Tensors are high-dimensional arrays that have many applications in science and engineering, including in image, video and signal processing, computer vision, and network analysis [11,12,16,17,18,19,20,26]. A new t-product based on third-order tensors proposed by Kilmer et al [1,2]. When using high-dimensional data, t-product shows a greater potential value than matricization, see [2,6,11,12,21,22,24,25,27]. The t-product has been found to have special value in many application fields, including image deblurring problems [1,6,11,12], image and video compression [26], facial recognition problems [2], etc.

In this paper, we consider the solution of large minimization problems of the form The Frobenius norm of singular tube of $\mathcal{A}$ rapidly attenuates to zero with the increase of the index number. In particular, $\mathcal{A}$ has ill-determined tubal rank. Many of its singular tubes are nonvanishing with tiny Frobenius norm of different orders of magnitude. Problems (1) with such a tensor is called the tensor discrete linear ill-posed problems. They arise from the restoration of color image and video, see e.g., [1,11,12]. Throughout this paper, the operation ∗ represents tensor t-product and ${\parallel ·\parallel }_F$ denotes the tensor Frobenius norm or the spectral matrix norm.

We assume that the observed tensor $\overrightarrow{\mathcal{B}}\in {\mathbb{R}}^{m\times 1\times n}$ is polluted by an error tensor $\overrightarrow{\mathcal{E}}\in {\mathbb{R}}^{m\times 1\times n}$, i.e., where ${\overrightarrow{\mathcal{B}}}_{true}\in {\mathbb{R}}^{m\times 1\times n}$ is an unknown and unavailable error-free tensor related to $\overrightarrow{\mathcal{B}}$. ${\overrightarrow{\mathcal{B}}}_{true}$ is determined by $\mathcal{A}\ast \overrightarrow{\mathcal{X}}={\overrightarrow{\mathcal{B}}}_{true}$, where ${\overrightarrow{\mathcal{X}}}_{true}$ represents the explicit solution of problems (1) that is to be found. We assume that the upper bound of the Frobenius norm of $\overrightarrow{\mathcal{E}}$ is known, i.e,  

Straightforward solution of (1) is usually meanless to get an approximation of ${\overrightarrow{\mathcal{B}}}_{true}$ because of the illposeness of $\mathcal{A}={\left[a\right]}_{i,j,k=1}^{l,m,n}$ and the error $\overrightarrow{\mathcal{E}}$ will be amplified severely. We use Tikhonov regularization to reduce this effect in this paper and replace (1) with penalty least-squares problems where $\mu$ is a regularization parameter. We assume that where $\mathcal{N}\left(\mathcal{A}\right)$ denotes the null space of $\mathcal{A}$, $\mathcal{I}$ is the identity tensor and $\overrightarrow{\mathcal{O}}\in {\mathbb{R}}^{m\times 1\times n}$ is a lateral slice whose elements are all zero. The normal equation of minimization problem (4) is then is the unique solution of the Tikhonov minimization problem (4) under the assumption (5).

There are many techniques to determine the regularization parameter $\mu$, such as the L-curve criterion, generalized cross validation (GCV), and the discrepancy principle. We refer to [4,5,8,9,10] for more details. In this paper, the discrepancy principle is extended to tensors based on t-product and is employed to determine a suitable $\mu$ in (4). The solution ${\overrightarrow{\mathcal{X}}}_{\mu }$ of (4) satisfies where $\eta >1$ is usually a user-specified constant and is independent of $\delta$ in (3). When ${\parallel \overrightarrow{\mathcal{E}}\parallel }_F$ is smaller enough, and $\delta$ approaches 0, result in ${\overrightarrow{\mathcal{X}}}_{\mu }\to {\overrightarrow{\mathcal{X}}}_{true}$. For more details on the discrepancy principle, see e.g., [7].

In this paper, we also consider the expansion of minimization problem (1) of the form where $\mathcal{B}\in {\mathbb{R}}^{m\times p\times n}$, $p>1$.

There are many methods for solving large-scale discrete linear ill-posed problems (1). Recently, a tensor Golub- Kahan bidiagonalization method [11] and a GMRES method [12] were introduced for solving large-scale linear ill-posed problems (4). The randomized tensor singular value decomposition (rt-SVD) method in [3] was presented for computing super large data sets, and has prospects in image data compression and analysis. Ugwu and Reichel [23] proposed a new random tensor singular value decomposition (R-tSVD), which improves the truncated tensor singular value decomposition (T-tSVD) in [1]. Kilmer et al. [2] presented a tensor Conjugate-Gradient (t-CG) method for tensor linear systems $\mathcal{A}\ast \overrightarrow{\mathcal{X}}=\overrightarrow{\mathcal{B}}$ corresponding to the least-squares problems. The regularization parameter in the t-CG method is user-specified. In this paper, we further discuss the automatical determinization of suitable regularization parameters of the tCG method by the discrepancy principle. The proposed method is called the tCG method with automatical determination of regularization parameters (auto-tCG). We also present a truncated auto-tCG method (auto-ttCG) to improve the auto-tCG method by reducing the computation. At last, a preprocessed version of the auto-ttCG method is proposed, which is abbreviated as auto-ttpCG.

The rest of this paper is organized as follows. Section 2 introduces some symbols and preliminary knowledge that will be used in the context. Section 3 presents the auto-tCG, auto-ttCG and auto-ttpCG methods for solving the minimization problems (4) and (9). Section 4 gives several examples on image and video restoration and Section 5 draws some conclusions.

This section gives some notations and definitions, and briefly summarizes some results that will be used later. For a third-order tensor $\mathcal{A}\in {\mathbb{R}}^{l\times m\times n}$, Figure 1 shows the frontal slices ${\mathcal{A}}_{(:,:,k)}$, lateral slices ${\mathcal{A}}_{(:,j,:)}$ and tube fibers ${\mathcal{A}}_{(i,j,:)}$. We abbreviate $A_k={\mathcal{A}}_{(:,:,k)}$ for simplication. An $ln\times m$ matrix is obtained by the operator $\mathbf{unfold}\left(\mathcal{A}\right)$, whereas the operator $\mathbf{fold}$ folds this matrix back to the tensor $\mathcal{A}$, i.e.,

The following remarks will be used in Section 3.

Let $F_n$ be an n-by-n unitary discrete Fourier transform matrix, i.e,   where $\omega =e^{\frac{-2\pi i}{n}}$, then we get the tensor $\widehat{\mathcal{A}}$ generated by using FFT along each tube of $\mathcal{A}$, i.e, where ⊗ is the Kronecker product, $F_n^{*}$ is the conjugate transposition of $F_n$ and ${\widehat{A}}_i$ denetes the frontal slices of $\widehat{\mathcal{A}}$. Thus the t-product of $\mathcal{A}$ and $\mathcal{B}$ in (10) can be expressed by and (10) is reformulated as It is easy to calculate (12) in MATLAB.

For a non-zero tensor $\overrightarrow{\mathcal{X}}\in {\mathbb{R}}^{m\times 1\times n}$, we can decompose it in the form where $\overrightarrow{\mathcal{D}}\in {\mathbb{R}}^{m\times 1\times n}$ is a normalized tensor; see, e.g., [6] and $d\in {\mathbb{R}}^{1\times 1\times n}$ is a tube scalar. Algorithm 1 summarizes the decomposition in (14).

Algorithm 1 Normalization

Input:$\overrightarrow{\mathcal{X}}\in {\mathbb{R}}^{m\times 1\times n}$ is a nonzero tensor

Output:$\overrightarrow{\mathcal{D}}$, $\mathit{d}$ with $\overrightarrow{\mathcal{X}}=\overrightarrow{\mathcal{D}}\ast \mathit{d}$, $\parallel \overrightarrow{\mathcal{D}}\parallel =1$

$\overrightarrow{\mathcal{D}}\leftarrow$ fft($\overrightarrow{\mathcal{X}}$,[ ],3)

for  $j=1,2,\cdots ,n$  do

${\mathit{d}}_j\leftarrow {\parallel \overrightarrow{{\mathcal{D}}_j}\parallel }_2$ (${\overrightarrow{\mathcal{D}}}_j$ is a vector)

if ${\mathit{d}}_j>tol$ then

$\overrightarrow{{\mathcal{D}}_j}\leftarrow \frac{1}{{\mathit{d}}_j}{\overrightarrow{\mathcal{D}}}_j$

else

${\overrightarrow{\mathcal{D}}}_j\leftarrow \mathbf{randn}(m,1)$; ${\mathit{d}}_j\leftarrow {\parallel \overrightarrow{{\mathcal{D}}_j}\parallel }_2$; $\overrightarrow{{\mathcal{D}}_j}\leftarrow \frac{1}{{\mathit{d}}_j}{\overrightarrow{\mathcal{D}}}_j$; ${\mathit{d}}_j\leftarrow 0$

end if

end for

$\overrightarrow{\mathcal{D}}\leftarrow$ ifft($\overrightarrow{\mathcal{D}}$,[ ],3); $\mathit{d}\leftarrow$ ifft($\mathit{d}$,[ ],3)

Given a tensor $\mathcal{A}\in {\mathbb{R}}^{l\times m\times n}$, the singular value decomposition (tSVD) of $\mathcal{A}$ is expressed as where $\mathcal{U}\in {\mathbb{R}}^{l\times l\times n}$ and $\mathcal{V}\in {\mathbb{R}}^{m\times m\times n}$ are orthogonal under t-product, is an upper triangular tensor with the singular tubes $s_j$ satisfying

The operators $\mathbf{squeeze}$ and $\mathbf{twist}$ [13] are expressed by

Figure 2 illustrates the transformation between a matrix and a tensor column by using $\mathbf{squeeze}$ and $\mathbf{twist}$. Generally, the operators ${\mathbf{multi}}_{-}\mathbf{squeeze}$ and ${\mathbf{multi}}_{-}\mathbf{twist}$ are defined for a third-order tensor to make it squeezed or twisted. For a tensor $\mathcal{D}\in {\mathbb{R}}^{m\times p\times n}$ with $p>1$, $\mathcal{C}={\mathbf{multi}}_{-}\mathbf{squeeze}\left(\mathcal{D}\right)$ means that all side slices of $\mathcal{D}$ are squeezed and stacked as front slices of $\mathcal{C}$, the operator ${\mathbf{multi}}_{-}\mathbf{twist}$ is the reverse operation of ${\mathbf{multi}}_{-}\mathbf{squeeze}$. Thus ${\mathbf{multi}}_{-}\mathbf{twist}\left({\mathbf{multi}}_{-}\mathbf{squeeze}\left(\mathcal{D}\right)\right)=\mathcal{D}$. We refer to Table 1 for more notations and definitions.

This section first discusses the automatical determination of a suitable regularization parameter for the tensor conjugate gradient (tCG) method presented by Kilmer et al. in [13]. We abbreviate the improved method as auto-tCG. A truncated auto-tCG method is developed to improve the auto-tCG method and is abbreviated as auto-ttCG. A preprocessed version of the auto-ttCG method is presented, which is abbreviated as auto-ttpCG.