Solving total-variation image super-resolution problems via proximal symmetric alternating direction methods

The single image super-resolution (SISR) problem represents a class of efficient models appealing in many computer vision applications. In this paper, we focus on designing a proximal symmetric alternating direction method of multipliers (SADMM) for the SISR problem. By taking full exploitation of the special structure, the method enjoys the advantage of being easily implementable by linearizing the quadratic term of subproblems in the SISR problem. With this linearization, the resulting subproblems easily achieve closed-form solutions. A global convergence result is established for the proposed method. Preliminary numerical results demonstrate that the proposed method is efficient and the computing time is saved by nearly 40% compared with several state-of-the-art methods.


Introduction
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 []. 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 []. 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 []. The learning-based algorithms recover the missing high frequency details by learning the relations between LR and HR image patches from a given database []. 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 [-]. 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 []. To reduce its computational complexity, a fast SR alternating direction method of multipliers (FSR-ADMM) based on the TV model is proposed in []. Considering the efficiency of a symmetric alternating direction method of multipliers (SADMM) [], 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: where the vector y ∈ R N l (N l = m l × n l ) corresponds to the LR observed image and x ∈ R N h (N h = m h × n h ) denotes the HR image with N h > N l . ν ∈ R N l represents a zero-mean additive white Gaussian noise (AWGN), ∈ R N l ×N h and H ∈ R N h ×N h stand for the downsampling and the blurring operations, respectively. Concisely, the SISR TV model corresponds to solving the following optimization problem: where y -Hx   stands for the data fidelity term while and α denotes the regularization parameter balancing the regularization term and the data fidelity term. The direct ADMM in [] is given hereinafter which first rewrites the problem (.) as and adopts the following iterative scheme: Recently, Zhao et al. [] proposed a fast single image super-resolution by adopting a new efficient analytical solution for  -norm regularized problems, which can reduce the number of iterations in each loop from five steps to three steps by tackling the downsampling operator and the blurring operator H simultaneously. By rewriting (.) as the following constrained optimization problem: they proposed the easier ADMM-type iterative scheme, i.e., FSR-ADMM: Note that (.a) is the classical least square problem and has the solution given by 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 T A = I. Then such formula can be solved efficiently by a Thomas algorithm in N  h flops []. 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 compu- . 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 [].
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. () Compute (H T T H + μ τ I N h ) - instead of computing (H T T H + μA T A) - so as to bypass the need of special condition of A. () Accelerate FSR-ADMM by introducing a new dual multiplier λ k+   . In light of the above analysis, this paper proposes the following iterative scheme based on semiproximal symmetric ADMM (FSR-SADMM): b In a nutshell, the contributions of this article can be summarized as follows: . We propose a customized semiproximal term especially suitable for SR imaging system which can avoid computing some boring matrix inversion such as Consequently, FSR-SADMM can be applied in wider scenarios without a strict condition of A. . 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., λ k+   ) 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 %. . We prove that the FSR-SADMM is convergent under mild conditions. The rest of this paper is organized as follows. In Section , we present some preliminaries which are useful for the subsequent analysis. In Section , we illustrate the FSR-SADMM to reconstruct the single image super-resolution. Section  provides convergence analysis of the proposed method. Section  presents extensive numerical results that evaluate the performance of the proposed reconstruction algorithm in comparison with some stateof-the-art algorithms. Finally, concluding remarks are provided in Section .

Variational reformulation of (1.4)
In this section, following He and Yuan's approach [], we reformulate the convex minimization model (.) as a variational form, which is useful for succedent algorithmic illustration and convergence analysis. Let us denote The Lagrangian function and augmented Lagrangian function of (.) can be written as That is, for any (z  , z  , λ), we have and Note that the mapping F(w) defined in (.) is affine with a skew-symmetric matrix, it is monotone. We denote by * the solution set of VI( , F, θ ).

Some notations
We use · to denote the -norm of a vector and let z  G = z T Gz for z ∈ R N and G ∈ R N×N . For a real symmetric matrix S, we denote S  (S ) if S is positive semidefinite (positive definite). For convenient analysis, we define the following matrices: Below we prove three assertions regarding the matrices just defined. These assertions make it possible to present our convergence analysis for the new algorithm compactly with alleviated notation. Proof We consider two cases.
(I) r ∈ (, ), s ∈ (, ). Since R , B  is assumed to be full column rank, we only need to check that Note that Then we havē H .
Therefore, the assertion H  is verified. Under the definition of the matrices H, M, and Q, by a simple manipulation, we get The second assertion HM = Q is proved. Consequently, we have Using (.), (.), and the above equation, we get Similarly, note that Therefore, the matrix G is positive definite.
(II) r ∈ (, ) and s = . Note that Thus H and G are positive definite and positive semidefinite, respectively. Here, we would emphasize that we do not require the positive definiteness of G. Instead, positive semidefiniteness of G is enough for our algorithmic analysis.

FSR-SADMM
In this section, we will present our new algorithm to solve (.). But we first present the iterative scheme by using the standard strictly contractive Peaceman-Rachford splitting method with two different relaxation factors: By introducing a customized semiproximal term especially for TV super-resolution imaging, our FSR-SADMM has the iterative scheme

Related methods
The methods proposed in [] and [] belong to the category of SADMM-type approaches with logarithmic-quadratic proximal regularization, which have larger step sizes compared with SADMM. The former is with the step sizes of r ∈ (, ), s ∈ (, ) while the latter s ∈ (, ), r ∈ (, s), which are different from r ∈ (, ), s ∈ (, ] considered in this paper. The parameters r and s in SADMM have also been studied intensively by He et al. [] and Gu et al. []. However, both of them cannot fully utilize the special structure of SISR scenarios involved in this paper. And computing (H T T H + μA T A) - cannot be avoided.

Global convergence
To make the analysis more elegant, we reformulate FSR-SADMM (.a)-(.d) into the form c Now we analyze the convergence for our proposed FSR-SADMM (.a)-(.d). We prove its global convergence from the contraction perspective. In order to further alleviate the notation in our analysis, we define an auxiliary sequencew k as where (z k+  , z k+  ) is produced by (.a) and (.c), and we immediately get and Moreover, we have the following relationship: which can be reformulated as a compact form under the notation of w k andw k : where M is defined in (.). Now we start to prove some properties for the sequence {w k } defined in (.). We are interested in estimating how accurate the pointw k is to a solution point w * of VI( , F, θ ). The main result is proved in Theorem .. To prove this main result, we require two lemmas. The first key lemma provides a lower bound on specially constructed functional in terms of a quadratic term involving the matrix Q defined in (.).

Lemma . For given w k ∈ , let w k+ be generated by FSR-SADMM (.a)-(.d) andw k be defined in (.). Then we havew ∈ and
where Q is defined in (.).
Proof From the first-order optimality condition of z  -subproblem in (.a), for any z  ∈ R N h , we obtain By the definition ofz k  andλ k in (.), the above inequality can be written as Similarly, from the first-order optimization condition of z  -subproblem in (.c), we achieve From the definitions of λ k+   ,z k  andλ k , we get Substituting this into (.) and considering the definition ofz k  in (.), we have In addition, as follows from (.) again, we have Combining (.), (.), and (.), we obtain By using the notation of Q in (.), and w and F in (.), the compact form of the above inequality is exactly (.).
In the next lemma, we further analyze the right-hand side of the inequality (.) and reformulate it as the sum of some quadratic terms. This new form is more convenient for our further analysis.
Lemma . For given w k ∈ , let w k+ be generated by FSR-SADMM (.a)-(.d) andw k be defined in (.). Then for any w ∈ , we get Proof By using Q = HM and M(w k -w k ) = (w kw k+ ) (see (.)), we have For any vectors a, b, c, d ∈ R N h and a matrix H ∈ R n×n , it follows that Applying the above identity with a = w, b =w k , c = w k , and d = w k+ gives Rearranging the last term in the above identity by using (.), (.), and (.), we obtain where the last inequality follows from the fact that w * ∈ * (see (.)). The proof is complete.
Recall that for the case  < r < ,  < s < , the matrix G is positive definite, while when  < r < , s = , G may not be positive definite. We need further investigate the term w k -w k  G . For the case  < r < , s = , we have Notice that Thus, we have Then we get Now, we treat the cross term in the above equation. For the previous iteration, we have Reconsidering s =  in (.d) and applying in (.), we achieve Similarly, we get Setting z  = z k+  and z  = z k  in (.) and (.), respectively, and then adding them, we get   of the Peppers, the wrinkles on the eyes of Lena, and also hair on Boboon's nose. FSR-SADMM can only get a little higher PSNR rather than Direct ADMM and FSR-ADMM for various tolerance. However, the results shown in Figure  indicate that, compared with FSR-ADMM, our FSR-SADMM saves roughly % computational time with high precision Tolerance =  - and can still save roughly % computational time under low precision Tolerance =  - . On the whole average, % computational time can be reduced, which efficiently accelerates the super-resolution imaging process. We also test the influence of relaxation parameter r for FSR-SADMM. We fix s = , μ = . and choose different values of r in the interval [., ] (more specifically, we choose r = {., ., ., ., }). For comparison purposes, we also plot for FSR-ADMM with μ = .. As shown in Figure , we see that the relaxation parameter r works well for a wide range of values. In particular, when r ≥ ., the PSNR cannot be improved and even decrease in the first  iterations, while the effectiveness of the improvement of   The best and second best for each algorithm are indicated by red and blue, respectively, with respect to r = 0.6.
PSNR is not obvious especially when r is less than or equal to .. As such, some offending values for the interval [., .] are preferred. From Table , our proposed FSR-SADMM outperforms ADMM and FSR-ADMM for all the tolerance. Especially, FSR-SADMM costs much less computing time and has a shorter iteration number for tolerance with higher precision such as  - .

Conclusions
For solving the SISR problem (.), we proposed a new algorithm based on the strictly contractive semiproximal Peaceman-Rachford splitting method in this paper. The global convergence of our algorithm is established. Then the computational results indicate that our algorithm achieves better performance compared with start-of-the-art methods including direct ADMM and FSR-ADMM. More specifically with Tolerance =  - , our algorithm can save computing time about % and % compared with direct ADMM and FSR-ADMM, respectively.