Linearized Bregman Iterations for Frame-Based Image Deblurring

Abstract

Abstract. Real images usually have sparse approximations under some tight frame systems derived from framelets, an oversampled discrete (window) cosine, or a Fourier transform. In this paper, we propose a method for image deblurring in tight frame domains. It is reduced to finding a sparse solution of a system of linear equations whose coefficient matrix is rectangular. Then, a modified version of the linearized Bregman iteration proposed and analyzed in [10,11,44,51] can be applied. Numerical examples show that the method is very simple to implement, robust to noise, and effective for image deblurring.

Blind motion deblurring using multiple images

“…The procedure is repeated and it converges to a sparse solution in the framelet domain. The algorithm is efficient and robust to noises as analyzed by [5] and we also have the following convergence results from [5]. See [4,5,6,32] for a more detailed analysis.…”

“…Step 2 is a non-blind image deblurring problem, which has been studied extensively in the literature; see, for instances, [1,22,8,7,18,5]. However, there is one more error source in Step 2 than the traditional non-blind deblurring problem has, that is, the error in the intermediate blur kernel p (k+1) i used for deblurring.…”

“…Inspired by recent non-blind deblurring techniques which are based on sparse approximation to the image under certain tight frame systems ( [7,4]), we also use the sparsity constraint on the clear image g under tight frame systems to regularize the non-blind deblurring. And we use a modified version of so-called linearized Bregman iteration (See [24,32,31,16,20,11,25,4,5,6,15]) to achieve impressive robustness to image noises, alignment errors, and, more importantly, perturbations on the given intermediate blur kernels. In our implementation, we choose the tight framelet system constructed in [12,30] as the choice of the tight frame system representing the clear image g.…”

“…More details and an improvement called "kicking" of the linearized Bregman iteration is described in [25], and a rigorous theory was given in [4,6]. The linearized Bregman iteration for frame-based image deblurring was proposed in [5]. Recently, a new type of iteration based on Bregman distance, called split Bregman iteration, was introduced in [15], which extended the utility of Bregman iteration and linearized Bregman iteration to minimizations of more general 1 -based regularizations including total variation, Besov norms and sums of such things.…”

Blind motion deblurring from a single image using sparse approximation

“…As we will show later, the perfect reconstruction property of tight framelet system allows the application of the so-called linearized Bregmen iteration (Osher et al [21]), which has been shown in recent literatures (Cai et al [6,5]) that it is more efficient than classic numerical algorithms (e.g. interior point method) do when solving this particular type of ℓ 1 norm minimization problems.…”