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Tensor Completion via Convolutional Sparse Coding Regularization

Tensor data often suffer from missing value problem due to the complex high-dimensional structure while acquiring them. To complete the missing information, lots of Low-Rank Tensor Completion (LRTC) methods have been proposed, most of which depend on the low-rank property of tensor data. In this way, the low-rank component of the original data could be recovered roughly. However, the shortcoming is that the detail information can not be fully restored, no matter the Sum of the Nuclear Norm (SNN) nor the Tensor Nuclear Norm (TNN) based methods. On the contrary, in the field of signal processing, Convolutional Sparse Coding (CSC) can provide a good representation of the high-frequency component of the image, which is generally associated with the detail component of the data. Nevertheless, CSC can not handle the low-frequency component well. To this end, we propose two novel methods, LRTC-CSC-I and LRTC-CSC-II, which adopt CSC as a supplementary regularization for LRTC to capture the high-frequency components. Therefore, the LRTC-CSC methods can not only solve the missing value problem but also recover the details. Moreover, the regularizer CSC can be trained with small samples due to the sparsity characteristic. Extensive experiments show the effectiveness of LRTC-CSC methods, and quantitative evaluation indicates that the performance of our models are superior to state-of-the-art methods.


I. INTRODUCTION
A tensor is often known as an extension of a 1D vector or 2D matrix, which can offer a high-dimensional storage structure for various data nowadays, such as color image, hyperspectral image, video, etc. Thus, it's widely leveraged in the field of computer vision, data mining [1], machine learning [2], and large-scale data analysis. However, it's common that the obtained tensor data is nevertheless undersampled when processing it, which derives many Low-Rank Tensor Completion (LRTC) [3][4] [5] algorithms to address the missing value problem. The general LRTC algorithm is formulated as: where X is the underlying tensor, T is the original data, Ω is the index set which implies the location corresponding to the observed entries. However, it's intractable to solve problem (1) which is NP-hard [6], due to the combinational nature of the function rank(·).
The definition of tensor rank is different from the known matrix's, because it is not unique. It depends on the tensor decomposition method used, e.g., CANDECOMP/PARAFAC (CP) [7], Tucker [8], Tensor-Train (TT) [9], Tensor Ring [10] etc. The definition of tensor rank, CP rank specifically, was first proposed by Kruskal et al. [11]. Immediately following, Tucker rank was adopted to approach the LRTC and gained unexpected achievements [3] [5]. However, the drawback of Tucker rank is that its components will be the ranks of highly unbalanced flattened matrices if the original tensor is high dimensional (N>3). To address above problem, A. Bengua [12] proposed a novel LRTC solver based on TT rank. Besides, they employ the Ket Augmentation (KA) to extend low dimensional tensors to higher ones and proves that TT rank is more capable of global information capturing when the dimension of a tensor is larger than three. Another novel definition of tensor rank named tubal rank [13] [14] is based on the recently proposed t-SVD decomposition [15], which is able to characterize the inherent low-rank structure of a tensor. Actually, no matter what rank is exploited to recover corrupted tensors, the core of these LRTC models is to find the relationship between missing entries and retained ones.
Despite the tensor rank estimation issue is intractable, nuclear norm [16] has been proved to be the most effective convex surrogate for the function rank(·) of the matrix. Liu et al. [3] proposed the sum of the nuclear norm (SNN) as a relaxation of the tensor rank. In addition, the tensor nuclear norm (TNN) [17] is proposed from t-product to keep consistent with the matrix cases on concepts (see details in Section II). No matter which nuclear norm is, the strong prior of the underlying tensor data is its low-rank property, and then the low-rank component can be recovered based on that.
Inevitably, the details of observation are overlooked by LRTC models. Thus, lots of regularizations are proposed to be employed as another prior to improve the details of recovered data, e.g., Total Variation (TV) [18] [19], framelets [20], wavelets [21] etc. Among these priors, TV is the most popular one in LRTC problems. Li et al. [22] combined TV and low-rankness to recover visual tensors and proposed a novel LRTC-TV model, where TV accounts for local piecewise priors. The starting point of LRTC-TV lies on the fact that the objects or edges in the spatial dimension will hold a smooth and piecewise structure. As a complementary prior arXiv:2012.00944v2 [cs.CV] 6 May 2021 to low rankness, TV can explore the local features properly. While Jiang et al. [23] combined TNN and an anisotropic TV to propose a TNN-3DTV model to exploit the correlations among the spatial and channel domains. The only fly in the ointment is that TV assumes the underlying tensor is piecewise smooth, resulting in undesired patch effects on the recovered tensors [24]. What's more, the regularizations above are all only based on the information within that image, which is fatal when the remaining entries are few. Zhang et al. [25] utilized Convolutional Neural Networks (CNN) to train a set of effective denoisers to solving the image denoising problem, which shows the importance of feature collections. Thus, it's necessary to introduce other prior information outer the image for a better result.
Another prominent paradigm in signal processing is sparse coding. By seeking a sparse representation under an overcomplete dictionary, the underlying data could be restored exceptionally. The fundamental of sparse coding is the dictionary learning, a process to obtain a dictionary D that can best represent a set of train data. As early as 2006, Elad et al. [26] leveraged dictionary learning-based method to tackle the image denoising problem. Later in 2010, Yang et al. [27] decided to sparsely code for each patch of the low-resolution input and then used the coefficients of learned representation to generate the high-resolution output. In this way, the image super-resolution could be represented by sparse entries.
For a fixed dictionary D ∈ R M ×N , given a signal X ∈ R N , the task of seeking its sparsest representation Γ ∈ R M is called sparse coding. Mathematically, the sparse coding problem can be formulated as: where Γ 0 denotes the number of non-zeros in Γ. There exists a convex relaxation of problem (2) in the form of Basis-Pursuit (BP) problem [28], formulated as: Correspondingly, there are several practical algorithms to solve problem (3), e.g., Orthogonal Matching Pursuit (OMP) [29] and the BP [30] [28]. However, most traditional dictionary learning methods are patch-based, which might ignore the consistency among the entries lied in different patches but have some common features in the image. Besides, the learned data will contain shifted versions of the same features, resulting in an over-redundant dictionary.
To tackle above problems, an alternative model, Convolutional Sparse Coding (CSC), has gained great attention in machine learning for image and video processing [31], [32] recently due to its shift-invariance property. In 2010, Zeiler et al. [33] first proposed the concept of CSC. To improve the computional effectiveness, Bristow et al. [34] and Heide et al. [32] proposed novel and fast solutions under the Alternating Direction Method of Multipliers (ADMM) framework in succession. Furthermore, Wohlberg [35] demonstrated that suitable penalties on the gradients of the coefficient maps will improve the impulse noise denoising performance. Papyan et al. [36] introduced the relationship between CSC and CNN and claimed that CNN can be analyzed theoretically via multilayer CSC.
In terms of applications, Papyan et al. [31] also proposed a slice-based dictionary learning algorithm, which was utilized in the CSC model to inpaint image as well as separate texture and cartoon. Recently, Zhang et al. [37] integrated the concept of low-rankness into CSC and addressed the rain streak removal. The work of Bao et al. [38] demonstrated that the CSC is able to process the high-frequency component of an image. In order to improve the capacity of the dictionary for learning multi-dimensional data, Bibi et al. [39] leveraged high order algebra to allow traditional CSC models encoding tensors. In addition, Xu et al. [40] decomposed the tensor dictionary and reconstructed it under the orthogonality-constrained convolutional factorization scheme to reduce computation costs. All in all, it can be argued that the success of CSC may attribute to the idea, "think globally and work locally".
Actually, the regularizations like TV, framelets, and wavelets only play the main role in a single image, which can be treated as forcing a hard constraint of physics on underlying data. The CNN-based regularization is often trained with thousands of data, which can be explained by feature collections in statistics. However, in real world, it's not always available to acquire such a large amount of samples in a limited time. Thus, we take CSC into consideration for its conveniency, which can be trained only with few-shot samples. The CSC is to extract the optimal representation dictionary in a limited training set. And this is different from the Deep learning (CNN etc.) based approaches, where the statistical feature should be learned from lots of samples.
Inspired by the capacity of CSC on extracting features locally with an overcomplete dictionary, two novel LRTC-CSC models are proposed to address the LRTC problem. By regarding CSC as a plug-and-play submodel in the optimization framework of the whole model, we effectively augment the high-frequency component of the underlying tensor. In this way, the global structure is handled by LRTC prior, and then CSC is leveraged to make up for another detail prior. Eventually, both global and local features can be well recovered. The main contributions of this paper are: 1) To tackle the missing value problem, we proposed two CSC regularized LRTC models, SNN-based and TNNbased, respectively. In both models, firstly, an overcomplete dictionary is pre-trained only with very small amount of data. And then the dictionary is used in CSC to restore the details of underlying tensor. As a result, both low-rank component and details are well recovered. 2) We came up with effective algorithms to solve the LRTC-CSC-I and LRTC-CSC-II model, which are based on the inexact ADMM method [41] and plug-and-play framework [42]. With the variable splitting techniques, the entire problem can be split into three subproblems and solved separately. 3) We tested our model on different kinds of datasets, including color images, MRI data and videos. Extensive experiments have verified the effectiveness of our model and claimed that the CSC regularization is suitable for different LRTC models. Furthermore, the performance of LRTC-CSC-II is superior to state-of-the-art models in the experiments.

A. Notations
Throughout this paper, the (i 1 i 2 · · · i n )-th element of a tensor X ∈ R I1×I2×···×In is denoted by x i1i2···in .Vectors are denoted by boldface lowercase letters, e.g. α and scalars by lowercase letters, e.g. α. We denote an n-mode tensor by calligraphic letters, e.g., X ∈ R I1×I2×···×In , where I k , k = 1, 2, · · · , n is the dimension of k-th mode and R is the field of real number. Let the upper case latters, e.g., X, denote the matrices. Especially, a mode-k matricization (also known as mode-k unfolding or flattening) of a tensor X is reshaping the tensor into a matrix X (k) ∈ R I k ×(I1···I k−1 I k+1 ···I N ) , which is defined as Tucker rank. For a 3-way tensor X ∈ R n1×n2×n3 , we denote its (i, j)-th mode-1, mode-2, and mode-3 fibers as X (:, i, j), X (i, :, j), and X (i, j, :). We use the Matlab notation X (i, :, :), X (:, i, :), and X (:, :, i) to denote the i-th horizontal, lateral and frontal slices, respectively. More often, X (i) and tube are used to represent X (:, :, i) and mode-3 fiber, respectively.
In addition, there are some mathematical operations of matrices and tensors used in this paper. The inner product of X ∈ R n×m and Y ∈ R n×m is defined as X, Y = tr(X H Y ), where X H is the conjugate transpose of the matrix X and tr(·) is matrix trace. The l 1 -norm is defined as X 1 = i1i2···in |x i1i2···in |, the Frobenius norm as X F = i1i2···in (x i1i2···in ) 2 and the nuclear norm of matrix as X * = i σ i , where σ i is the i-th largest singular value of matrix X. For a vector α, the l 2 -norm is α 2 = i α 2 i . For a tensor X ∈ R n1×n2×n3 , we useX to denote the result of discrete Fourier transformation of X along the 3-rd dimension, i.e.,X = fft(X , [], 3). Contrarily, we can compute X fromX using the inverse FFT, i.e., X = ifft(X , [], 3).
The work [43] give the first definition of the block circulation operation for a tensor X ∈ R n1×n2×n3 : The block diagonalization matrix of X is defined as . . .
The block circulant matrix can be block diagonalized, i.e., where F n3 ∈ C n3×n3 is the discrete Fourier transformation matrix, I n ∈ R n×n is an identity matrix, ⊗ denotes the Kronecker product. We also define the following operator . . .

B. Tensor Preliminaries Definition 1 (Tucker Decomposition and Tucker Rank)
[6] The Tucker decomposition is a form of higher-order PCA. It decomposes the tensor into a core tensor multiplied by a matrix along each mode. Given a tensor X ∈ R I×J×K , it can be decomposed as: where "× n " denotes mode-n product, "•" denotes the outer product, A ∈ R I×P , B ∈ R J×Q , C ∈ R K×R are factor matrices and G ∈ R P ×Q×R is the core tensor. A Tucker rank (also known as n-rank) of an N-order tensor is defined the rank of unfolding matrix along each mode, r = (r 1 , r 2 , . . . , r N ), where r n , n = 1, 2, . . . , N , denotes the rank of X (n) .

Definition 2 (Sum of Nuclear Norm (SNN)) [3]
The definition of tensor SNN is the weighted average of the nuclear norm of all matrices unfolded along each mode. Mathematically, it can be formulated as: where α i denotes the weight of matrix unfolded along i-th mode.
Using the above property, the t-product can be written as: Definition 4 (Other Special Tensor) [43] The identity tensor I ∈ R n×n×n3 is the tensor whose first frontal slice is the n × n identity matrix, and other frontal slices are all zeros. The orthogonal tensor Q ∈ R n×n×n3 is satisfies Theorem 1 (t-SVD) [43] Let X ∈ R n1×n2×n3 . Then it can be factored as where U ∈ R n1×n1×n3 , V ∈ R n2×n2×n3 are orthogonal, and S ∈ R n1×n2×n3 is an f-diagonal tensor. The t-SVD based on t-product, which can be efficiently obtained by computing a series of matrix SVDs in the Fourier domain.
Definition 5 (Tensor Tubal Rank) [13] The tensor tubal rank denoted as rank t (X ), is defined as the number of nonzero tubes of S, where S is from the t-SVD of X = U * S * V H , that is rank t (A) = # {i : S(i, :, :) = 0} .
Definition 6 (Tensor Nuclear Norm (TNN)) [13] Let X ∈ R n1×n2×n3 , the definition of tensor TNN is the sum of singular values of all the frontal slices ofX , it can be formulated as:

C. Convolutional Sparse Coding
Unlike the traditional sparse coding model, CSC replaces the general linear representation with a sum of convolutions between a set of filters d i (the set also known as a dictionary) and its corresponding feature maps Γ i . Besides, the patchbased dictionary restores the same information in the image several times, while CSC integrates these information only once. One assumes that an image X ∈ R N ×M admits a decomposition as X is the feature map corresponding to d i and * is the convolution operator. The most univesal form of CSC is Convolutional Basis Pursuit DeNoising (CBPDN), formulated as: In contrast to linear representation, convolutional dictionary can acquire shift-invariant features, which improves the efficiency of the model.

III. THE PROPOSED MODEL AND ALGORITHM
It's imperfect for LRTC models that they can only recover the low-rank component of underlying tensors. In the meanwhile, lots of details are ignored, which derives many composite algorithms with extra priors emerged to improve the missing details. These details often represent the local features or high-frequency component of an image. Previous studies about TV regularization have shown its over-patch (over-smooth) effects. To obtain better details when recovering underlying tensors, CSC is introduced to improve the highfrequency component of the underlying data. It's known that the high-frequency component often contains specific detailed information against the low-frequency one. Thus, it's a feasible choice to improve the high-frequency component solely. By using a pre-trained convolutional dictionary, the corresponding feature maps can be iteratively calculated to approximate the incomplete high-frequency component of tensor data, which is produced by LRTC models. Due to the overcompleteness of the dictionary, the convolution results of obtained feature maps and dictionary would be an improved version of the previous high-frequency component. In this section, we propose to incorporate CSC for different LRTC methods to verify the effectiveness of CSC regularization. Since the low-rank prior of LRTC can be approximated by SNN and TNN, two LRTC-CSC compound methods are developed.

Method LRTC-CSC-I
As the tratidional typical low-rank approximation method, SNN has gained unexpected achievements in LRTC problems. Even though, its promise is limited by the tensor mode-n flattening operator, which breaks the global structure of the whole tensor. To improve the performance of SNN-based model, we consider to impose a CSC regularization onto the detail component of underlying tensor, mathematically formulated as: where Φ(X ) denotes the regularization item, i.e. CSC, and λ is a trade-off parameter.
The constraint condition X Ω = T Ω can be rewritten as a function like: otherwise.
By introducing auxiliary variables {F k } N −1 k=1 to disentangle the relationship between {X (k) } N −1 k=1 and Z = X , the augmented Lagrangian function of (5) can be formulated as: where ω is the Lagrangian multiplier and β is the penalty parameter.
Under the iterative optimization framework of ADMM [44], the solution process of (7) can be concluded by solving three subproblems, i.e. [F, Z] ,X and ω.

A. Solving [F ,Z]-subproblem
In this part, we are going to solve the [F ,Z]-subproblem separately, since they are decoupled.
1) By fixing X and Z, rewrite the function of Fsubproblem: The iterative closed-form solution of (8) is: ) denotes the thresholding singular value decomposition (SVD) of X (k) and diag(·) denotes a diagnal matrix.
2) By fixing X and F , the Z-subproblem is: Actually as the plug-and-play framework suggested in [45], the problem (10) can be regarded as a new denoising problem. Fig. 1: Flowchart of LRTC-CSC models for low-rank tensor completion. The first step is to generate an initial tensor according to observations. Then a low-pass filter is utilized to divide the initial tensor into two components, a high-frequency and a low-frequency one. Next, the high-frequency component is processed by CSC model, in which the dictionary D (shown in the format of convolutional matrix) is pre-trained by a small-sample dataset. The stripe dictionary Ω, which is of size n × (2n − 1)m, is obtained by extracting the i-th patch from the global convolutional dictionary D. The stripe vector Γ i , which is the corresponding sparse respresent, contains all coefficients of atoms contributing to vec(X (k) ) i . At last, by merging the improved high-frequency component and the original low-frequency one, the final result is reconstructed.
Φ(Z) measures the degree of noise in Z, the smaller, the better. By treating " X − ω2 β2 " as a noisy image and "Z" as a clean image, CSC model becomes the denoiser to solve Zsubproblem and outputs a clean image for comparison. Thus, we use the noise image " X − ω2 β2 " to be the input of CSC model, which can be explicitly formulated as: where Input= X − ω2 β2 and Λ is the hyper-parameter. In consideration of a fact that only the high-frequency component of Input is processed by CSC [38], thus, we use a high-pass filter to acquire the high-frequency component of Input and in the rest of this section, Input represents its highfrequency component. Note that Input is a 3-rd tensor while M is a matrix. To keep consistency, Input is cut into three frontal slices for three different color channels in color images, repectively. And the following operations are completed at the level of matrices. At last, the underlying result can be obtained by summing the low-frequency component and recovered high-frequency component.
Because the inaccurate filters will lead to producing new artifacts or structure loss problems, an extra gradient constraint is adopted to suppress the outliers based on the original CSC model. What's more, as suggested in [35], the gradient constraint on the feature maps is superior to applying that on the image domain. Thus, we further utilize problem (11) with gradient constraint on the feature maps, and it can be explicitly formulated as: where g 0 and g 1 are the filters that compute the gradients along image rows and columns respectively, and τ is the hyperparameter. By introducing linear operators G j M i = g j * M i , the gradient constraint term of (12) can be rewritten as: Further more, considering the conception of block matrix, (13) can be simplified as: where and , the augmented Lagrangian function of (12) is: where B is introduced as an auxiliary variables satisfying B = M, Θ is the Lagrangian multiplier and ρ is the penalty parameter.
According to the inexact ADMM framework [46], the problem (16) can be solved by solving a sequence of subproblems.
1) Rewrite the M-subproblem: where C denotes the parameter item C = Θ ρ . By transforming (17) to the Fourier domain, it can be formulated as: whereD,M,Înput,B,Ĉ,Ψ 0 andΨ 1 denotes the corresponding expressions in the Fourier domain. A closed-form solution of (18) is: which can be solved by iterated application of the Sherman-Morrison formula [47]. After obtainingM, the corresponding M in spatial domain can be calcaulated by: where ifft(·) denotes the inverse Fourier transform.
2) Rewrite the B-subproblem: By using soft thresholding algorithm, the closed-form solution of (21) can be obtained as: 3) Update C by fixing M and B: where M * is the optimal solution of M. And after adding the low-frequency component of Input, an entire solution Z is obtained.

B. Solving X -subproblem
By fixing F and Z, the X -subproblem can be simplified and rewritten as: The closed-form solution of (25) can be obtained by: whereΩ is the complement set of Ω.

C. Solving ω-subproblem
By fixing X and F , X and Z, we can update the Lagrangian multiplier ω 1 and ω 2 , respectively: and The overall pseudocode of LRTC-CSC-I algorithm is summarized in Algorithm 1. Actually, our model can degenerate to HaLRTC model [3] by setting the hyperparameter λ = 0.

Method LRTC-CSC-II
Similar to method I, we use the TNN to approximate the low-rank prior of tensor, instead. The LRTC-CSC-II model is mathematically formulated as: The constraint condition can be rewritten as (6). By introducing auxiliary variables F = X and Z = X , the augmented Lagrangian function of (29) can be formulated as: In order to distinguish the k-th frontal slice and the k-th iteration times of a tensor, we use X (:, :, k) to denote the kth frontal slice and X k for the k-th iteration. We divide the problem (30) into three subproblems, i.e. [F, Z], X and ω, and solve the problem with the same optimization framework of ADMM.

A. Solving [F,Z]-subproblem
Similarly, we are going to solve the [F,Z]-subproblem separately, since the variables are decoupled.
1) By fixing X and Z, rewrite the function of Fsubproblem: arg min (31) The iterative closed-form solution of (31) is: where τ k = 1 β1 . After gettingF i+1 , we can get F i+1 by inverse Fourier transform 2) By fixing X and F , the Z-subproblem is: which is the same with Model I. The process of solving Zsubproblem has been introduced in detail in Model I, thus we omit them here for readability.

C. Solving ω-subproblem
By fixing X and F, X and Z, we can update the Lagrangian multiplier ω 1 and ω 2 , respectively: The overall pseudocode of LRTC-CSC-II algorithm is summarized in Algorithm 2. Actually, our model can degenerate to LRTC-TNN model [13] by setting the hyperparameter λ = 0.

IV. NUMERICAL EXPERIMENTS
In this section, we verified our LRTC-CSC models with popular color image datasets, such as Peppers, Lena, Starfish and so on. Besides, MRI data and video are also leveraged to our benchmark, which shows the generalization of LRTC-CSC model. Meanwhile, several state-of-the-art LRTC methods are compared in different situations:  1) HaLRTC [3]: It adopted SNN to approximate the lowrank prior of underlying tensors without any regularizations in the objective function. 2) LRTC-TNN [13]: It used TNN to describe the low-rank prior of underlying tensors without any regularizations in the objective function. 3) LRTC-TV [22]: It combined the Tucker rank-based SNN fidelity item and TV regularization to approximate the global low-rank structure and local smoothness, respectively. 4) TNN-3DTV [23]: It combined TNN and anisotopic TV regularization, inorder to extract the intrinsic structures of visual data and exploit the local smooth and piecewise priors, simultaneously. 5) MF-Framelet [20]: It leveraged matrix factorizationbased SNN to capture the global structure of underlying tensor with the Framelet regularization. 6) Framelet-TV [48]: It combined the TT rank-based SNN and the hybrid regularization of Framelet and TV, aiming at characterizing the global TT low-rankness, capturing the abundant details and enhancing the temporal smoothness of the tensor, respectively.
To measure the recovering performance of various models, both the Peak Signal to Noise Ratio (PSNR) and Structural SIMilarity Index (SSIM) are used. In consideration of the multiband structure of tensor data, we adopt the mean value of all bands as the evaluation metric.
All experiments are performed in MATLAB R2020a in Linux with an Inter(R) Core(TM) i7-9800X CPU at 3.80 GHz and 64GB RAM. To accelerate the running speed, we performed the CSC phase with a graphics processing GeForce RTX 2080 Ti.

A. Parameters
The convolutional filters are pre-trained with the popular fruit dataset [33] first in color image experiments as shown in Fig. 2. To make sure the best number of filters, we trained several dictionaries of different number of filters with a fruit dataset. And then these dictionaries were used to our benchmark for the sampling rate 20%. The experimental results are shown in Fig. 4. The best performance corresponds to the best number of filters. All the results in the left figure of Fig. 4 indicated that the proposed model would gain the best performance when a dictionary consists of 32 filters. Except for the Lena image, the other results in the right figure of Fig. 4 also show the superiority when the filter number is 32. Thus, we trained a dictionary of 32 filters with the fruit dataset beforehand. These filters are of size 16 × 16, Λ = 10, and τ = 0.06. It's worth noting that the computation of convolution is finished in the Fourier domain, which can be regarded as dot product, thus the size of filters can be even. Fig .3 shows the trained dictionary in visual.

B. Color Images
Before the experiments begin, we execute the missing process to original images. Specifically, to obtain observed data, all the pixels in color images will be erased randomly according to a setting ratio (same in the following MRI and video experiments).    by a comfortable margin in most situations. Its performance ends up second only to the proposed two models. Especially, the LRTC-CSC models have a larger lead at color images similar to peppers and fruits due to the dictionary trained by a fruit dataset, which also shows the importance of the relationship between underlying data and dataset. Thus why the results of LRTC-CSC-I on last two images are inferior to Framelet-TV can be explained by the few relationships between underlying tensors and dataset. In this way, the CSC plays a small role. Besides, when the missing ratio is 90%, other algorithms can hardly hold a stable SSIM value (especially the last three images), LRTC-CSC models are still standing at a relatively high level.

C. MRI dataset
In this subsection, an MRI dataset of size 180 × 216 × 30 is chosen to test our model. Except for the 30 bands used for the test, we select another 10 bands from the raw data as the dictionary training set. And different from the color image experiments, the dictionary used in this part is of size 16 × 16 × 30 for working well in most situations.
Color images own three channels only, while MRI data often include dozen or hundred bands. In this way, the relationship among different bands will be closer, which is very advantageous to the TNN-based method. However, our SNN-based model still shows its superiority after introducing the CSC prior. The same as color images, we randomly pick 10%, 20%, and 30% samples as the observations, and the results are shown in Fig. 6. It's intuitive that LRTC-CSC models achieve the best performance among these models, while HaLRTC achieves the worst performance, which verified the effectiveness of CSC regularization. In particular, LRTC-CSC models work well on detailed recovery comparing to TNN-3DTV and Framelet-TV. When the missing rate reaches 90%, our model can still hold a relatively clear result.   TNN-based methods. In other words, TNN can not handle the edge information due to lacking enough neighbors, which is determined by its predominant characteristic of global information capturing. The peformance of TT rank-based method is not satisfactory in MRI data recovering experiments because of imposing KA on the unbalanced tensor [48] [49].

D. Videos
In this subsection, we benchmarked our model on Suzie of size 144 × 176 × 150. The same as MRI experiments, we utilize 30 filters in a dictionary trained by only 10 frames of the raw video data. And except for the trained ones, another 30 frames of video are used for recovering task. By randomly selecting 10% samples, the recovered results are shown in Fig.  7. It is not hard to see that the results recovered by our LRTC-CSC models are the cleanest. Similar to MRI data experiment, TNN-3DTV is close to LRTC-CSC-I but it can not handle the edge frame well, e.g., the 30-th frame. It can also be observed that the SNN-based Framelet-TV is inferior to TNN-3DTV, which shows the low-rank approximating ability of TNN. Fig. 9 shows the recovery results of each frame in detail. Note that LRTC-CSC-II achieves the highest score in both PSNR and SSIM. However, at the beginning or end of the video, only the proposed LRTC-CSC-I is stable while the index curve of the TNN-based methods, including LRTC-CSC-II, drops rapidly, which shows the stability of SNN-based methods on edges, again. Compared with HaLRTC and LRTC-TNN, the proposed models have claimed the superiority of CSC regularization.

E. Convergency analysis
The numerical experiments have shown the remarkable performance of CSC-I and CSC-II. In order to demonstrate the convergence of algorithms in numerical, Fig. 10 shows the variation curves of the Relative Error (RE) values, i.e., RE = X −T F T F , and PSNR values of CSC-I and CSC-II algorithms with the number of iterations on three color images of Lena, Peppers and Fruits at M R = 70%. Factly, it can be observed that both the proposed algorithms converge to an optimal point.

V. CONCLUSION
Inspired by the capacity of CSC on high-frequency component processing, we introduced CSC into the traditional LRTC model. In this way, the details of the underlying tensor would be improved in addition to the low-rank component recovery. Comparing to other plug-and-play CNN prior to using thousands of samples, our method can achieve good performance with only a few samples. For the proposed models, we obtain effective algorithms based on the inexact ADMM method. And the effectiveness of LRTC-CSC-I and LRTC-CSC-II have been verified in color images, MRI data, and video data recovery experiments.
In future work, it would be of great interest to leverage the prior information of CSC to improve the performance of other models. For the marginal effects of the TNN-based method, CSC might be a promising method to strengthen the connection between the edge and subject.