Typical Nonlinear Evolution Equations Applied To Image Processing

With the fast development of nonlinear science, the research of nonlinear evolution equations (e.g. nonlinear ordinary derivative equations, nonlinear partial derivative equations, nonlinear difference equations and function equations etc.) and the application of it to the image processing have become the main content of the frontier of nonlinear science.The problem of the solving nonlinear evolution equations is not only age-old but also important in theory and practice. With the evolution of nonlinear science, solving nonlinear evolution equations has been indispensable to most physics, mechanics, geoscience, life sciences, applied mathematics and technical and scientific workers. Currently, many scholars are applying themselves to this field. On the other hand, nonlinear evolution equations have an abroad application in the image processing. Therefore, how to do the image processing better by use of nonlinear equations is an important research content for discussion currently.Based on the above important problems in the research field of nonlinear evolution equations, the solving problem of some typical nonlinear evolution equations and the application of nonlinear evolution equations to the image processing have been studied by the crossover and integration of multi-disciplines, such as mathematic mechanization, symbolic computation, physics, mechanics, signal processing and pattern recognition. The main contents of the dissertation are described as following:1. The algebraic method, which is based on the symbolic computation, has been applied to study new travelling wave solutions for the Boussinesq equation by means of Epsilon package in Maple. More new explicit travelling wave solutions are obtained, which contain solitons, triangular periodic, rational, Jacobi elliptic function periodic and Weierstrass elliptic function periodic solutions. Compared with the references [11, 43], the method put forward in this paper is more simple and convenient. More explicit travelling wave solutions can be obtained by using this method to solve other nonlinear evolution equations.2. Based on the section 1, a generalized algebraic method has been developed, and the problem of travelling wave solutions of nonlinearly dispersive Boussinesq equations (B(m,n) equations) has been investigated by using this method, and a variety of explicit exact travelling wave solutions, such as solitary wave, triangular periodic, rational, Jacobi and Weierstrass elliptic function periodic wave solutions were obtained formally, thus further enriching the solutions of B(m,ri) equations.3. A new blind gray-level watermarking algorithm in spatial domain based on chaotic sequences which were generated by the Logistic map which is one of the typical nonlinear evolution equations is proposed in this paper. By the introduction of characteristic string, algorithms based on bit-change and block-mean have been combined together quite well. Experimental results show that the proposed scheme is robust for some image operations such as noise adding, cropping, filtering and lossy JPEG compression.