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Depth Image Inpainting: Improving Low RankMatrix Completion with Low Gradient

RegularizationHongyang Xue, Shengming Zhang and Deng Cai, Member, IEEE

Abstract—We consider the case of inpainting single depthimages. Without corresponding color images, previous or nextframes, depth image inpainting is quite challenging. One naturalsolution is to regard the image as a matrix and adopt the lowrank regularization just as inpainting color images. However, thelow rank assumption does not make full use of the properties ofdepth images .

A shallow observation may inspire us to penalize the non-zerogradients by sparse gradient regularization. However, statisticsshow that though most pixels have zero gradients, there is stilla non-ignorable part of pixels whose gradients are equal to 1.Based on this specific property of depth images , we propose a lowgradient regularization method in which we reduce the penaltyfor gradient 1 while penalizing the non-zero gradients to allow forgradual depth changes. The proposed low gradient regularizationis integrated with the low rank regularization into the low ranklow gradient approach for depth image inpainting. We compareour proposed low gradient regularization with sparse gradientregularization. The experimental results show the effectivenessof our proposed approach.

Index Terms—Stereo image processing, Image restoration,Image inpainting, Depth image recovery.

I. INTRODUCTION

Image inpainting is an important research topic in the fieldsof computer vision and image processing[16], [37], [12], [42].A lot of approaches have been proposed to tackle inpaintingproblems for images of different categories [1], [11], [2].However, most research have been focused on natural imagesand medical images. The research amount on depth imageinpainting is relatively small.

The fast development of the RGB-D sensors, such asMicrosoft Kinect, ASUS Xtion Pro and Intel Leap Motion,enables a variety of applications based on the depth informa-tion by providing depth images of the scenes in real time.Together with the traditional multi-view stereo approaches,depth images are now playing a more and more important rolein computer vision research and applications [33], [34], [39],[38], yet the inpainting problem of them are not well-studied.

The main reason may be that most image inpainting tech-niques can be applied directly to depth images. Noting thatthere is only a simple mathematical relation between thedisparity value and the depth value, we will use disparity

H. Xue, S. Zhang, D. Cai are with State Key Lab of CAD&CG, Collegeof Computer Science, Zhejiang University, Hangzhou 310027, China.

E-mail: hyxue@outlook.com, michaelzhang@zju.edu.cn, deng-cai@cad.zju.edu.cn

instead of depth in our paper. In the remainder of the paper,depth and disparity will have the same meaning which refersto the disparity value. To obtain coarse inpainting results,we apply the low rank assumption and complete the depthimage with the low rank matrix completion approach [6]. Theinpainting results are not satisfactory enough. Depth imagesare textureless compared with natural images. The lack oftexture causes difficult for the low rank completion approach.In addition, the low rank completion approach usually resultsin excessive and spurious details in the inpainted areas (seeFigure I). Moreover, depth images have quite sparse gradients.In other words, gradients vanish at most places. Therefore,together with the textureless property, it is reasonable if oneregularizes the inpainting results with the sparse gradientprior. To improve depth inpainting results under the low rankassumption, the gradients can be regularized in the meanwhile.There have been work in recovering images (e.g. medicalimages [36] and natural images [9]) under the sparse gradientassumption.

However, statistics of depth image gradients show thatthe sparse gradient assumption is not accurate enough. Thegradients can be described more properly as low rather thansparse. In another word, at many places in the depth image,gradients are not always 0 but rather very small (see Figure4). This property has not been considered for image inpaintingbecause it is not universal in general images. For depth images,statistics show the universality of this property. Hence wepropose the low gradient regularization. We denote the lowgradient regularization as Lψ0 gradient regularization. Thenotation comes from the corresponding non-convex TVψ norm[21]. Relation between them will be explained in the remainderpart of this paper.

Unlike the L0 norm which penalizes the non-zero elementsequally (the norm is always 1 whether the element is 1 or100), our proposed Lψ0 measure reduces the penalty for smallelements. In depth images, our Lψ0 reduces the penalty forgradient 1 (all gradients are truncated into integer values) andthus allows for gradual depth changes.

Our main contributions are two-folds: First we proposethe low gradient regularization Lψ0 which well describes thestatistical property of the depth image gradients. Second wedevelop a solution to the Lψ0 gradient minimization problembased on [28]. We integrate our low gradient regularizationwith the low rank assumption into the LRL0ψ regularizationfor depth image inpainting. In the experiments we compareour LRL0ψ algorithm with two approaches, the low rank total

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Fig. 1. Left: Original disparity map. Middle: Corrupted disparity map with half pixels missing. Right: The inpainted result by low rank(LR) completion. Wecan see the inpainted map has much noise compared with original map.

variation (LRTV) [36] and the low rank L0 gradient (LRL0)approaches, which only enforce the sparse gradient constraint.

The remainder of the paper is organized as follows. SectionII describes related work in image inpainting and the sparsegradient. Section III introduces the LRTV [36] and the LRL0approaches which only consider the sparse gradient regular-ization. In section III we point out the defects of the sparsegradient regularization and then our main contribution, the lowgradient regularization, is described in section IV. In sectionV we perform experiments on our dataset and display theexperimental results. Finally We conclude our work in VI.

II. BACKGROUND

A. Low Rank Matrix Completion

Completing a matrix with missing observations is an in-triguing task in the machine learning and mathematics society[4], [7], [5], [24], [47]. Most work are based on the lowrank assumption of the underlying data. The low rank matrixcompletion is one of effective approaches for image inpainting.Candes et al. [6] first introduce the matrix completion problemby approximating the rank with nuclear norm. Zhang et al. [46]propose the truncated nuclear norm regularization and achieveexcellent results in image inpainting. Gu et al. [15] furtherpresent the weighted nuclear norm regularization and performthe image denoising task with outstanding results.

B. Depth Inpainting

Depth image inpainting has been considered mostly underthe situation of RGB-D inpainting or stereoscopic inpaintingproblems. Moreover, most depth inpainting approaches inpaintmissing regions of specific kinds (e.g. occlusions, missingcaused by sensor defects or holes caused by object-removal).Doria et al. [13] introduce a technique to fill holes in theLiDAR data sets. Wang et al. [43] present an algorithm forsimultaneous color and depth inpainting. They take the stereoimage pairs and the estimated disparity map as input andfill the holes cause by object removal. Lu et al. [26] clusterthe RGB-D image patches into groups and employ the lowrank matrix completion approach to enhance the depth imagesobtained from RGB-D sensors with the aid of correspondingcolor images. Zou et al. [48] consider inpainting RGB-Dimages by removing fence-like structures. Buyssens et al.[3] inpaint holes in the depth maps based on superpixel for

virtual view synthesizing in RGB-D scenes. Herrera et al. [20]inpaint incomplete depth maps produced by 3D reconstructionmethods under a second-order smoothness prior. As far aswe know, almost all depth inpainting approaches refer tocolor images. They are either color-guided depth inpaintingor simultaneous RGB-D inpainting. However, we considerinpainting only a single depth image.

C. TV and L0 Gradient Regularization

TV norm and its variations have been employed in imageprocessing for quite a long time [9], [14]. Perrone et al.[30] propose a blind deconvolution algorithm based on thetotal variation minimization. Guo et al. [17] extend the totalvariation norm to tensors for visual data recovery. Li et al. [25]use the total variation in their robust noisy image completionalgorithm.

Total variation refers to the integration of the norm ofgradients on the whole image. When a function is appliedto the norm of the gradients at each pixel before integral,the integration is called the total generalized variation. Hin-termuller et al. [22] propose a nonconvex TVq model for imagerestoration. They further propose a superlinearly solver forthe general concave generalized TV model in [21]. Ranftl etal. [31] utilize the total generalized variation for optical flowestimation.

As mentioned above, the TV norm is a relaxation of theL0 gradient. However, the TV norm also penalizes largegradient magnitudes. It may influence real image edges andboundaries [44]. Thus many algorithms directly solving L0

gradient minimization have been proposed. Xu et al. [44] adopta special alternating optimization strategy. They employ themethod for image deblurring [45]. Their work is later appliedfor mesh denoising by He et al. [18]. Nguyen et al. [28]propose a fast L0 gradient minimization approach based ona method named region fusion.

There are approaches that combine low rank regularizationwith the total variation minimization. Shi et al. [36] combineboth the low rank and the total variation regularization intoan LRTV algorithm for medical image super-resolution. Ji etal. [23] propose a tensor completion approach based on thetotal variation and the low-rank matrix completion. He et al.[19] employ the total variation regularization and the low-rankmatrix factorization for hyperspectral image restoration.

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III. LOW RANK SPARSE GRADIENT APPROACH

In this section, we will review the LRTV [36] algorithm andintroduce how it can be employed to inpaint depth images.Then the L0 gradient is used to replace the TV regularization.We briefly review the algorithm developed by Nguyen et al.[28] for the L0 gradient minimization. These are necessaryfor the explanation of our main contribution, the LRL0ψ

algorithm, which will be described in the next section.Given a corrupted disparity map or depth image D and its

inpainting mask Ω (missing areas), we hope to recover theoriginal image U. The recovered image U should match theobservations U|Ω = D|Ω. In conventional matrix completionimage inpainting scheme, the low rank prior is added asa regularization [6], [15], [46] and the unknown image isrecovered by solving

arg minU||U−D||2Ω + λ · rank(U) (1)

λ is a weight representing the importance of low rank. Theabove rank minimization problem is intractable due to thenon-convexity and discontinuous nature of the rank function.Theoretical studies show that the nuclear norm is the tightestconvex lower bound of the rank function of matrices [32].Therefore, rank is usually approximated by the nuclear norm[6]. We employ the nuclear norm [6] as the low rank prior.As discussed in the introduction, we also add the sparsegradient regularization for depth recovery. Altogether we havethe following formula

arg minU||U−D||2Ω + λr · ||U||∗ + λs · ||∇U||0 (2)

Notice that the third term in equation 2 is the L0 normof gradient. Minimization corresponding to the L0 norm isusually relaxed to L1 norm and thus the L0 gradient becomestotal variation [36], [19], [23]. The L0 norm is non-convex.The advantage of relaxing the L0 gradient to total variation isthat the problem becomes convex. We first employ the existedLRTV scheme as described in [36] for depth inpainting.

A. Total Variation

The L0 gradient norm is approximated by total variationand equation 2 now becomes

arg minU||U−D||2Ω + λr · ||U||∗ + λtv · TV (U) (3)

This problem has almost the same form as the super-resolution problem in [36]. Following the solution in [36], weemploy the ADMM alrgorithm [40] to solve the new equation

minU,M,Y||U−D||2Ω + λr||M||∗ + λtvTV (U)+ρ

2(||U−M + Y||2 − ||Y||2)

(4)

The minimization problem in equation 4 is broken into threesub-problems and the variables are iteratively updated.

The first subproblem needs to update Uk+1 by minimizingpart of equation 4 related with total variation.

arg minU||U−D||2Ω +λtvTV (U)+

ρ

2||U−Mk+Yk||2 (5)

This subproblem is solved by Bregman iteration [27]. In thesecond subproblem Mk+1 is updated by

arg minM||M||∗ +

ρ

2||Uk+1 −M + Yk||2 (6)

After the update of Uk+1 and Mk+1, Yk+1 is updated byYk+1 = Yk + (Uk+1 −Mk+1).

In our experiments, we initialize U as the coarse inpaintingresults obtained by the low rank matrix completion and Mand Y are set to 0.

The TV regularization improves the inpainting results com-pared with only low rank [6] (Figure 2). However, TV regu-larization has some drawbacks, it smoothes real depth edges[44]. We also observe that in depth inpainting results, theTV norm always becomes close and even lower than that ofthe groundtruth (see Figure 2). And even with a lower-than-groundtruth TV, the depth image remains noisy visually. Weobserve that even we optimize the results to have lower-than-groundtruth TV norm, the L0 norm of gradients is still farabove the groundtruth. Because the optimal solution underthe TV norm regularization is not exactly optimal for theL0 gradient , we decide to directly employ the L0 gradientregularization.

In theory, the L0 norm is the most suitable measure forsparsity. There are research on the gap between TV andL0 (e.g. [29], [8]). There also have been work on directlysolving L0 gradient minimization problems [44], [28] byapproximation strategies.

B. L0 Gradient

As mentioned above, the TV norm is widely used as theapproximation of the L0 norm of gradients for it enablesfast and tractable solutions. However, the drawbacks of thisapproximation are also studied [44]. Therefore, a lot of ap-proaches which directly and approximately minimize the L0

gradient have been proposed [10], [44], [41], [28]. Among theapproaches, Nguyen et al. [28] propose a region fusion methodfor L0 gradient minimization. Their approach achieves rathergood results while running the most efficiently compared withother L0 gradient minimization algorithms [10], [44], [41].

We replace the TV norm in subproblem 2 (equation 5) withL0 gradient

arg minU||U−D||2Ω +λl0||∇U||0 +

ρ

2||U−Mk+Yk||2 (7)

Following [28], we rewrite equation 7 as

arg minU

L∑i=1

||Ui −Di||2Ω+ρ

2||Ui −Mk

i + Y ki ||2

+λl02

∑j∈Ni

||Ui − Uj ||0(8)

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Fig. 2. Top left: Groundtruth (GT). Top right: Corrupted with 50% missing entries. Lower left: Low rank inpainting [6]. Its PSNR = 27.7596. Lower right:LRTV result. Its PSNR = 28.1319. The LRTV approach obtains better results than only low rank. We also compute the TV norm for GT, low rank, andLRTV. They are 642186, 833767 and 622276 respectively. We can see the LRTV result has lower TV than GT. The L0 norm of them are 63765, 117922and 81644. The LRTV result has much higher L0 norm than GT. In term of TV regularization, LRTV has achieved its optimal solution but in L0 norm it isstill far from optimal. The minimization of the TV norm does not necessarily mean the minimization of L0 gradients.

where L is the length of the signal (the number of pixels inthe image) and Ni the neighboring set (four connected pixels)of the ith pixel.

The bold symbol like U indicates matrix and the normalsymbol Ui, Mi, Di and Yi denote the values of U, M, D andY at the ith pixel or the mean value in the ith region.

In [28] they propose an algorithm which loops though allneighboring regions (groups) Gi and Gj . At first, all pixels arethemselves groups. For region Gi, Di, Mi and Yi denote themean value of G, D and Y in the ith region Gi. We rewritethe pairwise region cost for our objective function as

f = minUi,Uj

wi||Ui −Di||2Ω+wj ||Uj −Dj ||2Ω

+βci,j ||Ui − Uj ||0+ρwi2||Ui −Mk

i + Y ki ||2

+ρwj

2||Uj −Mk

j + Y kj ||2

(9)

where β is the auxiliary parameter (0 ≤ β ≤ λ) [28].The fusion criterion [28] derived from equation 9 which

decides whether region Gi and Gj should be fused nowbecomes

Ui, Uj =

A,A if fA ≤ fBBi, Bj otherwise

(10)

where A = (wiDi + wjDj + ρwiMki + ρwjM

kj − ρwiY ki −

ρwjYkj )/(wi + wj + ρwi + ρwj), Bi = (2wiDi + ρwi(M

ki −

Y ki ))/(2wi+ρwi), Bj = (2wjDj +ρwj(Mkj −Y kj ))/(2wj +

ρwj), fA is the value of equation 9 when Ui = Uj = A. fB isthe value of equation 9 when Ui = Bi and Uj = Bj . wi is thetotal number of pixels in region i while wi only counts pixelsin region i that are meanwhile not in the missing region Ω.

Then we modifty the region fusion minimization algorithmin [28] to solve the equation 8. In their original region fusion

minimization, the two regions will remain untouched if thecriterion equation 10 judges they should not be fused. In oursettings, we will update them by Ui = Bi and Uj = Bj .

The L0 gradient regularization leads to better results indepth inpainting results in most cases. However, it does notalways perform better than LRTV (see Figure 3). One ofthe reasons may be that the region fusion solution for L0

gradient minimization is only an approximation. But beyondthat, we have not fully utilized the features of the gradientmaps. We compute the gradients of depth images and find outthat the low L0 norm is not accurate enough to characterizethe property of depth image gradients (As shown in Figure4). Besides 0, a non-ignorable part of pixels have gradient 1.In L0 norm, all gradients larger than 0 are penalized equally.Based on the statistics of depth images, we hope to reduce thepenalty for gradient 1 so that gradual depth change is allowed.

IV. LOW RANK LOW GRADIENT APPROACH

In this section, we will describe our main contribution, theLRL0ψ algorithm. First we will point out our definition of thelow gradient. Then we will describe our low rank low gradientregularization algorithm.

A. Integral Gradient

The magnitude of gradient is usually a real number. How-ever, we will consider the gradients as integral values. Wedeal with depth images and disparity images with integraldepth values. Thus the gradients on each direction take in-tegral values. When doing statistics on the depth gradients,we truncate the magnitude of the gradients into integers.Denote the gradient as (∇x,∇y), the truncation of the gradient

magnitude is⌊√∇2x +∇2

y

⌋. Noting that the truncated value

takes 0 only and if only the gradients on both directions are

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Fig. 3. The first column are original depth images. The second column corresponds to the corrupted depth images. The third column displays the LRTVresults. The last column shows our LRL0 results. For first row (Adirondack), the PSNR of LRTV and LRL0 are 28.1319 and 28.6198 respectively. For secondrow (Jadeplant), they are 22.8231 and 22.7730. We can see LRL0 achieves better inpainting results than LRTV on Adirondack but fails on Jadeplant.

Fig. 4. Gradient magnitude histogram of groundtruth disparity map ofMiddlebury Adirondack dataset. We can see most pixels have gradientmagnitude 0 and a non-ignorable part have magnitude 1. This observationinspires us to employ low gradient regularization.

0. The truncated gradient magnitude takes 1 if and only if thegradients are (±1,±1), (0,±1) or (±1, 0). In other words, forzero gradients, the real values and integral values are exactlythe same. For gradient of value 1, it relates to 8 patterns(±1,±1), (0,±1) or (±1, 0). Our low gradient is defined onthe integral gradients.

B. Low Gradient

As discussed in previous sections, the gradients of depthimages cannot be simply depicted as sparse. The penalty ofthe small gradient value 1 should be reduced to allow forgradual depth changes (see Figure 4). There is a category ofgeneralized TVψ norm [21] where the penalty of gradientsare not increased linearly. Similar to the generalization of TV,we propose the Lψ0 norm so that the penalty for gradient 1 isreduced. Actually it is not a norm but rather a measure forthe property of data because it does not satisfy the absolutehomogeneity. Therefore we will call Lψ0 a measure. The Lψ0measure is as follows:

||X||Lψ0 = α#X = 1+ #X > 1 (11)

where 0 < α < 1 and #· denotes the number of elementsin the set.

We set α = 0.75 in all our experiments based on thestatistics of groundtruth gradients.

Thus our low gradient leads to the following optimizationproblem:

arg minU||U−D||2Ω + λl0ψ ||∇U||l0ψ +

ρ

2||U−Mk + Yk||2

(12)where λl0ψ is the weight of importance of the Lψ0 gradientterm.

We extend the region fusion minimization in section III-Bto solve Lψ0 gradient minimization problem.

In this case, the pairwise cost is as follows:

f = minUi,Uj

wi||Ui−Di||2Ω + wj ||Uj −Dj ||2Ω

+βci,j ||Ui − Uj ||Lψ0+ρwi2||Ui −Mk

i + Y ki ||2

+ρwj

2||Uj −Mk

j + Y kj ||2

(13)

The fusion criterion now contains three conditions. For groupsGi and Gj ,

• Ui = Uj . The optimal solution is Ui = Uj = A, where

A =A1

A2

A1 = 2(wiDi + wjDj) + ρwi(Mki − Y ki )

+ ρwj(Mkj − Y kj )

A2 = 2wi + 2wj + ρwi + ρwj

(14)

The function value of equation 13 under this condition isfA.

• |Ui−Uj | = 1, that is, Uj = Ui±1. The optimal solutionis Ui = B, Uj = B ± 1, where

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B =B1

A2

B1 = 2wiDi + 2wj(Dj ∓ 1)

+ ρwi(Mki − Y ki )

+ ρwj(Mkj − Y kj ∓ 1)

(15)

Denote the function value of equation 13 under thiscondition fB .

• |Ui − Uj | > 1. In this case Ui = Ci, Uj = Cj . Ci =(2wiDi+ρwi(M

ki −Y ki ))/(2wi+ρwi), Cj = (2wjDj+

ρwj(Mkj − Y kj ))/(2wj + ρwj). We denote the function

value as fC .The fusion criterion now becomes:

Ui, Uj =

A,A if fA ≤ fB and fA ≤ fCB,B ± 1 if fB < fA and fB ≤ fCCi, Cj otherwise

(16)

Based on this fusion criterion, we modify the region fusioniterations in [28] to solve the Lψ0 gradient minimizationproblem (see Algorithm 0). Our region fusion low gradientminimization algorithm is summarized in Algorithm 0. Noticethat compared with the region fusion minimization algorithmfor L0 in [28], our algorithm has several differences. Similarto the modification in L0 gradient minimization, when tworegions are not to be fused by the fusion criterion, we updatethem to the optimal solutions B,B±1 or Ci, Cj followingequation 16.

V. EXPERIMENTS AND RESULTS

A. Dataset

For there is no public dataset that aims at inpainting depthimages, we make a dataset for evaluating depth inpaintingapproaches. We convert the groundtruth disparity maps fromMiddlebury stereo dataset [35] to grayscale depth images.The groundtruth depth maps are from 14 images includingAdirondack, Jadeplant, Motorcycle, Piano, Playtable, Play-room, Recycle, Shelves, Teddy, Pipes, Vintage, MotorcycleE,PianoL and PlaytableP. The unknown values of the groundtruthdisparity maps are converted to 0s in depth images. To createdamaged images, we generate several masks including randommissing masks (see Figure V) and textual masks (see FigureV). The masks are provided in our dataset.

In addition to the depth images and masks, we also pro-vide our results which will be displayed in this sectiontogether with our codes for all the algorithms concernedin this paper. Our codes also enable generations of newrandom masks. New depth images can also be processedby our codes to produce inpainted results. Our dataset canbe accessed via http://www.cad.zju.edu.cn/home/dengcai/Data/depthinpaint/DepthInpaintData.html. Our code can be accessedvia https://github.com/xuehy/depthInpainting.

Algorithm 1 Region Fusion Minimization for Lψ0Input: image U with pixel number N , the level of sparseness

λ, missing region Ω, original missing image D, Y and M1: Initialize the regions as pixels themselves Gi ← i, Vi ←Ci, wi ← 1

2: wi ← 0 if i ∈ Ω, otherwise wi ← 13: Set Ni as the four-connected neighborhood of i4: Set ci,j = 1 if j ∈ Ni, otherwise ci,j = 05: β ← 0, iter ← 0, P ← N6: repeat7: i← 18: while i ≤ P do9: for all j ∈ Ni do

10: Compute fA, fB , fC following Section IV11: if fA ≤ fB and fA ≤ fC then12: Gi ← Gi ∪Gj13: Vi ← (wiVi + wjVj)/(wi + wj)14: wi ← wi + wj15: wi ← wi + wj16: Remove j in Ni and delete ci,j17: for all k ∈ Nj\i do18: if k ∈ Ni then19: ci,k ← ci,k + cj,k20: ck,i ← ci,k + cj,k21: else22: Ni ← Ni ∪ k23: Nk ← Nk ∪ i24: ci,k ← cj,k25: ck,i ← cj,k26: end if27: Remove j in Nk and delete ck,j28: end for29: Delete Gj ,Nj ,wj30: else if fB < fA and fB ≤ fC then31: if Ui > Uj then32: Vi ← B, Vj ← B − 133: else34: Vi ← B, Vj ← B + 135: end if36: end if37: P ← P − 1, i← i+ 138: end for39: end while40: iter ← iter + 141: β ← (iter/K)λ42: until β > λ43: for i = 1→ P do44: for all j ∈ Gi do45: Sj ← Vi46: end for47: end forOutput: filtered Image S

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Fig. 5. The left is an example of the random missing pattern. The right is an example of the textual missing mask.

Fig. 7. For the LRL0ψ algorithm we search for the best value of λl0ψfor the image Adirondack. The λl0ψ with highest PSNR is reported as theresult. This figure shows the results of LRL0ψ under different parameters. Thebest results of the other methods are reported here. We can see our LRL0ψachieves better results than the other approaches when the parameter getslarge enough.

B. Experiments

We first inpaint the depth images with the nuclear normregularization matrix completion approach. Then we apply theLRTV approach which is first used by Shi et al. [36] formedical image super-resolution. From the inpainted results,we can see LRTV reduces noise caused by the low rankcompletion. Later on, we employ the LRL0 algorithm. As wecan see, although LRL0 outperforms LRTV in most cases, itfails on some depth images. As discussed in section III-B,we propose the low gradient measure and the correspondingLRL0ψ algorithm based on the statistics of depth gradientmaps. In the end we inpaint the images with LRL0ψ . Weevaluate the inpainted results using PSNR and the PSNR iscomputed only on the missing area of the depth images.

C. Parameters and Setup

The experiments are performed on a PC of Intel i7 3.5GHzCPU with 16GB memory. Our parameter settings are listed asfollows. For all algorithms, we set the weight for the low rankterm λr = 10.

Mask type Adiron Jadepl Motor PianoMethod rand rand rand randLR 27.7596 21.9843 22.0838 18.2673LRTV 28.1319 22.8231 22.7546 18.5161LRL0 28.6198 22.7730 22.9303 19.5263LRL0ψ 28.8292 22.8725 23.1480 19.6801Mask type Playt Playrm Recyc ShelvsMethod rand rand rand randLR 22.8384 17.7783 23.5558 14.9259LRTV 23.2150 18.0073 23.9033 15.3300LRL0 24.3513 18.6153 24.7019 16.1661LRL0ψ 24.4886 18.7681 24.8112 16.2910Mask type Teddy Pipes Vintge MotorEMethod rand rand rand randLR 19.5329 22.1899 22.7924 22.3541LRTV 20.0866 22.9039 23.2368 23.2859LRL0 20.7570 23.4278 24.0400 23.1577LRL0ψ 20.8839 23.6760 24.2091 23.3850Mask type PianoL PlaytP adi tedMethod rand rand text textLR 19.8488 23.9959 28.2444 16.359LRTV 20.1330 23.2150 28.3886 16.9535LRL0 21.3011 25.6337 28.9347 17.0616LRL0ψ 21.4553 25.8008 29.2523 17.1101

TABLE IPSNR OF RESULTS FROM DIFFERENT INPAINTING METHODS ON OUR

DATASET. THE MASK TYPE INDICATES WHAT KIND OF MASK IS APPLIEDTO MAKE THE DEPTH IMAGE WITH MISSING AREAS. rand INDICATES

RANDOM MISSING AND text INDICATES TEXTUAL MASKS. NOTING THATOUR PROPOSED LRL0 APPROACH PERFORMS GENERALLY BETTER THAN

LRTV AND ACHIEVES RELATIVELY CLOSE RESULTS TO LRL0ψ .HOWEVER, LRL0 GETS WORSE-THAN-LRTV RESULTS ON MOTORE ANDJADEPLANT. OUR PROPOSED LRL0ψ APPROACH ALWAYS ACHIEVES THE

BEST RESULTS IN PSNR.

• For LRTV: we set the weight for TV term to λtv = 40.• For LRL0: we set the weight as λl0 = 30.• For LRL0ψ: we set the weight as λl0ψ = 100.For all algorithms, the iterations are stopped when the

the number of iterations exceeds 30 or the relative error ofsolutions is below 1× 10−3.

D. Results and Analysis

Some of the experimental results are shown in Figure 6and Figure 8. The PSNR of the results of the whole datasetare shown in Table I. The low rank results have obviousnoise (see Figure 6 and 8). Noting that the LRL0 approachperforms generally better than LRTV and achieves relatively

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Fig. 6. In this figure we show the results of Adirondack and Teddy with textual mask. For every image, the first row shows the original depth image, thedamaged image and the low rank result (from left to right). The second row displays the result of LRTV, LRL0 and LRL0ψ (from left to right). The PSNR ofAdirondack are 28.2444(Low rank), 28.3886(LRTV), 28.9374(LRL0) and 29.2523(LRL0ψ). The PSNR of Teddy are 16.3590, 16.9535, 17.0616 and 17.1101.

close results to LRL0ψ . However, LRL0 gets worse-than-LRTV results on MotorE and Jadepl. For high resolutionresults, please refer to our published dataset. Our proposedLRL0ψ approach always achieves the best results in PSNRbecause it allows for gradual pixel value variation which iscommon in depth images.

E. Parameter Analysis

We tune the parameter λl0ψ to see the effect of the weightof the low gradient regularization term. The resultant PSNRcurve is shown in Figure 7. When λl0ψ = 0, the LRL0ψ

algorithm degenerates to the LR algorithm. As λl0ψ increases,the low gradient term gets more important and the inpaintingresult improves over the LR algorithm. When λl0ψ gets largeenough(exceeds 80), the resultant PSNR gets stable. The

LRL0ψ algorithm outperforms the other approaches when theweight λl0ψ for the low gradient term gets large enough.

VI. CONCLUSIONS

We consider the problem of inpainting depth images. Thepopular image inpainting technique low rank matrix comple-tion always leads to noisy depth inpainting results. Based onthe gradient statistics of depth images, we propose the lowgradient regularization and combine it with the low rank priorinto the LRL0ψ approach. Then we extend the region fusionapproach in [28] for our Lψ0 gradient minimization problem.We compare our approach with the only low rank regular-ization and two other approaches, the LRTV and the LRL0approaches, which only enforce sparse gradient regularization.Experiments show that our proposed methods outperform theonly low rank method. Our proposed LRL0ψ approach also

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outperforms the LRTV and LRL0 approaches for it betterutilizes the gradient property.

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Hongyang Xue Hongyang Xue received the B.E.degree in Computer Science and Technology fromZhejiang University, China, in 2014. He is currentlya Ph.D. candidate in Computer Science at ZhejiangUniversity. His research include machine learning,computer vision and data mining.

Shengming Zhang Shengming Zhang is an internin the State Key Laboratory of CAD&CG, Under-graduate of Computer Science at Zhejiang Univer-sity, China. He will receive the Bachelor’s degreein Computer Science from Zhejiang University in2017. His research interests include computer vision,computer graphics and machine learning.

Deng Cai Deng Cai is a Professor in the StateKey Laboratory of CAD&CG, College of ComputerScience at Zhejiang University, China. He receivedthe Ph.D. degree in Computer Science from theUniversity of Illinois at Urbana Champaign in 2009.His research interests include machine learning, datamining and information retrieval.

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Fig. 8. In this figure we show the results of three images: Adirondack, Jadeplant and Vintage. For every image, the first row shows the original depth image,the damaged image and the low rank result (from left to right). The second row displays the result of LRTV, LRL0 and LRL0ψ (from left to right). ThePSNR of Adirondack are 27.7596(Low rank), 28.1319(LRTV), 28.6198(LRL0) and 28.8292(LRL0ψ). The PSNR of Jadeplant are 21.9843, 22.8231, 22.7730and 22.8725. The PSNR of Vintage are 22.7924, 23.2368, 24.0400 and 24.2091.