Tensor factorization via transformed tensor-tensor product for image alignment

Abstract

In this paper, we study the problem of a batch of linearly correlated image alignment, where the observed images are deformed by some unknown domain transformations, and corrupted by additive Gaussian noise and sparse noise simultaneously. By stacking these images as the frontal slices of a third-order tensor, we propose to utilize the tensor factorization method via transformed tensor-tensor product to explore the low-rankness of the underlying tensor, which is factorized into the product of two smaller tensors via transformed tensor-tensor product under any unitary transformation. The main advantage of transformed tensor-tensor product is that its computational complexity is lower compared with the existing literature based on transformed tensor nuclear norm. Moreover, the tensor $\ell _p$ $(0<p<1)$ norm is employed to characterize the sparsity of sparse noise and the tensor Frobenius norm is adopted to model additive Gaussian noise. A generalized Gauss-Newton algorithm is designed to solve the resulting model by linearizing the domain transformations, and a proximal Gauss-Seidel algorithm is developed to solve the corresponding subproblem. Furthermore, the convergence of the proximal Gauss-Seidel algorithm is established according to Kurdyka-Łojasiewicz property, whose convergence rate is also analyzed. Extensive numerical examples on real-world image datasets are carried out to demonstrate the superior performance of the proposed method as compared to several state-of-the-art methods in both accuracy and computational time.

Appendix

In this section, we give the definitions of the subdifferential and the Kurdyka-${\L}$ojasiewicz (KL) property of a function, respectively, which play a vital role for establishing the convergence and rate of convergence of the proximal Gauss-Seidel algorithm.

Definition 6

[32, Definition 8.3] Consider a function $f:\mathbb {R}^n\rightarrow (\infty ,+\infty ]$ and a point $\overline{\textbf{x}}$ with $f(\overline{\textbf{x}})$ finite. For any $\textbf{x}\in \mathbb {R}^n$, one says that

(i) $\textbf{v}$ is a regular subgradient of f at $\overline{\textbf{x}}$, written as $\textbf{v}\in \widehat{\partial } f(\overline{\textbf{x}})$, if

$$ {\lim \inf }_{\textbf{x}\rightarrow \overline{\textbf{x}},\textbf{x}\ne \overline{\textbf{x}}}\frac{f(\textbf{x})-f(\overline{\textbf{x}})-\langle \textbf{v}, \textbf{x}-\overline{\textbf{x}} \rangle }{\Vert \textbf{x}-\overline{\textbf{x}}\Vert }\ge 0. $$

(ii) $\textbf{v}$ is a subgradient of f at $\overline{\textbf{x}}$, written as $\textbf{v}\in \partial f(\overline{\textbf{x}})$, if there exist sequences $\textbf{x}^k\rightarrow \overline{\textbf{x}}$, $f(\textbf{x}^k)\rightarrow f(\overline{\textbf{x}})$, and $\textbf{v}^k\in \widehat{\partial } f(\textbf{x}^k)$ with $\textbf{v}^k\rightarrow \textbf{v}$.

Definition 7

[44, Definition 2.1] Let $f:\mathbb R^n\rightarrow \mathbb R\cup \{+\infty \}$ be a proper lower semicontinuous function. We say that f has the Kurdyka-${\L}$ojasiewicz (KL) property at point $x^{*}\in \text {dom}(\partial f)$, if there exist a neighborhood U of $x^{*}$, $\eta \in (0,+\infty ]$ and a continuous concave function $\varphi : [0,\eta )\rightarrow \mathbb R_{+}$ such that: (i) $\varphi (0)=0$; (ii) $\varphi$ is $C^1$ on $(0,\eta )$; (iii) for all $s\in (0,\eta )$, $\varphi {'}(s)>0$; (iv) and for all $x \text { in } U \cap [f(x^{*})<f<f(x^{*})+\eta ]$ the KL inequality holds:

$$\begin{aligned} \varphi {'}(f(x)-f(x^{*})) \ {\text {dist}}(0,\partial f(x)) \ge 1. \end{aligned}$$

(45)

If f satisfies the KL property at $x^*\in \text {dom}(\partial f)$, and the $\varphi (s)$ in (45) can be chosen as $\varphi (s)=\hat{c}s^{1-\theta }$ for some $\hat{c}>0$ and $\theta \in [0,1)$, then we say that f satisfies the KL property at point $x^{*}$ with exponent $\theta$.

A proper closed function f satisfying the KL property at every point in $\text {dom}(\partial f)$ is said to be a KL function, and a proper closed function f satisfying the KL property with exponent [0, 1) at every point in $\text {dom}(\partial f)$ is said to be a KL function with exponent $\theta$ [44].