A Single Image Deblurring Algorithm for Nonuniform Motion

Research ArticleA Single Image Deblurring Algorithm for Nonuniform MotionBlur Using Uniform Defocus Map Estimation

Chia-Feng Chang Jiunn-Lin Wu and Ting-Yu Tsai

Department of Computer Science and Engineering National Chung Hsing University Taichung 402 Taiwan

Correspondence should be addressed to Jiunn-Lin Wu jlwucsnchuedutw

Received 13 August 2016 Revised 23 December 2016 Accepted 15 January 2017 Published 13 February 2017

Academic Editor Abdel-Ouahab Boudraa

Copyright copy 2017 Chia-Feng Chang et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

One of the most common artifacts in digital photography is motion blur When capturing an image under dim light by using ahandheld camera the tendency of the photographerrsquos hand to shake causes the image to blur In response to this problem imagedeblurring has become an active topic in computational photography and image processing in recent years From the view of signalprocessing image deblurring can be reduced to a deconvolution problem if the kernel function of the motion blur is assumedto be shift invariant However the kernel function is not always shift invariant in real cases for example in-plane rotation of acamera or a moving object can blur different parts of an image according to different kernel functions An image that is degradedby multiple blur kernels is called a nonuniform blur image In this paper we propose a novel single image deblurring algorithm fornonuniform motion blur images that is blurred by moving object First a proposed uniform defocus map method is presented formeasurement of the amounts and directions of motion blurThese blurred regions are then used to estimate point spread functionssimultaneously Finally a fast deconvolution algorithm is used to restore the nonuniform blur image We expect that the proposedmethod can achieve satisfactory deblurring of a single nonuniform blur image

1 Introduction

Recently image deblurring has become an essential issue inphotography because of the popularity of handheld camerasand smartphones When an image has been captured underdim light that captured image often suffers from motionblur artifacts Handheld cameras often have insufficient lightand thus require excessive exposure times which causeblurring Numerous photographs capture unusual momentsthat cannot be repeated with different camera settings Math-ematically a blurred image 119861 can be modeled as

119861 = 119870 otimes 119871 + 119873 (1)

where 119870 is blur kernel otimes is convolution operator 119871 is latentunblurred image and 119873 is noise in blurred image Duringexposure the movement of the camera can be viewed as amotion blur kernel called the point spread function (PSF)However image deblurring is an ill-posed problem becausethe PSF can be unknown and images can be latent If thePSF is shift invariant an image deblurring problem can be

reduced to an image deconvolution problem According towhat is known regarding the PSF image deconvolution prob-lems can be divided into two categories nonblind imagedeconvolution and blind image deconvolution In nonblindimage deconvolution the PSF is already known or computedtherefore the problem focuses on how to recover a blurredimage by using a known PSF The Wiener filter [1] and Rich-ardson-Lucy deconvolution [2] are well-known nonblinddeconvolution algorithms that are effective at image deblur-ring for cases in which the PSF is not complicated Howeverin real cases the PSF is complex If a given PSF is not preciselyestimated ringing artifacts are generated in the deblurredresult To address this problem several blind image decon-volution methods [3ndash13] have been proposed Howeverthe blind deconvolution problem is even more confusinglyframed than the nonblind deconvolution problemThe blindproblem entails recovering the unblurred image and esti-mating the PSF simultaneously Image pair approaches havebeen proposed for image deblurring Leveraging additionalimages makes the blind deblurring problem more tractable

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 6089650 14 pageshttpsdoiorg10115520176089650

2 Mathematical Problems in Engineering

(a)

(b)

Figure 1 An example of a real-world image There are differentkinds of blur in the image The fallen ball is blurry and others arenot (a) Real-world image (b)Magnification red rectangle is blurredbasketball and blue rectangles are unblurred regions

Rav-Acha and Peleg used two motion blur images [9] Yuanet al recovered a blurred image from a noisemotion pairthat had been captured under low light conditions [10]Zhuo et al leveraged a flashmotion pair that provided clearstructural information for an image deblurring problem [11]Zhang et al applied varying image pairs to image deblurring[12] Nevertheless image pair approaches require additionalimages that must be captured by extra hardware Chang andWu proposed a new deconvolution method for deblurringa blurred image with uniform motion blur by using hyperLaplacian priors [13]

As shown in Figure 1 motion blur in a real-world imagecan be so complicated that the assumption of shift invarianceis not always held To deblur a nonuniformmotion blur imageis more difficult [14ndash23] The causes of nonuniform motionblur can be divided into two categories camera rotation[14 15] and target motion [16ndash23] To solve the target motionproblem nonuniform blur image deblurring methods havebeen proposed In 2006 Levin deblurred a nonuniformmotion blur image by segmenting nonuniform motion blurimages into blurred regions and unblurred regions by usingimage statistics [16] But this method works when the direc-tion of blur was vertical or horizontal In 2007 Cho et alproposed using multiple blurred images that had been cap-tured continuously [17] An image registration method wasthen applied to estimate the PSF but it required previous

discernment of the blurred object and additional imagesIn 2011 Tai et al proposed using a projective motion paththat was recorded with additional hardware to replace thetraditional plane blur kernel for image deblurring [14] Aprojectivemotion path describes themovement of the camerain a three-dimensional (3D) model therefore it can restorenonuniform motion blur images for which blur had beencaused by camera rotation However additional hardwarewas required to record the 3D blur kernel Several nonuni-form image deblurring methods with depth maps have beenproposed to handle this problem In 2012 Xu and Jia used adepth map to deblur blurred images with the same motionblur but different Gaussian blurs [22] In 2014 Hu et alproposed using depth estimation to deblur a nonuniformblur image [23]The authors separated the blurred image intolayers according to a depth map and then deblurred one layerbut maintained the others in a fixed state This deblurringmethod treats every object in the same depth as havingthe same blur amount However if there is an object witha different blur amount in the same layer deblurring failsbecause the blur amount of objects at the same depth is notalways the same

In this paper we propose a novel image deblurringalgorithm for nonuniform motion blur images To locateblurred parts we measure the blur amount for each edge inthe blurred image The measurement of blur is then appliedto propagate a uniform defocus map We use the proposeduniform defocus map to segment the nonuniform blur imageinto multiple blurred and unblurred portions Subsequentlywe estimate the PSF for each blurred portion to obtain the PSFof each portion All portions and the corresponding PSFs areentered as inputs to a fast deconvolution algorithm Finallyan unblurred result is obtained

The rest of the paper are organized as follows In Section 2we briefly review the related works on nonuniform imagedeblurring Section 3 presents our proposedmethod in detailThe experimental results are then designed for verifying theproposed method in Section 4 Finally we conclude thispaper in Section 5

2 Related Works

Nonuniform motion is an ill-posed problem that manystudies have been proposed to address this problem In 2006Levin removed nonuniform motion blur by separating ablurred image into unblurred regions and blurred regionsby using image statistics The authors located blurred andunblurred regions depending on the observation that thestatistics of derivative filters in images are substantiallychanged by partial motion blur They assumed that the blureffect resulted from motion at a constant velocity and thusmodeled the expected derivative distributions as functionsof the width of the blur kernel This was effective when thedirection ofmotion blur was vertical or horizontal or in otherwords the blur kernel was one-dimensional In 2007 Choet al proposed using multiple blurred images that were cap-tured continuously to remove nonuniform motion blur [17]These blurred images were captured for the same scene Theauthors applied an image registrationmethod to compute the

Mathematical Problems in Engineering 3

offsets of a moving object in two sequential images Oncethe moving object was found in the images they calculatedthe movement of the moving object from the tiny movementthat was computed from the image pair This movementwas the estimated PSF of the moving object The estimatedPSF was applied to recover an image of the blurred objectUsing multiple images to estimate the PSF made the nonuni-form deblurring problem more tractable However multiplesequential images are not always captured In 2011 Tai et alproposed using a projective motion path which was esti-mated using additional hardware to deblur a blurred imageThe projective motion path is a model that records everymovement of a camera through planar projective transforma-tion (ie homography) during the exposure time Homogra-phy provides not only planar information but also rotationalinformation for rotation blur whereas the traditional 2D PSFprovides only planar information Therefore homographycan recover a motion blur image with rotational blur suc-cessfully but it requires additional hardware to record theprojective motion path In 2014 Hu et al proposed a nonuni-form image deblurring approach that involves using a depthmap and applying depth estimation to an image deblurringalgorithm for a nonuniform blur imageThe approach entailsseparating the blurred image into layers according to esti-mated depth information Every object with the same depthestimate is classified into the same depth layer Hu et aldeblurred each layer butmaintained the others at a fixed levelThe depth map provided crucial information for deblurringa nonuniform blur image However the deblurred result willfail while there are different blur objects in the same depthlayer

Inspired by these nonuniform image deblurringmethodswe sought to locate blurred objects precisely in nonuniformmotion blur imagesThis is a vital task for deblurring spatiallyvarying blurred images We also sought to detect all types ofmotion blur objects in the blurred image automatically In thenext section we demonstrate a noteworthy characteristic ofblurred objects in nonuniform motion blur images explainthe dynamics of this characteristic and then propose ourmethod for solving this ill-posed image deblurring problem

3 Proposed Method

As mentioned in the previous section separating a nonuni-form blur image into unblurred and blurred portions is anessential task for recovering nonuniformmotion blur imagesAs shown in Figure 1 a nonuniform blur image containspartial motion blur that the fallen ball is blurry and the basketis not We notice that the ball and the basket are rigid objectsIf these rigid objects suffered from blur problem the pixelswithin a rigid object should suffer from same blur problemOnce the blurred objects were founded we can performdeblurring procedure for each blurred object therefore thenonuniform motion deblurring problem can be reduced toshift-invariant image deconvolution problem Through theproposed method we seek to measure the blur amount ofeach object in a blurred image and then apply that bluramount to find blurred objects in the blurred image Inspiredby Bae and Durand [18] we use the distance between the

0

Position

Inte

nsity

d1

d2

Blurred signal (120590 = 5)2nd derivative response model (120590 = 5)Blurred signal (120590 = 3)2nd derivative response model (120590 = 3)

Figure 2 An example of blur amount estimation

maximal value and minimal value of a second derivative todefine the blur amount for each pixel in the blurred objectConsider the blurred signals with varying sigma120590 in Figure 2A larger sigma yields a more blurred signal and a widerdistance between the maximum and minimum We considerthat the two signals include the blue signal that is extractedfrom the car and the red signal that is extracted from thebook as shown in Figure 3The blur amount of the blue signalin Figure 3(b) is greater than the blur amount of the red signalThe car is blurred and the book is not as shown in Figure 3(a)That is the distance between the second derivativemaximumand the local minimum can be viewed as the blur amount ofan object In fact the blur amount of all rigid object is thesame From this fact we unify the blur amount of an objectby using a 119896-means clustering algorithmThe flowchart of theproposed method is shown in Figure 4

31 Uniform Defocus Map According to Elder and Zucker[24] the edge regions that have notable frequency contentare suitable for blur amount measurement Step edges are themain edge type in a natural image therefore we considermeasuring the blur amount at edge pixels in this paper Inmathematics an ideal step edge can be formulated as follows

119891 (119909) = 119860119906 (119909) + 119861 (2)

where 119906(119909) is step function119860 is amplitude and 119861 is offset Ablurred step edge 119894(119909) can then be defined as the result of astep edge that convolutes the Gaussian function 119892(119909 120590)

119894 (119909) = 119891 (119909) otimes 119892 (119909 120590) (3)

where 119894(119909) is blurred step edge 119892(119909 120590) is Gaussian functionotimes is convolution operator and 120590 is the standard deviation ofGaussian function


(a)

0

Inte

nsity

Blurred signalSharp signal

10 20 30 40 50 60 70 80 900Position

(b)

Figure 3 An example of blurred edge and unblurred edge (a) A nonuniform blurred image (b) Signals of blurred edge and signal of sharpedge corresponding to the blue line and red line in (a) The 119909-axis is position and 119910-axis is intensity in gray level

311 BlurAmount Estimation In the formulation of a blurredstep edge 120590 can be represented as the blur amount We cancalculate 120590 by using a reblur method A reblur of the blurredstep edge 1198941(119909) is

1198941 (119909) = 119894 (119909) otimes 119892 (119909 120590)= 119860radic2120587 (1205902 + 12059020) exp(minus

11990922 (1205902 + 12059020)) (4)

where 1205900 is standard deviation of reblur Gaussian functionThen with blurred step edge divided by reblur blurred stepedge we can obtain the following equation

|nabla119894 (119909)|1003816100381610038161003816nabla1198941 (119909)1003816100381610038161003816= radic (1205902 + 12059020)1205902 exp(minus( 119909221205902 minus 11990922 (1205902 + 12059020)))

(5)

Let 119877 be the ratio of blurred step edge to reblur blurred stepedge When 119909 = 0 119877 has maximum value Then we have

119877 = |nabla119894 (119909)|1003816100381610038161003816nabla1198941 (119909)1003816100381610038161003816 =radic (1205902 + 12059020)1205902 (6)

Given a ratio 119877 the unknown 120590 can be computed using

120590 = 1radic1198772 minus 11205900 (7)

While 120590 is computed we use 120590 as the blur amount at edgepixel Blur measurement result is shown in Figure 5

312 Blur Amount Refinement After the blur amounts ofedges pixels have been computed we propagate the bluramount to nonedge pixels for which blur amounts have notpreviously been computed However phenomena such as

shadows and highlights may cause some outliers to appearin the measurement results These outliers may propagateincorrect results We should remove these outliers to obtainsparsemeasurement results Pixels in a sparsematrix are oftenzero Therefore instead of using a cross-bilateral filter suchas that applied in [18 20] we propose using a simple bilateralfilter for our sparse blur measurements The main idea is thatonly nonzero value pixels are considered in filter procedureThe definition of a simple bilateral filter which is applied onlyto nonzero pixels is as follows

119868filtered = 1119882119901 sum119909119894isinΩ

119868 (119909119894)119882119901 (119909119894) 119868 (119909) = 0 (8)

with

119882119901 = sum119909119894isinΩ

119882119877 (119909119894)119882119878 (119909119894) (9)

and the weighting functions119882119877 and119882119878 are defined as

119882119877 (119909119894) = 119891119903 (1003817100381710038171003817119868 (119909119894) minus 119868 (119909)1003817100381710038171003817) 119882119878 (119909119894) = 119892119904 (1003817100381710038171003817119909119894 minus 1199091003817100381710038171003817) (10)

where 119868filtered is the filtered result 119868 is the sparse blur mea-surement to be filtered 119909 are the coordinates of the currentpixel to be filtered Ω is the window centered in 119909 and 119868(119909119894)must be none zero value 119891119903 is the range kernel of Gaussianfunction for smoothing differences in intensities and119892119904 is thespatial kernel of Gaussian function for smoothing differencesin coordinates The intermediate result is shown in Figure 6Theword on the roof is smoother after application of a simplebilateral filter

313 Unify Blur Amount The blur amounts of pixels thatare located in the same rigid object are equal Therefore weuse 119896-means clustering to unify the measurement results119870-means clustering algorithms minimize the within-clustersums of squares Thus 119896-means methods cluster the pixels


Blur amountpropagation

Blur amountmeasurement

Remove outliers bysimple bilateral filter

Blind image deconvolution

Deblurred image

Unify blur

clusteringamount by k-means

Nonuniform blurimage

Figure 4 Flowchart of proposed method

with the same blur amounts together Labels of each pixel atthe edge region are calculated by minimizing the followingcost function

argmin119878

119896sum119894=1

sum119909119895isin119904119894

10038171003817100381710038171003817119909119895 minus 120583119894100381710038171003817100381710038172 (11)

where 119878 = 1199041 1199042 119904119896 is a set of clusters 119896 is the desirednumber of sets and 120583119894 is the mean of points in 119904119894 Theblur measurement result with 119896-means clustering which isshown in Figure 6 Close-up views in Figure 6(b) show thatthe blur amount of the car in our measurement result is

uniform but the amount for Zhuorsquos edgemap is notWe applymorphological operations to the refinement result and weobserve that 119896-means clustering successfully unifies the bluramount in a rigid object as shown in Figure 7

314 Blur Amount Propagation To propagate an estimatedblur amount to awhole image we apply thematting Laplacianinterpolation method to our estimated blur measurementThe estimated blur amounts from the edge regions arepropagated to other nonedge regions to form a defocus mapThe cost function of the matting Laplacian interpolationmethod is formulated as follows

119864 (119889) = 119889119879119871119889 + 120582 (119889 minus 119889)119879119863(119889 minus 119889) (12)

where 119889 is the full defocus map 119889 is the vector forms of thesparse defocus map L is the matting Laplacian matrix and119863 is diagonal matrix whose element 119863119894119894 is 1 at edge regionsand 0 otherwise and 120582 is a parameter which balances fidelityto the sparse depth map and smoothness of interpolationThe defocus map can be obtained by solving (12) [25] Theproposed defocus map and Zhuorsquos defocus map are shownin Figure 8 From Zhuorsquos defocus map in Figure 8(a) theestimated blur amounts are different at different parts of theblurred car therefore it was difficult to segment the blurredcar By contrast for the proposed defocus map shown inFigure 8(b) the estimated blur amount is uniform for the carThe key benefit from our uniform defocus map is that theblurred car can be easily segmented

32 Blind Image Deconvolution If the blurred object regionscan be segmented accurately the nonuniform blur problemcan be reduced to a uniform blur problemHence the blurredobjects are used as inputs for a uniform blur image deconvo-lution We applied blind image deconvolution to recover theblurred object regions

321 PSF Estimation In mathematics a blurred image canbe modeled as

119861 = 119868 lowast 119870 + 119873 (13)

where 119861 is blurred image lowast is convolution operator I islatent unblurred image 119870 is PSF and 119873 is noise in imageThe equation can then be represented as follows according toBayesrsquo theorem

119901 (119868 119870 | 119861) prop 119901 (119861 | 119868 119870) 119901 (119868) 119901 (119870) (14)

where 119901(119861|119868 119870) represents the likelihood and 119901(119868) and 119901(119870)denote the priors on the latent image and PSF In the PSFestimation step Bayesrsquo theorem can be transformed into thefollowing equations

1198701015840 = argmin119870

119870 lowast 119868 minus 119861 + 119901119870 (119870) 1198681015840 = argmin

119868

119870 lowast 119868 minus 119861 + 119901119868 (119868) (15)


(a) (b)

(c)

Figure 5 Intermediate result of applying simple bilateral filter to propose blur amount measure method (a) Blur measurement (b) Blurmeasurement with simple bilateral filter (c) Close-up views correspond to red rectangle in (a) and the same position in (b)

(a)

(b)

Figure 6 Effect of k-means method (a) Proposed defocus mapwith k-means clustering (b) Close-up views corresponding to redrectangle in (a) and in Figure 5(a) with the same position From (b)the blur amount of moving car in proposed defocus map is moreuniform than Zhuorsquos defocus map

Figure 7 Result of k-means clustering

We only consider the gradient of latent image and blurredimage while solving kernel estimation problem We rewrite(15) into the following energy function

119864 (119870) = sum1003817100381710038171003817119870 lowast 120597lowast119871 minus 120597lowast11986110038171003817100381710038172 + 120573 1198702 119864 (119871) = sum119870 lowast 119871 minus 1198612 + 120572 nabla1198712 (16)

where 120597lowast isin 120597119909 120597119910 120597119909119909 120597119909119910 120597119910119910 is the partial derivative oper-ators in different directions nabla is Sobel operator and 120572 and120573 are preset parameters


(a) (b)

Figure 8 Comparison of proposed defocus map and Zhuorsquos defocus map (a) Zhuorsquos defocus map (b) Proposed defocus map Comparing(a) and (b) proposed defocus map is more uniform in the rigid object such as car and book than Zhuorsquos defocus mapTherefore the blurredobjects in proposed defocus map can be easily segmented

We iteratively solve the preceding equations to obtainan accurate PSF To accelerate the PSF estimation processwe apply a shock filter before each PSF estimation step Theformulation of the shock filter is defined as follows

119868119905+1 = 119868119905 minus sign (Δ119868119905) 1003817100381710038171003817nabla1198681199051003817100381710038171003817 119889119905 (17)

where 119868119905 is the image at current iteration 119868119905+1 is the image atnext iterationΔ119868119905 is themap derived fromLaplacian operatorat iteration 119905 nabla119868119905 is the gradient of 119868 at current iteration and119889119905 is time step

322 Fast Adaptive Deconvolution When a sufficient PSF isobtained a fast adaptive deconvolution method is used forfinal deconvolution [26] Equation (18) is the minimizationproblem of the fast adaptive deconvolution method

argmin119868

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 2sum119896=1

1003816100381610038161003816(119891119896 lowast 119868)1198941003816100381610038161003816119902) (18)

where 119894 is an index running through all pixels In this paperwe use the value 23 as 119902 which is suggested by Krishnan andFergus [26] and 119891119896 is first-order derivative filter

1198911 = [1 minus1] 1198912 = [1 minus1]119879 (19)

We search for 119868 which minimizes the reconstruction error119870 lowast 119868 minus 1198612 with the image prior preferring 119868 to favor thecorrect sharp explanation

However 119902 lt 1 makes the optimization problem non-convex It becomes slow to solve the approximationUsing thehalf-quadratic splitting Krishnanrsquos fast algorithm introducestwo auxiliary variables 1205961 and 1205962 at each pixel to move the

(119891119896 lowast 119868)119894 terms outside the | sdot |119902 expression Thus (18) can beconverted to the following optimization problem

argmin119868120596

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 12057322sum119896=1

(119891119896 lowast 119868 minus 120596119896)2119894+ 2sum119896=1

1003816100381610038161003816(120596119896)1198941003816100381610038161003816119902) (20)

where (119891119896 lowast 119868 minus 120596119896)2 term is for constraint of 119891119896 lowast 119868 = 120596119896 and120573 is a control parameter that we will vary during the iterationprocess As 120573 parameter becomes large the solution of (20)converges to that of (18) This scheme also called alternatingminimization [26] where we adopt is a common techniquefor image restoration Minimizing (20) for a fixed 120573 can beperformed by alternating two steps This means that we solve120596 and 119868 respectively

To solve the 120596 subproblem first the input blurred image119861 is set to the initial 119868 Given a fixed 119868 finding the optimal 120596can be reduced to the following optimization problem

argmin120596

(|120596|119902 + 1205732 (120596 minus ])2) (21)

where the value ] = (119891119896lowast119868) For the case 119902 = 23120596 satisfyingthe above equation is the analytical solution of the followingquartic polynomial

1205964 minus 3]1205963 + 3]21205962 minus ]3120596 + 8271205733 = 0 (22)

To find and select the correct roots of the above quarticpolynomial we adopt Krishnanrsquos approach as detailed in[26]


(a) (b)

(c) (d)

Figure 9 Deblurred result of proposed method (a) Nonuniform motion blurred image (b) Deblurred result (c) Close-up views of the roofof car (d) PSF corresponding to blur object which is shown in Figure 7

Thenwe solve 119868 subproblem to get the latent image Givena fixed value of 120596 from previous iteration we obtain the

optimal 119868 by the following optimization problem Equation(20) is modified as

argmin119868

119870 lowast 119868 minus 1198612 + 1205732 ((1198911 lowast 119868 minus 1205961)2 + (1198912 lowast 119868 minus 1205962)2) (23)

By solving the above problems iteratively we can obtain thedeblurred result

In Figure 8 we show a comparison of the proposedmethod and Zhuorsquos method The blur amount is the same fora rigid object but the blur amount of the car is nonuniformin Zhuorsquos defocus map which is shown in Figure 8(a) Bycontrast the blur amount of the car in the proposed defocusmap is uniform Thus the car can be successfully detectedas a blurred object because of its uniform blur amount Thedeblurred result is shown in Figure 9 The movement of thecar is shown in Figure 9(d) In Figure 9(c) which can be com-pared with the blurred car the word on the roof of the car inthe deblurred image can be clearly discerned The deblurredresult shows a satisfactory view of the nonuniform motionblur image In the next sectionwe report experimental resultsthat demonstrate the effectiveness of the proposed nonuni-form deblurring algorithm

4 Experiments and Discussions

In this paper wemainly focus on the nonuniformmotion blurproblem which is caused by moving object Our proposedmethod was implemented with Visual C++NET softwareThe testing environment was a personal computer running

the 64-bit version of Windows 7 with an AMD PhenomII X4 945 34-GHz CPU and 8GB of RAM To show theeffectiveness of our proposed method we compared ourresult with the results of four state-of-the-art single imagedeblurring algorithms We used nonuniform motion blurimages as inputs These test images were taken with a high-end DSLR by using a long exposure

41 Comparison with Other Single Image DeblurringMethodsThefirst test image shows a basketball in the process of fallingWe sought to deblur the blurred basketball in this imageThis image of a basketball was a regular testing image it didnot show a complicated scene From Figure 10 a nonuniformmotion blur happened that the basketball was blurry but thecourt was not This nonuniform problem makes the deblur-ring problem more difficult to solve From Figures 10(b) and10(e) Fergus and Levinrsquos deblurringmethod favored a blurredresult because of nonuniformmotion blur And the basketballin the deblurred result of Shan and Chorsquo method is sharperbut ringing artifacts appeared around the line in basketballcourtThe proposedmethod segments the blurred object andthen deblurs it As a result the ringing artifacts around thecourt line do not appear and the blurred basketball can berecovered


(a) (b) (c)

(d) (e) (f)

(g)

Figure 10 A comparison between proposed method and state-of-the-art image deblurring algorithms (a) Blurred image (b) Fergus et al(c) Shan et al (d) Cho and Lee (e) Levin et al (f) Proposed method (g) Close-up views of (a) to (f) The first row is the patch at right topcorner And the second row is the patch containing details of basketball at center

42 Result fromaComplicated Scene Wenext used a differentnonuniform blur testing image that shows a complicatedscene namely a hoopoe standing amid weeds and hay Theweeds are luxuriant and itwas not easy to discern the hoopoeas shown in Figure 11(a) Figures 11(b) and 11(e) show the graylevel results of k-means clustering with different values of 119896A comparison of the two clustering results with different 119896values shows that the hoopoe in Figure 11(b) was not welldescribedThe foot of the hoopoe was classified as part of theweeds because of the complicated texture For small values ofk k-means clustering does not suffice to describe complicatedscenes By contrast Figure 11(h) shows that the blurred birdwas accurately described despite the same amount of blur

when the algorithm used a larger value of 119896 Therefore withan unusually complicated scene we can increase the value of119896 to attain a superior result In our experiment we used theparameter k = 7 for the regular testing image and k = 12 forthe test image of the complicated scene Comparing Figures11(h) and 11(i) the proposed uniform defocus map describesthe blur amount precisely but Zhoursquos defocus map fails todescribe the information of blur amount

The parameter Lambda 120582 balances fidelity to the sparsedepthmap and smoothness of interpolationWhen the bigger120582 is used the propagation result fit the given blur amountestimation map While 120582 is smaller the propagation result fitthe original image the result is shown in Figure 12 In our


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 11 Using different parameter 119896 for k-means clustering algorithm in proposed defocus map for a complicated scene (a) Blurred image(b) Clustering result (119896 = 7) (c) Close-up view of the hoopoe in (a) (d) Deblurred result (e) Clustering result (119896 = 12) (f) Close-up view ofthe hoopoe in (e) (g) Proposed defocus map (119896 = 7) (h) Proposed defocus map (119896 = 12) (e) Zhuorsquos defocus map

experiments we choose a fixed 120582 value 00001 for all exper-iments so that a soft constraint is put on estimated defocusmap to further refine small errors in our blur estimation

43 Robustness To verify the robustness of the proposedmethod for nonuniformmotion blur images we used a num-ber of real images with various moving objects and variousdegrees of blur All test images were taken with a high-endDSLR by using a long exposure For each blurred imageZhuorsquos defocus map the proposed uniform defocus map anda 119896-means clustering result are presented for comparisonFigure 13 shows a man waving his hand The waving hand isthe blurred object From Figure 13(e) the blur amount of theblurred object is uniform in the proposed map By contrastin Figure 13(d) the blur amount is overdefined in the blurredobject thus the blur amount in the hand is not uniformThisnonuniformity problem complicates the segmentation of theblurred hand Once the blurred hand had been located we

estimated the blur kernel of the blurred hand The estimatedPSF in Figure 13(f) shows that the hand of this man waswaving Some ringing occurred around the wrist of the manbecause of the influence of the gradient intensity of theshadow The next test image shows two students standingtogether the first student is beside the second student at thesame depth as shown in Figure 14The left hand side studentmoved from right to left but the other stood still The pro-posed defocusmap is shown in Figure 14(e)The blur amountof themoving student was brighter and uniform and the bluramount was darker for the motionless student Comparedwith the proposed defocusmap the blur amount was affectedby the color and texture in Zhuorsquos defocus map thereforeit was not uniform for the moving object As mentionedpreviously a nonuniform defocus map cannot separate theblurring precisely However the amount of blurring in ourproposed defocus map was uniform Therefore from Fig-ure 14(f) the graph on the T-shirt of the moving student


(a) (b) (c)

Figure 12 The result with different balance parameter 120582 in case of 119896 = 12 (a) 120582 = 0001 (b) 120582 = 00001 (c) 120582 = 000001

(a) (b) (c)

(d) (e)

(f) (g)

Figure 13 Deblurred result of image ldquoManrdquo (a) Blurred image with nonuniform motion blur (b) Deblurred result (c) Clustering result (d)Zhuorsquos defocus map (e) Proposed uniform defocus map (f) Close-up views of (a) and (b) (g) PSF corresponding to waving hand as shownin (a)

can be accurately recovered Next we used the image ldquoBallrdquowhich shows a sloped direction of motion blur and a smallmoving object The deblurred result is shown in Figure 15From Figure 15(a) it can be seen that the ball was falling fromthe top left to the bottom right this situation caused the ballto appear blurred A comparison of the proposed uniform

defocus map with Zhuorsquos defocus map shows that the bluramount was uniform for the blurred object in the proposedmap but Zhuorsquos defocus map failed to provide uniformityFigure 15(f) shows the estimated PSF corresponding to thefalling ball It precisely shows that the ball is falling from thetop left to the bottom right


(a) (b) (c)

(d) (e)

(f) (g)

Figure 14 Deblurred result of image ldquoStudentsrdquo (a) Blurred image with nonuniformmotion blur (b) Deblurred result (c) Clustering result(d) Zhuorsquos defocus map (e) Proposed uniform defocus map (f) Close-ups of (a) and (b) (g) PSF corresponding to moving student as shownin (a)

44 Limitation However we note that the shadow under theball was detected as a blurred object as shown in Figure 15(d)The reason for this phenomenon is that the border of theshadow region had a gradient of intensity From Figure 16(c)we can see that the signal shown in Figure 16(b) was similar tothe blurred signal shown in Figure 3 Therefore the shadowregion was recognized as a blurred object In this case a falsedeblurring result was generated

5 Conclusions

In this paper we propose a novel image deblurring algorithmfor nonuniform motion blur Because a rigid object has aconsistent amount of blur we propose a uniformdefocusmapfor image segmentation We segment the blurred image intoblurred regions and unblurred regions by using the proposeduniform defocus map Each blurred region is analyzed toobtain an estimate of its PSF Each blurred region and its PSF

are then entered as inputs to a uniform motion blur imagedeconvolution algorithm Finally an unblurred image isobtainedThe experiments showed that our deblurred resultshad a satisfactory visual perspective for any type of motionblur However for optimal results manual settings wererequired for numerous parameters Furthermore shadowstended to cause the algorithm to detect blurred objectsincorrectly

A possible future research direction is the automaticdeblurring of spatially varying motion blur images In futurework for automatic image deblurring it may be interestingto classify the blurred regions correctly for blurred images inwhich shadows exist This is expected to require an effectiveclassification method for selecting blurred objects correctly

Competing Interests

The authors declare that they have no competing interests


(a) (b) (c)

(d) (e) (f)

(g)

Figure 15 Deblurred result of image ldquoBallrdquo (a) Blurred image with nonuniform motion blur (b) Deblurred result (c) Clustering result (d)Zhuorsquos defocus map (e) Proposed uniform defocus map The gray rectangle is the detected blurred object (f) Close-up views of (a) and (b)on the basketball (g) The corresponding PSF to basketball as shown in (a)

(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)

Figure 16 Demonstration of the influence of shadow in the image with nonuniform motion blur (a) Blurred image (b) Shadow regioncorresponding to the red rectangle in (a) (c) A gray level signal corresponding to red line in (b) The 119909-axis is the position and 119910-axis isintensity in gray level We can observe that shadow region was detected as a blurred object in Figures 14(d) and 14(e) This phenomenon iscaused by a gradient intensity as shown in (c)

Acknowledgments

This research is supported by the Ministry of Science andTechnology Taiwan under Grants MOST 103-2221-E-005-073 and MOST 104-2221-E-005-090

References

[1] R C Gonzalez and R E Woods Digital Image ProcessingPrentice Hall 2nd edition 2002

[2] W H Richardson ldquoBayesian-based iterative method of imagerestorationrdquo Journal of the Optical Society of America vol 62no 1 pp 55ndash59 1972

[3] R Fergus B Singh A Hertzmann S T Roweis and W TFreeman ldquoRemoving camera shake from a single photographrdquoACMTransactions on Graphics vol 25 no 3 pp 787ndash794 2006

[4] A Levin R Fergus F Durand and W T Freeman ldquoImage anddepth from a conventional camera with a coded aperturerdquoACMTransactions on Graphics vol 26 no 3 Article ID 12764642007


[5] Q Shan J Jia and A Agarwala ldquoHigh-quality motion deblur-ring from a single imagerdquo ACM Transactions on Graphics vol27 no 3 article no 73 2008

[6] S Cho and S Lee ldquoFast motion deblurringrdquo ACM Transactionson Graphics vol 28 no 5 2009

[7] L Xu and J Jia ldquoTwo-phase Kernel estimation for robustmotion deblurringrdquo in Proceedings of the 11th European Confer-ence on Computer Vision (ECCV rsquo10) pp 157ndash170 HeraklionGreece September 2010

[8] A Levin Y Weiss F Durand and W T Freeman ldquoUnder-standing blind deconvolution algorithmsrdquo IEEETransactions onPattern Analysis and Machine Intelligence vol 33 no 12 pp2354ndash2367 2011

[9] A Rav-Acha and S Peleg ldquoTwo motion-blurred images arebetter than onerdquo Pattern Recognition Letters vol 26 no 3 pp311ndash317 2005

[10] L Yuan J Sun L Quan and H-Y Shum ldquoImage deblurringwith blurrednoisy image pairsrdquoACMTransactions onGraphicsvol 26 no 3 article 1 2007

[11] S Zhuo D Guo and T Sim ldquoRobust flash deblurringrdquo in Pro-ceedings of the IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo10) pp 2440ndash2447 San Francisco CalifUSA June 2010

[12] H Zhang D Wipf and Y Zhang ldquoMulti-observation blinddeconvolutionwith an adaptive sparse priorrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 36 no 8 pp1628ndash1643 2014

[13] C-F Chang and J-L Wu ldquoA new single image deblurringalgorithm using hyper laplacian priorsrdquo Frontiers in ArtificialIntelligence and Applications vol 274 pp 1015ndash1022 2015

[14] Y-WTai P Tan andM S Brown ldquoRichardson-Lucy deblurringfor scenes under a projective motion pathrdquo IEEE Transactionson Pattern Analysis and Machine Intelligence vol 33 no 8 pp1603ndash1618 2011

[15] O Whyte J Sivic A Zisserman and J Ponce ldquoNon-uniformdeblurring for shaken imagesrdquo International Journal of Com-puter Vision vol 98 no 2 pp 168ndash186 2012

[16] A Levin ldquoBlind motion deblurring using image statisticsrdquo inAdvances in Neural Information Processing Systems 19 (NIPSrsquo06) pp 841ndash848 MIT Press 2006

[17] S Cho Y Matsushita and S Lee ldquoRemoving non-uniformmotion blur from imagesrdquo in Proceedings of the IEEE 11thInternational Conference on Computer Vision (ICCV rsquo07) Riode Janeiro Brazil October 2007

[18] S Bae and F Durand ldquoDefocusmagnificationrdquo in Proceedings ofthe Annual Conference of the EuropeanAssociation for ComputerGraphics (EUROGRAPHICS rsquo07) pp 571ndash579 Prague CzechRepublic September 2007

[19] H-Y Lin K-J Li and C-H Chang ldquoVehicle speed detectionfrom a singlemotion blurred imagerdquo Image and Vision Comput-ing vol 26 no 10 pp 1327ndash1337 2008

[20] S Zhuo and T Sim ldquoDefocus map estimation from a singleimagerdquo Pattern Recognition vol 44 no 9 pp 1852ndash1858 2011

[21] M Hirsch C J Schuler S Harmeling and B Scholkopf ldquoFastremoval of non-uniform camera shakerdquo in Proceedings of theIEEE International Conference on Computer Vision (ICCV rsquo11)pp 463ndash470 Barcelona Spain November 2011

[22] L Xu and J Jia ldquoDepth-aware motion deblurringrdquo in Proceed-ings of the IEEE International Conference on ComputationalPhotography (ICCP rsquo12) pp 1ndash8 IEEE SeattleWash USAApril2012

[23] Z Hu L Xu and M-H Yang ldquoJoint depth estimation andcamera shake removal from single blurry imagerdquo in Proceedingsof the 27th IEEE Conference on Computer Vision and PatternRecognition (CVPR rsquo14) pp 2893ndash2900 IEEE Columbus OhioUSA June 2014

[24] J H Elder and SW Zucker ldquoLocal scale control for edge detec-tion and blur estimationrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 20 no 7 pp 699ndash716 1998

[25] A Levin D Lischinski and Y Weiss ldquoColorization using opti-mizationrdquo in Proceeding of the ACM Transactions on Graphics(ACM SIGGRAPH rsquo04) pp 689ndash694 Los Angeles Calif USAAugust 2004

[26] D Krishnan and R Fergus ldquoFast image deconvolution usinghyper-laplacian priorsrdquo in Proceedings of the 23rd AnnualConference onNeural Information Processing Systems (NIPS rsquo09)pp 1033ndash1041 Vancouver Canada December 2009

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


(a)

(b)

Figure 1 An example of a real-world image There are differentkinds of blur in the image The fallen ball is blurry and others arenot (a) Real-world image (b)Magnification red rectangle is blurredbasketball and blue rectangles are unblurred regions

Rav-Acha and Peleg used two motion blur images [9] Yuanet al recovered a blurred image from a noisemotion pairthat had been captured under low light conditions [10]Zhuo et al leveraged a flashmotion pair that provided clearstructural information for an image deblurring problem [11]Zhang et al applied varying image pairs to image deblurring[12] Nevertheless image pair approaches require additionalimages that must be captured by extra hardware Chang andWu proposed a new deconvolution method for deblurringa blurred image with uniform motion blur by using hyperLaplacian priors [13]

As shown in Figure 1 motion blur in a real-world imagecan be so complicated that the assumption of shift invarianceis not always held To deblur a nonuniformmotion blur imageis more difficult [14ndash23] The causes of nonuniform motionblur can be divided into two categories camera rotation[14 15] and target motion [16ndash23] To solve the target motionproblem nonuniform blur image deblurring methods havebeen proposed In 2006 Levin deblurred a nonuniformmotion blur image by segmenting nonuniform motion blurimages into blurred regions and unblurred regions by usingimage statistics [16] But this method works when the direc-tion of blur was vertical or horizontal In 2007 Cho et alproposed using multiple blurred images that had been cap-tured continuously [17] An image registration method wasthen applied to estimate the PSF but it required previous

discernment of the blurred object and additional imagesIn 2011 Tai et al proposed using a projective motion paththat was recorded with additional hardware to replace thetraditional plane blur kernel for image deblurring [14] Aprojectivemotion path describes themovement of the camerain a three-dimensional (3D) model therefore it can restorenonuniform motion blur images for which blur had beencaused by camera rotation However additional hardwarewas required to record the 3D blur kernel Several nonuni-form image deblurring methods with depth maps have beenproposed to handle this problem In 2012 Xu and Jia used adepth map to deblur blurred images with the same motionblur but different Gaussian blurs [22] In 2014 Hu et alproposed using depth estimation to deblur a nonuniformblur image [23]The authors separated the blurred image intolayers according to a depth map and then deblurred one layerbut maintained the others in a fixed state This deblurringmethod treats every object in the same depth as havingthe same blur amount However if there is an object witha different blur amount in the same layer deblurring failsbecause the blur amount of objects at the same depth is notalways the same

In this paper we propose a novel image deblurringalgorithm for nonuniform motion blur images To locateblurred parts we measure the blur amount for each edge inthe blurred image The measurement of blur is then appliedto propagate a uniform defocus map We use the proposeduniform defocus map to segment the nonuniform blur imageinto multiple blurred and unblurred portions Subsequentlywe estimate the PSF for each blurred portion to obtain the PSFof each portion All portions and the corresponding PSFs areentered as inputs to a fast deconvolution algorithm Finallyan unblurred result is obtained

The rest of the paper are organized as follows In Section 2we briefly review the related works on nonuniform imagedeblurring Section 3 presents our proposedmethod in detailThe experimental results are then designed for verifying theproposed method in Section 4 Finally we conclude thispaper in Section 5

2 Related Works

Nonuniform motion is an ill-posed problem that manystudies have been proposed to address this problem In 2006Levin removed nonuniform motion blur by separating ablurred image into unblurred regions and blurred regionsby using image statistics The authors located blurred andunblurred regions depending on the observation that thestatistics of derivative filters in images are substantiallychanged by partial motion blur They assumed that the blureffect resulted from motion at a constant velocity and thusmodeled the expected derivative distributions as functionsof the width of the blur kernel This was effective when thedirection ofmotion blur was vertical or horizontal or in otherwords the blur kernel was one-dimensional In 2007 Choet al proposed using multiple blurred images that were cap-tured continuously to remove nonuniform motion blur [17]These blurred images were captured for the same scene Theauthors applied an image registrationmethod to compute the




3 Proposed Method


0

Position

Inte

nsity

d1

d2





119891 (119909) = 119860119906 (119909) + 119861 (2)


119894 (119909) = 119891 (119909) otimes 119892 (119909 120590) (3)



(a)

0

Inte

nsity


10 20 30 40 50 60 70 80 900Position

(b)




11990922 (1205902 + 12059020)) (4)



(5)




120590 = 1radic1198772 minus 11205900 (7)





119868 (119909119894)119882119901 (119909119894) 119868 (119909) = 0 (8)

with

119882119901 = sum119909119894isinΩ

119882119877 (119909119894)119882119878 (119909119894) (9)


119882119877 (119909119894) = 119891119903 (1003817100381710038171003817119868 (119909119894) minus 119868 (119909)1003817100381710038171003817) 119882119878 (119909119894) = 119892119904 (1003817100381710038171003817119909119894 minus 1199091003817100381710038171003817) (10)








Deblurred image

Unify blur





argmin119878

119896sum119894=1

sum119909119895isin119904119894

10038171003817100381710038171003817119909119895 minus 120583119894100381710038171003817100381710038172 (11)




119864 (119889) = 119889119879119871119889 + 120582 (119889 minus 119889)119879119863(119889 minus 119889) (12)




119861 = 119868 lowast 119870 + 119873 (13)


119901 (119868 119870 | 119861) prop 119901 (119861 | 119868 119870) 119901 (119868) 119901 (119870) (14)


1198701015840 = argmin119870


119868

119870 lowast 119868 minus 119861 + 119901119868 (119868) (15)


(a) (b)

(c)


(a)

(b)







(a) (b)






argmin119868

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 2sum119896=1

1003816100381610038161003816(119891119896 lowast 119868)1198941003816100381610038161003816119902) (18)


1198911 = [1 minus1] 1198912 = [1 minus1]119879 (19)




argmin119868120596

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 12057322sum119896=1

(119891119896 lowast 119868 minus 120596119896)2119894+ 2sum119896=1

1003816100381610038161003816(120596119896)1198941003816100381610038161003816119902) (20)



argmin120596

(|120596|119902 + 1205732 (120596 minus ])2) (21)


1205964 minus 3]1205963 + 3]21205962 minus ]3120596 + 8271205733 = 0 (22)



(a) (b)

(c) (d)




argmin119868









(a) (b) (c)

(d) (e) (f)

(g)






(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












3 Proposed Method


0

Position

Inte

nsity

d1

d2





119891 (119909) = 119860119906 (119909) + 119861 (2)


119894 (119909) = 119891 (119909) otimes 119892 (119909 120590) (3)



(a)

0

Inte

nsity


10 20 30 40 50 60 70 80 900Position

(b)




11990922 (1205902 + 12059020)) (4)



(5)




120590 = 1radic1198772 minus 11205900 (7)





119868 (119909119894)119882119901 (119909119894) 119868 (119909) = 0 (8)

with

119882119901 = sum119909119894isinΩ

119882119877 (119909119894)119882119878 (119909119894) (9)


119882119877 (119909119894) = 119891119903 (1003817100381710038171003817119868 (119909119894) minus 119868 (119909)1003817100381710038171003817) 119882119878 (119909119894) = 119892119904 (1003817100381710038171003817119909119894 minus 1199091003817100381710038171003817) (10)








Deblurred image

Unify blur





argmin119878

119896sum119894=1

sum119909119895isin119904119894

10038171003817100381710038171003817119909119895 minus 120583119894100381710038171003817100381710038172 (11)




119864 (119889) = 119889119879119871119889 + 120582 (119889 minus 119889)119879119863(119889 minus 119889) (12)




119861 = 119868 lowast 119870 + 119873 (13)


119901 (119868 119870 | 119861) prop 119901 (119861 | 119868 119870) 119901 (119868) 119901 (119870) (14)


1198701015840 = argmin119870


119868

119870 lowast 119868 minus 119861 + 119901119868 (119868) (15)


(a) (b)

(c)


(a)

(b)







(a) (b)






argmin119868

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 2sum119896=1

1003816100381610038161003816(119891119896 lowast 119868)1198941003816100381610038161003816119902) (18)


1198911 = [1 minus1] 1198912 = [1 minus1]119879 (19)




argmin119868120596

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 12057322sum119896=1

(119891119896 lowast 119868 minus 120596119896)2119894+ 2sum119896=1

1003816100381610038161003816(120596119896)1198941003816100381610038161003816119902) (20)



argmin120596

(|120596|119902 + 1205732 (120596 minus ])2) (21)


1205964 minus 3]1205963 + 3]21205962 minus ]3120596 + 8271205733 = 0 (22)



(a) (b)

(c) (d)




argmin119868









(a) (b) (c)

(d) (e) (f)

(g)






(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

0

Inte

nsity


10 20 30 40 50 60 70 80 900Position

(b)




11990922 (1205902 + 12059020)) (4)



(5)




120590 = 1radic1198772 minus 11205900 (7)





119868 (119909119894)119882119901 (119909119894) 119868 (119909) = 0 (8)

with

119882119901 = sum119909119894isinΩ

119882119877 (119909119894)119882119878 (119909119894) (9)


119882119877 (119909119894) = 119891119903 (1003817100381710038171003817119868 (119909119894) minus 119868 (119909)1003817100381710038171003817) 119882119878 (119909119894) = 119892119904 (1003817100381710038171003817119909119894 minus 1199091003817100381710038171003817) (10)








Deblurred image

Unify blur





argmin119878

119896sum119894=1

sum119909119895isin119904119894

10038171003817100381710038171003817119909119895 minus 120583119894100381710038171003817100381710038172 (11)




119864 (119889) = 119889119879119871119889 + 120582 (119889 minus 119889)119879119863(119889 minus 119889) (12)




119861 = 119868 lowast 119870 + 119873 (13)


119901 (119868 119870 | 119861) prop 119901 (119861 | 119868 119870) 119901 (119868) 119901 (119870) (14)


1198701015840 = argmin119870


119868

119870 lowast 119868 minus 119861 + 119901119868 (119868) (15)


(a) (b)

(c)


(a)

(b)







(a) (b)






argmin119868

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 2sum119896=1

1003816100381610038161003816(119891119896 lowast 119868)1198941003816100381610038161003816119902) (18)


1198911 = [1 minus1] 1198912 = [1 minus1]119879 (19)




argmin119868120596

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 12057322sum119896=1

(119891119896 lowast 119868 minus 120596119896)2119894+ 2sum119896=1

1003816100381610038161003816(120596119896)1198941003816100381610038161003816119902) (20)



argmin120596

(|120596|119902 + 1205732 (120596 minus ])2) (21)


1205964 minus 3]1205963 + 3]21205962 minus ]3120596 + 8271205733 = 0 (22)



(a) (b)

(c) (d)




argmin119868









(a) (b) (c)

(d) (e) (f)

(g)






(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Deblurred image

Unify blur





argmin119878

119896sum119894=1

sum119909119895isin119904119894

10038171003817100381710038171003817119909119895 minus 120583119894100381710038171003817100381710038172 (11)




119864 (119889) = 119889119879119871119889 + 120582 (119889 minus 119889)119879119863(119889 minus 119889) (12)




119861 = 119868 lowast 119870 + 119873 (13)


119901 (119868 119870 | 119861) prop 119901 (119861 | 119868 119870) 119901 (119868) 119901 (119870) (14)


1198701015840 = argmin119870


119868

119870 lowast 119868 minus 119861 + 119901119868 (119868) (15)


(a) (b)

(c)


(a)

(b)







(a) (b)






argmin119868

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 2sum119896=1

1003816100381610038161003816(119891119896 lowast 119868)1198941003816100381610038161003816119902) (18)


1198911 = [1 minus1] 1198912 = [1 minus1]119879 (19)




argmin119868120596

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 12057322sum119896=1

(119891119896 lowast 119868 minus 120596119896)2119894+ 2sum119896=1

1003816100381610038161003816(120596119896)1198941003816100381610038161003816119902) (20)



argmin120596

(|120596|119902 + 1205732 (120596 minus ])2) (21)


1205964 minus 3]1205963 + 3]21205962 minus ]3120596 + 8271205733 = 0 (22)



(a) (b)

(c) (d)




argmin119868









(a) (b) (c)

(d) (e) (f)

(g)






(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c)


(a)

(b)







(a) (b)






argmin119868

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 2sum119896=1

1003816100381610038161003816(119891119896 lowast 119868)1198941003816100381610038161003816119902) (18)


1198911 = [1 minus1] 1198912 = [1 minus1]119879 (19)




argmin119868120596

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 12057322sum119896=1

(119891119896 lowast 119868 minus 120596119896)2119894+ 2sum119896=1

1003816100381610038161003816(120596119896)1198941003816100381610038161003816119902) (20)



argmin120596

(|120596|119902 + 1205732 (120596 minus ])2) (21)


1205964 minus 3]1205963 + 3]21205962 minus ]3120596 + 8271205733 = 0 (22)



(a) (b)

(c) (d)




argmin119868









(a) (b) (c)

(d) (e) (f)

(g)






(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)






argmin119868

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 2sum119896=1

1003816100381610038161003816(119891119896 lowast 119868)1198941003816100381610038161003816119902) (18)


1198911 = [1 minus1] 1198912 = [1 minus1]119879 (19)




argmin119868120596

119873sum119894=1

((119870 lowast 119868 minus 119861)2119894 + 12057322sum119896=1

(119891119896 lowast 119868 minus 120596119896)2119894+ 2sum119896=1

1003816100381610038161003816(120596119896)1198941003816100381610038161003816119902) (20)



argmin120596

(|120596|119902 + 1205732 (120596 minus ])2) (21)


1205964 minus 3]1205963 + 3]21205962 minus ]3120596 + 8271205733 = 0 (22)



(a) (b)

(c) (d)




argmin119868









(a) (b) (c)

(d) (e) (f)

(g)






(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)

(c) (d)




argmin119868









(a) (b) (c)

(d) (e) (f)

(g)






(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)

(g)






(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)

(g) (h) (i)






(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)


(a) (b) (c)

(d) (e)

(f) (g)





(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e)

(f) (g)



5 Conclusions




Competing Interests



(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b) (c)

(d) (e) (f)

(g)


(a) (b)0 5 10 15 20 25 30 35 40

6065707580859095

100

(c)


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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