Discrete Dynamics in Nature and Society
Volume 2018 (2018), Article ID 2598160, 12 pages
https://doi.org/10.1155/2018/2598160

Research Article

Enhancing Matrix Completion Using a Modified Second-Order Total Variation

Wendong Wang and Jianjun Wang

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

Correspondence should be addressed to Jianjun Wang; wjj@swu.edu.cn

Received 17 March 2018; Revised 20 July 2018; Accepted 14 August 2018; Published 12 September 2018

Academic Editor: Seenith Sivasundaram

Copyright © 2018 Wendong Wang and Jianjun Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we propose a new method to deal with the matrix completion problem. Different from most existing matrix completion methods that only pursue the low rank of underlying matrices, the proposed method simultaneously optimizes their low rank and smoothness such that they mutually help each other and hence yield a better performance. In particular, the proposed method becomes very competitive with the introduction of a modified second-order total variation, even when it is compared with some recently emerged matrix completion methods that also combine the low rank and smoothness priors of matrices together. An efficient algorithm is developed to solve the induced optimization problem. The extensive experiments further confirm the superior performance of the proposed method over many state-of-the-art methods.

1. Introduction

Matrix Completion (MC) refers to the problem of filling in missing entries of an unknown matrix from a limited sampling of its entries. This problem has received widespread attentions of researchers in many real applications; see, e.g., [1–3]. Generally, it is impossible to exactly recover arbitrary matrices from such a problem without some additional information since the problem is extraordinarily ill-posed [4] and hence has infinitely many solutions. However, in many realistic scenarios, very often the matrix we wish to recover is low-rank or can be approximated by a low-rank matrix, as illustrated in Figure 1. Therefore, an immediate idea to model MC is to pursue the low rank of matrices by solving the following optimization problem:

where and the entries of in location set () are given while the remaining entries are missing.

(a) An image example

(b) The singular values

Figure 1: (a) Lenna image (). (b) All the singular values of Lenna image are sorted from larger to smaller. A simple calculation will lead to the fact that the top 40 singular values contribute more than 80% information of all singular values.

Unfortunately, problem (1) is of little practical use since it is NP-hard in general. To circumvent this issue, a nuclear norm minimization method was suggested in [4, 5], which solves

where and denotes the th largest singular value of . Especially in [4], Candès and Recht have proved theoretically that any low-rank matrices can be exactly recovered by solving problem (2) under some necessary assumptions. Recently, some new and deep results on the above problem (2) have been obtained to tackle the theoretical defects in [4]; see, e.g., [6, 7]. Many algorithms have also been proposed to solve problem (2) and its variants, including the Accelerated Proximal Gradient (APG) algorithms [8, 9], the Iterative Reweighted Least Squares (IRLS) algorithm [10], and the Singular Value Thresholding (SVT) algorithms [11, 12].

Nuclear norm is the tightest convex approximation of rank function, but it is far from the closest one [13]. The relationship between nuclear norm and the rank function of matrices is similar to that between norm and norm of vectors [4, 14]. To get a much closer approximation to the rank function, many nonconvex functions used to approximate the norm, see, e.g., [15–19], have been successfully extended to replace the nuclear norm [20–22]. Two representative ones are the Schatten- quasi-norm [20, 23, 24] and the truncated nuclear norm [21, 25] which is also called the partial sum of singular values [26]. The induced methods based on the above two approximate functions have been well investigated to deal with the MC problem [20, 21] and obtained much better performance than previous nuclear norm minimization method. Besides the above, there still exist many other methods used to tackle the MC problem; see, e.g., [27, 28] for the matrix factorization based methods and [29, 30] for the greedy methods.

Overall, almost all the existing MC methods are designed to approach the rank function and thus to induce the low rank. To some degree, low rank only characterizes the global prior of a matrix. In some instances, however, besides the low rank the matrices often have some additional structural priors. We still take the matrix (grey image) shown in Figure 1(a) for example. It is rich in smoothness features (priors). In other words, an entry and its neighboring entries in such a matrix often have small difference in their values. When dealing with such matrices, most existing MC methods in fact can not well capture their smoothness priors and hence may result in a poor performance. On the other hand, when the entries of a matrix reach an incredibly high missing ratio or the high-quality recovered matrices are in urgent need, one has no choice but to take their additional priors into consideration since it will become very difficult to exactly/robustly recover any matrices only by using their low rank prior. Therefore, how to mine more available priors of underlying matrices and integrate them into MC becomes a very crucial problem.

In this paper, we propose a new method that combines the low rank and smoothness priors of underlying matrices together to deal with the MC problem. Our method is not the first work on this topic, but by using a modified second-order total variation of matrices it becomes very competitive. In summary, the contributions of this paper are stated as follows:

(i)A modified second-order total variation and nuclear norm of matrix are combined to characterize the smoothness and low-rank priors of underling matrices, respectively, which makes our method much more competitive for MC problem.(ii)An efficient algorithm is developed to deal with the optimization problem and the extensive experiments testify to the effectiveness of the proposed method over many state-of-the-art MC methods.

The remainder of this paper is organized as follows. In Section 2, we review some MC methods that simultaneously optimize both the low rank and smoothness priors of underlying matrices. Since matrices can be considered as the second-order tensors, this indicates that most tensor based completion methods can also be applied to the MC problem. Two related tensor completion methods are also included and introduced in this section. In Section 3, we present the proposed method and design an efficient algorithm to solve the induced optimization problem. Experimental results are presented in Section 4. Finally, the conclusion and future work are given in Section 5.

2. A Review on MC with Smoothness Priors

The low rank and smoothness priors of underlying matrices have been well studied in MC and visual data processing communities [31, 32], respectively. However, their combined work for MC is rarely reported. To the best of our knowledge, Han et al. [33] gave the first such work for MC and proposed a Linear Total Variation approximation regularized Nuclear Norm (LTVNN) minimization method, which takes the form

where is a penalty parameter. In LTVNN method, the smoothness priors of underlying matrices are constrained by a Linear Total Variation Approximation (LTVA), which is defined as

where

It has been shown in [33] that this LTVNN method is superior to many state-of-the-art MC methods, such as previous SVT method [11] and TNNR method [21], that only pursue the low rank. However, LTVNN method may result in staircase effect due to the use of linear total variation. On the other hand, the induced LTVNN algorithm does not solve problem (3) directly, but solves it by dividing problem (3) into two subproblems firstly, then solving these two subproblems independently and outputting the final results based on the solutions to these two subproblems lastly. Obviously, such an algorithm strategy is not consistent with their proposed optimization problem (3).

Recently, such consideration has been extended to deal with the tensor completion (TC) problem. From the perspective of tensor, vectors and matrices can be considered as the first- and second-order tensors, respectively. Therefore, most existing TC methods can still be used to deal with the MC problem. In 2014, Chen et al. [34] proposed a Tucker decomposition based Simultaneous Tensor Decomposition and Completion (STDC) method for TC. This STDC method solves the following optimization problem:

where , is the Tucker decomposition of tensor , is a unified factor matrix that contains all factor matrices, is the Kronecker product, and is a preset matrix designed by some priors. Similar to the matrix case, the elements of tensor in location set ( are given while the rest are missing. In problem (6), , i.e., the first term of the objective function, characterizes the low rank of tensor , and its second term enforces the similarity between individual components, which provides some smoothed results for . Thus, STDC can be considered as a tensor extension of low-rank and smooth matrix completion. Later, based on another PARAFAC decomposition of tensors, Yokota et al. [35] proposed a Smooth PARAFAC tensor Completion (SPC) method that integrates low rank and smoothness priors of tensor by solving

where is the PARAFAC decomposition of tensor , is the outer product, stands for the complement set of , is the classical norm on vectors, and the matrix is a smoothness constraint matrix, defined as

SPC method integrates low rank and smoothness priors into a single term (i.e., the second term of the objective function in problem (7)), and it is totally different from the STDC method. Besides, it has been proved numerically in [35] that SPC method performs much better than the SDTC method. More discussion on tensor and TC problems is beyond the scope of this paper, we refer interested readers to [35, 36] and the references within.

3. Proposed New Method

In this section, we propose a new method that combines the low rank and smoothness priors of underlying matrices to deal with the MC problem. In our method, the nuclear norm is used to characterize the low rank of underlying matrices while their smoothness priors are characterized by a modified second-order total variation (MSTV), which is defined as

where

and

Obviously, such a modified second-order total variation is a new metric for smoothness priors of matrices, which is not different from the classical second-order total variation [37], or from previous linear total variation approximation. However, from a functional perspective, it not only inherits the geometrical structure of second-order total variation which avoids the staircase effect caused by linear total variation approximation, but also yields a smooth function that allows more efficient numerical algorithms. Figure 2 shows the geometrical differences between the linear total variation approximation and our modified second-order total variation. Besides, equation (9) can be further written in matrix form as

where for

(a) LTVA

(b) MSTV

Figure 2: The black points denote the local but nonedged entries, the red points denote the corresponding neighboring entries concerned in LTVA and our MSTV, and the grey points denote the remaining entries of matrices. Obviously, our MSTV takes more informative neighboring entries into consideration and hence yields a better performance.

Now, the proposed methods that are based on nuclear norm and a modified second-order total variation can be modeled as the following optimization problem:

where is a penalty parameter. Although in problem (14) nuclear norm is used to characterize the low rank of matrices, it can be replaced by some other approximate functions of rank function, such as Schatten- quasi-norm and truncated nuclear norm. On the other hand, both problem (14) and the aforementioned problem (3) share some similarities, but one will see later in the experimental part that the induced algorithm by proposed problem (14) performs much better than that of problem (3).

To solve problem (14), we adopt the Alternating Direction Method of Multipliers (ADMM) [38], which has been widely used to solve many application problems. To use ADMM, we first rewrite problem (14) as

Then the corresponding augmented Lagrangian function of (15) is presented as

where is the dual variable and is a penalty parameter associated with the augmentation. According to ADMM, the iterative procedure can be described as follows:

(1) Fixing and , solve for by

 which is equivalent to

 where . According to [11], problem (18) has a closed-form solution, which can be obtained by the singular value thresholding operator , i.e.,

 where is the singular value decomposition of .(2) Fixing and , solve for by

 which is equivalent to

 where . By setting the derivative of with respect to to zero, we have

 where is an identity matrix. Equation (22) is the well known Sylvester equation [39]. In this paper, we resort to the Matlab command to solve it; i.e.,

 After we obtain , i.e., the solution to (22), we can hence approximately obtain by

(3) Update by

We summarize the iterative procedure in Algorithm 1. Note that we do not use a fixed but adopt an adaptive updating strategy for (see the 7th step in Algorithm 1). Such a setting is mainly inspired by the recent studies on ADMM [21, 26, 40]. In fact, both the theoretical and applied results have shown that such a setting can help achieve quick convergence for ADMM based algorithms and hence avoid high computation cost. Without loss of generality, we simply set and for Algorithm 1. It should also be noted that the performance of the proposed algorithm can be further improved if one can optimize these two parameters.

Input: , , , = = , = 0, , , and

Output:

 1:   Initialize

 2:   repeat

 3:     Update by (19);

 4:     Update by (23);

 5:     Update by (24);

 6:     Update by (25);

 7:     Update ;

 8:     Update .

 9:   until a certain stopping criterion is satisfied.

 10: return .

Algorithm 1: Solving problem (14) via ADMM.

4. Numerical Experiments

In this section, some experiments are conducted to evaluate the performance of the proposed method. They mainly involve solving the randomly masked image completion problem and the text masked image reconstruction problem. The former problem aims to complete the incomplete images generated by masking their entries randomly, and the latter one aims to reconstruct the images with text. In our experiments, the Lenna image shown in Figure 1(a) and 20 widely used test images shown in Figure 3 are used to generate the desired test matrices. To evaluate the quality of the reconstructed images, Structural SIMilarity (SSIM) and Peak Signal-to-Noise Ratio (PSNR) indices are adopted. We refer readers to [41] for SSIM index. As to PSNR index, suppose Mean Squared Error (MSE) is defined as , where counts the number of elements in and and are the original image and the reconstructed image, respectively; then we can calculate the PSNR value by . The missing ratio of an incomplete image is always defined as . All the experiments are implemented in Matlab and the codes can be downloaded from https://github.com/DongSylan/LIMC.

Figure 3: The test images. All are and will be numbered in order from 1 to 20, from left to right, and from top to bottom.

4.1. Convergence Behavior

In this part, we conduct some experiments to examine the convergence behavior of our Algorithm 1. The desired test matrices are generated by masking the entries in Lenna image randomly with a 60% missing ratio. Before the convergence analysis, we have to determine a proper penalty parameter since it plays a vital role in our Algorithm 1. Figure 4 plots the relationship between and the PSNR and SSIM results obtained by our Algorithm 1. It is easy to see that is the best selection. In above experiments, we initialize and set the stopping criterion as

where, here and throughout, is called the Relative Neighboring Iteration Error (RNIE). Similar to RNIE, we define Relative True Iteration Error (RTIE) as . In what follows, we always set and for our Algorithm 1.

(a) versus PSNR

(b) versus SSIM

Figure 4: Penalty parameter selection.

Figure 5 plots the convergence results of our Algorithm 1. One can easily see that both the RNIE and RTIE decrease rapidly and consistently. Specifically, in terms of RNIE, when exceeds 200 it tends to be smaller than . However, under the same condition, RTIE tends to be a constant. These observations to some degree indicate that the stopping criterion (26) is reasonable for our Algorithm 1.

(a) RNIE

(b) RTIE

Figure 5: Convergence behavior of Algorithm 1.

4.2. Randomly Masked Image Completion

To illustrate the effectiveness of our method, we compare our Algorithm 1 with 5 other state-of-the-art algorithms. They are IRLS algorithm [24], PSSV algorithm [26], LTVNN algorithm [33], and the SPC algorithms [35] with TV and QV smoothing, i.e., the SPC-TV and SPC-QV algorithms. In our experiments, we set , , and for IRLS algorithm, for PSSV algorithm, and for LTVNN algorithm, for SPC-TV algorithm, and for SPC-QV algorithm. We also uniformly set their maximum iterations to 500.

We start with completing the Lenna image masked randomly with several different missing ratios . The obtained PSNR and SSIM results are presented in Table 1, where the highest PSNR and SSIM values are underlined and marked in bold, while their second highest values are underlined only. One can directly see that, when the missing ratio decreases or in other words the known information increases, all the algorithms tend to perform better. However, our Algorithm 1 performs better than the rest of the algorithms in both PSNR and SSIM perspectives whenever the missing ratio ranges. Specifically, when compared to the IRLS and PSSV algorithms that only pursue the low rank of underlying matrices, our Algorithm 1 improves their PSNR and SSIM performance by more than 5dB and 0.2, respectively. Figure 6 shows the reconstructed images obtained by these algorithms. It is easy to find that our Algorithm 1 not only avoids the staircase effect resulting from LTVNN algorithm, but also leads to a better smoothness performance than SPC-TV and SPC-QV algorithms.

Table 1: PSNRSSIM results on Lenna image with different missing ratios by different algorithms.

70%
65%
60%
55%
50%

IRLS
21.890.595
22.490.661
22.910.707
23.130.747
23.340.774

PSSV
21.800.585
22.600.646
23.210.696
23.900.744
24.610.791

LTVNN
23.890.793
24.420.824
24.910.853
25.410.881
25.900.900

SPC-TV
24.290.743
24.880.776
25.240.801
25.720.829
26.060.852

SPC-QV
26.660.864
27.150.883
27.500.898
27.890.912
28.180.925

Ours
27.870.906
28.460.921
28.960.936
29.540.948
29.890.957

(a) Original image

(b) Masked image

(c) IRLS

(d) PSSV

(e) LTVNN

(f) SPC-TV

(g) SPC-QV

(h) Ours

Figure 6: Visual results on Lenna image with 60% missing ratio by different algorithms. (a) Original image. (b) Randomly masked image. (c) IRLS: PSNR=22.91dB, SSIM=0.707. (d) PSSV: PSNR=23.21dB, SSIM=0.696. (e) LTVNN: PSNR=24.91dB, SSIM=0.853. (f) SPC-TV: PSNR=25.24dB, SSIM=0.801. (g) SPC-QV: PSNR=27.50dB, SSIM=0.898. (h) Ours: PSNR=28.96dB, SSIM=0.936. The figure is better viewed in zoomed PDF.

To further confirm the effectiveness of our method, we apply our Algorithm 1, together with other 5 algorithms, to 20 widely used test images (Figure 3). The obtained results are presented in Table 2. An overall impression observed from Table 2 is that our Algorithm 1 achieves the highest SSIM in all cases and the highest PSNR in almost all cases. Although in terms of PSNR our Algorithm 1 performs weaker than SPC-QV algorithm in some cases, but these cases only occupy a small proportion (13/80) of all the cases, and the PSNR difference in these cases is also very small (less than 0.8dB).

Table 2: PSNRSSIM results on 20 test images with different missing ratios by different algorithms.

Missing ratio=70%
Missing ratio=60%

IRLS
PSSV
LTVNN
SPC-TV
SPC-QV
Ours
IRLS
PSSV
LTVNN
SPC-TV
SPC-QV
Ours

1
27.580.802
27.880.821
29.320.902
29.140.864
30.590.910
31.960.936
29.110.869
29.350.882
30.410.935
29.840.888
30.940.924
33.550.965

2
21.340.586
21.430.593
23.370.768
23.500.715
25.290.819
25.250.838
22.610.712
22.670.700
24.220.833
24.340.784
26.100.868
26.230.888

3
22.030.596
22.120.606
24.290.804
24.700.761
27.370.879
28.730.922
23.120.709
23.800.730
25.440.868
25.780.825
28.270.911
29.960.949

4
19.870.470
20.190.487
21.950.645
21.550.592
22.640.692
22.600.728
20.900.610
20.790.596
22.490.736
22.010.681
23.190.768
23.210.800

5
22.770.630
23.250.644
24.670.755
24.710.735
26.150.816
26.040.829
23.980.735
24.220.742
25.440.824
25.410.803
26.800.864
27.000.882

6
22.720.623
22.560.609
24.550.766
24.860.742
26.940.838
27.130.861
23.690.721
23.940.716
25.530.833
25.690.802
27.610.875
27.850.895

7
20.760.582
20.910.592
22.640.804
23.110.734
24.940.852
25.240.887
21.980.704
22.210.702
23.570.865
24.040.797
25.790.890
26.450.929

8
24.260.683
24.140.689
25.480.839
26.860.808
28.620.879
27.850.882
25.560.782
26.210.788
26.680.886
28.110.852
29.810.906
29.140.917

9
20.400.558
20.330.557
22.010.687
22.140.667
23.780.762
23.680.784
21.370.674
21.280.665
22.730.769
22.850.748
24.450.824
24.550.846

10
23.390.621
23.250.614
25.090.730
25.340.731
27.250.816
27.540.838
24.260.717
24.480.725
25.960.805
26.190.801
28.080.867
28.610.888

11
19.190.539
19.600.571
21.690.798
21.530.690
23.230.810
23.290.850
20.690.669
20.680.662
22.520.856
22.360.753
23.960.853
24.070.894

12
17.870.525
17.920.528
19.800.696
19.950.658
22.040.780
22.410.800
18.890.642
19.120.654
20.660.784
20.890.746
23.070.845
23.570.865

13
24.690.619
24.640.616
27.070.783
26.830.742
28.700.833
29.220.864
25.690.731
25.730.719
27.810.843
27.510.805
29.260.875
30.110.906

14
24.000.668
23.890.666
25.530.777
25.940.766
27.960.847
27.920.861
25.120.760
25.250.765
26.490.842
26.960.830
28.830.889
29.030.905

15
23.440.595
23.710.605
25.640.735
25.380.707
26.920.791
27.110.814
24.480.704
24.490.700
26.220.800
25.930.773
27.500.842
28.040.870

16
19.730.545
19.720.546
21.430.750
21.920.690
24.060.825
24.540.868
20.850.666
21.040.658
22.410.826
22.810.764
24.930.872
25.670.913

17
19.760.514
19.590.504
21.940.704
21.960.653
24.100.787
24.740.835
20.760.634
20.940.630
22.860.789
22.890.738
24.960.844
25.750.885

18
20.120.555
19.960.543
22.110.748
22.670.708
25.710.846
27.100.898
20.980.659
21.760.675
23.390.827
23.840.783
26.750.889
28.430.932

19
21.830.604
21.720.599
23.630.809
24.350.758
26.930.889
27.760.927
22.780.715
23.640.721
24.950.877
25.590.822
28.020.920
29.130.953

20
20.270.577
20.380.584
22.010.748
22.210.703
23.840.806
23.200.812
21.410.695
21.630.693
22.890.823
23.190.779
24.850.861
24.220.868

Missing ratio=55%
Missing ratio=50%

IRLS
PSSV
LTVNN
SPC-TV
SPC-QV
Ours
IRLS
PSSV
LTVNN
SPC-TV
SPC-QV
Ours

1
29.450.887
29.980.905
30.870.946
30.060.899
31.040.930
34.030.973
29.900.901
30.740.923
31.470.957
30.460.910
31.450.937
34.760.978

2
22.910.750
23.210.745
24.530.859
24.750.815
26.440.887
26.650.906
23.150.782
23.800.787
24.880.881
25.150.842
26.840.905
27.100.922

3
23.360.743
24.580.780
25.880.890
26.200.850
28.660.924
30.470.958
23.590.772
25.420.824
26.350.910
26.690.873
29.040.936
30.940.965

4
21.150.658
21.030.643
22.650.769
22.180.718
23.340.796
23.440.826
21.280.696
21.260.687
22.820.800
22.360.753
23.560.823
23.680.851

5
24.170.766
24.610.783
25.710.851
25.680.830
27.060.883
27.300.902
24.360.793
25.020.819
25.950.873
25.930.854
27.310.900
27.610.918

6
23.950.758
24.540.763
25.940.859
26.050.829
27.900.892
28.120.909
24.080.783
25.010.799
26.260.880
26.370.852
28.120.905
28.330.920

7
22.310.743
22.980.752
24.040.889
24.550.824
26.210.905
27.000.944
22.370.772
23.480.790
24.280.906
24.780.848
26.400.917
27.360.954

8
25.810.810
27.130.830
27.110.906
28.580.875
30.040.918
29.390.931
26.040.831
28.080.861
27.620.922
29.140.892
30.490.927
29.800.942

9
21.670.715
21.770.714
23.050.803
23.200.783
24.770.849
24.930.871
21.840.749
22.190.757
23.360.833
23.550.815
25.090.872
25.250.890

10
24.470.750
25.070.772
26.380.836
26.620.832
28.450.888
29.010.906
24.620.778
25.660.812
26.710.860
27.020.858
28.810.905
29.450.922

11
21.000.710
21.150.702
22.890.878
22.720.781
24.290.872
24.420.911
21.170.743
21.600.742
23.090.896
22.920.806
24.460.887
24.650.923

12
19.090.683
19.660.705
20.970.814
21.270.782
23.440.868
23.970.887
19.310.721
20.380.758
21.390.846
21.820.818
23.990.892
24.560.909

13
25.940.764
26.250.760
28.170.865
27.830.830
29.580.891
30.550.920
26.110.792
26.700.796
28.400.884
28.080.851
29.800.906
30.820.932

14
25.350.790
25.920.807
26.930.868
27.410.857
29.190.906
29.520.922
25.520.813
26.650.843
27.260.887
27.730.876
29.490.919
29.900.934

15
24.750.744
24.910.746
26.540.831
26.240.806
27.850.867
28.520.893
24.940.773
25.250.781
26.770.855
26.520.832
28.120.887
28.860.911

16
21.120.706
21.640.707
22.770.853
23.240.795
25.290.891
26.100.928
21.280.738
22.220.752
23.160.878
23.660.825
25.690.908
26.560.941

17
20.940.670
21.510.684
23.200.821
23.190.770
25.200.865
26.080.903
21.090.706
22.000.732
23.520.848
23.550.804
25.510.885
26.440.918

18
21.240.696
22.640.733
23.960.856
24.390.816
27.180.905
28.910.943
21.360.727
23.500.783
24.400.880
24.880.844
27.600.920
29.490.953

19
23.000.750
24.390.770
25.430.898
26.090.848
28.440.931
29.610.962
23.190.777
25.200.815
25.880.917
26.530.871
28.670.941
29.940.968

20
21.640.734
22.130.740
23.200.848
23.600.813
25.230.883
24.580.890
21.770.764
22.670.783
23.490.872
23.930.840
25.580.900
25.000.909

4.3. Text Masked Image Reconstruction

Compared to the randomly masked image completion problem, text masked image reconstruction is a relatively hard task since the entries masked by the text are not randomly distributed in the matrix and the text may mask some important texture information. However, this problem can still be transformed into an MC problem by checking the position of the text first and regarding the corresponding entries as missing values.

Figure 7 shows the visual results obtained by above-mentioned algorithms on reconstruction of the Lenna image with text. In terms of PSNR and SSIM values, our Algorithm 1 and the SPC-QV algorithm rank first and second, respectively, followed by the rest of the algorithms. In terms of visual quality, the proposed Algorithm 1 well reconstructs the original image and its local feature structure is also well kept without seeing any text traces. In contrast, the images reconstructed by IRLS and PSSV algorithms are covered with the obvious text traces. The LTVNN algorithm suffers from staircase effect again. The image reconstructed by the SPC-QV algorithm is better than that by SPC-TV algorithm, but still has some faint text traces. Based on previous 20 test images, we generate another 20 desired images with text (Figure 8) to compare our Algorithm 1 with 5 other algorithms in dealing with the text masked image reconstruction problem. The obtained PSNR and SSIM results are reported in Table 3, which confirms again the excellent performance of our Algorithm 1.

Table 3: PSNRSSIM results on reconstruction of 20 test images with text by different algorithms.

IRLS
PSSV
LTVNN
SPC-TV
SPC-QV
Ours

1
28.560.956
30.470.975
30.910.981
29.780.965
30.740.973
34.170.992

2
22.130.911
23.960.930
24.430.955
24.620.939
26.130.962
26.650.972

3
22.270.895
25.450.944
25.410.963
25.780.948
27.700.971
30.190.987

4
21.140.897
21.680.904
22.690.933
22.500.920
23.430.939
23.580.950

5
23.600.919
25.180.943
25.330.951
25.400.945
26.620.960
26.960.969

6
22.850.902
24.940.932
25.740.956
25.560.941
27.300.964
27.980.973

7
21.140.906
23.300.933
23.200.959
23.700.937
25.000.963
26.480.983

8
24.930.926
29.080.962
27.330.969
28.720.956
30.120.971
31.020.979

9
21.120.905
22.700.926
22.940.940
23.380.935
24.660.953
25.070.962

10
23.840.912
26.490.948
26.040.949
26.760.950
28.190.964
29.340.974

11
20.200.895
21.320.911
22.210.958
22.070.924
23.410.951
23.720.971

12
18.120.892
20.510.926
20.640.943
20.970.933
22.620.956
23.610.967

13
25.320.920
27.090.938
27.950.959
27.720.947
29.280.965
30.720.977

14
24.310.920
26.740.951
26.670.958
27.180.953
28.580.968
29.210.977

15
24.300.913
25.410.929
26.370.948
26.060.938
27.420.955
28.500.968

16
20.190.890
22.560.920
22.410.949
22.970.931
24.800.961
26.330.977

17
19.760.874
21.800.908
22.770.942
22.610.925
24.270.953
25.570.969

18
19.770.872
23.900.933
23.720.953
23.970.937
26.480.966
29.240.984

19
21.430.895
24.760.931
25.190.967
25.120.943
27.250.971
29.550.988

20
20.690.902
22.710.934
22.800.951
23.020.941
24.440.961
24.180.967

(a) Original image

(b) Masked image

(c) IRLS

(d) PSSV

(e) LTVNN

(f) SPC-TV

(g) SPC-QV

(h) Ours

Figure 7: Visual results on Lenna image with text by different algorithms. (a) Original image. (b) Text masked image. (c) IRLS: PSNR=22.14dB, SSIM=0.903. (d) PSSV: PSNR=24.41dB, SSIM=0.929. (e) LTVNN: PSNR=25.33dB, SSIM=0.962. (f) SPC-TV: PSNR=25.33dB, SSIM=0.941. (g) SPC-QV: PSNR=27.07dB, SSIM=0.966. (h) Ours: PSNR=29.12dB, SSIM=0.984. The figure is better viewed in zoomed PDF.

Figure 8: The desired text masked images. They are numbered in the same manner as we stated in Figure 3.

5. Conclusion and Future Work

This paper proposed a new method that combines the low rank and smoothness priors of matrices to tackle the matrix completion problem. Different from previous LTVNN method, our proposed method characterizes the smoothness of matrices by using a modified second-order total variation, which not only avoids the staircase effect caused by LTVNN method, but also leads to a better performance. Even compared to the recently emerged smooth PARAFAC tensor completion method, our method is still highly competitive in both PSNR and SSIM perspectives. The extensive experiments further confirm the excellent performance of our proposed method. Potential future work includes replacing the nuclear norm used in our method with other (nonconvex) low-rank promoting functions, integrating more available priors of matrices to enhance the performance of existing matrix completion methods, and applying our modified second-order total variation to the tensor completion problems.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the Natural Science Foundation of China under Grants Nos. 61273020 and 61673015.

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