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A new reweighted {l}_{1} minimization algorithm for image deblurring
Journal of Inequalities and Applications volume 2014, Article number: 238 (2014)
Abstract
In this paper, a new reweighted {l}_{1} minimization algorithm for image deblurring is proposed. The algorithm is based on a generalized inverse iteration and linearized Bregman iteration, which is used for the weighted {l}_{1} minimization problem {min}_{u\in {\mathbb{R}}^{n}}\{{\parallel u\parallel}_{\omega}:Au=f\}. In the computing process, the effective using of signal information can make up the detailed features of image, which may be lost in the deblurring process. Numerical experiments confirm that the new reweighted algorithm for image restoration is effective and competitive to the recent stateoftheart algorithms.
1 Introduction
Image deblurring is a fundamental problem in image processing, since many reallife problems can be modeled as deblurring problems [1]. In this paper, a new reweighted {l}_{1} minimization algorithm for image deblurring is proposed. The algorithm is obtained based on a generalized inverse iteration and a linearized Bregman iteration.
Simply, we shall denote images as vectors in {\mathbb{R}}^{n} by concatenating their columns. Let u\in {\mathbb{R}}^{n} be the underlying image. Then the observed blurred image f\in {\mathbb{R}}^{n} is given by
where \eta \in {\mathbb{R}}^{n} is an additive noise and A\in {\mathbb{R}}^{m\times n} is a linear blurring operator. This problem is illposed due to the large condition number of the matrix A. Any small perturbation on the observed blurred image f may cause the direct solution {A}^{1}f, which is very difficult to obtain from the original image u [2]. This is a widely studied subject and many corresponding approaches have been developed, and one of them is to minimize some cost functionals [1]. The simplest method is a Tikhonov regularization, which minimizes an energy consisting of a data fidelity term and an {l}_{2} norm regularization term. A is a convolution, which can solve the problem in the Fourier domain. In this case, the method is called a Wiener filter [3], this is a linear method, and the edges of restored image are usually smeared. To overcome this, a total variation (TV)based regularization was proposed by Rudin et al. in [4], which is known as the ROF model. Due to its virtue of preserving edges, it is widely used in image processing, such as blind deconvolution, inpainting, and superresolution; see [1]. However, as we know, for the TV yields staircasing [5, 6], these TVbased methods do not preserve the fine structures, details, and textures. To avoid these drawbacks, nonlocal methods were proposed for denoising [7, 8], and then extended to deblurring [9]. Also, the Bregman iteration, introduced to image science [10], was shown to improve TVbased blind deconvolution [11–13]. Recently, a nonlocal TV regularization was invented based on graph theory [14] and applied to image deblurring [15]. Another approach for deblurring is the waveletbased method, etc. [16].
Normally, the original image u\in {\mathbb{R}}^{n} will be found by solving the following constrained minimization problem:
where J(u) is a continuous convex function, and when J(u) is strictly or strongly convex, the solution of (1.2) is unique.
This constrained optimization problem (1.2) arise in many applications, like in image compression, reconstruction, inpainting, segmentation, compressed sensing, etc. The problem (1.2) can be transformed into a linear programming problem, and then solved by a conventional linear programming solver in many cases. Recently, fixedpoint continuation method [17] and Bregman iteration [18] are very popular. Specially, Bregman iterative regularization was proposed by Osher et al. [10]. In the past few years, a series of new methods have been developed, and among them, the linearized Bregman method [19–22] and the split Bregman method [23–26] got most attention.
Specially, when J(u)={\parallel u\parallel}_{1}, the problem (1.2) becomes
Obviously, the problem (1.3) is an {l}_{1}norm minimization problem. Since many practical problems related to the sparsity of the solution make the problem (1.3) stay on focus for years, like in signal processing, compressive sensing etc. [18, 19]. Similar to the problem (1.2), the problem (1.3) also can be transformed into a linear program and then solved by conventional linear programming solvers. However, such solvers are not tailored for the matrix A that is largescale and completely dense. Fortunately, the problem (1.3) can be solved very effectively by the linearized Bregman method [19–22, 27]. The computing speed of its simplified form with soft threshold operator is faster [19, 21, 22]. The corresponding convergence analysis was discussed in [20].
In this paper we highlight numerical computation of coefficient in sparse reconstruction methods for image deblurring, described by an operator \mathrm{\Phi}:X\to Y between Hilbert spaces X and Y. We seek sparse solutions in an orthogonal basis {\{{\psi}_{j}\}}_{j\in N}. The standard approach is the weighted {\ell}_{1} minimization (1.3):
Here {\ell}_{\omega}^{1}(N) denotes the space of coefficients {u}_{j} such that {\sum}_{j}{\omega}_{j}{u}_{j}<\mathrm{\infty}. In order to simplify the notation we introduce the operator A:{\ell}^{2}(N)\to Y, ({u}_{j})\to {\sum}_{j}{u}_{j}\mathrm{\Phi}{\psi}_{j}. Moreover, we will assume that {\{{\omega}_{j}\}}_{j\in \mathbb{N}} entail positive weights and there is a constant {\omega}_{0}>0 such that {\omega}_{j}\ge {\omega}_{0} for all j\in N. Hence {\sum}_{j}{\omega}_{j}{u}_{j} is really a norm on {\ell}^{1}(N), denoted by {\parallel u\parallel}_{\omega}. Then the {\ell}_{1} minimization can be rewritten as
Naturally one can set {\omega}_{k+1}(i)=\frac{1}{{u}_{k}(i)}. Then we can see the weighted {\ell}_{1} norm as a kind of approximation to {\ell}_{0} norm, but we can easily note that when {u}_{k}(i)=0, {\omega}_{k+1}(i) is not well defined. The good news is we can regularize it as {\omega}_{k+1}(i)=\frac{1}{{u}_{k}(i)+\u03f5}, where \u03f5>0 is a small number [28]. So in this paper we set
On this basis, the authors propose a new reweighted {l}_{1} minimization method to solve the problem (1.5) and illustrate by numerical experiments.
The rest of the paper is organized as follows. In Section 2, we summarize the existing methods for solving the constrained problem (1.3). In Section 3, the generalized shrinkage operator is proposed. The new algorithm is proposed in Section 4. Numerical results are shown in Section 5. Finally, we draw some conclusions in Section 6.
2 Preliminaries
2.1 Generalized inverse
We are interested in the iterative formula of the generalized inverse, because it is used by our new algorithm. Therefore, before we give a detailed discussion, we first give some definitions and lemmas.
Definition 2.1 [29]
Let A\in {\mathbb{C}}^{m\times n}, then X is called the pseudoinverse of A and denoted by {A}^{\u2020}. If X satisfies the following properties, i.e., the MoorePenrose conditions:
Remark 2.1 The inner inverse is not unique. In general, the set of the inner inverses of the matrix A is denoted {A}^{}.
Definition 2.2 [29]
Let A,B\in {\mathbb{C}}^{n\times m}, the set
is called the range of (A,B).
Lemma 2.1 [30]
Let A\in {\mathbb{C}}^{m\times n}\ne 0; if initial matrix {V}_{0} satisfies
where I is an identity matrix with the same dimension as matrix A and {A}^{\ast} is the conjugate transpose of matrix A. Then the sequence {\{{V}_{q}\}}_{q\in \mathbb{N}} generated by
is convergent to {A}^{\u2020}.
2.2 Linearized Bregman iteration
The Bregman distance [31], based on the convex function J, between points u and v, is defined by
where p\in \partial J(v) is an element in the subgradient set of J at the point v. In general {D}_{J}^{p}(u,v)\ne {D}_{J}^{p}(v,u) and the triangle inequality is not satisfied, so {D}_{J}^{p}(u,v) is not a distance in the usual sense. For details, see [31].
To solve (1.3), in [19] the linearized Bregman iteration is generated by
where δ is a constant and {p}^{0}={u}^{0}=0. Hereafter, we use \parallel \cdot \parallel ={\parallel \cdot \parallel}_{2} to denote the {l}_{2} norm.
When J(u)={\parallel u\parallel}_{1}, algorithm (2.7) can be rewritten as
where {u}^{0}={v}^{0}=0, and
is the soft thresholding operator [18] with
Namely, the algorithm (2.8) is called an {A}^{T} linearized Bregman iteration.
Subsequently, when A is any matrix, the constraint condition Au=f of the problem (1.3) is not satisfied. So the conditions will be extended to solve the leastsquares problem {min}_{u\in {\mathbb{R}}^{n}}{\parallel Auf\parallel}^{2}, and the algorithm becomes the following {A}^{\u2020} linearized Bregman iteration [22]:
where {A}^{\u2020} is generalized inverse of matrix A.
3 The generalized shrinkage operator
Theorem 3.1 {T}_{\mu}(v)=arg{min}_{u\in {\mathbb{R}}^{n}}\{\mu {\parallel u\parallel}_{1}+\frac{1}{2}{\parallel uv\parallel}^{2}\}.
Proof Let f(u)=\mu {\parallel u\parallel}_{1}+\frac{1}{2}{\parallel u{v}^{k}\parallel}^{2}=\mu {\sum}_{i=1}^{n}{u}_{i}+\frac{1}{2}{\sum}_{i=1}^{n}{({v}_{i}^{k}{u}_{i})}^{2}, then we have
Case 1: {v}_{i}^{k}>\mu >0.

(1)
If {u}_{i}>0, and notice that \frac{\partial f(u)}{\partial {u}_{i}}=0 then {u}_{i}={v}_{i}^{k}\mu >0, for this case f(u) gets its minimum at point {u}_{i}={v}_{i}^{k}\mu along the direction {e}_{i} and the minimum is
f(u){}_{{u}_{i}={v}_{i}^{k}\mu}=\mu ({v}_{i}^{k}\mu )+\frac{1}{2}{\mu}^{2}+{\delta}_{1}\phantom{\rule{0.25em}{0ex}}(>0)={\mathrm{\Delta}}_{1}+{\delta}_{1}.(3.2) 
(2)
If {u}_{i}<0, and notice that \frac{\partial f(u)}{\partial {u}_{i}}={u}_{i}{v}_{i}^{k}\mu <0, again we find that f(u) decreases along the direction {e}_{i}:
f(u){}_{{u}_{i}=0}=\frac{1}{2}{\left({v}_{i}^{k}\right)}^{2}+{\delta}_{1}\phantom{\rule{0.25em}{0ex}}(>0)={\mathrm{\Delta}}_{2}+{\delta}_{1}.(3.3)
Since {\mathrm{\Delta}}_{2}{\mathrm{\Delta}}_{1}=\frac{1}{2}{({v}_{i}^{k})}^{2}(\mu {v}_{i}^{k}\frac{1}{2}{\mu}^{2})=\frac{1}{2}{({v}_{i}^{k}\mu )}^{2}>0, along the direction {e}_{i} we find that the minimizer of f(u) is {u}_{i}={v}_{i}^{k}\mu.
Case 2: {v}_{i}^{k}<\mu <0.

(1)
If {u}_{i}>0, since \frac{\partial f(u)}{\partial {u}_{i}}={u}_{i}{v}_{i}^{k}+\mu >0, f(u) increases along the direction {e}_{i}:
f(u){}_{{u}_{i}=0}=\frac{1}{2}{\left({v}_{i}^{k}\right)}^{2}+{\delta}_{3}={\mathrm{\Delta}}_{3}+{\delta}_{3}.(3.4) 
(2)
If {u}_{i}<0, since \frac{\partial f(u)}{\partial {u}_{i}}=0 we have {u}_{i}={v}_{i}^{k}+\mu <0, the minimizer of f(u) along the direction {e}_{i} is {u}_{i}={v}_{i}^{k}+\mu and the corresponding minimum is
f(u){}_{{u}_{i}={v}_{i}+\mu}=\mu ({v}_{i}^{k}+\mu )+\frac{1}{2}{\mu}^{2}+{\delta}_{3}={\mathrm{\Delta}}_{4}+{\delta}_{3}.(3.5)
Since {\mathrm{\Delta}}_{3}{\mathrm{\Delta}}_{4}=\frac{1}{2}{({v}_{i}^{k})}^{2}+\mu ({v}_{i}^{k}+\mu )\frac{1}{2}{\mu}^{2}=\frac{1}{2}{({v}_{i}^{k}+\mu )}^{2}>0, we can get the minimizer of f(u) at {u}_{i}={v}_{i}^{k}+\mu along the direction {e}_{i}.
Case 3: \mu \le {v}_{i}^{k}\le \mu.

(1)
If {u}_{i}>0, since \frac{\partial f(u)}{\partial {u}_{i}}={u}_{i}{v}_{i}^{k}+\mu >0, f(u) increases along the direction {e}_{i}:
f(u){}_{{u}_{i}=0}=\frac{1}{2}{\left({v}_{i}^{k}\right)}^{2}+\delta .(3.6) 
(2)
If {u}_{i}<0, since \frac{\partial f(u)}{\partial {u}_{i}}={u}_{i}{v}_{i}^{k}\mu <0, f(u) decreases along the direction {e}_{i}:
f(u){}_{{u}_{i}=0}=\frac{1}{2}{\left({v}_{i}^{k}\right)}^{2}+\delta ,(3.7)
when {u}_{i}=0, the minimum of f(u) along the direction {e}_{i} is f(u)=\frac{1}{2}{({v}_{i}^{k})}^{2}+\delta.
In conclusion, we have the following soft shrinkage operator:
The minimizer of the minimization problem is given by
□
The unknown variable u is componentwise separable in the problem
for any v\in {\ell}^{2}(N)\cap {\ell}_{\omega}^{1}(N) and \omega >0. Then each of its components {u}_{i} can be independently obtained by the shrinkage operation, which is also referred as soft thresholding [32]:
For {v}_{i}, {\omega}_{i} and \mu \in R, we define {u}_{i}\in R
The generalized shrinkage operator leads to the sparse solution and removes noises. Hence, the algorithm with the generalized shrinkage operator converges to a sparse solution and is robust to noises.
4 The new reweighted {l}_{1} minimization algorithm
The sequence \{{u}^{k}\} given by {A}^{\u2020} linearized Bregman iteration converges to an optimal solution of the problem (1.3). The computation of generalized inverse {A}^{\u2020} is time consuming; to overcome this, a method called chaotic iterative algorithm is proposed combined with (2.5). In this algorithm we just need matrixvector multiplication, so the generalized inverse {A}^{\u2020} can be computed efficiently. In order to understand the algorithm better, we give a brief description of this method as follows:
where {y}^{0}={V}_{0}{f}^{0}, {V}_{0}=\alpha {A}^{\ast} and 0<\alpha <\frac{2}{{\parallel A\parallel}^{2}}. The corresponding sequence \{{u}^{k}\} also converges to an optimal solution of the problem (1.3).
Here we first study an iteratively reweighted leastsquares (IRLS) method [33] for robust statistical estimation. Considering a regression problem Ax=b where the observation matrix A is underdetermined; it was noticed as regards a standard leastsquares regression, in which {\parallel r\parallel}_{2} is minimized where r=Axb is the residual vector. To overcome the problem of lacking of robustness of the algorithm, IRLS was proposed as an iterative method to
where \rho (\cdot ) is a penalty function such as the {\ell}_{1} norm. This minimization can be accomplished by solving a sequence of weighted leastsquares problems where the weights \{{w}_{i}\} depend on the previous residuals {w}_{i}={\rho}^{\prime}({r}_{i})/{r}_{i}. The typical choice of ρ is inversely proportional to the residual, so that the large residuals will be penalized less in the subsequent iterations. Then an IRLS involving an iteratively reweighted {\ell}_{2}norm can be better approximated by an {\ell}_{1}like criterion. Inspired by the above idea, in order to better approximate an {\ell}_{0}like criterion [34], our algorithm involves the iteratively reweighted {\ell}_{1}norm.
Since that reweighted minimization can enhance the sparsity and the chaotic iterative algorithm can reduce the computational complexity of the generalized inverse {A}^{\u2020}, we iteratively solve the following weighted {\ell}_{1} minimization problem:
We refine the chaotic iterative algorithm, and obtain a new reweighted {l}_{1} minimization algorithm as follows:
where {y}^{0}={V}_{0}{f}^{0}, {V}_{0}=\alpha {A}^{\ast}, and 0<\alpha <\frac{2}{{\parallel A\parallel}^{2}}.
5 Numerical experiments
In this section, we test the reweighted {l}_{1} minimization algorithm for the problem (4.3). We used Word image. Here Word is a 256\times 256 sparse image. In our experiments we tested several kinds of blurring kernels including disk, Gaussian, and motion. We compare different algorithms through both visual effects and quality measurements. Here, the quality of restoration is measured by the signaltonoise ratio (SNR), defined by
where {u}^{\ast}, {u}^{0}, and mean(\cdot ) are the restored image, original image, and average operator, respectively.
Our code is written in MATLAB and run on a Windows PC with a Intel(R) Core(TM) 2 Duo CPU T8100 @ 2.10 GHz 2.10 GHz and 1.5 GB memory. The MATLAB version is 7.1.
Reweighted {l}_{1} minimization algorithm: Step 1. Set {u}^{0}=0, {f}^{0}=0, {y}^{0}={V}_{0}{f}^{0}, {V}_{0}=\alpha {A}^{T}, 0<\alpha <\frac{2}{{\parallel A\parallel}_{2}^{2}}, 0<\delta <1, \mu =\mathrm{parameter}.
Step 2. The sequence {\{{u}^{k}\}}_{k\in \mathbb{N}} generated by (4.4).
Step 3. Until \frac{\parallel {u}^{k+1}{u}^{k}\parallel}{\parallel {u}^{k}\parallel}<\u03f5.
We demonstrate the performance of the reweighted {l}_{1} minimization algorithm, the chaotic iterative algorithm, the {A}^{T} Bregman iteration, and the {A}^{\u2020} Bregman iteration with pinv(A) in MATLAB.
In the first experiment, the images we used were blurred with a ‘disk’ kernel of \mathrm{hsize}=15. The blurry and restored images are presented in Figure 1. By comparing these three algorithms, it is clear that the reweighted {l}_{1} minimization algorithm performs better in terms of SNR than the chaotic iterative algorithm, and the {A}^{T} Bregman iteration lemma is a little slower than the chaotic iterative algorithm and the {A}^{T} Bregman iteration, which is still acceptable.
In the second experiment the images were blurred with a ‘Gaussian’ kernel of \mathrm{hsize}=7. The results are shown in Figure 2. The comparison of the restored effect and the computing time is basically the same as the first one.
In the third experiment we used a part of the Word image blurred with a 3\times 5 ‘motion’ kernel to better show the local information of the recovered image. The restored small sparse Word images after using the reweighted {l}_{1} minimization algorithm, the chaotic iterative algorithm, the {A}^{T} Bregman iteration, and the {A}^{\u2020} Bregman iteration are plotted in Figure 3. Again we obtain a similar conclusion to the above experiments.
In fact, the complexity analysis also shows comparative results of several methods. Set the same loop number is K. So, the workload of the {A}^{\u2020} algorithm (2.11) is two parts. They are the workload of the {A}^{\u2020} and the loop of the (2.11). The workload is O({n}^{3}) during the computation of A=USV, {A}^{\u2020}={V}^{T}{S}^{\u2020}{U}^{T}, when m<n, because of the singular value decomposition involving multiplication of the matrix and matrix and eigenvalue calculation. The workload of the loop of the (2.11) is O(m\ast n\ast K), because the loop only contains multiplication of matrix and vector. Therefore, the total workload of the {A}^{\u2020} algorithm (2.11) is O({n}^{3})+O(m\ast n\ast K). The workload of the chaotic iteration (4.1), the reweighted {l}_{1} minimization algorithm (4.4) and the {A}^{T} Bregman iteration (2.8) are O(m\ast n\ast K), respectively. Obviously, K<m\ll n, the workload of the {A}^{\u2020} algorithm (2.11) is bigger than the other three algorithms.
All the experiment data are listed in Table 1. In summary, for the restored quality of the three methods we have \mathrm{Reweighted}>\mathrm{Chaotic}>{A}^{\u2020}\gg {A}^{T}, while for the computing time the order of magnitude is about 1:1:{10}^{2}:1. The numerical examples illustrate that the new reweighted {l}_{1} minimization algorithm is fast and efficient for deblurring the image. It is a very useful method.
6 Conclusion
In this paper, we propose the reweighted {l}_{1} minimization algorithm for image deblurring. Above all, we can see that the recovery of the image effect is obvious. Especially in the case of a large degree of blurring and difficult to recover details, it is stable and effective. In addition, we can improve the efficiency of this reweighted {l}_{1} minimization algorithm combining with the ‘kicking’ technology. Because of the scale factor and efficiency of the algorithm {A}^{\u2020}, the new method proposed in this paper can be used in a parallel operation to get a better algorithm.
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Acknowledgements
This research was partly supported by Fund of Oceanic Telemetry Engineering and Technology Research Center, State Oceanic Administration (grant no. 2012003), the NSFC (grant nos. 60971132,61101208) and Fundamental Research Funds for the Central Universities (grant no. 13CX02086A).
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Qiao, T., Wu, B., Li, W. et al. A new reweighted {l}_{1} minimization algorithm for image deblurring. J Inequal Appl 2014, 238 (2014). https://doi.org/10.1186/1029242X2014238
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DOI: https://doi.org/10.1186/1029242X2014238
Keywords
 reweighted {l}_{1} minimization
 generalized inverse
 linearized Bregman iteration
 image deblurring