039814000 02280 03166 03347 03867 03890 040175000 02301 03195 03372 03861 03950 04105

Size 25 σ 163000 02705 03730 03854 03860 03996 042954000 02733 03805 03894 03876 04040 044075000 02760 03910 03903 03861 04185 04447

Butterfly

Size 5 σ 163000 03940 04709 05535 05597 05768 058874000 03967 04815 05577 05614 05909 060285000 03982 04933 05602 05635 06074 06102

Size 15 σ 22000 02925 03818 04622 04717 05018 051243000 02957 03877 04661 04849 05228 052844000 02991 04002 04713 04967 05335 05390

Size 25 σ 163000 03471 04119 05117 05292 05478 056134000 03492 04189 05142 05325 05488 056305000 03507 04263 05163 05358 05660 05800

Zebra

Size 5 σ 163000 03913 04007 04905 05192 04913 052064000 03979 04038 04957 05278 05106 053025000 04020 04070 04992 05109 05179 05355

Size 15 σ 23000 02832 03348 03982 04094 04295 044904000 02878 03387 04026 04155 04394 045905000 02911 03429 04071 04219 04448 04622

Size 25 σ 163000 03402 03697 04066 04712 04763 049334000 03463 03731 04100 04789 04850 050165000 03504 03768 04136 04857 04959 05082




Table 2 Continued


House

Size 5 σ 163000 03842 04028 04472 05040 05062 051564000 03898 04081 04563 05049 05103 051725000 03924 04127 04659 05047 05133 05320

Size 15 σ 23000 02914 03620 04378 04357 04334 045594000 02929 03688 04380 04373 04370 046805000 02946 03753 04404 04375 04389 04749

Size 25 σ 163000 03429 03743 04653 04719 04522 049464000 03496 03783 04741 04724 04686 050375000 03495 03823 04752 04663 04703 05091

(a) (b) (c)

(g)(f)(e)(d)

(h)








(a) (b) (c)

(g)(f)(e)(d)

(h)



0 1000 2000 3000 4000

325

300

275

250

225

200

175


(a)

0 100 200 300 400 500

300

275

250

225

200

175


(b)

Figure 5 Continued

(a) (b) (c)

(g)(f)(e)(d)

(h)





0 50 100 150 2000000

0002

0004

0006


(c)

50 100 150 20000000

00001

00002

00003

00004

00005


(d)


(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)






4 Conclusions


(c)(b)(a)

(f)(e)(d)

(g)




Data Availability




Acknowledgments


References










































E(x H) minxH


T(x minus f(x))

+β2x0 +

τ2H

22

(7)




λ2x

T(x minus f(x)) +

β2x0 +

τ2H

22

st d1 Hx x Fθ(z)

(8)



μ2


22 +

c


22 +

λ2x

T(x minus f(x)) +

α2

x minus v22 +

β2v0 +

τ2H

22 (9)


minθ

μ2


22 +

c


22 (10)



minH

μ2


22 +

τ2H

22 (11)


Hk+1

μ d1


TFθ(z)k+1

μFθ(z)k+1 Fθ(z)k+11113872 1113873

T+ τ

(12)


mind1



22 (13)



dk+11

12μ

μHk+1

Fθ(z)k+1

+ μuk1 minus 11113872 1113873

+


2+ 4μy

1113970

(14)




minx

c


22 +

λ2x

T(x minus f(x)) +

α2

x minus v22

(15)


xk+1

xk

minus c c xk

minus Fθ(z)k+1

minus u21113872 1113873 + λ xk

minus f xk

1113872 11138731113872 1113873 + α xk

minus v1113872 11138731113960 1113961

(16)



uk+11 u

k1 minus d

k+11 + H

k+1Fθ(z)

k+1 (17)



uk+12 u

k2 minus x

k+1+ Fθ(z)

k+1 (18)


vk+1

xk+1 xk+1

111386811138681113868111386811138681113868111386811138682 ge

βα

0 otherwise

⎧⎪⎪⎨

⎪⎪⎩(19)














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




Lena

Size 5 σ 163000 2484 2506 2522 2557 2580 26914000 2487 2507 2525 2562 2582 26945000 2490 2511 2529 2568 2585 2711

Size 15 σ 163000 2436 2501 2512 2526 2581 26334000 2438 2503 2516 2528 2592 26475000 2440 2507 2520 2529 2609 2657

Size 25 σ 23000 2299 2370 2444 2455 2457 24614000 2300 2372 2449 2457 2459 24645000 2302 2375 2451 2459 2461 2466

Cameraman

Size 5 σ 163000 2363 2538 2668 2673 2712 27464000 2366 2551 2673 2675 2720 28055000 2367 2569 2685 2692 2740 2814

Size 15 σ 23000 2173 2339 2402 2409 2424 24594000 2175 2343 2415 2417 2438 24645000 2176 2348 2420 2423 2447 2471

Size 25 σ 163000 2283 2439 2529 2537 2551 26294000 2284 2449 2532 2540 2574 26305000 2288 2466 2535 2542 2592 2650

Butterfly

Size 5 σ 163000 2288 2450 2523 2552 2556 25954000 2290 2463 2526 2555 2562 26015000 2291 2478 2527 2558 2566 2612

Size 15 σ 22000 2057 2196 2299 2329 2347 23983000 2060 2204 2303 2347 2359 24054000 2063 2222 2307 2357 2369 2413

Size 25 σ 163000 2181 2317 2396 2433 2446 24814000 2183 2333 2399 2436 2452 24895000 2184 2348 2400 2439 2459 2498


Table 1 Continued


Zebra

Size 5 σ 163000 2358 2481 2589 2650 2687 27174000 2361 2486 2596 2656 2696 27245000 2364 2492 2602 2662 2705 2730

Size 15 σ 23000 2169 2395 2505 2532 2638 26764000 2171 2403 2512 2543 2657 26855000 2173 2411 2520 2554 2669 2692

Size 25 σ 163000 2268 2441 2551 2604 2678 27224000 2271 2447 2557 2617 2690 27355000 2273 2454 2565 2629 2703 2742

House

Size 5 σ 163000 2881 3008 3071 3043 3244 32714000 2893 3017 3091 3048 3254 32885000 2899 3028 3110 3060 3261 3318

Size 15 σ 23000 2611 2712 2769 2752 2836 28474000 2614 2720 2770 2757 2837 28545000 2618 2733 2773 2762 2839 2862

Size 25 σ 163000 2771 2870 2961 2941 3029 30874000 2784 2878 2973 2949 3047 31085000 2785 2887 2981 2956 3056 3115



Lena

Size 5 σ 163000 04482 04310 04513 05147 05182 057794000 04553 04327 04559 05122 05131 059255000 04603 04369 04614 05147 05136 05984

Size 15 σ 163000 04026 04307 04298 04938 05178 055854000 04074 04338 04346 04942 05282 056835000 04116 04393 04404 04946 05410 05745

Size 25 σ 23000 03484 03946 04761 04949 05130 052904000 03531 03973 04812 04962 05116 052975000 03566 04003 04842 04969 05290 05373

Cameraman

Size 5 σ 163000 03086 04045 04332 04336 04513 045334000 03107 04133 04436 04395 04563 046755000 03140 04233 04450 04458 04573 04723

Size 15 σ 23000 02255 03132 03260 03860 03743 039814000 02280 03166 03347 03867 03890 040175000 02301 03195 03372 03861 03950 04105

Size 25 σ 163000 02705 03730 03854 03860 03996 042954000 02733 03805 03894 03876 04040 044075000 02760 03910 03903 03861 04185 04447

Butterfly

Size 5 σ 163000 03940 04709 05535 05597 05768 058874000 03967 04815 05577 05614 05909 060285000 03982 04933 05602 05635 06074 06102

Size 15 σ 22000 02925 03818 04622 04717 05018 051243000 02957 03877 04661 04849 05228 052844000 02991 04002 04713 04967 05335 05390

Size 25 σ 163000 03471 04119 05117 05292 05478 056134000 03492 04189 05142 05325 05488 056305000 03507 04263 05163 05358 05660 05800

Zebra

Size 5 σ 163000 03913 04007 04905 05192 04913 052064000 03979 04038 04957 05278 05106 053025000 04020 04070 04992 05109 05179 05355

Size 15 σ 23000 02832 03348 03982 04094 04295 044904000 02878 03387 04026 04155 04394 045905000 02911 03429 04071 04219 04448 04622

Size 25 σ 163000 03402 03697 04066 04712 04763 049334000 03463 03731 04100 04789 04850 050165000 03504 03768 04136 04857 04959 05082




Table 2 Continued


House

Size 5 σ 163000 03842 04028 04472 05040 05062 051564000 03898 04081 04563 05049 05103 051725000 03924 04127 04659 05047 05133 05320

Size 15 σ 23000 02914 03620 04378 04357 04334 045594000 02929 03688 04380 04373 04370 046805000 02946 03753 04404 04375 04389 04749

Size 25 σ 163000 03429 03743 04653 04719 04522 049464000 03496 03783 04741 04724 04686 050375000 03495 03823 04752 04663 04703 05091

(a) (b) (c)

(g)(f)(e)(d)

(h)








(a) (b) (c)

(g)(f)(e)(d)

(h)



0 1000 2000 3000 4000

325

300

275

250

225

200

175


(a)

0 100 200 300 400 500

300

275

250

225

200

175


(b)

Figure 5 Continued

(a) (b) (c)

(g)(f)(e)(d)

(h)





0 50 100 150 2000000

0002

0004

0006


(c)

50 100 150 20000000

00001

00002

00003

00004

00005


(d)


(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)






4 Conclusions


(c)(b)(a)

(f)(e)(d)

(g)




Data Availability




Acknowledgments


References


















































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




Lena

Size 5 σ 163000 2484 2506 2522 2557 2580 26914000 2487 2507 2525 2562 2582 26945000 2490 2511 2529 2568 2585 2711

Size 15 σ 163000 2436 2501 2512 2526 2581 26334000 2438 2503 2516 2528 2592 26475000 2440 2507 2520 2529 2609 2657

Size 25 σ 23000 2299 2370 2444 2455 2457 24614000 2300 2372 2449 2457 2459 24645000 2302 2375 2451 2459 2461 2466

Cameraman

Size 5 σ 163000 2363 2538 2668 2673 2712 27464000 2366 2551 2673 2675 2720 28055000 2367 2569 2685 2692 2740 2814

Size 15 σ 23000 2173 2339 2402 2409 2424 24594000 2175 2343 2415 2417 2438 24645000 2176 2348 2420 2423 2447 2471

Size 25 σ 163000 2283 2439 2529 2537 2551 26294000 2284 2449 2532 2540 2574 26305000 2288 2466 2535 2542 2592 2650

Butterfly

Size 5 σ 163000 2288 2450 2523 2552 2556 25954000 2290 2463 2526 2555 2562 26015000 2291 2478 2527 2558 2566 2612

Size 15 σ 22000 2057 2196 2299 2329 2347 23983000 2060 2204 2303 2347 2359 24054000 2063 2222 2307 2357 2369 2413

Size 25 σ 163000 2181 2317 2396 2433 2446 24814000 2183 2333 2399 2436 2452 24895000 2184 2348 2400 2439 2459 2498


Table 1 Continued


Zebra

Size 5 σ 163000 2358 2481 2589 2650 2687 27174000 2361 2486 2596 2656 2696 27245000 2364 2492 2602 2662 2705 2730

Size 15 σ 23000 2169 2395 2505 2532 2638 26764000 2171 2403 2512 2543 2657 26855000 2173 2411 2520 2554 2669 2692

Size 25 σ 163000 2268 2441 2551 2604 2678 27224000 2271 2447 2557 2617 2690 27355000 2273 2454 2565 2629 2703 2742

House

Size 5 σ 163000 2881 3008 3071 3043 3244 32714000 2893 3017 3091 3048 3254 32885000 2899 3028 3110 3060 3261 3318

Size 15 σ 23000 2611 2712 2769 2752 2836 28474000 2614 2720 2770 2757 2837 28545000 2618 2733 2773 2762 2839 2862

Size 25 σ 163000 2771 2870 2961 2941 3029 30874000 2784 2878 2973 2949 3047 31085000 2785 2887 2981 2956 3056 3115



Lena

Size 5 σ 163000 04482 04310 04513 05147 05182 057794000 04553 04327 04559 05122 05131 059255000 04603 04369 04614 05147 05136 05984

Size 15 σ 163000 04026 04307 04298 04938 05178 055854000 04074 04338 04346 04942 05282 056835000 04116 04393 04404 04946 05410 05745

Size 25 σ 23000 03484 03946 04761 04949 05130 052904000 03531 03973 04812 04962 05116 052975000 03566 04003 04842 04969 05290 05373

Cameraman

Size 5 σ 163000 03086 04045 04332 04336 04513 045334000 03107 04133 04436 04395 04563 046755000 03140 04233 04450 04458 04573 04723

Size 15 σ 23000 02255 03132 03260 03860 03743 039814000 02280 03166 03347 03867 03890 040175000 02301 03195 03372 03861 03950 04105

Size 25 σ 163000 02705 03730 03854 03860 03996 042954000 02733 03805 03894 03876 04040 044075000 02760 03910 03903 03861 04185 04447

Butterfly

Size 5 σ 163000 03940 04709 05535 05597 05768 058874000 03967 04815 05577 05614 05909 060285000 03982 04933 05602 05635 06074 06102

Size 15 σ 22000 02925 03818 04622 04717 05018 051243000 02957 03877 04661 04849 05228 052844000 02991 04002 04713 04967 05335 05390

Size 25 σ 163000 03471 04119 05117 05292 05478 056134000 03492 04189 05142 05325 05488 056305000 03507 04263 05163 05358 05660 05800

Zebra

Size 5 σ 163000 03913 04007 04905 05192 04913 052064000 03979 04038 04957 05278 05106 053025000 04020 04070 04992 05109 05179 05355

Size 15 σ 23000 02832 03348 03982 04094 04295 044904000 02878 03387 04026 04155 04394 045905000 02911 03429 04071 04219 04448 04622

Size 25 σ 163000 03402 03697 04066 04712 04763 049334000 03463 03731 04100 04789 04850 050165000 03504 03768 04136 04857 04959 05082




Table 2 Continued


House

Size 5 σ 163000 03842 04028 04472 05040 05062 051564000 03898 04081 04563 05049 05103 051725000 03924 04127 04659 05047 05133 05320

Size 15 σ 23000 02914 03620 04378 04357 04334 045594000 02929 03688 04380 04373 04370 046805000 02946 03753 04404 04375 04389 04749

Size 25 σ 163000 03429 03743 04653 04719 04522 049464000 03496 03783 04741 04724 04686 050375000 03495 03823 04752 04663 04703 05091

(a) (b) (c)

(g)(f)(e)(d)

(h)








(a) (b) (c)

(g)(f)(e)(d)

(h)



0 1000 2000 3000 4000

325

300

275

250

225

200

175


(a)

0 100 200 300 400 500

300

275

250

225

200

175


(b)

Figure 5 Continued

(a) (b) (c)

(g)(f)(e)(d)

(h)





0 50 100 150 2000000

0002

0004

0006


(c)

50 100 150 20000000

00001

00002

00003

00004

00005


(d)


(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)






4 Conclusions


(c)(b)(a)

(f)(e)(d)

(g)




Data Availability




Acknowledgments


References









































Table 1 Continued


Zebra

Size 5 σ 163000 2358 2481 2589 2650 2687 27174000 2361 2486 2596 2656 2696 27245000 2364 2492 2602 2662 2705 2730

Size 15 σ 23000 2169 2395 2505 2532 2638 26764000 2171 2403 2512 2543 2657 26855000 2173 2411 2520 2554 2669 2692

Size 25 σ 163000 2268 2441 2551 2604 2678 27224000 2271 2447 2557 2617 2690 27355000 2273 2454 2565 2629 2703 2742

House

Size 5 σ 163000 2881 3008 3071 3043 3244 32714000 2893 3017 3091 3048 3254 32885000 2899 3028 3110 3060 3261 3318

Size 15 σ 23000 2611 2712 2769 2752 2836 28474000 2614 2720 2770 2757 2837 28545000 2618 2733 2773 2762 2839 2862

Size 25 σ 163000 2771 2870 2961 2941 3029 30874000 2784 2878 2973 2949 3047 31085000 2785 2887 2981 2956 3056 3115



Lena

Size 5 σ 163000 04482 04310 04513 05147 05182 057794000 04553 04327 04559 05122 05131 059255000 04603 04369 04614 05147 05136 05984

Size 15 σ 163000 04026 04307 04298 04938 05178 055854000 04074 04338 04346 04942 05282 056835000 04116 04393 04404 04946 05410 05745

Size 25 σ 23000 03484 03946 04761 04949 05130 052904000 03531 03973 04812 04962 05116 052975000 03566 04003 04842 04969 05290 05373

Cameraman

Size 5 σ 163000 03086 04045 04332 04336 04513 045334000 03107 04133 04436 04395 04563 046755000 03140 04233 04450 04458 04573 04723

Size 15 σ 23000 02255 03132 03260 03860 03743 039814000 02280 03166 03347 03867 03890 040175000 02301 03195 03372 03861 03950 04105

Size 25 σ 163000 02705 03730 03854 03860 03996 042954000 02733 03805 03894 03876 04040 044075000 02760 03910 03903 03861 04185 04447

Butterfly

Size 5 σ 163000 03940 04709 05535 05597 05768 058874000 03967 04815 05577 05614 05909 060285000 03982 04933 05602 05635 06074 06102

Size 15 σ 22000 02925 03818 04622 04717 05018 051243000 02957 03877 04661 04849 05228 052844000 02991 04002 04713 04967 05335 05390

Size 25 σ 163000 03471 04119 05117 05292 05478 056134000 03492 04189 05142 05325 05488 056305000 03507 04263 05163 05358 05660 05800

Zebra

Size 5 σ 163000 03913 04007 04905 05192 04913 052064000 03979 04038 04957 05278 05106 053025000 04020 04070 04992 05109 05179 05355

Size 15 σ 23000 02832 03348 03982 04094 04295 044904000 02878 03387 04026 04155 04394 045905000 02911 03429 04071 04219 04448 04622

Size 25 σ 163000 03402 03697 04066 04712 04763 049334000 03463 03731 04100 04789 04850 050165000 03504 03768 04136 04857 04959 05082




Table 2 Continued


House

Size 5 σ 163000 03842 04028 04472 05040 05062 051564000 03898 04081 04563 05049 05103 051725000 03924 04127 04659 05047 05133 05320

Size 15 σ 23000 02914 03620 04378 04357 04334 045594000 02929 03688 04380 04373 04370 046805000 02946 03753 04404 04375 04389 04749

Size 25 σ 163000 03429 03743 04653 04719 04522 049464000 03496 03783 04741 04724 04686 050375000 03495 03823 04752 04663 04703 05091

(a) (b) (c)

(g)(f)(e)(d)

(h)








(a) (b) (c)

(g)(f)(e)(d)

(h)



0 1000 2000 3000 4000

325

300

275

250

225

200

175


(a)

0 100 200 300 400 500

300

275

250

225

200

175


(b)

Figure 5 Continued

(a) (b) (c)

(g)(f)(e)(d)

(h)





0 50 100 150 2000000

0002

0004

0006


(c)

50 100 150 20000000

00001

00002

00003

00004

00005


(d)


(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)







(c)(b)(a)

(f)(e)(d)

(g)






4 Conclusions


(c)(b)(a)

(f)(e)(d)

(g)




Data Availability




Acknowledgments


References











































Table 2 Continued


House

Size 5 σ 163000 03842 04028 04472 05040 05062 051564000 03898 04081 04563 05049 05103 051725000 03924 04127 04659 05047 05133 05320

Size 15 σ 23000 02914 03620 04378 04357 04334 045594000 02929 03688 04380 04373 04370 046805000 02946 03753 04404 04375 04389 04749

Size 25 σ 163000 03429 03743 04653 04719 04522 049464000 03496 03783 04741 04724 04686 050375000 03495 03823 04752 04663 04703 05091

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4 Conclusions


(c)(b)(a)

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Data Availability




Acknowledgments


References














































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4 Conclusions


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Data Availability




Acknowledgments


References









































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Figure 5 Continued

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4 Conclusions


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Data Availability




Acknowledgments


References











































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4 Conclusions


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Data Availability




Acknowledgments


References













































(c)(b)(a)

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4 Conclusions


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Data Availability




Acknowledgments


References













































(c)(b)(a)

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4 Conclusions


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Data Availability




Acknowledgments


References












































4 Conclusions


(c)(b)(a)

(f)(e)(d)

(g)




Data Availability




Acknowledgments


References










































Data Availability




Acknowledgments


References






















































Documents

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