扩散方程的动力学分析

THE DYNAMICAL ANALYSIS OF DIFFUSION

扩散方程──热力学中用于描述热量分布及其变化规律的方程，已在另一门完全不同的学科──计算机视觉中获得了广泛的应用．扩散方程及其变化形式，可用来产生尺度空间及检测边缘．本文从数值解法及动力学分析的角度，分析了线性扩散方程、非线性扩散方程及偏置的非线性扩散方程．扩散方程的数值迭代式解事实上就是一个动力学系统的映射函数．所以扩散过程的稳态就是该动力学系统的不动点，扩散方程所产生的尺度空间就是该动力学系统的轨道．本文指出了选代式的不动点数目及其吸引域的分布决定了扩散过程的行为，并给出了一个性能良好的，有边缘增强作用的扩散方程的流量所需满足的充分条件．

Diffusion equations have been applied to generating scale space and detecting edges in computer vision. This paper analyses isotropic diffusion, anisotropic diffusion and biased anisotropic diffusion from the point of view of numerical analysis and dynamical system. The iteration function defined by the numerical solution of a diffusion equation corresponds to the mapping function of a dynamical system. The steady state of diffusion is thus the fixed point of this dynamical system, and the scale space is the orbit. It is shown that the number of the fixed points, their distribution and the corresponding attracting basin of the iteration function dominate the behavior of the diffusion. Sufficient conditions on the flow for a well-behaved anisotropic diffusion process having edge enhancement property are given.