典型非线性演化方程及其应用于图像处理的研究

【作者】 高亮；

【导师】 徐伟；

【作者基本信息】 西北工业大学 ， 应用数学， 2006， 硕士

【摘要】 随着非线性科学的迅猛发展,非线性演化方程(包括非线性常微分方程、非线性偏微分方程、非线性差分方程和函数方程等)及其应用于图像处理方面的研究己成为非线性科学前沿领域的主要研究内容。 非线性演化方程的求解问题是古老而在理论和实际上都很重要的研究课题。随着非线性科学的进展,非线性演化方程的求解成为广大物理学、力学、地球科学、生命科学、应用数学和工程技术科学工作者研究非线性问题所不可缺少的。目前,国内外许多学者都致力于这方面的研究。另一方面,非线性演化方程在图像处理方面有着广泛的应用。因此,如何更好的利用非线性方程进行数字图像处理,成为当前重要的研究课题。 本文围绕非线性演化方程领域的上述重要研究命题,运用数学机械化、符号计算、物理学、力学、信号处理及模式识别等学科交叉综合技术,研究某些典型非线性演化方程的求解问题及非线性演化方程在图像处理方面的应用,主要研究内容包括: 1.利用一种基于符号计算的代数方法,结合Maple环境中的Epsilon软件包,求解了Boussinesq方程,获得了若干其它方法不曾给出的形式更为丰富的新的显式行波解,其中包括孤波解、三角函数解、有理函数解、Jacobi椭圆函数周期解和Weierstrass椭圆函数周期解.与文献[11,43]相比,本文的方法更为简便、易行,将该方法应用于其它非线性演化方程(组)中,可获得更多的显式行波解。 2.在1的基础上,发展了一种推广的代数方法,并借助该方法求解了非线性色散Boussinesq方程(B(m,n)方程),获得了其多种显式精确行波解,包括孤波解、三角函数解、有理函数解、Jacobi椭圆函数周期解和Weierstrass椭圆函数周期解,进一步丰富了B(m,n)方程的解。 3.空间域上的水印算法往往不能同时有效的抵抗高斯噪声和椒盐噪声的攻击,而仅对其中的一种表现出较强的健壮性。本文借助于典型非线性演化方程——Logistic映射产生混沌序列,然后基于混沌序列提出了一种新的空间域盲水印算法,通过引入特征串的概念,算法有效的克服了空间域水印算法的上述缺点,一系列水印攻击实验表明该算法能够有效的抵抗高斯噪声和椒盐噪声的攻击,而且对诸如图像增强、锐化、滤波、剪切、JPEG压缩等一些常见的图像处理也都表现出了较好的健壮性。更多还原

【Abstract】 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.更多还原