The Split Bregman Method And Its Applications In Image Processing

Image processing is a rapidly developing subject and been broadly applied in many fields, such as scientific research, industrial production, medical and health, education, management and communication. In addition, the rapid development of computer science and the widespread of image display devices have provided a favorable external environment and become a major driving force for image processing. In no doubt, image processing has become one of the most important applied science.In image processing and computer vision systems, the systematic use of partial differential equations began in 1980s. Image segmentation and image filtering are the two branches that directly impact the formation of the subject [5]. The proposition of axiom system for AMSS (Affine Morphological Scale Space) equation [7] is a sign of the formation of image processing based on partial differential equations. At present, benefited from the partial differential equations theory, image processing is rapidly developing. In turn, the new issues in the image processing promote the development of the partial differential equations theory, such as the viscosity solution theory [8].With its rapid development, a large number of models and algorithms were proposed. The whole subject has a tendency to flourish. The traditional methods of image restoration are established on linear system, such as Winner filters [15], regularized filters [16]. etc. Due to its simplicity, it is used for many years, but in general it is not satisfactory. Therefore, based on the regularation filter, Rudin, Osher and Fatemi proposed the well-known ROF model [18]. Due to the good performance of the ROF model, a lot of work has been done. In order to be able to solve it quickly, besides the modified model.a number of algorithms were proposed. However, ROF model is seriously non-linear and non-differentiable and hard to solve efficiently. The classical methods to solve it are time marching schemes [18], fixed-point iteration [32], primal-dual Newton method [35],graph cut algorithm [36, 38, 39], etc. Image segmentation is another important aspect in the low-level image processing, its problem is maily from two aspects [1], one is that the image segmentation is of the ill-pose character, the other is that a natural image usually contains too much information to use a unified approach to represent an object. In addition to traditional methods, methods based on the variation/PDE have become popular in the past nearly 20 years. Most of these models can be divided into two categories,namely geodesic active contour/snakes model and the Mumford-Shah model. Unfortunately, all these models suffer from the problem of local minima and slow convergence,therefore, several convex models about image segmentation were proposed, such as the GCS model proposed by Chan, etc.Basically, the above problems can be summarized as a class of L₁ problem. The split Bregman method [9], which is introduced in detail in this paper, may be the most efficient method for dealing with such problems. Its accuracy and efficiency soon attracted a number of follow-up study [10, 11, 12, 13]. This mainly involves the applications of several important aspects in low-level image processing, such as image restoration and image segmentation. This article expand this work. The basic idea and steps of split Bregman method are introduced in detail. The basic idea is that, by introducing a equal constraint, the more complex item,L₁ ofΦ(u),is substituted by a new function.Then theΦ(u) is put in the differentiable second-order item by Bregman iterative.Obviously, the above substitution is a critical step, which is the so-called split method. Then with the relaxation method, another new function is introduced, and the more complex Bregman iteration formula becomes a simple form used by the split Bregman method. The split Bregman is the following three stepsFor Step1,with the properties of P, the appropriate algorithm is sellected, such as Gauss-Seidel iteration [54],Fourier transform, and so on. In the numerical experiments we have used the Gauss-Seidel iteration and found that a large number of iteration is not required and the good effect is acquired by selecting 1 or 2 iteration times, which saves time greatly. For Step2, we can efficiently compute the optimal value by using shrinkage operators. Step3 is the updating of relaxation item. Stones from other hills may serve to polish jade, the idea of splitting the Bregman method has referential significance.The main work of this paper is to use split Bregman method in a few of more specific image processing problems. Firstly, split Bregman method is used in the weighted denoising ROF[9] and high-order model [26]. Then the specific algorithm of recovery model of vector image TV [1] is given the. Finally, optimization problem proposed in weighted GCS model [46,9] is solved by the use of Bregman split method. It shows that the adaptability of Bregman split method is very strong. The convergence order of split Bregman method is lack of theoretical analysis. However, from,the text of the appendix A and a large number of numerical experiments, the converging speed of Bregman method is very fast. According to the experimental results, both time and space complexity of the split Bregman method are linear. Parallel analysis of split Bregman method is a useful exploration of the author. The split Bregman method is very easy to be paralleled, and can get a linear speedup, therefore, split Bregman method is appropriate to deal with large-size images.It can be anticipated that in the near future split Bregman method will have rapid development in the study of theoretical basis, methods, relations with other optimization methods and parallelization.