# Blur Kernel Estimation by Structure Sparse Prior
## 1 Introduction  
Image blur is very common in the daily life, which is an undesirable quality degradation of clear images. It is usually caused by several reasons, such as atmospheric turbulence, defocusing, or object motion. The blurry images present bad visual quality and degrade the performance of further computer vision tasks [1, 2]. Therefore, it is of great importance to remove blur effectively from a single image in practical applications. Recently, many image deblurring methods have been explored. Many previous methods made restrictive assumptions on the blur kernel, such as a parameterized linear motion in a single or multiple images. [PERSON] et al. (2006) [3] recovered a latent image by incorporating an anti-reflective boundary condition on PDE model. [PERSON] et al. (2007) avoided ringing artifacts in deconvolution by using a sparse derivative prior [4]. [PERSON] et al. (2008) introduced an effective method which can reduce ringing artifacts by using bilateral filtering and residual deconvolution [5]. These methods try to constrain the form of the kernel or put constraints on the underlying sharp image, which is not suitable for image deblurring in many practical scenarios. To reduce the complexity of blind image deblurring, many regularization and image priors are introduced [6, 7, 8]. The regularization based methods incorporate various priors of the latent image or kernel to remove the blur artifacts. The regularization terms are usually \(l_{1}\) norm [6, 9] or \(l_{2}\)[7, 10] norm. However, more commonly, the _total variation_ (TV) norm and its variations are adopted. [PERSON] et al. [11] proposed an edge detector to handle unconstrained blurs. [PERSON] et al. [8] introduced a new regularization term, namely \(l_{1}/l_{2}\) norm, which gives the lowest cost for the true sharp image. These methods can stabilize the result and are effective in weakening the influence of noise, although they tend to produce over smoothed results [12]. Bayesian model has been widely used in blind deconvolution. By introducing some prior about the image and blur, such methods impose constraints on the estimates and act as regularizers [13]. Variational bayesian_ (VB) aims at obtaining approximations to the posterior distributions of the unknowns. This variational approximation method in a Bayesian formulation has been adopted in blind deconvolution problem [14; 15; 16]. VB has some advantages over conventional Bayesian blind deconvolution methods, but it needs a pre-specified, decreasing sequence of noise values to achieve a reasonable result. It is difficult for VB to make transparent analysis [17]. Sparse representation utilizes the sparsity as the prior to recover the latent image, which has been a popular method in blind image deblurring. [PERSON] et al. proposed a method by adapting the sparseness in a learned regression model [18]. [PERSON] et al. proposed a blind deblurring method by utilizing the sparsity of the blur kernel and natural images to estimate the blur kernel [19]. [PERSON] and [PERSON] introduced the over-complete dictionary in image denoising [20], which was adopted in image deblurring. [PERSON] et al. proposed a method that combines blind image restoration and recognition with sparse representation prior [21]. [PERSON] et al. proposed a non-blind deblurring method by combining sparse representation prior and sparse gradient prior [22]. However, the sparse representation-based methods usually adopt the \(l_{1}\)-sparsity penalty term, which assumes that the coefficients are identical independent distributed. The conventional \(l_{1}\)-sparsity model has been proved to be unstable, especially when the observed image is degraded [12]. In recent years, sparsity has been a widely used prior for image restoration, such as image denoising, inpainting, super-resolution, and deblurring. For example, the sparse representation model (SRM) based image deblurring methods assume that natural images can be represented by a dictionary and a sparse coefficient [23].