Restricted Singular Value Decomposition for a Tensor Triplet under T-Product and Its Applications

Abstract

We investigate and discuss in detail the structure of the restricted singular value decomposition for a tensor triplet under t-product (T-RSVD). The algorithm is provided with a numerical example illustrating the main result. For applications, we consider color image watermarking processing with T-RSVD.

1. Introduction

A great number of problems lead to decompositions of higher-order tensors; examples include color image processing, genomic signals, higher-order statistics, pattern recognition, chemometrics, aerospace engineering, etc., (see e.g., [1,2,3,4,5,6,7,8] and references therein). There have been many papers discussing tensor decompositions under various tensor products [3,9,10,11,12,13].

Some well-known multiplications of tensors are the n-mode, Kronecker, Khatri–Rao and Einstein products. The t-product is a new type of tensor multiplication that can be used in imaging (see e.g., [3,14,15,16]). Kilmer and Martin [17] investigated the tensor singular value decomposition under t-product (T-SVD) for third-order tensors. Martin et al. [15] extended the T-SVD to order-p tensors. Very recently, He et al. [3] considered the generalized singular value decompositions for two tensors via the t-product.

To our knowledge, there is little information on the decompositions for three tensors under t-product. Motivated by the wide applications of tensor decompositions in order to improve the theoretical development of tensor decompositions, we consider the restricted singular value decomposition (T-RSVD) for three tensors under t-product. One goal of this article is to investigate and discuss the structure and algorithm of T-RSVD.

Another goal is to give an application for color image watermarking processing. In the context of information security, watermarking provides a way of tracing and identifying the source of digital content, which is important in cases of digital piracy or copyright infringement. We refer readers to [18] for more details and applications. Especially, the research on simultaneously embedding multiple watermarks is of great significance and motivation. As the digital world expands, the need for multiple watermarks in a single piece of content becomes more evident. It can provide enhanced protection. For instance, as Mintzer and Braudaway [19] mentioned using one watermark to convey ownership information, another to verify the integrity of the content, and a third to transmit the title or description of the content. With the help of T-RSVD, we can consider the simultaneous embedding of three watermarks.

The remainder of the paper is organized as follows. In Section 2, we review some definitions, notations and background used throughout the paper. In Section 3, we derive the structure and properties of T-RSVD. We present an algorithm and a numerical example to illustrate the main result. We give an application for color image watermarking processing in Section 4.

2. Preliminaries

An order three dimension $n_1\times n_2\times n_3$ tensor $\mathcal{A}=\left(a_{i_1{i}_2{i}_3}\right)\in {\mathbb{C}}^{n_1\times n_2\times n_3}$ is a multidimensional array with $n_1{n}_2{n}_3$ entries. Let ${\mathbb{C}}^{n_1\times n_2\times n_3}$ be the set of all order three dimension $n_1\times n_2\times n_3$ tensors over the complex number field $\mathbb{C}$.

If $\mathcal{A}=\left(a_{i_1{i}_2{i}_3}\right)\in {\mathbb{C}}^{n_1\times n_2\times n_3}$ with $n_1\times n_2$ frontal slices $A_1,\dots ,A_{n_3}$, then

$$\begin{array}{c}\hfill unfold\left(\mathcal{A}\right)=\left[\begin{array}{c}{A}_1\\ A_2\\ ⋮\\ A_{n_3}\end{array}\right],\end{array}$$

(1)

$$\begin{array}{c}\hfill bcirc\left(\mathcal{A}\right)=\left[\begin{array}{ccccc}{A}_1& A_{n_3}& A_{n_3-1}& \cdots & A_2\\ A_2& A_1& A_{n_3}& \cdots & A_3\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ A_{n_3}& A_{n_3-1}& A_{n_3-2}& \cdots & A_1\end{array}\right],\end{array}$$

(2)

where $A_i=\mathcal{A}(:,:,i)$ for $i=1,\dots ,n_3.$ The operations $fold(·)$ and ${bcirc}^{-1}(·)$ take (1) and (2) back to an $n_1\times n_2\times n_3$ tensor. That means

$$\begin{array}{c}\hfill fold(unfold\left(\mathcal{A}\right))=\mathcal{A},{bcirc}^{-1}(bcirc\left(\mathcal{A}\right))=\mathcal{A}.\end{array}$$

Note that

$$\begin{array}{c}\hfill (F_{n_3}\otimes I_{n_1})bcirc\left(\mathcal{A}\right)(F_{n_3}^{*}\otimes I_{n_2})=diag(D_1,\dots ,D_{n_3}),\end{array}$$

(3)

where $F_{n_3}^{*}$ is the conjugate transpose of $n_3\times n_3$ normalized discrete Fourier transformation (DFT) matrix $F_{n_3}$, and ⊗ denotes the Kronecker product of matrices. The matrices $D_1,\dots ,D_{n_3}$ are diagonal (sub-diagonal, upper-triangular, lower-triangular) if and only if the matrices $A_1,\dots ,A_{n_3}$ are diagonal (sub-diagonal, upper-triangular, lower-triangular) [20], and

$$\begin{array}{c}\hfill A_i=\sum _{j=1}^{n_3}{\omega }^{(j-1)(i-1)}{D}_j,i=1,\dots ,n_3,\end{array}$$

(4)

where $\omega =e^{-2\pi \mathbf{i}/n_3}$ is a primitive $n_3$th root of unity.

The definitions of t-product of two tensors, identity tensor, conjugate transpose, inverse tensor and unitary tensor are given as follows.

Definition 1

(t-product of third-order tensors [17]). Let $\mathcal{A}\in {\mathbb{C}}^{n_1\times n_2\times n_3}$ and $\mathcal{B}\in {\mathbb{C}}^{n_2\times l\times n_3}$ be given. Then the t-product $\mathcal{A}\ast \mathcal{B}\in {\mathbb{C}}^{n_1\times l\times n_3}$ is defined as

$$\begin{array}{c}\hfill \mathcal{A}\ast \mathcal{B}=fold(bcirc(\mathcal{A})unfold(\mathcal{B}\left)\right).\end{array}$$

(5)

Definition 2

(Identity tensor [17]). The identity tensor $\mathcal{I}\in {\mathbb{C}}^{m\times m\times n}$ is defined to be a tensor whose first frontal slice is the $m\times m$ identity matrix and whose other frontal slices are zero matrices, i.e., $bcirc\left(\mathcal{I}\right)=I_{mn}$.

Definition 3

(Conjugate transpose [17]). For a given tensor $\mathcal{A}\in {\mathbb{C}}^{l\times m\times n}$, ${\mathcal{A}}^{*}$ is the $m\times l\times n$ tensor obtained by conjugate transposing each of the frontal slices and then reversing the order of transposed frontal slices 2 through n, i.e.,

$$\begin{array}{c}\hfill {\mathcal{A}}^{*}(:,:,1)=A_1^{*},{\mathcal{A}}^{*}(:,:,i)=A_{n+2-i}^{*},i=2,\dots ,n,\end{array}$$

where $A_k=\mathcal{A}(:,:,k)$, $k=1,\dots ,n$.

Definition 4

(Tensor inverse [17]). An $m\times m\times n$ tensor $\mathcal{A}$ has an inverse $\mathcal{B}\in {\mathbb{C}}^{m\times m\times n}$, provided that $\mathcal{A}\ast \mathcal{B}=\mathcal{I}$ and $\mathcal{B}\ast \mathcal{A}=\mathcal{I},$ i.e., $bcirc\left(\mathcal{B}\right)=bcirc{\left(\mathcal{A}\right)}^{-1}.$

Definition 5

(Unitary tensor [17]). An $m\times m\times n$ tensor $\mathcal{A}$ is unitary, provided that $\mathcal{A}\ast {\mathcal{A}}^{*}=\mathcal{I}$ and ${\mathcal{A}}^{*}\ast \mathcal{A}=\mathcal{I},$ i.e., $bcirc\left(\mathcal{A}\right)$ is a unitary matrix.

3. Restricted Singular Value Decomposition for Three Tensors

In this section, we consider the restricted singular value decomposition for three tensors under t-product (T-RSVD). The following lemma gives the restricted singular value decomposition (RSVD) for matrices.

Lemma 1

(RSVD for matrices [21]). Let $A\in {\mathbb{C}}^{m\times n},B\in {\mathbb{C}}^{m\times p}$ and $C\in {\mathbb{C}}^{q\times n}$ be given. There exist unitary matrices $U\in {\mathbb{C}}^{q\times q}$ and $V\in {\mathbb{C}}^{p\times p}$, and nonsingular matrices $P\in {\mathbb{C}}^{m\times m}$ and $Q\in {\mathbb{C}}^{n\times n}$ such that

$$\begin{array}{c}\hfill A=P{S}_{a}Q,B=P{S}_{b}V,C=U{S}_{c}Q,\end{array}$$

(6)

where

$$\left(\begin{array}{cc}{S}_a& S_b\\ S_c\end{array}\right)=\begin{array}{cc}& \begin{array}{ccccccccccc}{r}_1& r_2& r_3& r_4& r_6& & & & r_3& r_4& r_5\end{array} \\ \begin{array}{c}{r}_1\\ r_2\\ r_3\\ r_4\\ r_5\\ \\ \\ \\ r_2\\ r_4\\ r_6\end{array}& \left(\begin{array}{ccccccccccc}I& 0& 0& 0& 0& 0& & 0& 0& 0& 0\\ 0& I& 0& 0& 0& 0& & 0& 0& 0& 0\\ 0& 0& I& 0& 0& 0& & 0& I& 0& 0\\ 0& 0& 0& S_{abc}& 0& 0& & 0& 0& I& 0\\ 0& 0& 0& 0& 0& 0& & 0& 0& 0& I\\ 0& 0& 0& 0& 0& 0& & 0& 0& 0& 0\\ & & & & & & & & & \\ 0& 0& 0& 0& 0& 0& & & & & \\ 0& I& 0& 0& 0& 0& & & & & \\ 0& 0& 0& I& 0& 0& & & & & \\ 0& 0& 0& 0& I& 0& & & & & \end{array}\right)\end{array},$$

(7)

and $S_{abc}=diag\{{\sigma }_1,\dots ,{\sigma }_{r_4}\}$ is a square nonsingular diagonal matrix with positive diagonal elements, ${\sigma }_1,\dots ,{\sigma }_{r_4}$ are defined to be the non-trivial restricted singular values of the matrix triplet $A,B,C$. Expressions for the integers $r_1,\dots ,r_6$ are the following:

$$\begin{array}{c}\hfill r_1=r\left(\begin{array}{cc}A& B\\ C& 0\end{array}\right)-r\left(B\right)-r\left(C\right),\end{array}$$

(8)

$$\begin{array}{c}\hfill r_2=r(A,B)+r\left(C\right)-r\left(\begin{array}{cc}A& B\\ C& 0\end{array}\right),\end{array}$$

(9)

$$\begin{array}{c}\hfill r_3=r\left(B\right)+r\left(\begin{array}{c}A\\ C\end{array}\right)-r\left(\begin{array}{cc}A& B\\ C& 0\end{array}\right),\end{array}$$

(10)

$$\begin{array}{c}\hfill r_4=r\left(A\right)+r\left(\begin{array}{cc}A& B\\ C& 0\end{array}\right)-r\left(\begin{array}{c}A\\ C\end{array}\right)-r(A,B),\end{array}$$

(11)

$$\begin{array}{c}\hfill r_5=r(A,B)-r\left(A\right),r_6=r\left(\begin{array}{c}A\\ C\end{array}\right)-r\left(A\right),\end{array}$$

(12)

where the symbol $r\left(A\right)$ stands for the rank of the matrix A.

The following theorem presents the RSVD for three tensors under t-product.

Theorem 1

(T-RSVD). Let $\mathcal{A}\in {\mathbb{C}}^{m\times p\times n_3},\mathcal{B}\in {\mathbb{C}}^{m\times q\times n_3},$ and $\mathcal{C}\in {\mathbb{C}}^{t\times p\times n_3}$. Then $\mathcal{A},\mathcal{B},$ and $\mathcal{C}$ can be factored as

$$\begin{array}{c}\hfill \mathcal{A}=\mathcal{P}\ast {\mathcal{S}}_a\ast \mathcal{Q},\mathcal{B}=\mathcal{P}\ast {\mathcal{S}}_b\ast \mathcal{V},\mathcal{C}=\mathcal{U}\ast {\mathcal{S}}_c\ast \mathcal{Q},\end{array}$$

(13)

where $\mathcal{U}\in {\mathbb{C}}^{t\times t\times n_3}$ and $\mathcal{V}\in {\mathbb{C}}^{q\times q\times n_3}$ are unitary, $\mathcal{P}\in {\mathbb{C}}^{m\times m\times n_3},\mathcal{Q}\in {\mathbb{C}}^{p\times p\times n_3}$ are invertible, ${\mathcal{S}}_a$ is a $m\times p\times n_3$ f-diagonal tensor, ${\mathcal{S}}_b$ and ${\mathcal{S}}_c$ are $m\times q\times n_3$ and $t\times p\times n_3$ tensors whose frontal slices are, respectively, block lower-triangular and block upper-triangular matrices with the following forms

$$\begin{array}{c}\hfill {\left(S_b\right)}_i=\sum _{j=1}^{n_3}{\omega }^{(j-1)(i-1)}\left(\begin{array}{cc}0& 0\\ 0& I_{s_j}\\ 0& 0\end{array}\right),{\left(S_c\right)}_i=\sum _{j=1}^{n_3}{\omega }^{(j-1)(i-1)}\left(\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& I_{s_j^{\prime }}& 0& 0& 0\\ 0& 0& 0& I_{s_j^{\prime \prime }}& 0\end{array}\right),i=1\dots ,n_3.\end{array}$$

Proof. 

Let $\tilde{A}=bcirc\left(\mathcal{A}\right)\in {\mathbb{C}}^{\left(m{n}_3\right)\times \left(p{n}_3\right)},\tilde{B}=bcirc\left(\mathcal{B}\right)\in {\mathbb{C}}^{\left(m{n}_3\right)\times \left(q{n}_3\right)},$ and $\tilde{C}=bcirc\left(\mathcal{C}\right)\in {\mathbb{C}}^{\left(t{n}_3\right)\times \left(p{n}_3\right)}.$ It follows from (3) that

$$\begin{array}{c}\hfill (F_{n_3}\otimes I_m)\tilde{A}(F_{n_3}^{*}\otimes I_p)=diag(\widetilde{A_1},\dots ,\widetilde{A_{n_3}}),\end{array}$$

$$\begin{array}{c}\hfill (F_{n_3}\otimes I_m)\tilde{B}(F_{n_3}^{*}\otimes I_q)=diag(\widetilde{B_1},\dots ,\widetilde{B_{n_3}}),\end{array}$$

$$\begin{array}{c}\hfill (F_{n_3}\otimes I_t)\tilde{C}(F_{n_3}^{*}\otimes I_p)=diag(\widetilde{C_1},\dots ,\widetilde{C_{n_3}}),\end{array}$$

where $F_{n_3}$ is $n_3\times n_3$ normalized DFT matrix, $\widetilde{A_i},\widetilde{B_i},$ and $\widetilde{C_i}$ are $m\times p,m\times q,$ and $t\times p$ matrices, $i=1,\dots ,n_3.$ Observe that $\widetilde{A_i}$ and $\widetilde{B_i}$ have the same number of columns, meanwhile, $\widetilde{A_i}$ and $\widetilde{C_i}$ have the same number of rows. Applying RSVD to each matrix triplet $\widetilde{A_i},\widetilde{B_i},$ and $\widetilde{C_i}$ gives $\widetilde{A_i}=P_i{\Omega }_{A_i}{Q}_i$, $\widetilde{B_i}=P_i{\Omega }_{B_i}{V}_i$, and $\widetilde{C_i}=U_i{\Omega }_{C_i}{Q}_i$, $i=1,2,\dots ,n_3$, i.e.,

$$\begin{array}{c}\hfill \left(\begin{array}{c}\begin{array}{c}\widetilde{A_1}\\ & \ddots \\ & & \widetilde{A_{n_3}}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{P}_1\\ & \ddots \\ & & P_{n_3}\end{array}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{\Omega }_{A_1}\\ & \ddots \\ & & {\Omega }_{A_{n_3}}\end{array}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{Q}_1\\ & \ddots \\ & & Q_{n_3}\end{array}\end{array}\right),\end{array}$$

(14)

$$\begin{array}{c}\hfill \left(\begin{array}{c}\begin{array}{c}\widetilde{B_1}\\ & \ddots \\ & & \widetilde{B_{n_3}}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{P}_1\\ & \ddots \\ & & P_{n_3}\end{array}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{\Omega }_{B_1}\\ & \ddots \\ & & {\Omega }_{B_{n_3}}\end{array}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{V}_1\\ & \ddots \\ & & V_{n_3}\end{array}\end{array}\right),\end{array}$$

(15)

and

$$\begin{array}{c}\hfill \left(\begin{array}{c}\begin{array}{c}\widetilde{C_1}\\ & \ddots \\ & & \widetilde{C_{n_3}}\end{array}\end{array}\right)=\left(\begin{array}{c}\begin{array}{c}{U}_1\\ & \ddots \\ & & U_{n_3}\end{array}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{\Omega }_{C_1}\\ & \ddots \\ & & {\Omega }_{C_{n_3}}\end{array}\end{array}\right)\left(\begin{array}{c}\begin{array}{c}{Q}_1\\ & \ddots \\ & & Q_{n_3}\end{array}\end{array}\right).\end{array}$$

(16)

According to Lemma 1, $\left(\begin{array}{cc}{\Omega }_{A_i}& {\Omega }_{B_i}\\ {\Omega }_{C_i}\end{array}\right)$ is in the form of (7), $U_i$ and $V_i$ are unitary, and $P_i$ and $Q_i$ are nonsingular, $i=1,2,\dots ,n_3$. We multiply $(F_{n_3}^{*}\otimes I_m),(F_{n_3}^{*}\otimes I_m),$ and $(F_{n_3}^{*}\otimes I_t)$ to the left of each of the block diagonal matrices in (14)–(16), respectively, and $(F_{n_3}\otimes I_p),$$(F_{n_3}\otimes I_q),$ and $(F_{n_3}\otimes I_p)$ to the right of each of the block diagonal matrices in (14)–(16). Then, we obtain

$$\begin{array}{cc}\hfill bcirc\left(\mathcal{A}\right)=& \left[(F_{n_3}^{*}\otimes I_m)\left(\begin{array}{c}\begin{array}{c}{P}_1\\ & \ddots \\ & & P_{n_3}\end{array}\end{array}\right)(F_{n_3}\otimes I_m)\right]\hfill \\ & \left[(F_{n_3}^{*}\otimes I_m)\left(\begin{array}{c}\begin{array}{c}{\Omega }_{A_1}\\ & \ddots \\ & & {\Omega }_{A_{n_3}}\end{array}\end{array}\right)(F_{n_3}\otimes I_p)\right]\hfill \\ & \left[(F_{n_3}^{*}\otimes I_p)\left(\begin{array}{c}\begin{array}{c}{Q}_1\\ & \ddots \\ & & Q_{n_3}\end{array}\end{array}\right)(F_{n_3}\otimes I_p)\right]\hfill \\ \hfill =& bcirc\left(\mathcal{P}\right)bcirc\left({\mathcal{S}}_a\right)bcirc\left(\mathcal{Q}\right),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill bcirc\left(\mathcal{B}\right)=& \left[(F_{n_3}^{*}\otimes I_m)\left(\begin{array}{c}\begin{array}{c}{P}_1\\ & \ddots \\ & & P_{n_3}\end{array}\end{array}\right)(F_{n_3}\otimes I_m)\right]\hfill \\ & \left[(F_{n_3}^{*}\otimes I_m)\left(\begin{array}{c}\begin{array}{c}{\Omega }_{B_1}\\ & \ddots \\ & & {\Omega }_{B_{n_3}}\end{array}\end{array}\right)(F_{n_3}\otimes I_q)\right]\hfill \\ & \left[(F_{n_3}^{*}\otimes I_q)\left(\begin{array}{c}\begin{array}{c}{V}_1\\ & \ddots \\ & & V_{n_3}\end{array}\end{array}\right)(F_{n_3}\otimes I_q)\right]\hfill \\ \hfill =& bcirc\left(\mathcal{P}\right)bcirc\left({\mathcal{S}}_b\right)bcirc\left(\mathcal{V}\right),\hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill bcirc\left(\mathcal{C}\right)=& \left[(F_{n_3}^{*}\otimes I_t)\left(\begin{array}{c}\begin{array}{c}{U}_1\\ & \ddots \\ & & U_{n_3}\end{array}\end{array}\right)(F_{n_3}\otimes I_t)\right]\hfill \\ & \left[(F_{n_3}^{*}\otimes I_t)\left(\begin{array}{c}\begin{array}{c}{\Omega }_{C_1}\\ & \ddots \\ & & {\Omega }_{C_{n_3}}\end{array}\end{array}\right)(F_{n_3}\otimes I_p)\right]\hfill \\ & \left[(F_{n_3}^{*}\otimes I_p)\left(\begin{array}{c}\begin{array}{c}{Q}_1\\ & \ddots \\ & & Q_{n_3}\end{array}\end{array}\right)(F_{n_3}\otimes I_p)\right]\hfill \\ \hfill =& bcirc\left(\mathcal{U}\right)bcirc\left({\mathcal{S}}_c\right)bcirc\left(\mathcal{Q}\right),\hfill \end{array}$$

(19)

which yields

$$\mathcal{A}=\mathcal{P}\ast {\mathcal{S}}_a\ast \mathcal{Q},$$

(20)

$$\mathcal{B}=\mathcal{P}\ast {\mathcal{S}}_b\ast \mathcal{V},$$

(21)

and

$$\mathcal{C}=\mathcal{U}\ast {\mathcal{S}}_c\ast \mathcal{Q}.$$

(22)

Note that $\mathcal{U}$ and $\mathcal{V}$ are unitary, $\mathcal{P}$ and $\mathcal{Q}$ are invertible. It follows from (4) that the frontal slices of ${\mathcal{S}}_a,{\mathcal{S}}_b,$ and ${\mathcal{S}}_c$ have the following forms

$${\left(S_a\right)}_i=\sum _{j=1}^{n_3}{\omega }^{(j-1)(i-1)}{\Omega }_{A_j},{\left(S_b\right)}_i=\sum _{j=1}^{n_3}{\omega }^{(j-1)(i-1)}{\Omega }_{B_j},$$

$${\left(S_c\right)}_i=\sum _{j=1}^{n_3}{\omega }^{(j-1)(i-1)}{\Omega }_{C_j},i=1\dots ,n_3.$$

□

We provide the algorithm for T-RSVD.

The cost of Algorithm 1 is not less than $\mathcal{O}\left({(n_1\times n_2\times n_3)}^3\right)$.

Algorithm 1: Compute the T-RSVD of tensors $\mathcal{A},\mathcal{B},$ and $\mathcal{C}$

Input:$\mathcal{A}\in {\mathbb{C}}^{m\times p\times n_3},\mathcal{B}\in {\mathbb{C}}^{m\times q\times n_3},\mathcal{C}\in {\mathbb{C}}^{t\times p\times n_3}.$

Output:${\mathcal{S}}_a\in {\mathbb{C}}^{m\times p\times n_3},{\mathcal{S}}_b\in {\mathbb{C}}^{m\times q\times n_3},{\mathcal{S}}_c\in {\mathbb{C}}^{t\times p\times n_3},\mathcal{U}\in {\mathbb{C}}^{t\times t\times n_3},\mathcal{V}\in {\mathbb{C}}^{q\times q\times n_3},$

$\mathcal{P}\in {\mathbb{C}}^{m\times m\times n_3},\mathcal{Q}\in {\mathbb{C}}^{p\times p\times n_3}.$

1. $\tilde{A}=bcirc\left(\mathcal{A}\right),\tilde{B}=bcirc\left(\mathcal{B}\right),\tilde{C}=bcirc\left(\mathcal{C}\right).$

2. $(F_{n_3}\otimes I_m)\tilde{A}(F_{n_3}^{*}\otimes I_p)=diag\left(\widetilde{A_j}\right),(F_{n_3}\otimes I_m)\tilde{B}(F_{n_3}^{*}\otimes I_q)=diag\left(\widetilde{B_j}\right),$

$(F_{n_3}\otimes I_t)\tilde{C}(F_{n_3}^{*}\otimes I_p)=diag\left(\widetilde{C_j}\right),j=1,2,\dots ,n_3.$

3. for $j=1,\dots ,n_3$ do

% Give the RSVD of $\widetilde{A_j},\widetilde{B_j},$ and $\widetilde{C_j}$ by the method in [22],

$\widetilde{A_j}=P_j{\Omega }_{A_j}{Q}_j,$

$\widetilde{B_j}=P_j{\Omega }_{B_j}{V}_j,$

$\widetilde{C_j}=U_j{\Omega }_{C_j}{Q}_j,$

end for

4. ${\Omega }_A=diag\left({\Omega }_{A_j}\right),{\Omega }_B=diag\left({\Omega }_{B_j}\right),{\Omega }_C=diag\left({\Omega }_{C_j}\right),U=diag\left(U_j\right),V=diag\left(V_j\right),$

$P=diag\left(P_j\right),Q=diag\left(Q_j\right).$

5. ${\tilde{S}}_a=(F_{n_3}^{*}\otimes I_m){\Omega }_A(F_{n_3}\otimes I_p),{\tilde{S}}_b=(F_{n_3}^{*}\otimes I_m){\Omega }_B(F_{n_3}\otimes I_q),{\tilde{S}}_c=(F_{n_3}^{*}\otimes I_t){\Omega }_C(F_{n_3}\otimes I_p),$

$\tilde{U}=(F_{n_3}^{*}\otimes I_t)U(F_{n_3}\otimes I_t),\tilde{V}=(F_{n_3}^{*}\otimes I_q)V(F_{n_3}\otimes I_q),$

$\tilde{P}=(F_{n_3}^{*}\otimes I_m)P(F_{n_3}\otimes I_m),\tilde{Q}=(F_{n_3}^{*}\otimes I_p)Q(F_{n_3}\otimes I_p).$

6. ${\mathcal{S}}_a={bcirc}^{-1}\left({\tilde{S}}_a\right),{\mathcal{S}}_b={bcirc}^{-1}\left({\tilde{S}}_b\right),{\mathcal{S}}_c={bcirc}^{-1}\left({\tilde{S}}_c\right),$

$\mathcal{U}={bcirc}^{-1}\left(\tilde{U}\right),\mathcal{V}={bcirc}^{-1}\left(\tilde{V}\right),\mathcal{P}={bcirc}^{-1}\left(\tilde{P}\right),\mathcal{Q}={bcirc}^{-1}\left(\tilde{Q}\right).$

Example 1.

Let $\mathcal{A}\in {\mathbb{C}}^{5\times 4\times 3}$, $\mathcal{B}\in {\mathbb{C}}^{5\times 6\times 3}$ and $\mathcal{C}\in {\mathbb{C}}^{7\times 4\times 3}$ be tensors with the following forms:

$$\begin{array}{c}\hfill A_1=\left(\begin{array}{c}\begin{array}{cccc}0.5303+0.5248\mathbf{i}& 0.4221+0.7424\mathbf{i}& 0.5356+0.7391\mathbf{i}& 0.6216+0.6644\mathbf{i}\\ 0.7312+1.0116\mathbf{i}& 0.1918+0.7801\mathbf{i}& 1.0096+0.6131\mathbf{i}& 0.6995+1.0930\mathbf{i}\\ 0.7828+0.9208\mathbf{i}& 0.5561+0.4061\mathbf{i}& 0.8372+0.1792\mathbf{i}& 0.8905+0.9073\mathbf{i}\\ 0.7793+0.9972\mathbf{i}& 1.2191+0.6961\mathbf{i}& 0.3713+0.5136\mathbf{i}& 1.1552+1.0565\mathbf{i}\\ 0.5657+0.2247\mathbf{i}& 0.4331+0.7622\mathbf{i}& 0.5833+0.8704\mathbf{i}& 0.6561+0.4125\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill A_2=\left(\begin{array}{c}\begin{array}{cccc}0.7495+1.2373\mathbf{i}& 0.1073+0.3358\mathbf{i}& 1.0968-0.0209\mathbf{i}& 0.6810+1.1587\mathbf{i}\\ 0.4769-0.2104\mathbf{i}& 1.1669+0.9282\mathbf{i}& -0.0650+1.2320\mathbf{i}& 0.8769+0.0870\mathbf{i}\\ 0.5336+0.7049\mathbf{i}& 0.0139+0.7114\mathbf{i}& 0.8243+0.6365\mathbf{i}& 0.4596+0.8100\mathbf{i}\\ 0.5427+1.2123\mathbf{i}& -0.2579+0.5388\mathbf{i}& 1.0273+0.2411\mathbf{i}& 0.3576+1.1957\mathbf{i}\\ 0.5218+0.8433\mathbf{i}& 1.1630+0.5181\mathbf{i}& 0.0079+0.3463\mathbf{i}& 0.9135+0.8730\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill A_3=\left(\begin{array}{c}\begin{array}{cccc}0.8390+0.3249\mathbf{i}& -0.0225+0.4260\mathbf{i}& 1.3270+0.4157\mathbf{i}& 0.7047+0.4016\mathbf{i}\\ 0.6379+0.3782\mathbf{i}& 1.4253+0.5669\mathbf{i}& 0.0071+0.5724\mathbf{i}& 1.1182+0.4880\mathbf{i}\\ 0.8143+0.7469\mathbf{i}& 0.1163+0.2199\mathbf{i}& 1.1919+0.0088\mathbf{i}& 0.7398+0.7044\mathbf{i}\\ 0.7343+0.6969\mathbf{i}& 0.2639+0.6815\mathbf{i}& 0.9643+0.6020\mathbf{i}& 0.7312+0.7945\mathbf{i}\\ 0.2706-0.1724\mathbf{i}& 0.7472+0.8606\mathbf{i}& -0.0959+1.1341\mathbf{i}& 0.5319+0.1001\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill B_1=\left(\begin{array}{c}\begin{array}{ccccccc}0.9678+1.0857\mathbf{i}& 1.9658+2.4999\mathbf{i}& 1.8365+2.4539\mathbf{i}& 2.3199+2.7399\mathbf{i}& 1.8102+2.4752\mathbf{i}& 0.8597+1.3350\mathbf{i}& 1.2342+1.4315\mathbf{i}\\ 1.5437+1.4667\mathbf{i}& 2.7275+3.0925\mathbf{i}& 2.5159+2.8494\mathbf{i}& 2.7909+4.0341\mathbf{i}& 2.1110+3.0183\mathbf{i}& 0.4634+1.7009\mathbf{i}& 1.1811+2.3560\mathbf{i}\\ 1.4646+1.0428\mathbf{i}& 2.9047+2.0090\mathbf{i}& 2.7081+1.7155\mathbf{i}& 3.3544+3.0897\mathbf{i}& 2.6059+1.9291\mathbf{i}& 1.1449+1.1409\mathbf{i}& 1.7323+1.9562\mathbf{i}\\ 1.1014+1.3758\mathbf{i}& 2.8630+2.8554\mathbf{i}& 2.7241+2.5984\mathbf{i}& 4.0353+3.8373\mathbf{i}& 3.2509+2.7793\mathbf{i}& 2.3711+1.5791\mathbf{i}& 2.6126+2.2774\mathbf{i}\\ 1.0417+0.8922\mathbf{i}& 2.0977+2.2487\mathbf{i}& 1.9584+2.3345\mathbf{i}& 2.4568+2.0245\mathbf{i}& 1.9140+2.2563\mathbf{i}& 0.8852+1.1671\mathbf{i}& 1.2936+0.8882\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill B_2=\left(\begin{array}{c}\begin{array}{ccccccc}1.6302+1.1994\mathbf{i}& 2.7995+2.1430\mathbf{i}& 2.5751+1.6989\mathbf{i}& 2.7673+3.7497\mathbf{i}& 2.0757+2.0271\mathbf{i}& 0.3098+1.2518\mathbf{i}& 1.0915+2.4986\mathbf{i}\\ 0.4485+0.7437\mathbf{i}& 1.7340+2.2485\mathbf{i}& 1.6849+2.5580\mathbf{i}& 2.9096+1.2503\mathbf{i}& 2.4047+2.3085\mathbf{i}& 2.2296+1.1078\mathbf{i}& 2.1604+0.1857\mathbf{i}\\ 1.1941+1.1835\mathbf{i}& 1.9959+2.6001\mathbf{i}& 1.8307+2.4705\mathbf{i}& 1.9049+3.1328\mathbf{i}& 1.4161+2.5553\mathbf{i}& 0.1049+1.4102\mathbf{i}& 0.6937+1.7458\mathbf{i}\\ 1.3604+1.3769\mathbf{i}& 2.0418+2.6560\mathbf{i}& 1.8506+2.2706\mathbf{i}& 1.6530+4.0758\mathbf{i}& 1.1719+2.5509\mathbf{i}& -0.3966+1.5076\mathbf{i}& 0.3426+2.5781\mathbf{i}\\ 0.5517+1.0956\mathbf{i}& 1.9021+2.2274\mathbf{i}& 1.8386+1.9933\mathbf{i}& 3.0645+3.1096\mathbf{i}& 2.5188+2.1601\mathbf{i}& 2.2291+1.2407\mathbf{i}& 2.2119+1.8821\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill B_3=\left(\begin{array}{c}\begin{array}{ccccccc}1.9015+0.6399\mathbf{i}& 3.1403+1.4586\mathbf{i}& 2.8768+1.4222\mathbf{i}& 2.9489+1.6319\mathbf{i}& 2.1830+1.4419\mathbf{i}& 0.0829+0.7815\mathbf{i}& 1.0316+0.8654\mathbf{i}\\ 0.6725+0.8131\mathbf{i}& 2.3250+1.8862\mathbf{i}& 2.2478+1.8607\mathbf{i}& 3.7500+2.0357\mathbf{i}& 3.0827+1.8697\mathbf{i}& 2.7315+1.0049\mathbf{i}& 2.7087+1.0514\mathbf{i}\\ 1.7713+0.7406\mathbf{i}& 3.0417+1.3392\mathbf{i}& 2.7979+1.0752\mathbf{i}& 3.0065+2.2965\mathbf{i}& 2.2550+1.2699\mathbf{i}& 0.3361+0.7787\mathbf{i}& 1.1856+1.5190\mathbf{i}\\ 1.5120+1.1491\mathbf{i}& 2.7359+2.5126\mathbf{i}& 2.5295+2.3793\mathbf{i}& 2.8771+3.0553\mathbf{i}& 2.1902+2.4674\mathbf{i}& 0.5972+1.3649\mathbf{i}& 1.2812+1.7124\mathbf{i}\\ 0.2089+0.7056\mathbf{i}& 0.9802+2.1076\mathbf{i}& 0.9596+2.3849\mathbf{i}& 1.7398+1.2161\mathbf{i}& 1.4483+2.1608\mathbf{i}& 1.4224+1.0417\mathbf{i}& 1.3392+0.2141\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill C_1=\left(\begin{array}{c}\begin{array}{cccc}7.1881+6.9753\mathbf{i}& 5.1342+6.9579\mathbf{i}& 7.6688+6.1966\mathbf{i}& 8.1884+7.9916\mathbf{i}\\ 4.8046+4.6507\mathbf{i}& 2.4018+4.0024\mathbf{i}& 5.8412+3.3377\mathbf{i}& 5.0574+5.1448\mathbf{i}\\ 6.5231+6.3144\mathbf{i}& 4.6489+5.9511\mathbf{i}& 6.9664+5.1761\mathbf{i}& 7.4267+7.1342\mathbf{i}\\ 5.0711+4.5700\mathbf{i}& 5.6075+5.0339\mathbf{i}& 4.0314+4.6524\mathbf{i}& 6.5785+5.3729\mathbf{i}\\ 5.6683+6.0800\mathbf{i}& 3.7989+5.4458\mathbf{i}& 6.2208+4.6295\mathbf{i}& 6.3563+6.7875\mathbf{i}\\ 3.9585+3.9067\mathbf{i}& 3.1749+4.4012\mathbf{i}& 3.9818+4.0993\mathbf{i}& 4.6496+4.6212\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill C_2=\left(\begin{array}{c}\begin{array}{cccc}4.8070+4.3309\mathbf{i}& 4.0376+4.7113\mathbf{i}& 4.7088+4.3351\mathbf{i}& 5.7199+5.0747\mathbf{i}\\ 4.7909+3.8380\mathbf{i}& 3.6841+4.9274\mathbf{i}& 4.9292+4.7796\mathbf{i}& 5.5635+4.7139\mathbf{i}\\ 5.0686+4.9549\mathbf{i}& 2.5796+4.3957\mathbf{i}& 6.1302+3.7199\mathbf{i}& 5.3538+5.5192\mathbf{i}\\ 6.0918+6.4247\mathbf{i}& 4.4663+5.7944\mathbf{i}& 6.4191+4.9416\mathbf{i}& 6.9861+7.1838\mathbf{i}\\ 4.6481+4.4668\mathbf{i}& 2.9568+3.9805\mathbf{i}& 5.2112+3.3757\mathbf{i}& 5.1483+4.9807\mathbf{i}\\ 5.6523+5.9783\mathbf{i}& 3.4304+4.8733\mathbf{i}& 6.4517+3.9518\mathbf{i}& 6.1939+6.5352\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill C_3=\left(\begin{array}{c}\begin{array}{cccc}5.8659+5.3457\mathbf{i}& 4.4210+6.0571\mathbf{i}& 6.0977+5.6524\mathbf{i}& 6.7756+6.3334\mathbf{i}\\ 5.4631+5.1432\mathbf{i}& 2.8010+5.0694\mathbf{i}& 6.5931+4.4929\mathbf{i}& 5.7788+5.8750\mathbf{i}\\ 7.2234+7.1743\mathbf{i}& 6.2645+7.0710\mathbf{i}& 6.9389+6.2668\mathbf{i}& 8.6748+8.1950\mathbf{i}\\ 5.5242+5.4475\mathbf{i}& 6.0286+5.3725\mathbf{i}& 4.4471+4.7627\mathbf{i}& 7.1340+6.2235\mathbf{i}\\ 4.8612+5.0317\mathbf{i}& 2.0329+3.6799\mathbf{i}& 6.1858+2.8003\mathbf{i}& 4.9566+5.3788\mathbf{i}\\ 5.3460+5.1352\mathbf{i}& 4.7360+5.8883\mathbf{i}& 5.0663+5.5168\mathbf{i}& 6.4605+6.1042\mathbf{i}\end{array}\end{array}\right).\end{array}$$

It follows from Theorem 1 and Algorithm 1 that

$$\begin{array}{c}\hfill \mathcal{A}=\mathcal{P}\ast {\mathcal{S}}_a\ast \mathcal{Q},\mathcal{B}=\mathcal{P}\ast {\mathcal{S}}_b\ast \mathcal{V},\mathcal{C}=\mathcal{U}\ast {\mathcal{S}}_c\ast \mathcal{Q},\end{array}$$

(23)

where $\mathcal{U}\in {\mathbb{C}}^{t\times t\times n_3}$ and $\mathcal{V}\in {\mathbb{C}}^{q\times q\times n_3}$ are unitary, $\mathcal{P}\in {\mathbb{C}}^{m\times m\times n_3},\mathcal{Q}\in {\mathbb{C}}^{p\times p\times n_3}$ are invertible, and

$$\begin{array}{c}\hfill P_1=\left(\begin{array}{c}\begin{array}{ccccc}0.6408-0.5877\mathbf{i}& -0.0936-0.0107\mathbf{i}& -0.4060+0.5214\mathbf{i}& 2.2421+5.0044\mathbf{i}& 0.2949+0.0390\mathbf{i}\\ -0.1613+0.2793\mathbf{i}& 0.4269-0.0810\mathbf{i}& 0.4228-1.2971\mathbf{i}& 2.6692+5.3346\mathbf{i}& 0.3346+0.0419\mathbf{i}\\ 0.4351-0.4090\mathbf{i}& -0.6897+0.1941\mathbf{i}& -0.7972+0.1545\mathbf{i}& 2.3592+5.8345\mathbf{i}& -0.6301-0.0906\mathbf{i}\\ 0.4874-0.7717\mathbf{i}& -0.9595+0.3726\mathbf{i}& -0.2353+1.3777\mathbf{i}& 2.0832+8.1176\mathbf{i}& 0.1698+0.0205\mathbf{i}\\ 0.1609-0.1738\mathbf{i}& 0.8428-0.1838\mathbf{i}& 0.9707-0.3759\mathbf{i}& 1.4278+6.5459\mathbf{i}& -0.4527-0.0595\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill P_2=\left(\begin{array}{c}\begin{array}{ccccc}-0.2512+0.3184\mathbf{i}& -0.8487+0.5414\mathbf{i}& -1.1722-1.0526\mathbf{i}& 3.0836+7.1109\mathbf{i}& -0.1229-0.0429\mathbf{i}\\ -0.3248+0.4861\mathbf{i}& 1.0281-0.4627\mathbf{i}& 1.7064+1.6331\mathbf{i}& 5.3474+7.9779\mathbf{i}& -0.0898-0.0413\mathbf{i}\\ -0.2539+0.3055\mathbf{i}& -0.0275-0.3210\mathbf{i}& -0.4593-0.2713\mathbf{i}& 3.7814+5.1409\mathbf{i}& -0.0982+0.0136\mathbf{i}\\ -0.3560+0.0728\mathbf{i}& -0.4230-0.1031\mathbf{i}& -0.7313-1.1984\mathbf{i}& 4.1175+6.0276\mathbf{i}& -0.0816-0.0287\mathbf{i}\\ 0.3311-0.3674\mathbf{i}& -0.4491+0.6748\mathbf{i}& 0.5431+0.8102\mathbf{i}& 2.2330+5.9796\mathbf{i}& -0.0161+0.0261\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill P_3=\left(\begin{array}{c}\begin{array}{ccccc}-0.3059+0.3610\mathbf{i}& 0.5091-0.4685\mathbf{i}& -0.5184-0.9038\mathbf{i}& 4.2031+6.9941\mathbf{i}& 0.1701+0.0145\mathbf{i}\\ 0.6811-0.5568\mathbf{i}& -0.3334+0.5769\mathbf{i}& 0.9122+1.3152\mathbf{i}& 2.8736+7.2549\mathbf{i}& 0.1871+0.0129\mathbf{i}\\ -0.1097+0.1755\mathbf{i}& -0.3939+0.0771\mathbf{i}& -0.8263-0.9132\mathbf{i}& 3.7283+8.3036\mathbf{i}& 0.1302+0.0584\mathbf{i}\\ -0.5188+0.2754\mathbf{i}& 0.1895-0.4533\mathbf{i}& -0.0753-0.1377\mathbf{i}& 2.3868+9.6501\mathbf{i}& 0.1436+0.0154\mathbf{i}\\ -0.5290+0.5041\mathbf{i}& 0.9653-0.4482\mathbf{i}& 1.4892+1.1565\mathbf{i}& 3.9351+5.1095\mathbf{i}& -0.0641+0.0167\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Q_1=\left(\begin{array}{c}\begin{array}{cccc}-0.0152+0.0008\mathbf{i}& 0.0052-0.0003\mathbf{i}& 0.0042-0.0002\mathbf{i}& 0.0085-0.0004\mathbf{i}\\ 0.3822-0.8686\mathbf{i}& -0.5113+0.7359\mathbf{i}& 0.0025+0.6981\mathbf{i}& 0.1330-0.5357\mathbf{i}\\ -1.3744+0.1132\mathbf{i}& 2.2744-2.3223\mathbf{i}& -0.5762+3.5012\mathbf{i}& -0.3227-0.8922\mathbf{i}\\ 16.3877-9.0211\mathbf{i}& 15.0754-5.6503\mathbf{i}& 14.7485-9.8723\mathbf{i}& 18.7357-10.3581\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Q_2=\left(\begin{array}{c}\begin{array}{cccc}-0.0159+0.0167\mathbf{i}& 0.0055-0.0058\mathbf{i}& 0.0044-0.0047\mathbf{i}& 0.0089-0.0093\mathbf{i}\\ -0.1900-0.4386\mathbf{i}& 0.2039+0.4691\mathbf{i}& 0.0773+0.1913\mathbf{i}& -0.0979-0.2206\mathbf{i}\\ -0.0433-0.1733\mathbf{i}& 0.7916+0.1304\mathbf{i}& -1.0174+0.1131\mathbf{i}& 0.2912-0.1051\mathbf{i}\\ 17.1512-7.2539\mathbf{i}& 15.0898-4.4501\mathbf{i}& 16.2790-8.6406\mathbf{i}& 19.4136-8.4856\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill Q_3=\left(\begin{array}{c}\begin{array}{cccc}-0.0155-0.0167\mathbf{i}& 0.0054+0.0058\mathbf{i}& 0.0043+0.0047\mathbf{i}& 0.0087+0.0093\mathbf{i}\\ -0.2820-0.3994\mathbf{i}& 0.2413+0.4149\mathbf{i}& 0.2093+0.1722\mathbf{i}& -0.1650-0.1771\mathbf{i}\\ 0.4003+0.0073\mathbf{i}& 0.4382+0.0189\mathbf{i}& -1.5852+0.0118\mathbf{i}& 0.5705+0.0154\mathbf{i}\\ 19.0878-3.7427\mathbf{i}& 17.2905-0.7168\mathbf{i}& 17.6647-5.5544\mathbf{i}& 22.0257-4.3068\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill U_1=\left(\begin{array}{c}\begin{array}{cccccc}0.1054+0.2152\mathbf{i}& 0.0949-0.0731\mathbf{i}& 0.0209-0.0258\mathbf{i}& 0.2220+0.0214\mathbf{i}& 0.0051-0.1728\mathbf{i}& -0.0446+0.1700\mathbf{i}\\ -0.1830-0.0106\mathbf{i}& 0.3933-0.2579\mathbf{i}& 0.0145+0.1753\mathbf{i}& -0.0275+0.1724\mathbf{i}& -0.1577-0.2441\mathbf{i}& 0.1679+0.0948\mathbf{i}\\ -0.1066-0.1453\mathbf{i}& -0.0150+0.0924\mathbf{i}& -0.0448+0.0555\mathbf{i}& -0.0960+0.0102\mathbf{i}& -0.0641-0.1135\mathbf{i}& 0.2924+0.0744\mathbf{i}\\ 0.1422+0.0776\mathbf{i}& 0.1950-0.1345\mathbf{i}& 0.2179-0.0819\mathbf{i}& -0.1872+0.0065\mathbf{i}& 0.2229+0.3532\mathbf{i}& 0.1096+0.1109\mathbf{i}\\ 0.3893+0.1686\mathbf{i}& 0.0423+0.0307\mathbf{i}& -0.0311-0.0038\mathbf{i}& -0.1386-0.0652\mathbf{i}& -0.1125+0.0171\mathbf{i}& -0.1152+0.1255\mathbf{i}\\ -0.1001+0.0303\mathbf{i}& 0.3373-0.1433\mathbf{i}& -0.1972-0.1863\mathbf{i}& 0.0041+0.0017\mathbf{i}& 0.0424+0.0948\mathbf{i}& 0.2399+0.0643\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill U_2=\left(\begin{array}{c}\begin{array}{cccccc}-0.0271+0.0574\mathbf{i}& -0.4738+0.1835\mathbf{i}& 0.1470-0.1615\mathbf{i}& -0.0016+0.0967\mathbf{i}& 0.0458+0.0943\mathbf{i}& 0.4059+0.0489\mathbf{i}\\ -0.1138+0.0479\mathbf{i}& 0.2263-0.0749\mathbf{i}& 0.0146+0.1956\mathbf{i}& 0.3062-0.1033\mathbf{i}& 0.0124+0.1959\mathbf{i}& -0.0104+0.1229\mathbf{i}\\ 0.0058-0.1376\mathbf{i}& -0.1306+0.0788\mathbf{i}& 0.0147+0.0614\mathbf{i}& 0.1488-0.1841\mathbf{i}& -0.0683-0.1907\mathbf{i}& 0.2169+0.0885\mathbf{i}\\ -0.3325-0.0111\mathbf{i}& -0.0530-0.0575\mathbf{i}& 0.0370+0.1065\mathbf{i}& 0.1376-0.1857\mathbf{i}& 0.1034-0.1806\mathbf{i}& -0.0858+0.1292\mathbf{i}\\ -0.0897+0.0322\mathbf{i}& -0.2217-0.0040\mathbf{i}& -0.2887+0.2487\mathbf{i}& -0.2355+0.0075\mathbf{i}& -0.1306-0.0206\mathbf{i}& 0.2770+0.0877\mathbf{i}\\ 0.0872-0.0300\mathbf{i}& -0.0489+0.0535\mathbf{i}& -0.3116-0.0649\mathbf{i}& -0.1771+0.2345\mathbf{i}& -0.0770-0.2311\mathbf{i}& -0.1628+0.2071\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill U_3=\left(\begin{array}{c}\begin{array}{cccccc}0.2309+0.0505\mathbf{i}& -0.1479+0.1096\mathbf{i}& -0.1576+0.2048\mathbf{i}& 0.2842-0.0579\mathbf{i}& 0.0791+0.0871\mathbf{i}& -0.1743+0.1854\mathbf{i}\\ 0.2495-0.1127\mathbf{i}& -0.1168+0.1075\mathbf{i}& 0.2459-0.0555\mathbf{i}& 0.1508+0.0639\mathbf{i}& -0.1420-0.2668\mathbf{i}& -0.0037+0.1078\mathbf{i}\\ -0.4449-0.2074\mathbf{i}& -0.2231+0.0297\mathbf{i}& 0.1255-0.0258\mathbf{i}& -0.2640+0.1617\mathbf{i}& 0.0398+0.2650\mathbf{i}& -0.3123+0.2548\mathbf{i}\\ 0.2953+0.0149\mathbf{i}& 0.1418+0.0568\mathbf{i}& -0.0295+0.1790\mathbf{i}& -0.2309+0.0448\mathbf{i}& 0.2169+0.2944\mathbf{i}& 0.1615+0.1433\mathbf{i}\\ 0.0385+0.1177\mathbf{i}& 0.1669-0.0418\mathbf{i}& 0.2708-0.3069\mathbf{i}& -0.2373+0.0416\mathbf{i}& -0.2154-0.2440\mathbf{i}& -0.0079+0.1162\mathbf{i}\\ -0.0826-0.1074\mathbf{i}& -0.0056-0.0327\mathbf{i}& -0.0647-0.3652\mathbf{i}& 0.3285-0.2423\mathbf{i}& 0.1011+0.1815\mathbf{i}& 0.0755+0.0775\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill V_1=\left(\begin{array}{c}\begin{array}{ccccccc}-0.3415+0.1010\mathbf{i}& 0.0300-0.2416\mathbf{i}& 0.1807+0.1893\mathbf{i}& 0.0558-0.1649\mathbf{i}& -0.0886+0.1240\mathbf{i}& -0.1395-0.1271\mathbf{i}& 0.0734+0.2405\mathbf{i}\\ 0.2890+0.0317\mathbf{i}& -0.3653-0.0220\mathbf{i}& -0.0635-0.1062\mathbf{i}& 0.1117+0.1213\mathbf{i}& 0.2928+0.0604\mathbf{i}& -0.0117+0.0348\mathbf{i}& -0.1816-0.1726\mathbf{i}\\ 0.0874+0.2393\mathbf{i}& 0.2122-0.0524\mathbf{i}& 0.0238-0.0688\mathbf{i}& 0.0848-0.0480\mathbf{i}& -0.4611+0.0309\mathbf{i}& 0.3490+0.0988\mathbf{i}& -0.1682-0.0256\mathbf{i}\\ 0.3112-0.2922\mathbf{i}& -0.0508-0.0055\mathbf{i}& 0.2356+0.2536\mathbf{i}& -0.3086+0.0646\mathbf{i}& -0.2476-0.1863\mathbf{i}& 0.1613-0.0736\mathbf{i}& 0.3172+0.0636\mathbf{i}\\ 0.1966-0.1029\mathbf{i}& 0.0407+0.1393\mathbf{i}& -0.0874+0.3287\mathbf{i}& 0.2014-0.4036\mathbf{i}& -0.1963+0.2617\mathbf{i}& -0.3328+0.1715\mathbf{i}& 0.0904-0.3965\mathbf{i}\\ -0.1763+0.3055\mathbf{i}& -0.0640+0.3116\mathbf{i}& 0.0264+0.2972\mathbf{i}& -0.1110-0.1227\mathbf{i}& 0.1530-0.0536\mathbf{i}& 0.3092-0.5074\mathbf{i}& -0.0026-0.3738\mathbf{i}\\ 0.0089-0.1151\mathbf{i}& 0.0295-0.2376\mathbf{i}& 0.0286-0.2216\mathbf{i}& 0.0459-0.2883\mathbf{i}& 0.0381-0.2235\mathbf{i}& 0.0328-0.1129\mathbf{i}& 0.0325-0.1580\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill V_2=\left(\begin{array}{c}\begin{array}{ccccccc}-0.1297-0.1675\mathbf{i}& 0.1617-0.0854\mathbf{i}& -0.0165+0.0411\mathbf{i}& -0.1585+0.1909\mathbf{i}& 0.0221+0.0319\mathbf{i}& -0.1052-0.0653\mathbf{i}& 0.2073-0.1539\mathbf{i}\\ -0.1460-0.0398\mathbf{i}& 0.3177+0.0404\mathbf{i}& -0.2142-0.0586\mathbf{i}& -0.1842-0.1175\mathbf{i}& 0.1187+0.1507\mathbf{i}& -0.1224-0.1249\mathbf{i}& 0.1848+0.1410\mathbf{i}\\ 0.2523+0.0383\mathbf{i}& 0.1733-0.0432\mathbf{i}& -0.1905-0.0448\mathbf{i}& -0.0875+0.1668\mathbf{i}& -0.0881-0.0385\mathbf{i}& 0.2080+0.0938\mathbf{i}& -0.0417-0.2169\mathbf{i}\\ -0.0506+0.1462\mathbf{i}& 0.0356-0.1132\mathbf{i}& -0.1192-0.0120\mathbf{i}& 0.0142-0.0373\mathbf{i}& 0.1457+0.0954\mathbf{i}& -0.0786+0.0005\mathbf{i}& -0.0253+0.0135\mathbf{i}\\ -0.0858-0.0843\mathbf{i}& -0.0128-0.0194\mathbf{i}& 0.0478+0.0345\mathbf{i}& -0.1027-0.0854\mathbf{i}& 0.0893+0.0816\mathbf{i}& 0.1421+0.1412\mathbf{i}& -0.0558-0.0372\mathbf{i}\\ -0.0696-0.0422\mathbf{i}& -0.0683-0.0288\mathbf{i}& -0.0686-0.0144\mathbf{i}& 0.0534-0.0184\mathbf{i}& 0.0216+0.0195\mathbf{i}& 0.1322+0.0649\mathbf{i}& 0.1099+0.0117\mathbf{i}\\ 0.0581-0.0545\mathbf{i}& 0.1136-0.1148\mathbf{i}& 0.1071-0.1088\mathbf{i}& 0.1218-0.1328\mathbf{i}& 0.0985-0.1081\mathbf{i}& 0.0376-0.0536\mathbf{i}& 0.0579-0.0704\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill V_3=\left(\begin{array}{c}\begin{array}{ccccccc}-0.1006+0.1861\mathbf{i}& 0.4325+0.0891\mathbf{i}& -0.2664-0.1355\mathbf{i}& -0.0683-0.0691\mathbf{i}& -0.0932-0.0146\mathbf{i}& 0.0766+0.1185\mathbf{i}& -0.0017-0.0174\mathbf{i}\\ -0.0680-0.1072\mathbf{i}& 0.1545+0.0189\mathbf{i}& -0.3050+0.0072\mathbf{i}& -0.0057+0.0969\mathbf{i}& 0.2728-0.0343\mathbf{i}& -0.1181-0.0068\mathbf{i}& -0.0462-0.0838\mathbf{i}\\ 0.0460-0.1893\mathbf{i}& 0.0507-0.0294\mathbf{i}& -0.2515+0.1265\mathbf{i}& 0.0121-0.0096\mathbf{i}& 0.2423+0.0099\mathbf{i}& -0.0580-0.1324\mathbf{i}& -0.0801+0.1028\mathbf{i}\\ 0.1875-0.0357\mathbf{i}& 0.0615+0.1229\mathbf{i}& -0.0588-0.0883\mathbf{i}& -0.3116+0.0288\mathbf{i}& 0.1080-0.0421\mathbf{i}& -0.0381+0.0434\mathbf{i}& 0.2962-0.0591\mathbf{i}\\ -0.0765-0.0494\mathbf{i}& 0.0254-0.0175\mathbf{i}& 0.1073+0.0083\mathbf{i}& -0.1587-0.0325\mathbf{i}& 0.1164+0.0431\mathbf{i}& 0.1268+0.0862\mathbf{i}& -0.1286-0.0023\mathbf{i}\\ -0.0533-0.0247\mathbf{i}& -0.1041-0.0263\mathbf{i}& -0.1371-0.0283\mathbf{i}& 0.1006+0.0248\mathbf{i}& -0.0428+0.0062\mathbf{i}& 0.0822+0.0482\mathbf{i}& 0.1601+0.0455\mathbf{i}\\ 0.1279+0.0821\mathbf{i}& 0.2705+0.1885\mathbf{i}& 0.2540+0.1794\mathbf{i}& 0.3278+0.2372\mathbf{i}& 0.2590+0.1932\mathbf{i}& 0.1360+0.1169\mathbf{i}& 0.1811+0.1386\mathbf{i}\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\left(S_a\right)}_1=\left(\begin{array}{c}\begin{array}{cccc}1& 0& 0& 0\\ 0& 0.4583& 0& 0\\ 0& 0& 0.1069& 0\\ 0& 0& 0& 0.0161\\ 0& 0& 0& 0\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\left(S_a\right)}_2=\left(\begin{array}{c}\begin{array}{cccc}0& 0& 0& 0\\ 0& -0.0940-0.0121\mathbf{i}& 0& 0\\ 0& 0& -0.0106+0.0028\mathbf{i}& 0\\ 0& 0& 0& -0.0070+0.0002\mathbf{i}\\ 0& 0& 0& 0\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\left(S_a\right)}_3=\left(\begin{array}{c}\begin{array}{cccc}0& 0& 0& 0\\ 0& -0.0940+0.0121\mathbf{i}& 0& 0\\ 0& 0& -0.0106-0.0028\mathbf{i}& 0\\ 0& 0& 0& -0.0070-0.0002\mathbf{i}\\ 0& 0& 0& 0\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\left(S_b\right)}_1=\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\right),{\left(S_b\right)}_2=\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\right),{\left(S_b\right)}_3=\left(\begin{array}{c}\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill {\left(S_c\right)}_1=\left(\begin{array}{c}\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\end{array}\right),{\left(S_c\right)}_2=\left(\begin{array}{c}\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\end{array}\right),{\left(S_c\right)}_3=\left(\begin{array}{c}\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\end{array}\right).\end{array}$$

As special cases of T-RSVD, we obtain T-PSVD and T-QSVD for two tensors.

Corollary 1

(T-PSVD [3]). Let $\mathcal{A}\in {\mathbb{C}}^{m\times n_2\times n_3}$ and $\mathcal{B}\in {\mathbb{C}}^{h\times n_2\times n_3}$. Then $\mathcal{A}$ and $\mathcal{B}$ can be factored as

$$\begin{array}{c}\hfill \mathcal{A}=\mathcal{U}\ast {\mathcal{S}}_a\ast \mathcal{P},\mathcal{B}=\mathcal{V}\ast {\mathcal{S}}_b\ast {\mathcal{P}}^{-*},\end{array}$$

where $\mathcal{U}\in {\mathbb{C}}^{m\times m\times n_3}$ and $\mathcal{V}\in {\mathbb{C}}^{h\times h\times n_3}$ are unitary, and $\mathcal{P}\in {\mathbb{C}}^{n_2\times n_2\times n_3}$ is invertible, ${\mathcal{S}}_a$ is a $m\times n_2\times n_3$ f-diagonal tensor, ${\mathcal{S}}_b$ is a $h\times n_2\times n_3$ tensor whose frontal slices have the following form

$$\begin{array}{c}\hfill {\left(S_b\right)}_i=\sum _{j=1}^{n_3}{\omega }^{(j-1)(i-1)}\left(\begin{array}{cccc}{S}_{r_{1j}}& 0& 0& 0\\ 0& 0& I_{r_{3j}}& 0\\ 0& 0& 0& 0\end{array}\right),i=1,\dots ,n_3.\end{array}$$

Corollary 2

(T-QSVD [3]). Let $\mathcal{A}\in {\mathbb{C}}^{m\times n_2\times n_3}$ and $\mathcal{B}\in {\mathbb{C}}^{h\times n_2\times n_3}$. Then, $\mathcal{A}$ and $\mathcal{B}$ can be factored as

$$\begin{array}{c}\hfill \mathcal{A}=\mathcal{U}\ast {\mathcal{S}}_a\ast \mathcal{P},\mathcal{B}=\mathcal{V}\ast {\mathcal{S}}_b\ast \mathcal{P},\end{array}$$

where $\mathcal{U}\in {\mathbb{C}}^{m\times m\times n_3}$ and $\mathcal{V}\in {\mathbb{C}}^{h\times h\times n_3}$ are unitary, $\mathcal{P}\in {\mathbb{C}}^{n_2\times n_2\times n_3}$ is invertible, ${\mathcal{S}}_a$ is a $m\times n_2\times n_3$ f-diagonal tensor, ${\mathcal{S}}_b$ is a $h\times n_2\times n_3$ tensor whose frontal slices have the following forms

$$\begin{array}{c}\hfill {\left(S_b\right)}_i=\sum _{j=1}^{n_3}{\omega }^{(j-1)(i-1)}\left(\begin{array}{ccc}0& 0& 0\\ 0& I_{r_j}& 0\end{array}\right),i=1\dots ,n_3.\end{array}$$

4. An Application from Color Image Watermarking Processing

In this section, we use a third-order tensor, $\mathcal{A}\in {\mathbb{C}}^{m\times n\times 3}$ stands for a color image, where the number “3” represents the three channels of RGB, and m and n denote the height and width of the image. Then, we present the T-RSVD-based color image watermarking schemes. At present, the research on the algorithm of adding a gray or color watermark to a color image is very rich. Moreover, B. Harjito et al. proposed the method of adding two gray watermarks to a gray image [23]. However, to our knowledge, there has been little work on adding three color watermarks to one color image. Based on T-RSVD, we implant three color watermarks into a color image at the same time and only save two keys to ensure the extraction process.

Let $\mathcal{F}$ be a color host image of size $M\times M\times 3$. Three color watermark images of the same size $N\times N\times 3$, i.e., $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are simultaneously inserted into the color host image. The formal procedures of the T-RSVD-based color watermark embedding are given as follows.

A1. 

T-RSVD-based decomposes three color watermark images $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$,

$$\mathcal{A}=\mathcal{P}\ast {\mathcal{S}}_a\ast \mathcal{Q},$$

$$\mathcal{B}=\mathcal{P}\ast {\mathcal{S}}_b\ast \mathcal{V},$$

$$\mathcal{C}=\mathcal{U}\ast {\mathcal{S}}_c\ast \mathcal{Q},$$

where $\mathcal{P}$ and $\mathcal{U}$ are the secret keys and are saved to extract the implanted watermarks.

A2. 

Calculate the main components of each color watermark image,

$${\mathcal{A}}_w\Leftarrow {\mathcal{S}}_a\ast \mathcal{Q},{\mathcal{B}}_w\Leftarrow {\mathcal{S}}_b\ast \mathcal{V},{\mathcal{C}}_w\Leftarrow {\mathcal{S}}_c\ast \mathcal{Q}.$$

A3. 

Orthogonal transformation divides the color host image $\mathcal{F}$ into several non-overlapping color image blocks of size $N\times N\times 3$, and ${\mathcal{T}}_{ij}\in {\mathbb{C}}^{N\times N\times 3}$ is the orthogonal transformed image on block position $(i,j)$, $i,j=1,\dots ,m$, where $m=\frac{M}{N}$.

A4. 

Implant the main components ${\mathcal{A}}_w$, ${\mathcal{B}}_w$ and ${\mathcal{C}}_w$ into the transformed color host image blocks ${\mathcal{T}}_{11}$, ${\mathcal{T}}_{21}$ and ${\mathcal{T}}_{12}$:

$${\mathcal{T}}_{11}\Leftarrow {\mathcal{T}}_{11}+\alpha {\mathcal{A}}_w,$$

$${\mathcal{T}}_{21}\Leftarrow {\mathcal{T}}_{21}+\alpha {\mathcal{B}}_w.$$

$${\mathcal{T}}_{12}\Leftarrow {\mathcal{T}}_{12}+\alpha {\mathcal{C}}_w,$$

where $\alpha$ is a scale factor used to control watermarking strength [23]. When the value of $\alpha$ is actually confirmed, the larger $\alpha$ is, the stronger the robustness of the watermark is, and the weaker the invisibility of the watermark is. The balance between robustness and invisibility needs to be considered. Save ${\mathcal{T}}_{11}$, ${\mathcal{T}}_{21}$ and ${\mathcal{T}}_{12}$ to guarantee the extraction.

A5. 

Implement inverse orthogonal transformation for all image blocks ${\mathcal{T}}_{ij}$, and gain the watermarked color image ${\mathcal{F}}_w$.

The formal procedure of the T-RSVD-based color watermark extraction process is given as follows:

B1. 

Split the watermarked color image ${\mathcal{F}}_w$ into several non-overlapping color image blocks ${\widehat{\mathcal{T}}}_{ij}$ with size $N\times N\times 3$, $i,j=1,\dots ,m$.

B2. 

Extract the main components of each color watermark image as follows:

$${\widehat{\mathcal{A}}}_w\Leftarrow \frac{1}{\alpha }({\widehat{\mathcal{T}}}_{11}-{\mathcal{T}}_{11}),$$

$${\widehat{\mathcal{B}}}_w\Leftarrow \frac{1}{\alpha }({\widehat{\mathcal{T}}}_{21}-{\mathcal{T}}_{21}),$$

$${\widehat{\mathcal{C}}}_w\Leftarrow \frac{1}{\alpha }({\widehat{\mathcal{T}}}_{12}-{\mathcal{T}}_{12}).$$

B3. 

Calculate the extracted watermarks $\widehat{\mathcal{A}}$, $\widehat{\mathcal{B}}$, and $\widehat{\mathcal{C}}$:

$$\widehat{\mathcal{A}}\Leftarrow \mathcal{P}\ast {\widehat{\mathcal{A}}}_w,$$

$$\widehat{\mathcal{B}}\Leftarrow \mathcal{P}\ast {\widehat{\mathcal{B}}}_w,$$

$$\widehat{\mathcal{C}}\Leftarrow \mathcal{U}\ast {\widehat{\mathcal{C}}}_w.$$

The imperceptibility of the proposed schemes can be measured by the Peak-Signal-to-Noise-Ratio (PSNR) [23], when the value of PSNR is greater than 30, the signal distortion is less. Now, we present the basic framework for implanting three color watermarks concurrently into a color image, as well as a frame diagram for the extraction of the color watermarked image in Figure 1 and Figure 2.

In this experimentation, we use the color image “Trees” of $512\times 512$ pixels as the host image and three color images “SHU Logo”, “NUS Logo” and “Flower” as the watermarks to be implanted, which have the same size $128\times 128$. Choosing a scaling factor $\alpha =0.05$, we utilize the above framework to address the practical color image watermarking problem by 2-level Haar Discrete Wavelet Transform (DWT) [24]. To examine the robustness, we add Gaussian noise with a variance of $0.001$ to watermarked images during the watermark extraction process. Additionally, we apply the same framework to T-SVD for the comparison. Since T-SVD can only compute the decomposition of one tensor at a time, in both A1–5 and B1–3, we only process one watermark at a time, and finally repeat the whole process of watermarking three times to complete the embedding/extracting of watermarks. For this reason, we mainly consider the following three orders.

• Order 1: SHU Logo →NUS Logo →Flower;
• Order 2: NUS Logo →Flower →SHU Logo;
• Order 3: Flower →NUS Logo →SHU Logo.

As for the T-RSVD-based algorithm, since it embeds watermarks at one time, for different orders we only change the embedding position of watermarks, i.e., in Order 1 we embed the watermark “SHU Logo” in $T_{11}$, while in Order 2 we embed the watermark “NUS Logo” in $T_{11}$. Similarly, we also set $\alpha =0.05$ for the T-SVD-based algorithm, and adjust the variance of Gaussian noise based on the approximate PSNR for a fair comparison. The results are shown in

Table 1. Comparison with T-SVD-based image watermarking.

	Method	PSNR (dB)				Time (s)
		Trees	SHU Logo	NUS Logo	Flower	Embedding	Extracting
Order 1	T-RSVD	36.5237	34.0030	33.9725	33.9886	1.1875	0.0937
	T-SVD	25.2867	24.1887	24.7375	73.9727	1.9218	0.0937
Order 2	T-RSVD	36.5237	34.0198	33.9450	33.9722	1.1250	0.0937
	T-SVD	12.4073	73.9771	52.6427	52.1781	2.1562	0.1250
Order 3	T-RSVD	36.5237	33.9610	33.9461	33.9553	1.2187	0.0781
	T-SVD	17.6431	73.9833	25.3310	24.7176	2.1093	0.0937

and Figure 3 (these algorithms have been coded in MATLAB R2018b and executed on a laptop computer with Core 2 Duo-2.60 GHz CPU).

Table 1 demonstrates that our T-RSVD-based watermarking algorithm has more reliable imperceptibility, robustness and less watermark embedding time than that based on T-SVD. It can also be seen that images processed by our proposed algorithm remain highly consistent regardless of the order of watermark embedding, whereas T-SVD-based ones yield significant variations depending on the orders: the PSNR values of watermarked images “Trees” differ, and the last embedded watermark has the highest PSNR value, which differs greatly from the other two. It shows the fact, as Mintzer and Braudaway [19] pointed out, that the order of watermark embedding is of paramount importance when multiple watermarks need to meet different application requirements. The process of embedding a watermark can potentially alter the content, and hence, influence the effectiveness of subsequent watermark insertions. Our work tackles this issue head-on. It should also be noted when extracting the watermark added Gaussian noise with a variance of $0.00001$ (instead of $0.001$) to watermarked images. This is because the T-SVD-based framework constructed in this paper (embedding watermark three times repeatedly) is sensitive to noise, which can be easily observed from Figure 3. In particular, since there is not much difference in the images generated by T-RSVD, we only present the results by T-RSVD with Order 1 in Figure 3.

The watermarking framework based on T-RSVD has the following advantages. First, three color watermarks can be handled concomitantly. Secondly, only two keys need to be stored, and three color watermarks can be extracted (in contrast, T-SVD requires storing three keys). Furthermore, through the numerical experiment, it can be seen that the proposed methods have responsible imperceptibility and security.

5. Conclusions and Prospects

In this paper, we first establish the theoretical foundation of restricted singular value decomposition for three tensors under t-product (T-RSVD). Following this, we present an algorithm for computing T-RSVD and provide a numerical example to verify the accuracy and effectiveness of the algorithm. Additionally, we derive two corollaries from T-RSVD, namely T-PSVD and T-QSVD, which are SVDs for two tensors under t-product.

One key tool of our approach is Formula (3). Indeed, this is due to the structure, i.e., the circulant matrix, exhibited by the tensor when applying t-product. This characteristic of t-product has prompted us to apply T-RSVD to color image processing. As we mentioned earlier, a color image $\mathcal{F}$ can correspond to a third-order tensor $\mathcal{A}$, and each column (or row) of $bcirc\left(\mathcal{A}\right)$, which consists of all three frontal slices of $\mathcal{A}$, corresponds to the data from the RGB channels of $\mathcal{F}$. It implies that performing a t-product on two third-order tensors is equivalent to transforming two color images into one image. It is noteworthy that this kind of transformation occurs simultaneously on the RGB three channels, which is of great significance for preserving the information characteristics of images. Consequently, we propose a T-RSVD-based watermarking algorithm, which embeds three watermarks into a given host color image simultaneously. Meanwhile, as far as we know, there is limited research on embedding three watermarks simultaneously, which demonstrates the superiority and novelty of our proposed watermarking algorithm. Nevertheless, the watermarking framework based on the T-RSVD that we proposed is still preliminary and foundational. Further improvements are needed, such as determining the value of watermarking strength factor $\alpha$, selecting the orthogonal transformation method and so on. We plan to introduce more model comparisons in our future work and consider a wider range of applications.

Particularly, note that quaternions are also an essential tool for color image processing. This observation paves the way for our future work, namely, studying the SVD for quaternion tensors under t-product and some other similar decompositions, as well as their applications, such as embedding dynamic watermarks into color videos. However, it should be emphasized that due to the existence of the fundamental quaternion units $\mathsf{i},\mathsf{j}$, and $\mathsf{k}$, Formula (3) no longer holds in this case. Therefore, further research is required. On the other hand, very recently, Chen, Wang, and Xie (2024) [25] presented the solvability and the expression of general solutions to the classical matrix equation $AXB=C$ regarding dual quaternions. In a parallel development, Zhang, Wang, and Xie (2024) [26] investigated the Hermitian solution of a system of matrix equations over commutative quaternions, which consists of $AXB=C$, one generalized Sylvester equation $DXE+FYG+HZ+WI=J$, and some additional equations of the forms $KX=L$ and $XM=N$. It is well-known that the matrix equation $AXB=C$ plays a crucial role in image processing. These may also provide an opportunity for further research on image processing based on generalized quaternions.

We hope that our work will be helpful to those who are researching multiple watermarking algorithms.