SISR is a technique that aims at restoring a high-resolution (HR) image from a single degraded low-resolution (LR) image. Since the HR image contains more details than the LR one, the image of HR is preferred in many practical cases, such as video surveillance, the hyperspectral technique, and remote sensing. Super-resolution (SR) is a typical ill-posed inverse problem since a multiplicity of solutions exist for any input LR pixel [
1]. To tackle such a problem, most of the SR methods reduce the size of the solution space by incorporating strong prior information, which can be obtained by training data, using various regularization methods, and capturing specific image features [
2]. Motivated by those ideas, the SISR schemes can be broadly divided into three categories: interpolation-based methods, learning-based methods, and reconstruction-based methods.
Interpolation-based methods such as the bicubic approach are simple and easy to implement but tend to blur the high frequency details and produce aliasing artifacts at salient edges [
3]. The learning-based algorithms recover the missing high frequency details by learning the relations between LR and HR image patches from a given database [
4]. However, they highly rely on the similarity between the training set and the test images and generally have high computational complexity. Reconstruction-based methods that are considered in this paper belong to the third category of SISR schemes [
5‐
7]. As SISR essentially is a highly ill-posed problem, the performance of such approaches mainly relies on the prior knowledge. Among all the priors, smoothness priors such as the total variation (TV) prior have been widely used in many image processing applications [
8]. To reduce its computational complexity, a fast SR alternating direction method of multipliers (FSR-ADMM) based on the TV model is proposed in [
9]. Considering the efficiency of a symmetric alternating direction method of multipliers (SADMM) [
10], our paper aims at constructing a fast SR symmetric alternating direction method of multipliers (FSR-SADMM).
In the SISR problem, the observed LR image is modeled as a noisy version of the blurred and downsampled HR image estimated as follows:
$$ y=\Phi Hx+\nu, $$
(1.1)
where the vector
\(y \in\mathbb{R}^{N_{l}}\) (
\(N_{l}=m_{l}\times n_{l}\)) corresponds to the LR observed image and
\(x \in\mathbb{R}^{N_{h}}\) (
\(N_{h}= m_{h}\times n_{h}\)) denotes the HR image with
\(N_{h}> N_{l}\).
\(\nu\in\mathbb{R}^{N_{l}}\) represents a zero-mean additive white Gaussian noise (AWGN),
\(\Phi \in\mathbb{R}^{N_{l}\times N_{h}}\) and
\(H \in\mathbb {R}^{N_{h}\times N_{h}}\) stand for the downsampling and the blurring operations, respectively.
Concisely, the SISR TV model corresponds to solving the following optimization problem:
$$ \min_{x} \frac{1}{2} \underbrace{\|y - \Phi Hx \|_{2}^{2}}_{\text{data fidelity}} + \alpha \underbrace{\|x \|_{\mathrm{TV}}}_{\text{TV regularization}}, $$
(1.2)
where
\(\|y - \Phi Hx\|_{2}^{2}\) stands for the data fidelity term while
\(\|x\|_{\mathrm{TV}}=\phi(Ax)= \sqrt{\|\nabla_{h}x\| _{2}^{2}+\|\nabla_{\nu}x\|_{2}^{2}}\) represents the regularization
1 prior with
\(A=[\nabla _{h},\nabla_{\nu}]^{\mathsf{T}}\in \mathbb{R}^{2N_{h}\times N_{h}}\), and
α denotes the regularization parameter balancing the regularization term and the data fidelity term. The direct ADMM in [
6] is given hereinafter which first rewrites the problem (
1.2) as
$$\begin{aligned} &\min_{x,z,u} \frac{1}{2} \|y - \Phi {z} \|_{2}^{2} + \alpha \phi(u) \\ &\quad \text{s.t. } H x = z, \\ &\hphantom{\quad \text{s.t. }}A x = u, \end{aligned}$$
and adopts the following iterative scheme:
$$\begin{aligned}& x^{k+1} = \mathop{\operatorname {argmin}}_{x} \frac{\mu}{2}\bigl\Vert {H x}- z^{k} +d^{k}_{1}\bigr\Vert _{2}^{2} + \frac{\mu}{2}\bigl\Vert Ax - u^{k} +d^{k}_{2}\bigr\Vert _{2}^{2}, \end{aligned}$$
(1.3a)
$$\begin{aligned}& z^{k+1} = \mathop{\operatorname {argmin}}_{z} \frac{1}{2} \Vert y - \Phi z \Vert _{2}^{2} +\frac{\mu}{2}\bigl\Vert Hx^{k+1} - z +d^{k}_{1} \bigr\Vert _{2}^{2}, \end{aligned}$$
(1.3b)
$$\begin{aligned}& u^{k+1} = \mathop{\operatorname {argmin}}_{u} \alpha \phi(u) + \frac{\mu}{2}\bigl\Vert Ax^{k+1} - u +d^{k}_{2} \bigr\Vert _{2}^{2}, \end{aligned}$$
(1.3c)
$$\begin{aligned}& d^{k+1}_{1}= d^{k}_{1} +\bigl(\Phi x^{k+1} - z^{k+1}\bigr), \end{aligned}$$
(1.3d)
$$\begin{aligned}& d^{k+1}_{2}= d^{k}_{2} + \bigl(Ax^{k+1} - u^{k+1}\bigr). \end{aligned}$$
(1.3e)
Recently, Zhao
et al. [
9] proposed a fast single image super-resolution by adopting a new efficient analytical solution for
\(\ell_{2}\)-norm regularized problems, which can reduce the number of iterations in each loop from five steps to three steps by tackling the downsampling operator Φ and the blurring operator
H simultaneously. By rewriting (
1.2) as the following constrained optimization problem:
$$ \begin{aligned} &\min_{x,u} \frac{1}{2} \|y - \Phi Hx \|_{2}^{2} + \alpha \phi(u) \\ &\quad \text{s.t. } A x = u, \end{aligned} $$
(1.4)
they proposed the easier ADMM-type iterative scheme,
i.e., FSR-ADMM:
$$\begin{aligned}& x^{k+1}=\mathop{\operatorname {argmin}}_{x} \|y-\Phi Hx \|_{2}^{2}+\mu\bigl\| Ax-u^{k}+d^{k} \bigr\| _{2}^{2}, \end{aligned}$$
(1.5a)
$$\begin{aligned}& u^{k+1} = \mathop{\operatorname {argmin}}_{u} \alpha \phi(u) + \frac{\mu}{2}\bigl\| Ax^{k+1}-u+d^{k} \bigr\| _{2}^{2}, \end{aligned}$$
(1.5b)
$$\begin{aligned}& d^{k+1} = d^{k} + \bigl(Ax^{k+1}-u^{k+1} \bigr). \end{aligned}$$
(1.5c)
Note that (
1.5a) is the classical least square problem and has the solution given by
$$ x^{k+1}=\bigl(H^{\mathsf{T}}\Phi^{\mathsf{T}}\Phi H+ \mu A^{\mathsf{T}}A\bigr)^{-1}\bigl(H^{\mathsf{T}}\Phi^{\mathsf{T}}y+\mu A^{\mathsf{T}}\bigl(u^{k}-d^{k} \bigr)\bigr). $$
(1.6)
As to such an expensive inversion process, the methods to alleviate the computational burden can be roughly categorized into two main categories, one is to ideally assume
\(A^{\mathsf{T}}A=I\). Then such formula can be solved efficiently by a Thomas algorithm in
\(32N_{h}^{2}\) flops [
11]. The other option is to establish a block circulant matrix with circulant blocks (BCCB)
A. Under such conditions, the Woodbury formula can be utilized to decrease the order of computation complexity from
\(\mathcal{O} (8N_{h}^{3})\) to
\(\mathcal{O}(2N_{h}\log2N_{h})\) [
9]. Nevertheless, in realistic settings the problem is that one does not necessarily know in advance if the BCCB assumption is good for SISR because it may lead to the appearance of ringing artifacts emanating from the boundaries, which is a well-known mismatch and degradation consequence of such deconvolved images [
12].
To alleviate the above dilemma and further apply FSR-ADMM into wider scenarios with proper matrix A, we propose a new method with two-fold solution. (1) Compute \((H^{\mathsf{T}}\Phi^{\mathsf{T}}\Phi H+\frac{\mu}{\tau }I_{N_{h}})^{-1}\) instead of computing \((H^{\mathsf{T}}\Phi^{\mathsf{T}}\Phi H+ {\mu} A^{\mathsf{T}}A)^{-1}\) so as to bypass the need of special condition of A. (2) Accelerate FSR-ADMM by introducing a new dual multiplier \(\lambda ^{k+\frac{1}{2}} \).
In light of the above analysis, this paper proposes the following iterative scheme based on semiproximal symmetric ADMM (FSR-SADMM):
2
$$\begin{aligned}& x^{k+1}= \biggl(H^{\mathsf{T}}\Phi^{\mathsf{T}}\Phi H+ \frac{\mu}{\tau}I_{N_{h}}\biggr)^{-1} \biggl(H^{\mathsf{T}}\Phi^{\mathsf{T}}y+\frac{\mu}{\tau}x^{k}-\mu A^{\mathsf{T}}\biggl(Ax^{k}-u^{k}-\frac {\lambda^{k}}{\mu}\biggr) \biggr), \end{aligned}$$
(1.7a)
$$\begin{aligned}& \lambda^{k+\frac{1}{2}} = \lambda^{k} - r\mu\bigl(Ax^{k+1}- \lambda^{k}\bigr), \end{aligned}$$
(1.7b)
$$\begin{aligned}& u^{k+1} = \mathop{\operatorname {argmin}}_{u} \alpha \phi(u) + \frac{\mu}{2}\bigl\Vert Ax^{k+1}-u-\lambda^{k+\frac{1}{2}}/\mu \bigr\Vert _{2}^{2}, \end{aligned}$$
(1.7c)
$$\begin{aligned}& \lambda^{k+1} = \lambda^{k}- s\mu\bigl(Ax^{k+1}- \lambda^{k+1}\bigr). \end{aligned}$$
(1.7d)
In a nutshell, the contributions of this article can be summarized as follows:
1.
We propose a customized semiproximal term especially suitable for SR imaging system which can avoid computing some boring matrix inversion such as
\((H^{\mathsf{T}}\Phi^{\mathsf{T}}\Phi H+\mu A^{\mathsf{T}}A)^{-1}\) existing in (
1.5a). Consequently, FSR-SADMM can be applied in wider scenarios without a strict condition of
A.
2.
Based on the iterative scheme of strictly contractive semiproximal Peaceman-Rachford splitting method, our proposed FSR-SADMM can significantly reduce the total iteration count while only involving additional dual update (i.e., \(\lambda^{k+\frac {1}{2}}\)) which added negligible computational burden for each iteration. As a result, our proposed FSR-SADMM prefers to maintain a better convergence rate which can experimentally reduce the computing time by 40%.
3.
We prove that the FSR-SADMM is convergent under mild conditions.
The rest of this paper is organized as follows. In Section
2, we present some preliminaries which are useful for the subsequent analysis. In Section
3, we illustrate the FSR-SADMM to reconstruct the single image super-resolution. Section
4 provides convergence analysis of the proposed method. Section
5 presents extensive numerical results that evaluate the performance of the proposed reconstruction algorithm in comparison with some state-of-the-art algorithms. Finally, concluding remarks are provided in Section
6.