Low-Rank Tensor Recovery Based on Nonconvex Geman Norm and Total Variation

Abstract

Tensor restoration finds applications in various fields, including data science, image processing, and machine learning, where the global low-rank property is a crucial prior. As the convex relaxation to the tensor rank function, the traditional tensor nuclear norm is used by directly adding all the singular values of a tensor. Considering the variations among singular values, nonconvex regularizations have been proposed to approximate the tensor rank function more effectively, leading to improved recovery performance. In addition, the local characteristics of the tensor could further improve detail recovery. Currently, the gradient tensor is explored to effectively capture the smoothness property across tensor dimensions. However, previous studies considered the gradient tensor only within the context of the nuclear norm. In order to better simultaneously represent the global low-rank property and local smoothness of tensors, we propose a novel regularization, the Tensor-Correlated Total Variation (TCTV), based on the nonconvex Geman norm and total variation. Specifically, the proposed method minimizes the nonconvex Geman norm on singular values of the gradient tensor. It enhances the recovery performance of a low-rank tensor by simultaneously reducing estimation bias, improving approximation accuracy, preserving fine-grained structural details and maintaining good computational efficiency compared to traditional convex regularizations. Based on the proposed TCTV regularization, we develop TC-TCTV and TRPCA-TCTV models to solve completion and denoising problems, respectively. Subsequently, the proposed models are solved by the Alternating Direction Method of Multipliers (ADMM), and the complexity and convergence of the algorithm are analyzed. Extensive numerical results on multiple datasets validate the superior recovery performance of our method, even in extreme conditions with high missing rates.

1. Introduction

In the era of big data, high-order data have become ubiquitous across various scientific and engineering fields (such as signal processing [1], computer vision [2], machine learning [3] and data mining [4]). This increasing abundance of high-order data has motivated the widespread use of tensors, which serve as multidimensional extensions of vectors (first-order tensors) and matrices (second-order tensors) for modeling complex data structures. Tensor recovery is a classical inverse problem aimed at reconstructing a tensor $\mathcal{T}\in {\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$ with certain structural priors from corrupted observations $\mathcal{Y}=\mathsf{\Phi }\left(\mathcal{T}\right)$. Due to inherent limitations in signal acquisition equipment, including sensor sensitivity and photon effects, tensor data collected often exhibit noticeable degradation, which manifests as missing values or corruptions. For observation tensors with massive amounts of information loss, this issue is often solved using tensor recovery methods based on low-rank tensor priors, leading to the formulation of tensor completion models (TC) [5,6]. In the case of observation tensors contaminated with substantial noise, tensor robust principal component analysis models (TRPCA) [7,8] are commonly employed.

In the field of low-rankness (L) priors for tensors, an unavoidable concern revolves around how to characterize the rank of tensors. Although a tensor is a higher-order extension of a matrix, the definition of tensor rank remains ambiguous, making it challenging to directly tackle the problem of tensor rank minimization. Currently, there exist numerous definitions of tensor rank based on different tensor decomposition methods, such as the CP rank [9], Tucker rank [10], multi-rank [11], and tubal-rank [12] methods, among others. However, it is NP-hard to directly solve the minimization problem for CP rank and Tucker rank. The tensor nuclear norm (TNN), widely adopted as a convex surrogate for tubal rank, is computationally efficient but fails to faithfully approximate the true tensor rank. Extensive studies have shown that the nonconvex approximation of the rank function can offer better estimation accuracy than the nuclear norm. To better approximate tensor rank, various nonconvex alternatives have been proposed, such as the $L_p$ norm [13], the $L_{1/2}$ norm [14], the $L_g$ norm [15] and so on. Researchers are committed to designing more effective regularizers to characterize low-rankness prior information about the underlying tensor, and the corresponding tensor recovery model can be constructed as follows:

$$\underset{\mathcal{X}}{min}\mathcal{L}\left(\mathcal{X}\right),\mathrm{s}.\mathrm{t}.{\mathcal{X}}_0=\mathsf{\Phi }\left(\mathcal{X}\right),$$

(1)

where $\mathcal{X}$ denotes the tensor to be recovered, ${\mathcal{X}}_0$ denotes the observed tensor, $\mathcal{L}\left(·\right)$ is the regularizer characterizing the low-rankness prior of the tensor, and $\mathsf{\Phi }(·)$ is the operator describing the degradation process. Chen et al. [16] propose a denoising method that integrates the low-rankness prior with a Difference of Gaussian filter to enhance denoising performance. However, the effectiveness of the filter relies on appropriate parameter tuning to balance noise suppression and detail preservation.

To further enhance the accuracy of tensor restoration, researchers have focused on exploring the local features of tensors, thereby introducing the smoothness (S) characteristic. The smoothness prior describes the structural properties of a tensor, where adjacent pixels typically show similar properties or values. For $\mathcal{X}\in {\mathbb{R}}^{n_1\times \dots \times n_d}$ with t-SVD rank R and L and S priors, the following inequalities are induced: $R-1\le {\mathrm{rank}}_{\mathrm{t}-\mathrm{SVD}}\left({\mathcal{G}}_k\right)\le R$, where ${\mathcal{G}}_k$ denotes the gradient tensor along its k-th mode. This implies that the L structure between the gradient tensor and the original one remains consistent, and constraining the L structure of the gradient tensor can indirectly induce the low-rank property of the original structure. The smoothness property is often characterized by the low energy of $\mathcal{X}$’s gradient tensor. Since it was proven in [17] that the difference operation changes neither low-rankness nor boundness, the L and S priors of the gradient tensor can be captured by the specific regularizer we designed.

To more effectively characterize the global low-rank properties and local smooth features of tensors, we propose a novel regularization, the Tensor-Correlated Total Variation (TCTV), based on gradient tensors and the nonconvex Geman norm. Meanwhile, based on the TCTV regularizer, we develop TC-TCTV and TRPCA-TCTV models for completion and denoising tasks and solve them using the ADMM framework. Experiments validate the outstanding performance of the proposed models in both completion and denoising tasks. The specific contributions are as follows:

• We introduce a novel nonconvex regularizer termed the Tensor-Correlated Total Variation (TCTV), which is based on gradient tensors and the nonconvex Geman norm. The TCTV captures both the low-rankness and smoothness priors of the gradient tensor simultaneously, thereby alleviating the inconvenience of choosing a balancing parameter.
• Building upon the TCTV regularization, we propose two novel models, namely TC-TCTV and TRPCA-TCTV. To solve these models efficiently, we design Alternating Direction Method of Multipliers (ADMM) algorithms, which can obtain closed-form solutions for each variable. Additionally, we test the sensitivity of the parameters and the convergence of the proposed algorithms.
• Extensive experimental results demonstrate the robust tensor recovery capability of our model. As depicted in Figure 1, the image recovery performance remains impressive even with a missing rate as high as 99.5%.

The remainder of this paper is organized as follows. Section 2 introduces some notations and preliminaries under the high-order t-SVD framework. The tensor completion (TC-TCTV) model and tensor robust principal component analysis (TRPCA-TCTV) model are shown in Section 3. In Section 4, extensive results are presented. We draw some conclusions about our work in Section 5.

2. Preliminaries

2.1. Notations

In this paper, we use the following notations, and please refer to [18,19,20,21] for more details. Matrices and tensors are represented by the bold capital letter $\mathbf{X}\in {\mathbb{R}}^{n_1\times n_2}$ and bold calligraphic letter $\mathcal{X}\in {\mathbb{R}}^{n_1\times \dots \times n_d}$, respectively. Scalars and vectors are denoted by the lowercase letter x and bold lowercase letter $\mathbf{x}\in {\mathbb{R}}^n$. For a d-way tensor $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$, $\mathcal{X}(i_1,i_2,\dots ,i_d)$ denotes its $(i_1,i_2,\dots ,i_d)$-th entry, and $\mathcal{X}(:,:,i_3,\dots ,i_d)$ is its $(i_3,\dots ,i_d)$-th face slice, which can also be written as ${\mathcal{X}}^{(i_3,\dots ,i_d)}$ for brevity.

2.2. High-Order t-Singular-Value Decomposition Framework

Under the t-singular-value decomposition (t-SVD) framework, we introduce some definitions as follows for use.

Definition 1

(invertible transform P [21]). For $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$ with invertible transform P [21], its transform form is defined as follows:

$${\mathcal{X}}_{\mathsf{P}}:=\mathsf{P}\left(\mathcal{X}\right)=\mathcal{X}{\times }_3{\mathrm{U}}_{n_3}{\times }_4\dots {\times }_d{\mathrm{U}}_{n_d},$$

(2)

where ${\mathrm{U}}_{n_j}\in {\mathbb{R}}^{n_j\times n_j}(j=3,...,d)$ denotes the invertible transform matrix (like the discrete Fourier transform matrix), and ${\times }_j$ denotes the mode-j product ($\mathcal{C}=\mathcal{A}{\times }_j\mathcal{B}$ means $\mathcal{C}(\dots ,i_{j-1},:,i_{j+1},\dots )=\mathcal{B}·\mathcal{A}(\dots ,i_{j-1},:,i_{j+1},\dots )$). This satisfies the conditions that ${\mathsf{P}}^{-1}\left(\mathsf{P}\left(\mathcal{X}\right)\right)=\mathcal{X}$ and ${\mathsf{P}}^{-1}\left(\mathcal{X}\right):=\mathcal{X}{\times }_3{\mathrm{U}}_{n_3}^{-1}{\times }_4\dots {\times }_d{\mathrm{U}}_{n_d}^{-1}$.

Definition 2

(tensor product [21]). For $\mathcal{A}\in {\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$ and $\mathcal{B}\in {\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$, the tensor product is defined as follows:

$$\mathcal{A}{\ast }_{\mathsf{P}}\mathcal{B}={\mathsf{P}}^{-1}\left(\mathsf{P}\left(\mathcal{A}\right)\mathsf{\Delta }\mathsf{P}\left(\mathcal{B}\right)\right),$$

(3)

where Δ denotes the face-wise product ($\mathcal{C}=\mathcal{A}\mathsf{\Delta }\mathcal{B}$ means ${\mathrm{C}}^{(i_3,\dots ,i_d)}={\mathrm{A}}^{(i_3,\dots ,i_d)}{\mathrm{B}}^{(i_3,\dots ,i_d)}$ for each frontal slice).

Definition 3

(transpose [21]). The transpose of an order-d tensor $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times \dots \times n_d}$ is denoted as ${\mathcal{X}}^{\mathrm{T}}\in {\mathbb{R}}^{n_2\times n_1\times n_3\times \dots \times n_d}$ and satisfies ${\mathcal{X}}_{\mathsf{P}}^{\mathrm{T}}(:,:,i_3,\dots ,i_d)={\mathcal{X}}_{\mathsf{P}}{(:,:,i_3,\dots ,i_d)}^{\mathrm{T}}$ for each frontal slice.

Definition 4

(orthogonal tensor [21]). For an orthogonal tensor $\mathcal{X}\in {\mathbb{R}}^{n\times n\times n_3\times \dots \times n_d}$, it follows that ${\mathcal{X}}^{\mathrm{T}}{\ast }_{\mathsf{P}}\mathcal{X}=\mathcal{X}{\ast }_{\mathsf{P}}{\mathcal{X}}^{\mathrm{T}}={\mathcal{I}}_n$.

Definition 5

(f-diagonal tensor [21]). For an f-diagonal tensor $\mathcal{X}\in {\mathbb{R}}^{n\times n\times n_3\times \dots \times n_d}$, it follows that all frontal slices are diagonal.

Theorem 1

(t-SVD [21]). Any tensor $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times \dots \times n_d}$ can be factorized as follows by t-SVD:

$$\mathcal{X}=\mathcal{U}{\ast }_{\mathsf{P}}\mathcal{S}{\ast }_{\mathsf{P}}{\mathcal{V}}^{\mathrm{T}},$$

(4)

where $\mathcal{U}\in {\mathbb{R}}^{n_1\times n_1\times \dots \times n_d}$ and $\mathcal{V}\in {\mathbb{R}}^{n_2\times n_2\times \dots \times n_d}$ are both orthogonal and $\mathcal{S}\in {\mathbb{R}}^{n_1\times n_2\times \dots \times n_d}$ is f-diagonal.

Definition 6

(t-SVD rank [21]). For any tensor $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times \dots \times n_d}$ factorized as $\mathcal{X}=\mathcal{U}{\ast }_{\mathsf{P}}\mathcal{S}{\ast }_{\mathsf{P}}{\mathcal{V}}^{\mathrm{T}}$ by t-SVD, the t-SVD rank can be defined as follows:

$${\mathrm{rank}}_{\mathrm{t}-\mathrm{SVD}}\left(\mathcal{X}\right):=♯\{i:\mathcal{S}(i,i,:,\dots ,:)\ne \mathbf{0}\},$$

(5)

where ♯ is used to denote the cardinality of a set, which refers to the number of elements within the set.

Definition 7

(tensor Geman norm [15]). For any tensor $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times \dots \times n_d}$ with transform $\mathsf{P}$, the tensor Geman norm ($L_g$) can be defined as follows:

$${\parallel \mathcal{X}\parallel }_{\mathsf{P}}:={\displaystyle \frac{1}{\ell }}\sum _{i_3=1}^{n_3}\dots \sum _{i_d=1}^{n_d}{\parallel {\mathcal{X}}_{\mathsf{P}}(:,:,i_3,\dots ,i_d)\parallel }_g,$$

(6)

where ${\parallel ·\parallel }_g$ denotes the Geman norm of a matrix and can be defined as follows:

$${∥{\mathcal{X}}_{\mathsf{P}}(:,:,i_3,\dots ,i_d)∥}_g:={∥\mathbf{X}∥}_g=\lambda \sum _{t=1}^{min\{p,q\}}{\displaystyle \frac{{\sigma }_t\left(\mathbf{X}\right)}{{\sigma }_t\left(\mathbf{X}\right)+g}},\forall \mathbf{X}\in {\mathbb{R}}^{p\times q},$$

(7)

where ${\sigma }_t\left(\mathbf{X}\right)$ denotes the t-th biggest singular value of X, and g is a positive parameter. λ is an adaptive parameter that is set as a large initial value and gradually decreases to the target value.

Theorem 2

(t-SVT [21]). For any tensor $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times \dots \times n_d}$ factorized by t-SVD, its tensor singular value thresholding (t-SVT) is defined as follows:

$${\mathit{\text{t-SVT}}}_{\tau }\left(\mathcal{X}\right):=\mathcal{U}{\ast }_{\mathsf{P}}{\mathcal{S}}_{\tau }{\ast }_{\mathsf{P}}{\mathcal{V}}^{\mathrm{T}},$$

(8)

where ${\mathcal{S}}_{\tau }={\mathsf{P}}^{-1}\left({({\mathcal{S}}_{\mathsf{P}}-\tau )}_{+}\right)$, $t_{+}=max(0,t)$, and ${\mathit{\text{t-SVT}}}_{\tau }\left(\mathcal{X}\right)$ obeys the following equation:

$${\mathit{\text{t-SVT}}}_{\tau }\left(\mathcal{K}\right)=arg\underset{\mathcal{K}}{min}{\tau \parallel \mathcal{K}\parallel }_{\mathsf{P}}+{\displaystyle \frac{1}{2}}{\parallel \mathcal{K}-\mathcal{X}\parallel }_{\mathrm{F}}^2.$$

(9)

Definition 8

(gradient tensor). The gradient tensor of an order-d tensor $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times \dots \times n_d}$ along the k-th mode is defined as follows:

$${\mathcal{G}}_k:={\nabla }_k\left(\mathcal{X}\right)=\mathcal{X}{\times }_k{\mathrm{D}}_{n_k},(k=1,2,\dots ,d),$$

(10)

where ${\mathrm{D}}_{n_k}$ is a row-circulant matrix composed by $\begin{array}{c}\hfill (-1,1,0,\dots ,0)\end{array}$.

3. Tensor Recovery with Nonconvex TCTV Norm

3.1. TCTV Norm

We propose a novel nonconvex norm, termed the TCTV, which integrates the tensor nuclear norm based on differences with the nonconvex Geman norm. This design leverages the complementary strengths of both components: the tensor nuclear norm ensures structural integrity and low-rank representation, while the Geman norm enhances accuracy in rank approximation and reduces estimation bias. By combining these advantages, the TCTV achieves superior performance in preserving fine-grained details, making it highly effective for challenging tensor recovery tasks. The definition of the TCTV is presented as follows:

For any tensor $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times \dots \times n_d}$, we define the TCTV norm based on the t-CTV norm [17]. The main change required to convert between the two norms is to replace the convex nuclear norm with the nonconvex Geman norm.

$$\begin{array}{c}\hfill {\parallel \mathcal{X}\parallel }_{\mathrm{TCTV}}:={\displaystyle \frac{1}{m}}\sum _{k\in \mathsf{\Gamma }}{\parallel {\mathcal{G}}_k\parallel }_{\mathsf{P}},\end{array}$$

(11)

where ${\mathcal{G}}_k$ represents correlated tensors of $\mathcal{X}$, $\mathsf{\Gamma }$ denotes a prior set composed of directions along which $\mathcal{X}$ equips low-rankness and smoothness priors, and  m denotes the cardinality of $\mathsf{\Gamma }$. Through comparative analysis in ablation experiments, we observed that the use of the Geman norm achieves favorable performance. While other nonconvex norms also demonstrated promising results, this study focuses on the Geman function as a representative case for experimental validation. The generalization of nonconvex TCTV norms will be further explored in future work to provide a more comprehensive framework.

3.2. Models

For the TC model, we suppose that $\mathcal{X}\in {\mathbb{R}}^{n_1\times n_2\times n_3\times \dots \times n_d}$ is the underlying unknown tensor with the L and S priors. The TC-TCTV model is established as follows, using the TCTV norm to characterize the joint L and S priors of $\mathcal{X}$:

$$\underset{\mathcal{X}}{min}{\parallel \mathcal{X}\parallel }_{\mathrm{TCTV}}\mathrm{s}.\mathrm{t}.{\mathcal{P}}_{\mathsf{\Omega }}\left(\mathcal{X}\right)={\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right),$$

(12)

where $\mathcal{P}(·)$ denotes the projection operator and $\mathsf{\Omega }$ denotes the index set of observed entries and obey random Bernoulli sampling ($\mathsf{\Omega }\sim \mathrm{Ber}\left(p\right)$). Then, we reformulate the TC-TCTV model as follows:

$$\begin{array}{c}\underset{\mathcal{X},{\mathcal{G}}_k,\mathcal{K}}{min}{\displaystyle \frac{1}{m}}\sum _{k\in \mathsf{\Gamma }}{\parallel {\mathcal{G}}_k\parallel }_{\mathsf{P}}+{\delta }_{\mathsf{\Omega }}\left(\mathcal{K}\right)\hfill \\ \mathrm{s}.\mathrm{t}.{\mathcal{G}}_k={\nabla }_k\left(\mathcal{X}\right),\mathcal{X}+\mathcal{K}={\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right),\hfill \end{array}$$

(13)

where ${\mathcal{X}}_0$ denotes the observed tensor, ${\nabla }_k(·)$ denotes the difference operator along a certain mode, $\mathcal{K}$ is restricted in ${\mathsf{\Omega }}^{\perp }$ to compensate for the missing entries of $\mathcal{X}$, and ${\delta }_{\mathsf{\Omega }}\left(\mathcal{K}\right)$ is an indicative function that can be defined as follows:

$${\delta }_{\mathsf{\Omega }}\left(\mathcal{K}\right)=\left\{\begin{array}{cc}0,\hfill & {\mathcal{P}}_{\mathsf{\Omega }}\left(\mathcal{K}\right)=\mathcal{O},\hfill \\ +\infty ,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(14)

For the TRPCA model, we assume that the underlying tensor $\mathcal{X}$ is corrupted by noise $\mathcal{E}$. Its corruption obeys $\mathcal{M}=\mathcal{X}+\mathcal{E}$, and $\mathcal{M}$ is the observation. Thus, the TRPCA-TCTV model is established as follows:

$$\begin{array}{c}\underset{\mathcal{X},{\mathcal{G}}_k,\mathcal{E}}{min}{\displaystyle \frac{1}{m}}\sum _{k\in \mathsf{\Gamma }}\parallel {\mathcal{G}}_k{\parallel }_{\mathsf{P}}+\beta {\parallel \mathcal{E}\parallel }_1\hfill \\ \mathrm{s}.\mathrm{t}.{\mathcal{G}}_k={\nabla }_k\left(\mathcal{X}\right),\mathcal{X}+\mathcal{E}=\mathcal{M}.\hfill \end{array}$$

(15)

3.3. Optimization Algorithms

3.3.1. Optimization to TC-TCTV

The augmented Lagrange function of (13) is

$$\begin{array}{cc}\hfill \mathcal{L}(\mathcal{X},\{{\mathcal{G}}_k,k\in \mathsf{\Gamma }\},\mathcal{K},\{{\mathsf{\Lambda }}_k,k\in \mathsf{\Gamma }\},\mathsf{{\rm Y}})& =\sum _{k\in \mathsf{\Gamma }}\left({\displaystyle \frac{1}{m}}\parallel {\mathcal{G}}_k{\parallel }_{\mathsf{P}}+\langle {\mathsf{\Lambda }}_k,{\nabla }_k\left(\mathcal{X}\right)-{\mathcal{G}}_k\rangle +{\displaystyle \frac{{\mu }_t}{2}}{\parallel {\nabla }_k\left(\mathcal{X}\right)-{\mathcal{G}}_k\parallel }_{\mathrm{F}}^2\right)\hfill \\ \hfill & +{\delta }_{\mathsf{\Omega }}\left(\mathcal{K}\right)+\langle \mathsf{{\rm Y}},{\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)-\mathcal{X}-\mathcal{K}\rangle +{\displaystyle \frac{{\mu }_t}{2}}{\parallel {\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)-\mathcal{X}-\mathcal{K}\parallel }_{\mathrm{F}}^2,\hfill \end{array}$$

(16)

where ${\mathsf{\Lambda }}_k$ and $\mathsf{{\rm Y}}$ are Lagrange multipliers, and ${\mu }_t$ is a penalty parameter. Then, (16) can be rewritten as follows:

$$\mathcal{L}=\sum _{k\in \mathsf{\Gamma }}\left({\displaystyle \frac{1}{m}}\parallel {\mathcal{G}}_k{\parallel }_{\mathsf{P}}+{\displaystyle \frac{{\mu }_t}{2}}{\parallel {\nabla }_k\left(\mathcal{X}\right)-{\mathcal{G}}_k+{\mathsf{\Lambda }}_k/{\mu }_t\parallel }_{\mathrm{F}}^2\right)+{\delta }_{\mathsf{\Omega }}\left(\mathcal{K}\right)+{\displaystyle \frac{{\mu }_t}{2}}{\parallel {\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)-\mathcal{X}-\mathcal{K}+\mathsf{{\rm Y}}/{\mu }_t\parallel }_{\mathrm{F}}^2.$$

(17)

Step 1. Update ${\mathcal{X}}^{t+1}$ from $\mathcal{L}$:

The following system is obtained by taking the derivative of (17) with respect to $\mathcal{X}$:

$$\begin{array}{c}\hfill \left(\mathcal{I}+\sum {\nabla }_k^{\mathrm{T}}{\nabla }_k\right)\left(\mathcal{X}\right)={\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)-{\mathcal{K}}^t+{\mathsf{{\rm Y}}}^t/{\mu }_t+\sum {\nabla }_k^{\mathrm{T}}({\mathcal{G}}_k^t-{\mathsf{\Lambda }}_k^t/{\mu }_t).\end{array}$$

(18)

According to [22], we adopt a multidimensional FFT to diagonalize the difference tensors, ${\mathcal{D}}_k$, corresponding to ${\nabla }_k(·)$. Then, we can obtain the optimal solution of (18) as follows:

$$\begin{array}{cc}\hfill {\mathcal{X}}^{t+1}& ={\mathcal{F}}^{-1}\left({\displaystyle \frac{\mathcal{F}({\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)-{\mathcal{K}}^t+{\mathsf{{\rm Y}}}^t/{\mu }_t)+\mathcal{H}}{\mathbf{1}+{\sum }_{k\in \mathsf{\Gamma }}\mathcal{F}{\left({\mathcal{D}}_k\right)}^{\ast }\odot \mathcal{F}\left({\mathcal{D}}_k\right)}}\right),\hfill \end{array}$$

(19)

where $\mathbf{1}$ is a tensor with all elements equal to 1, and $\mathcal{H}={\sum }_{k\in \mathsf{\Gamma }}\mathcal{F}{\left({\mathcal{D}}_k\right)}^{\ast }\odot \mathcal{F}({\mathcal{G}}_k^t-{\mathsf{\Lambda }}_k^t/{\mu }_t)$.

Step 2. Update ${\mathcal{G}}_k^{t+1}$ from $\mathcal{L}$: For each $k\in \mathsf{\Gamma }$, we extract the part related to ${\mathcal{G}}_k$:

$$\begin{array}{cc}\hfill {\mathcal{G}}_k^{t+1}& =arg\underset{{\mathcal{G}}_k}{min}\parallel {\mathcal{G}}_k{\parallel }_{\mathsf{P}}+{\displaystyle \frac{m{\mu }_t}{2}}{\parallel {\nabla }_k\left({\mathcal{X}}^{t+1}\right)-{\mathcal{G}}_k+{\displaystyle \frac{{\mathsf{\Lambda }}_k^t}{{\mu }_t}}\parallel }_F^2.\hfill \end{array}$$

(20)

Let ${\mathcal{P}}_k^{t+1}={\nabla }_k\left({\mathcal{X}}^{t+1}\right)+{\displaystyle \frac{{\mathsf{\Lambda }}_k^t}{{\mu }_t}}$, and we consider ${\mathcal{G}}_k$ and ${\mathcal{P}}_k$ as the compositions of all face slices.  Equation (20) can then be rewritten as follows:

$$\begin{array}{c}\hfill \{{\mathbf{G}}_k^{t+1}\left(1\right),\dots ,{\mathbf{G}}_k^{t+1}\left(M\right)\}=arg\underset{\{{\mathbf{G}}_k\left(1\right),\dots ,{\mathbf{G}}_k\left(M\right)\}}{min}\sum _{j=1}^M\left(\parallel {\mathbf{G}}_k{\left(j\right)\parallel }_g+{\displaystyle \frac{m{\mu }_t}{2}}{\parallel {\mathbf{G}}_k\left(j\right)-{\mathbf{P}}_k^{t+1}\left(j\right)\parallel }_F^2\right),\end{array}$$

(21)

where $M=n_3\dots n_d$ denotes the number of face slices of ${\mathcal{G}}_k$. For each $j\in M$, we extract the subproblem as follows:

$$\begin{array}{c}\hfill {\mathbf{G}}_k^{t+1}=arg\underset{{\mathbf{G}}_k}{min}\parallel {\mathbf{G}}_k{\parallel }_g+{\displaystyle \frac{m{\mu }_t}{2}}{\parallel {\mathbf{G}}_k-{\mathbf{P}}_k^{t+1}\parallel }_F^2.\end{array}$$

(22)

Let ${\sigma }_1^t\ge {\sigma }_2^t\ge \dots \ge {\sigma }_{q_n}^t$ denote the singular values of ${\mathbf{G}}_k^t\in {\mathbb{R}}^{r_n\times s_n}$ with $q_n=min\{r_n,s_n\}$, $\nabla \varphi \left({\sigma }^t\right)$ denote the gradient of $\varphi \left(x\right)={\displaystyle \frac{\lambda x}{x+\gamma }}$ at ${\sigma }^t$, and  $f\left({\mathbf{G}}_k\right)={\displaystyle \frac{1}{2}}{∥{\mathbf{G}}_k-{\mathbf{P}}_k^{t+1}∥}_F^2$. By setting the Lipschitz constant to 1, it is easy to prove that the gradient of $f\left({\mathbf{G}}_k\right)$ is Lipschitz-continuous. Following [23], we consider the nonascending order of singular values and the antimonotone property of the gradient of the nonconvex Geman function to obtain two inequalities:

$$\begin{array}{cc}\hfill & 0\le \nabla \varphi \left({\sigma }_1^t\right)\le \nabla \varphi \left({\sigma }_2^t\right)\le \dots \le \nabla \varphi \left({\sigma }_{q_n}^t\right),\hfill \\ \hfill & \varphi \left({\sigma }_i\left({\mathbf{G}}_k\right)\right)\le \varphi \left({\sigma }_i^t\right)+\nabla \varphi \left({\sigma }_i^t\right)\left({\sigma }_i\left({\mathbf{G}}_k\right)-{\sigma }_i^t\right),\hfill \end{array}$$

(23)

where $i=1,2,\dots ,q_n$. Then, the subproblem with respect to ${\mathbf{G}}_k$ can be rewritten as follows through relaxation:

$$\begin{array}{c}arg\underset{{\mathbf{G}}_k}{min}{\displaystyle \frac{1}{m{\mu }_t}}\sum _{n=1}^{q_n}\varphi \left({\sigma }_n^t\right)+\nabla \varphi \left({\sigma }_n^t\right)\left({\sigma }_n\left({\mathbf{G}}_k\right)-{\sigma }_n^t\right)+f\left({\mathbf{G}}_k\right)\hfill \\ =arg\underset{{\mathbf{G}}_k}{min}{\displaystyle \frac{1}{m{\mu }_t}}\sum _{n=1}^{q_n}\nabla \varphi \left({\sigma }_n^t\right){\sigma }_n\left({\mathbf{G}}_k\right)+{\displaystyle \frac{1}{2}}{∥{\mathbf{G}}_k-{\mathbf{P}}_k^{t+1}∥}_{\mathrm{F}}^2.\hfill \end{array}$$

(24)

Based on [23,24], we solve (24) through generalized-weight singular value thresholding (WSVT) [25].

Lemma 1.

For any ${\displaystyle \frac{1}{m{\mu }_t}}>0$, the auxiliary variable ${\mathcal{P}}_k^{t+1}={\nabla }_k\left({\mathcal{X}}^{t+1}\right)+{\displaystyle \frac{{\mathsf{\Lambda }}_k^t}{{\mu }_t}}$ and $0\le \nabla \varphi \left({\sigma }_1^t\right)\le \nabla \varphi \left({\sigma }_2^t\right)\le \dots \le \nabla \varphi \left({\sigma }_{q_n}^t\right)$. Therefore, the global optimal solution, ${\mathbf{G}}_k^{\ast }$, to (24) can be obtained as follows:

$${\mathbf{G}}_k^{t+1}=\mathit{WSVT}\left({\mathbf{P}}_k^{t+1},{\displaystyle \frac{\nabla \varphi }{m{\mu }_t}}\right)=\mathbf{U}{\mathbf{S}}_{{\displaystyle \frac{\nabla \varphi }{m{\mu }_t}}}\left(\mathsf{\Sigma }\right){\mathbf{V}}^T.$$

(25)

$${\mathbf{S}}_{{\displaystyle \frac{\nabla \varphi }{m{\mu }_t}}}\left(\mathsf{\Sigma }\right)=\mathrm{Diag}\left\{max\left({\mathsf{\Sigma }}_{ii}-{\displaystyle \frac{\nabla \varphi \left({\sigma }_i^k\right)}{m{\mu }_t}},0\right)\right\},(i=1,2,\dots ,q_n).$$

(26)

Then, we can easily obtain ${\mathcal{G}}_k^{t+1}$ by concatenating $\{{\mathbf{G}}_k^{t+1}\left(1\right),\dots ,{\mathbf{G}}_k^{t+1}\left(M\right)\}$ along a certain mode. The closed-form solution of (20) is shown as follows:

$${\mathcal{G}}_k^{t+1}=\mathrm{Concat}\{{\mathbf{G}}_k^{t+1}\left(1\right),\dots ,{\mathbf{G}}_k^{t+1}\left(M\right)\}.$$

(27)

Step 3. Update ${\mathcal{K}}^{t+1}$ from $\mathcal{L}$:

$${\mathcal{K}}^{t+1}={\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)-{\mathcal{X}}^{t+1}+{\mathsf{{\rm Y}}}^t/{\mu }_t,{\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{K}}^{t+1}\right)=\mathcal{O}.$$

(28)

Step 4. Update ${\mathsf{\Lambda }}_k^{t+1}$ and ${\mathsf{{\rm Y}}}^{t+1}$ from $\mathcal{L}$: Based on the ADMM algorithm, ${\mathsf{\Lambda }}_k^{t+1}$ and ${\mathsf{{\rm Y}}}^{t+1}$ have the following update equations:

$${\mathsf{\Lambda }}_k^{t+1}={\mathsf{\Lambda }}_k^t+{\mu }_t({\nabla }_k\left({\mathcal{X}}^{t+1}\right)-{\mathcal{G}}_k^{t+1}),$$

(29)

$${\mathsf{{\rm Y}}}^{t+1}={\mathsf{{\rm Y}}}^t+{\mu }_t({\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)-{\mathcal{X}}^{t+1}-{\mathcal{K}}^{t+1}).$$

(30)

Finally, the penalty parameter, ${\mu }_{t+1}$, is updated by ${\mu }_{t+1}=\rho {\mu }_t$, where the constant $\rho >1$. The entire optimization scheme for the TC-TCTV is described in Algorithm 1.

Algorithm 1: ADMM for TC-TCTV.

Input: observation ${\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)$, priori set $\mathsf{\Gamma }$ and transform $\mathsf{P}$.

Initialize ${\mathcal{G}}_k^0={\nabla }_k\left({\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)\right),{\mathcal{K}}^0={\mathsf{{\rm Y}}}^0={\mathsf{\Lambda }}_k^0=\mathcal{O}.$   Repeat   Update ${\mathcal{X}}^{t+1}$ by (19);   Update ${\mathcal{G}}_k^{t+1}$ by (27) for each $k\in \mathsf{\Gamma };$   Update ${\mathcal{K}}^{t+1}$ by (28);   Update ${\mathsf{\Lambda }}_k^{t+1}$ and ${\mathsf{{\rm Y}}}^{t+1}$ by (29), (30);   Let ${\mu }_{t+1}=\rho {\mu }_t;t=t+1.$   Until converged

Output: The optimal solution $\widehat{\mathcal{X}}={\mathcal{X}}^{t+1}.$

3.3.2. Optimization to TRPCA-TCTV

The processes for optimizing to the TRPCA-TCTV and TC-TCTV are quite similar, except that some variables are replaced in the latter. Specifically, $\mathcal{K}$ is replaced by sparse additive noise, $\mathcal{E}$, and the observation ${\mathcal{P}}_{\mathsf{\Omega }}\left({\mathcal{X}}_0\right)$ is modified to $\mathcal{M}$. Thus, the ADMM optimization process for TRPCA-TCTV can be deduced as follows:

$${\mathcal{X}}^{t+1}={\mathcal{F}}^{-1}\left({\displaystyle \frac{\mathcal{F}(\mathcal{M}-{\mathcal{E}}^t+{\mathsf{{\rm Y}}}^t/{\mu }_t)+\mathcal{H}}{1+{\sum }_{k\in \mathsf{\Gamma }}\mathcal{F}{\left({\mathcal{D}}_k\right)}^{\ast }\odot \mathcal{F}\left({\mathcal{D}}_k\right)}}\right);$$

(31)

$${\mathcal{G}}_k^{t+1}=\mathrm{Concat}\{{\mathbf{G}}_k^{t+1}\left(1\right),\dots ,{\mathbf{G}}_k^{t+1}\left(M\right)\};$$

(32)

$${\mathcal{E}}^{t+1}={\mathcal{S}}_{1/m{\mu }_t}(\mathcal{M}-{\mathcal{X}}^{t+1}+{\mathsf{{\rm Y}}}^t/{\mu }_t);$$

(33)

$${\mathsf{\Lambda }}_k^{t+1}={\mathsf{\Lambda }}_k^t+{\mu }_t({\nabla }_k\left({\mathcal{X}}^{t+1}\right)-{\mathcal{G}}_k^{t+1});$$

(34)

$${\mathsf{{\rm Y}}}^{t+1}={\mathsf{{\rm Y}}}^t+{\mu }_t(\mathcal{M}-{\mathcal{X}}^{t+1}-{\mathcal{E}}^{t+1}).$$

(35)

Here, $\mathcal{S}(·)$ denotes the soft-thresholding operator. The whole optimization scheme for the TRPCA-TCTV is summarized in Algorithm 2. It begins with initializing several tensors to zero, including ${\mathcal{G}}_k^0$, ${\mathcal{E}}^0$, ${\mathsf{{\rm Y}}}^0$, and ${\mathsf{\Lambda }}_k^0$. Then, it enters a repeat loop where it iteratively updates these tensors and a parameter $\mu$ using specific equations. The updates involve refining the estimates of $\mathcal{X}$, ${\mathcal{G}}_k$, $\mathcal{E}$, ${\mathsf{\Lambda }}_k$, and $\mathsf{{\rm Y}}$. The parameter $\mu$ is increased by a factor of $\rho$ after each iteration, and the process continues until convergence is achieved. The final output is the optimal solution $\widehat{\mathcal{X}}$, representing the refined data estimate after the iterative process has converged.

Algorithm 2: ADMM for TRPCA-TCTV.

Input: observation $\mathcal{M}$, priori set $\mathsf{\Gamma }$ and transform $\mathsf{P}$.

Initialize ${\mathcal{G}}_k^0={\mathcal{E}}^0={\mathsf{{\rm Y}}}^0={\mathsf{\Lambda }}_k^0=\mathcal{O}.$   Repeat   Update ${\mathcal{X}}^{t+1}$ by (31);   Update ${\mathcal{G}}_k^{t+1}$ by (32) for each $k\in \mathsf{\Gamma };$   Update ${\mathcal{E}}^{t+1}$ by (33);   Update ${\mathsf{\Lambda }}_k^{t+1}$ and ${\mathsf{{\rm Y}}}^{t+1}$ by (34), (35);   Let ${\mu }_{t+1}=\rho {\mu }_t;t=t+1.$   Until converged

Output: The optimal solution $\widehat{\mathcal{X}}={\mathcal{X}}^{t+1}.$

3.4. Computational Complexity Analysis

For Algorithm 1, we mainly consider the cost of calculating ${\mathcal{X}}^{t+1}$, ${\mathcal{G}}_k^{t+1}$, ${\mathcal{K}}^{t+1}$, ${\mathsf{\Lambda }}_k^{t+1}$, and ${\mathsf{{\rm Y}}}^{t+1}$ in each iteration. The time complexity of computing ${\mathcal{X}}^{t+1}$ is $\begin{array}{c}\hfill O(n_1\dots n_{d}log(n_1\dots n_d))\end{array}$. The cost of calculating ${\mathcal{G}}_k^{t+1}$ is $\begin{array}{c}\hfill O(n_1\dots n_d{r}_n^2)\end{array}$. Calculating ${\mathcal{K}}^{t+1}$, ${\mathsf{\Lambda }}_k^{t+1}$, and ${\mathsf{{\rm Y}}}^{t+1}$ have the same complexity $\begin{array}{c}\hfill O(n_1\dots n_d)\end{array}$. Therefore, the per-iteration complexity of Algorithm 1 is $\begin{array}{c}\hfill O(n_1\dots n_{d}log(n_1\dots n_d)+n_1\dots n_d{r}_n^2+3n_1\dots n_d)\end{array}$.

For Algorithm 2, it is easy to see that the only difference exists in calculating ${\mathcal{E}}^{t+1}$, which involves the soft-thresholding operator. The cost of computing ${\mathcal{E}}^{t+1}$ is $\begin{array}{c}\hfill O(n_1\dots n_d)\end{array}$, and the total complexity is the same as in Algorithm 1.

3.5. Convergence Analysis

Since the constraints in the proposed models are separable, they can be transformed into a standard two-block ADMM by using the method given in [26]. The convergence of Algorithms 1 and 2 can be proven by using the general conclusion given in [26]. To provide an intuitive visualization of the error variation after each iteration, curves of the relative error against the number of iterations are plotted in Figure 2. It is easy to see that the gap between ${\mathcal{X}}^{t+1}$ and ${\mathcal{X}}_0$ deacreases with increasing iterations. After 50 iterations, the errors gradually decrease and become closer to zero, which validates the convergence of the algorithm intuitively.

3.6. Sensitivity Analysis

Considering that $\rho$ and ${\mu }_0$ are two crucial parameters in Algorithms 1 and 2, we conducted a sensitivity analysis by adjusting the value of each parameter. The effects of $\rho$ and ${\mu }_0$ on the PSNR at various sampling rates are shown in Figure 3. We can observe that a convincing performance in terms of the PSNR is achieved when $\rho$ = 1.1 and ${\mu }_0$ = ${10}^{-5}$.

4. Experimental Results

In this section, we conduct numerous experiments on public datasets to evaluate the performance of the proposed models. We compare our models with different methods (

Table 1. TC and TRPCA methods for comparison.

Types	Methods
TC−L prior	SNN [5]	KBR [27]	IRTNN [28]	TNN [21]
TC−L+S priors	MF-TV [29]	SPC-TV [30]	TNN-TV [31]	t-CTV [17]
TRPCA−L+S priors	LRTV [32]	LRTDTV [30]	TLRHTV [33]	t-CTV

) considering L or S priors of tensors. All experiments were performed on MATLAB (R2021a); the CPU of the computer was AMD Ryzen 7 5800H with Radeon Graphics (16 CPUs), and the memory was 16 GB.

For a quantitative comparison, three picture quality indices (PQIs) were employed, including the peak signal-to-noise ratio (PSNR [34]), structural similarity (SSIM [35]), and feature similarity (FSIM [36]). The larger the PSNR, SSIM, and FSIM, the better the performance of the recovery model. All the parameters used in these models were used as given in the reference papers or adjusted optimally.

Parameter setting: First, we give brief explanations of important variables in the models. $\mathcal{X}$, ${\mathcal{G}}_k$, and $\mathcal{E}$ denote the recovered tensor, gradient tensor, and noise, respectively. $\mathcal{K}$ is restricted in ${\mathsf{\Omega }}^{\perp }$ to compensate for the missing entries of $\mathcal{X}$. ${\mathsf{\Lambda }}_k$ and $\mathsf{{\rm Y}}$ are Lagrange multipliers. For all competing approaches, we either adopted the parameters suggested in the released code or adjusted some to obtain better results. We set $\rho$ = 1.1, ${\mu }_0$ = ${10}^{-5}$, max(${\mu }_t$) = ${10}^{10}$, and ${\mu }_{t+1}$ = min($\rho \ast {\mu }_t$, max(${\mu }_t$)) in Algorithms 1 and 2. In (7), $g=100$ and ${\lambda }_0$ are set to a large value first, and then $\lambda$ decreases by $\lambda ={\eta }^k{\lambda }_0$ with $\eta =0.1$ until reaching a target value of 1. Since the nonconvex norm plays an important role in two models, we conducted ablation experiments and present the results in

Table 2. Recovery performances of different nonconvex norms.

SR	Norms	Laplace	Geman	ETP	Capped ${\mathit{L}}_1$	MCP	Logarithm	SCAD	${\mathit{L}}_{\mathit{p}}$
70%	PSNR	38.847	38.854	38.853	38.849	38.852	38.853	38.846	38.849

. In subsequent experiments, we will use only the Geman norm as an example.

4.1. Color Image Completion

In this experiment, we randomly choose 20 color images (sized $\begin{array}{c}\hfill n_1\times n_2\times 3\end{array}$) from the BSD and USC-SIPI databases, and all original images are shown in Figure 4. The images are corrupted with different sampling rates (from 5% to 30%). The completion results of quantitative metrics as averages are presented in

Table 3. PQI comparison of color image completion under various sampling rates (SRs). Results are averaged over 20 randomly selected images, and the best result is highlighted in bold.

Methods	PQIs	OBS	SNN	KBR	IRTNN	TNN	MF-TV	SPC-TV	TNN-TV	t-CTV	TCTV
5% SR	PSNR	6.404	18.168	16.625	14.672	17.734	8.612	18.068	20.459	21.763	23.912
	SSIM	0.017	0.315	0.240	0.148	0.273	0.075	0.317	0.450	0.566	0.647
	FSIM	0.567	0.668	0.666	0.668	0.678	0.642	0.652	0.607	0.825	0.855
10% SR	PSNR	6.640	20.028	18.457	18.201	19.946	10.879	19.970	22.366	25.086	26.319
	SSIM	0.030	0.429	0.337	0.289	0.411	0.129	0.422	0.569	0.726	0.765
	FSIM	0.621	0.731	0.734	0.736	0.754	0.657	0.727	0.725	0.891	0.906
15% SR	PSNR	6.887	21.548	20.312	20.368	21.630	13.442	21.576	23.877	27.213	28.290
	SSIM	0.042	0.535	0.447	0.428	0.527	0.228	0.527	0.665	0.810	0.835
	FSIM	0.643	0.786	0.794	0.798	0.811	0.706	0.788	0.804	0.925	0.935
20% SR	PSNR	7.150	22.927	22.176	22.203	23.089	16.007	23.003	25.125	29.003	29.981
	SSIM	0.054	0.625	0.558	0.548	0.624	0.336	0.618	0.738	0.863	0.883
	FSIM	0.653	0.831	0.844	0.844	0.852	0.749	0.836	0.857	0.946	0.954
30% SR	PSNR	7.728	25.470	26.869	25.943	25.865	23.969	25.450	27.292	32.242	33.271
	SSIM	0.079	0.764	0.778	0.740	0.769	0.677	0.754	0.838	0.928	0.939
	FSIM	0.663	0.896	0.924	0.913	0.910	0.887	0.899	0.919	0.971	0.976
	TIME	0	9.313	78.002	204.069	8.652	734.033	84.287	39.760	75.598	52.739

. The results show that the proposed TC-TCTV model achieved higher PSNR, SSIM, and FSIM values than the other methods. We also drew a bar chart of the PSNR values of the images recovered by the different methods (as shown in Figure 5), which presents an intuitive comparison. Meanwhile, our method strikes a balance between the quality of the recovery results and computational efficiency. As indicated in Table 3, the runtime of our algorithm is entirely acceptable. Due to space limitations, a partial selection of images and sampling conditions was made for an intuitive visual comparison, as shown in Figure 6. We magnified specific local regions in the recovery image and then computed the differences between the ground truth image and the recovered images from competing methods. This approach provides a straightforward visual assessment of the restoration performance achieved by the different methods.

We further tested all the completion models on extreme conditions, when sampling rates were set to 3%, 1%, and 0.5%. Due to space limitations, the recovery results of some typical cases are given in

Table 4. Averaged PQIs of completion results under 3%, 1%, and 0.5% sampling rates.

Methods	PQIs	OBS	SNN	KBR	IRTNN	TNN	MF-TV	SPC-TV	TNN-TV	t-CTV	TCTV
0.5% SR	PSNR	6.858	9.371	16.095	8.796	7.493	7.300	7.563	9.780	18.403	19.466
	SSIM	0.501	0.571	0.624	0.528	0.528	0.520	0.542	0.664	0.708	0.746
	FSIM	0.672	0.763	0.731	0.752	0.725	0.748	0.726	0.694	0.801	0.795
1% SR	PSNR	7.240	13.181	16.196	10.563	9.092	7.971	14.002	15.105	19.057	21.422
	SSIM	0.668	0.735	0.740	0.688	0.699	0.681	0.766	0.810	0.806	0.847
	FSIM	0.742	0.804	0.781	0.792	0.787	0.797	0.809	0.775	0.852	0.853
3% SR	PSNR	5.616	18.014	17.060	12.631	16.300	7.195	17.680	20.747	21.412	23.957
	SSIM	0.671	0.776	0.747	0.700	0.744	0.683	0.780	0.838	0.849	0.898
	FSIM	0.763	0.819	0.801	0.796	0.808	0.794	0.814	0.821	0.903	0.929

and Figure 1 and Figure 7. In comparison, our method shows strong restoration capabilities, while other methods almost fail.

4.2. Denoising Task

In this part, we applied the TRPCA-TCTV model for the denoising task to remove noise from the observations. Due to space limitations, we only present a few examples. Table 1 presents seven relevant methods. Likewise, we chose the parameters suggested in the release codes or tuned them finely to obtain better results.

Specifically, we chose salt and pepper noise (N) at different levels. The datasets comprise color images and hyperspectral images (hsi_PaC, PaviaU, Salinas). From the results given in

Table 5. Denoising performance of competing methods under various levels of salt and pepper noise.

Methods	PQIs	OBS	SNN	KBR	TNN	LRTV	LRTDTV	TLR-HTV	t-CTV	TCTV
0.2 N	PSNR	11.287	42.253	36.509	45.598	39.374	38.762	35.593	46.952	49.949
	SSIM	0.077	0.848	0.947	0.987	0.956	0.941	0.926	0.989	0.990
	FSIM	0.475	0.940	0.959	0.989	0.965	0.958	0.933	0.992	0.993
0.4 N	PSNR	8.285	23.368	35.363	28.999	36.682	37.751	33.267	43.954	45.765
	SSIM	0.029	0.452	0.931	0.584	0.928	0.934	0.886	0.984	0.983
	FSIM	0.328	0.623	0.947	0.772	0.939	0.953	0.900	0.989	0.989

and Figure 8, it can be seen that our method achieves the best results when noise levels are at 20% and 40%.

5. Conclusions and Future Work

In this paper, we devised the TCTV as a regularizer to capture the low-rankness and smoothness priors of the gradient tensor. Specifically, the nuclear norm was replaced by the nonconvex Geman norm, which showed better performance in tensor rank estimation. Based on this, we constructed the TC-TCTV and TRPCA-TCTV models, which showed superior performance. Even under extreme conditions like when 99.5% of the pixel values are missing, the TC-TCTV model can still achieve impressive recovery results, while the other methods all fail.

There are still various interesting studies to carry out. We plan to develop general nonconvex TC and TRPCA models by using different nonconvex norms. It would also be meaningful to test more recovery tasks like super-resolution, flash–no flash reconstruction, and motion deblurring. We also plan to incorporate deep priors into our future work to build a more generalized model to solve various recovery tasks such as super-resolution and deblurring.