# Deblurring by Solving a TVp-Regularized Optimization Problem Using Split Bregman Method

Authors: Su Xiao
Journal: Advances in Multimedia
 (2014)

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## Abstract

Image deblurring is formulated as an unconstrained minimization problem, and its penalty function is the sum of the error term and TVp-regularizers
with 0<p<1. Although TVp-regularizer is a powerful tool that can significantly promote the sparseness of image gradients, it is neither convex nor smooth, thus making the
presented optimization problem more difficult to deal with. To solve this minimization
problem efficiently, such problem is first reformulated as an equivalent constrained minimization problem by introducing new variables and new constraints. Thereafter, the split
Bregman method, as a solver, splits the new constrained minimization problem into subproblems. For each subproblem, the corresponding efficient method is applied to ensure
the existence of closed-form solutions. In simulated experiments, the proposed algorithm
and some state-of-the-art algorithms are applied to restore three types of blurred-noisy
images. The restored results show that the proposed algorithm is valid for image deblurring and is found to outperform other algorithms in experiments.


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## Body

## 1. Introduction

Over the past half century, image deblurring has been intensively studied and extensively applied in many fields, such as biometric identification, remote sensing, and video surveillance, among others. As an ill-posed and inverse problem, image deblurring requires a stable solution. For this purpose, Tikhonov et al. [1] propose regularization technology that initially uses ∥Cu∥22 as the regularizer, with C as the Tikhonov operator. However, the Tikhonov regularized image deblurring algorithms favor reconstructing oversmoothing images such that the most important edges are lost. To preserve edges, Rudin et al. [2] propose the total variation (TV) model: ∥Au-f∥22/2+TV(u), where TV(u) is a TV-regularizer that has two forms, namely, isotropic TV-regularizer (∥∇u∥1) and anisotropic TV-regularizer (∥∇1u∥1+∥∇2u∥1). From the piecewise continuity of natural images, the values of the elements in the gradients of images are mostly equal to zero; that is, the gradients of natural images are sparse. As a powerful tool, the TV-regularizer has been employed by many state-of-the-art image deblurring algorithms [3–5], but recent research reveals that for modeling the sparseness of image gradient, the lp-norm (∥·∥pp) with 0<p<1 is more suitable than the l1-norm (∥·∥1) of TV-regularizer [6]. Moreover, the TVp-regularizer (∥∇u∥pp) outperforms the TV-regularizer in terms of edge preservation and noise suppression.From the nonconvexity and nonsmoothness ofTVp-norm, imaging inverse problems involving TVp-regularizer or lp-regularizer are generally confronted with challenges in the existence of solutions and low efficiencies of corresponding algorithms. For TVp-regularized image deblurring with specific values of p, Levin et al. [7] originally use iterative reweighted least squares (IRLS) to decompose the deblurring problem into equivalent subproblems, for which the conjugate gradient (CG) is then employed. Zhuo et al. [8] then propose a similar algorithm for blur identification in blind deblurring, which also adopts IRLS method with CG iteration. When applying IRLS to image deblurring problems involving TVp-regularizers, hundreds of CG iterations are usually needed, thus giving rise to negative effects on the speed of image deblurring. For lp-regularized compressive sensing, a commonly used method is interior point, which can solve the corresponding optimization problems in polynomial time under certain conditions [9]. However, the interior-point method is inapplicable to optimization problems with equality constraints, and the initial points are difficult to choose. These factors can be regarded as the two primary inherent drawbacks of the interior-point method.Variable splitting technology has recently been proposed and extensively applied to inverse problems [10, 11], including TVp-regularized image deblurring problems [12]. By introducing new variables under the framework of variable splitting, the optimization tasks of imaging inverse problems are transformed into subtasks that are easier to address. For each subtask, the corresponding solver should be carefully selected to ensure that the additional computational costs are low.AlthoughTVp-regularizer is more appropriate than TV-regularizer in promoting the sparseness of image gradients, statistics show that very few image deblurring algorithms have considered the TVp-regularizer. The most likely reason is that when TVp-regularizers are involved, image deblurring problems are neither convex nor smooth, and efficient algorithms are difficult to develop. To take full advantage of TVp-regularizer, this study models image deblurring as an unconstrained minimization problem, the penalty function of which is combined with the error term and the sparse TVp-regularizers with 0<p<1. For convenience of image deblurring, the unconstrained minimization problem is reexpressed as an equivalent constrained minimization problem with new variables and constraints. Thereafter, the constrained minimization problem is decomposed into subproblems by the split Bregman method [13]. For these subproblems, generalized shrinkage/thresholding (GST) functions [14], fast Fourier transforms (FFTs), and other methods are introduced. By restoring different types of blurred-noisy images, the proposed algorithm shows effectiveness and better performance than some similar state-of-the-art algorithms (Algorithm 1). The proposed algorithm is empirically found to converge to satisfactory solutions with only several iterations, as verified by experiments.Algorithm 1:Proposed algorithm: SBMTVp.
(1) Input:λ, γ, K and J
(2) Initialize:{wi0}i=12 and {bi0}i=12
(3) Precompute:u~k+1=(ATA+γ∑i=12‍∇iT∇i)-1 and ATf
(4) fork = 0 to K.
(a)uk+1=u~k+1(ATf+γ∑i=12‍∇iTdik)
(b) if||uk+1-u||F>||uk-u||F
(i) break;
(c) end
(d) Computew1k+1 and w2k+1 according to (20) to (22)
(e) Computeb1k+1 and b2k+1 according to (17) and (18)
(5) end
(6)  Output:uk+1


## 2. Family of Bregman Methods

As variants of the standard Bregman iteration method, the split Bregman method (SBM), linearized Bregman method (LBM) [15], and Bregmanized operator splitting (BOS) [3] are the key members of the family of Bregman methods. SBM is proposed for the following minimization problem:
(1)min⁡xJ(Gx),s.t.lly=Hx,
where J(x) is generally nonsmooth, and G is a linear operator that may be invertible or not. Involving new variables and constraints, problem (1) is reformulated in the following equivalent form:
(2)min⁡x,zJz+y-Hx222,s.t.llz=Gx.
SBM is employed to solve problem (2) by splitting it into the following subproblems:
(3)xk+1=argminxy-Hx22+ρzk-Gx-dk22,(4)zk+1=argminzJ(z)+ρz-Gxk+1-dk222,(5)dk+1=dk+Gxk+1-zk+1,
where the constant ρ>0. For subproblems (3) to (5), if J(z) is a convex function, closed-form solutions generally exist. In particular, when J(z)=∥z∥1 and J(z)=∥z∥0, the solutions to subproblem (4) are the popular soft-thresholding function and hard-thresholding function, respectively. When solving some other types of optimization problems, SBM may also be called the alternating direction method of multipliers (ADMM) or Douglas-Rachford splitting method. More details on the connections among SBM, ADMM, and the Douglas-Rachford splitting method can be found in the literature [16].When applied to optimization problems, such asl2-TV optimization problem or l2-l1 optimization problem, SBM is generally faster than the two-step iterative thresholding methods [17] and Nesterov’s methods [18, 19]. Therefore, the applications of SBM to imaging inverse problems have become active research fields. Through SBM, many classic image deblurring/denoising problems [13, 20, 21] have been efficiently solved. Aside from SBM, LBM and BOS are also popular methods for imaging inverse problems. LBM, which can be regarded as the combination of the standard Bregman iteration and fixed-point methods, is initially proposed for compressed sensing problems, and its application is later extended to image deblurring. To solve problem (1), with x0=q0=0, LBM generates the following subproblems:
(6)xk+1=argminxDJqkx,xk+α<HTHxk-y,x>+12βx-xk22,qk+1=qk-αHTHxk-y-1βxk+1-xk,
where the distance DJqk(x,xk) is defined as
(7)DJqkx,xk=JGx-JGxk-qk,x-xk.Ifxk+1=xk in iteration, “kicking” [22] is used in LBM. However, LBM cannot be applied when problem (1) involves TV-regularizer or multiple l1-regularizers. BOS is proposed for TV and nonlocal TV (NLTV) models in imaging inverse problems. BOS decomposes the problem (1) into the following subproblems:
(8)xk+1=min⁡xδJ(Gx)+12yk-Hx22,(9)yk+1=yk+y-Hxk+1.
The forward-backward operator splitting (FBOS) [23] is then used to solve problem (8) as follows:
(10)zk+1,i+1=xk,i-μHTHxk+1,i-yk,xk+1,i+1=min⁡xδJGx+12μx-zk+1,i+122.
BOS can be regarded as Bregman iteration plus FBOS because (8) and (9) represent the standard Bregman iteration. Compared with SBM, which can usually obtain good results with only one inner iteration, LBM and BOS are more dependent on the inner solvers or inner iterations. Therefore, when applied to image deblurring problems, both LBM and BOS show lower efficiencies than SBM. In addition to the fast convergent rate, the parameters of SBM need not vary with iteration. Thus, the number of conditions can be reduced by carefully selecting the parameters. In particular, SBM is more suitable for multiregularized image deblurring problems.


## 3. The Proposed Algorithm: SBMTVp

The degradation of images can be modeled as follows:(11)f=Au+n,
where f∈RM, A∈RM×N, u∈RN, and n∈RM represent blurred-noisy image, blur operator, the unknown sharp image in vector form, and Gaussian additive noise, respectively. u cannot obviously be estimated according only to (11), especially when A is heavily ill-posed. Therefore, as a rule, image deblurring problems are transformed into optimization problems. This study casts image deblurring as follows:
(12)min⁡uf-Au222+λ∇1upp+∇2upp,
where ∇1∈RL×N and ∇2∈RL×N denote the vertical and horizontal gradient operators, respectively; ∥∇1u∥pp+∥∇2u∥pp=∑i=1L|(∇1u)i|p+∑i=1L|(∇2u)i|p is TVp-regularizer with 0<p<1; and the constant λ>0 controls the trade-off between data fidelity and regularization. By introducing the variables w1 and w2 and the constraints w1=∇1u and w2=∇2u, problem (12) can be reformulated as follows:
(13)min⁡u,w1,w2f-Au222+λw1pp+w2pps.tkkklw1=∇1u,w2=∇2u.From the nonsmoothness oflp-norm, problem (13) cannot be analytically solved in a direct manner. If we let x=u, z=w1T,w2TT, G=∇1T,∇2TT, ∥y-Hx∥22=∥f-Au∥22, and J(z)=(∥w1∥pp+∥w2∥pp) in problem (2), then we can find that problems (2) and (13) are the same problems with the same structures. Therefore, this study uses SBM as an “indirect” method of solving problem (13). By SBM, problem (13) is split into the following subproblems:
(14)uk+1=argminuAu-f22+γ∇1u-w1k-b1k22+∇2u-w2k-b2k22,(15)w1k+1=argminw1γ2λ∇1uk+1-w1-b1k22+w1pp,(16)w2k+1=argminw2γ2λ∇2uk+1-w2-b2k22+w2pp,(17)b1k+1=b1k+∇1uk+1-w1k+1,(18)b2k+1=b2k+∇2uk+1-w2k+1,
where the constant γ>0.Except for subproblems (15) and (16), other subproblems can be directly computed, and subproblem (14) (i.e., uk+1 subproblem) has a closed-form solution:
(19)uk+1=ATA+γ∑i=12‍∇iT∇i-1ATf+γ∑i=12‍∇iTdik,
where d1k=w1k+b1k and d2k=w2k+b2k. According to the theory of Tikhonov regularization, (19) provides a regularized solution that is an approximation of the exact solution u*. Choosing the proper values for γ, with k→∞, we find that uk+1 is approaching the exact solution [1]. With the help of FFTs in (19), ATA, {∇iT∇i}i=12, ATf, and {∇iTdik}i=12 can all be computed with the computational cost of O(nlog⁡⁡n). After precomputing ATA, ∑i=12‍∇iT∇i, ATf, and ∑i=12‍∇iTdik, and with the equal help of FFTs, uk+1 can be obtained with the cost of O(nlog⁡⁡n). Several numerical methods, such as IRLS and interior-point method, have been applied to the minimization of subproblems (15) and (16) with specific values of p, but these methods cannot usually converge to satisfactory solutions, especially for large-scale problems (e.g., image deblurring). To guarantee the convergence of minimization of subproblems (15) and (16), as well as to reduce the costs of computing them, GST functions are employed as the solvers. When we let i=1 to approach L and let (w1k+1)i and (w2k+1)i represent any elements of w1k+1 and w2k+1, respectively, then (w1k+1)i and (w2k+1)i can be analytically computed by
(20)w1k+1i=GST∇1uk+1-b1ki,λγ,p,w2k+1i=GST∇2uk+1-b2ki,λγ,p.The GST function in (20) is defined as
(21)GST(x,y,p)=sgn⁡xTpj+1x,yifx>τyp0ifx≤τyp,
where sgn⁡ is the signum function; Tp0(|x|,y)=|x|; Tpj+1(|x|,y) is iteratively computed by
(22)Tpj+1x,y=x-ypTpjx,yp-1withj=0,1,2,…,J;
and the thresholding value τyp=(2y(1-p))1/(2-p)+yp(2y(1-p))(p-1)/(2-p). When p=1, the GST function is
(23)GST(x,y,1)=sgn⁡(x)(x-y)ifx>y0ifx≤y,
which is the famous soft-thresholding function. When p=0, the GST function is given by(24)GST(x,y,0)=xifx>2y1/20ifx≤2y1/2,
which is the famous hard-thresholding function. Therefore, soft-thresholding and hard-thresholding functions can be regarded as the special cases of the GST function. The computational costs of both functions are O(n) because w1k+1 and w2k+1 are computed element-by-element.The proposed iterative algorithm is generated from (17) to (22). The proposed algorithm is called “SBMTVp” because it solves TVp-regularized image deblurring with the use of the split Bregman method. The efficiency of the proposed algorithm is mainly dependent on the computations of subproblems uk+1, w1k+1, and w2k+1, and all have low computational costs as briefly analyzed above. Therefore, the efficiency of the proposed algorithm is high, as will be verified by the experiments. As mentioned in [14], the GST function has the following four properties: (1)  GST(-x,y,p)=-GST(x,y,p), (2)  GST(x,y,p)≤GST(x′,y,p) if x≤x′, (3)  limx→∞GST(x,y,p)=∞, and (4)  0≤GST(x,y,p)≤∞ for 0≤x≤∞. Thus, according to the proof in [24], the convergence of GST is guaranteed; that is, subproblems w1k+1 and w2k+1 converge with iteration.


## 4. Experiments and Results

As shown in Figure1, four standard gray images of different types are chosen as the sharp images for experiments. To create the blurred-noisy images shown in Figure 2, according the degradation model in (11), sharp images are firstly filtered by blur operators in Table 1, wherefspecial is MATLAB function, and then Gaussian noises of different levels are added to these images. In Figure 2, blurred signal-to-noise ratio (BSNR) is defined as
(25)BSNR=10×log⁡10⁡uk+1-u22Nδ2,
where δ is the variance of noise. BSNR is employed as the objective criterion of degradation. The proposed SBMTVp is employed to restore the blurred-noisy images in Figure 2 to verify its effectiveness. The algorithms in [3, 7] are introduced for comparison in terms of restoring the same blurred-noisy images to verify the superiority of SBMTVp. The experiments are conducted on a laptop with Windows XP, Intel Duo 2 CPU @ 2.10 GHz, and 2 GB of RAM, and all the algorithms are implemented in MATLAB R2011b platform. For SBMTVp, the values of J, λ, and γ are set as 2, 7e-9, and 5e-5, respectively. With only several iterations, SBMTVp can obtain satisfactory results. Thus, K is set to be 10. As for the algorithms in [3, 7], all parameters are left at the default settings. The stopping criteria of DeblurSps and {TV,NLTV}_BOS are “kouter>2 and kinner>200” and “k≥30 or ∥Auk-f∥2≤0.99σ (σ is the noise level),” respectively, which are the default criteria adopted by their references. kouter and kinner denote outer iterations and inner iterations (CG iterations), respectively. The restored results of all algorithms are shown in Figures 3 to 5 and Tables 3, 4, and 5, with peak signal-to-noise ratio (PSNR) defined as
(26)PSNR=10×log⁡10⁡N×2552uk+1-u22.
PSNR is employed as the objective criterion for the quality of restored images. For p, several typical values, that is, 1/4, 1/3, and 4/5, are considered. “DeblurSps” represents the algorithm in [7], which solves the TVp-regularized (p=0.8) image deblurring problem using IRLS and CG. “TV_BOS” and “NLTV_BOS” represent the algorithms in [3], which apply the Bregmanized operator splitting to the TV and NLTV models. To compare the speeds of all algorithms, each of them is run for 10 times to restore the blurred-noisy images, and the average time of each algorithm is shown in Table 2.Table 1
Blur operators.
Types
Kernels
Gaussian
fspecial (“gaussian”, [11, 11], 9)
Motion
fspecial (“motion”, 20, 20)
Average
fspecial (“average”, [11, 11])Table 2
Average time of each algorithm to restore a blurred-noisy image.
Algorithms
DeblurSps
TV_BOS
NLTV_BOS
SBMTV1/4
SBMTV1/3
SBMTV4/5
Average time
50.38 s
23.12 s
333.19 s
0.53 s
0.55 s
0.63 sTable 3
PSNRs of images in Figure3.
Algorithms
PSNR values
DeblurSps
Figure3(a)
Figure3(b)
Figure3(c)
Figure3(d)
29.26 dB
25.51 dB
29.11 dB
27.94 dB
TV_BOS
Figure3(e)
Figure3(f)
Figure3(g)
Figure3(h)
27.37 dB
24.64 dB
27.56 dB
27.56 dB
NLTV_BOS
Figure3(i)
Figure3(j)
Figure3(k)
Figure3(l)
27.80 dB
24.76 dB
27.98 dB
27.18 dB
SBMTV1/4
Figure3(m)
Figure3(n)
Figure3(o)
Figure3(p)
29.59 dB
27.11 dB
30.55 dB
29.20 dB
SBMTV1/3
Figure3(q)
Figure3(r)
Figure3(s)
Figure3(t)
29.68 dB
27.02 dB
30.75 dB
29.61 dB
SBMTV4/5
Figure3(u)
Figure3(v)
Figure3(w)
Figure3(x)
29.57 dB
27.05 dB
30.01 dB
29.66 dBTable 4
PSNRs of images in Figure4.
Algorithms
PSNR values
DeblurSps
Figure4(a)
Figure4(b)
Figure4(c)
Figure4(d)
30.04 dB
26.70 dB
29.64 dB
27.31 dB
TV_BOS
Figure4(e)
Figure4(f)
Figure4(g)
Figure4(h)
28.60 dB
26.18 dB
28.75 dB
28.15 dB
NLTV_BOS
Figure4(i)
Figure4(j)
Figure4(k)
Figure4(l)
28.90 dB
26.33 dB
29.07 dB
27.84 dB
SBMTV1/4
Figure4(m)
Figure4(n)
Figure4(o)
Figure4(p)
31.65 dB
28.83 dB
31.27 dB
28.52 dB
SBMTV1/3
Figure4(q)
Figure4(r)
Figure4(s)
Figure4(t)
31.76 dB
29.20 dB
31.49 dB
28.95 dB
SBMTV4/5
Figure4(u)
Figure4(v)
Figure4(w)
Figure4(x)
31.64 dB
29.22 dB
31.57 dB
29.65 dBTable 5
PSNRs of images in Figure5.
Algorithms
PSNR values
DeblurSps
Figure5(a)
Figure5(b)
Figure5(c)
Figure5(d)
29.23 dB
25.58 dB
29.09 dB
27.95 dB
TV_BOS
Figure5(e)
Figure5(f)
Figure5(g)
Figure5(h)
27.55 dB
24.84 dB
27.71 dB
27.63 dB
NLTV_BOS
Figure5(i)
Figure5(j)
Figure5(k)
Figure5(l)
27.86 dB
24.73 dB
28.05 dB
27.30 dB
SBMTV1/4
Figure5(m)
Figure5(n)
Figure5(o)
Figure5(p)
29.97 dB
27.47 dB
30.75 dB
29.88 dB
SBMTV1/3
Figure5(q)
Figure5(r)
Figure5(s)
Figure5(t)
30.37 dB
27.49 dB
31.12 dB
30.61 dB
SBMTV4/5
Figure5(u)
Figure5(v)
Figure5(w)
Figure5(x)
29.99 dB
27.51 dB
30.23 dB
30.15 dBSharp images. (a) Boat image with a resolution of512×512. (b) Cameraman image with a resolution of 256×256. (c) Hill image with a resolution of 512×512. (d) Peppers image with a resolution of 256×256.
(a)
(b)
(c)
(d)Blurred-noisy images. ((a) to (d)) Gaussian blurred-noisy images with BSNR = 40 dB; ((e) to (h)) motion blurred-noisy images with BSNR = 40 dB; ((i) to (l)) average blurred-noisy images with BSNR = 40 dB.
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(l)Restored Gaussian blurred-noisy images obtained by ((a)–(d)) DeblurSps, ((e)–(h)) TV_BOS, ((i)–(l)) NLTV_BOS, ((m)–(p))SBMTV1/4, ((q)–(t)) SBMTV1/3, and ((u)–(x)) SBMTV4/5.
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(x)Restored motion blurred-noisy images obtained by ((a)–(d)) DeblurSps, ((e)–(h)) TV_BOS, ((i)–(l)) NLTV_BOS, ((m)–(p))SBMTV1/4, ((q)–(t)) SBMTV1/3, and ((u)–(x)) SBMTV4/5.
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(x)Restored average blurred-noisy images obtained by ((a)–(d)) DeblurSps, ((e)–(h)) TV_BOS, ((i)–(l)) NLTV_BOS, ((m)–(p))SBMTV1/4, ((q)–(t)) SBMTV1/3, and ((u)–(x)) SBMTV4/5.
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(x)The results in Figures3 to 5 and Tables 3 to 5 demonstrate that SBMTVp can efficiently restore different types of blurred-noisy images and displays superior performance over other algorithms, especially in terms of speed. As an approximation method, IRLS cannot ensure that DeblurSps will obtain the optimal solution to the image deblurring problem. In addition, to solve a sequence of subproblems generated by IRLS, hundreds of CG iterations are needed. Therefore, the restored results of DeblurSps are worse than those of SBMTVp. Moreover, this algorithm is slower than SBMTVp and TV_BOS. However, as a TVp-regularized algorithm, DeblurSps outperforms TV_BOS and NLTV_BOS on the restored results. When BOS is applied to TV-regularized and NLTV-regularized image deblurring problems, the resulting subproblems cannot be solved by simple methods (e.g., shrinkage methods), such that the process of image deblurring becomes more complex with the loss of efficiency. In particular, for the NLTV-regularized image deblurring problem, additional time is required to update the weight function. Therefore, as shown in Tables 2 to 5 and Figures 3 to 5, the algorithms in [3] are inferior to other algorithms in terms of their performances on the restored results and their speed. The convergence of SBMTVp cannot be theoretically proven. Therefore, to analyze its convergence, Figure 6 shows the evolution of ∥uk+1-u∥F in the procedure of iteration, where ∥·∥F denotes the Frobenius norm. The curves clearly demonstrate that with iteration, uk+1 is approaching u, that is, the SBMTVp converges. In general, with two to four iterations, SBMTVp can obtain satisfactory solutions.Evolution of∥uk+1-u∥F of SBMTVp in the procedure of iteration; (a) evolution of ∥uk+1-u∥F when restoring Gaussian blurred-noisy images; (b) evolution of ∥uk+1-u∥F when restoring motion blurred-noisy images; (c) evolution of ∥uk+1-u∥F when restoring average blurred-noisy images.
(a)
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(c)


## 5. Conclusions

This study proposes a novel image deblurring algorithm that solves aTVp-regularized minimization problem using SBM. To obtain the closed-form solutions to the subproblems generated by SBM efficiently, the FFTs and GST functions are introduced. In the experiment, three types of blurred noisy images are restored by SBMTVp and several state-of-the-art algorithms. Comparison results show that SBMTVp outperforms other algorithms in terms of restored results and speed. Although this study does not prove the convergence of SBMTVp, Figure 6 clearly illustrates the fact.



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*Source: 906464-2014-12-16.xml*