Dept. of Electr. Eng., Minnesota Univ., Minneapolis, MN.
IEEE Trans Image Process. 1996;5(11):1539-53. doi: 10.1109/83.541424.
In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik (1990). The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator.
在本文中，我们分析了 Perona 和 Malik（1990）提出的各向异性扩散模型的行为。主要思想是将各向异性扩散方程表示为来自某个优化问题，因此可以根据相应的能量曲面的形状来分析其行为。我们证明各向异性扩散是求解能量最小化问题的最速下降法。当能量泛函存在唯一的全局最小值时，各向异性扩散是适定的，而当各向异性扩散不适定时，则是因为其能量泛函存在无限多个全局最小值，这些全局最小值在图像空间中密集分布。我们给出了一个充分条件，用于确定各向异性扩散是适定的，以及一个充分必要条件，用于确定其由于全局最小值密集分布而不适定。通过将扩散算子分解成两个部分，我们说明了各向异性扩散的平滑和边缘增强机制：一个部分沿边缘切向扩散，因此作为各向异性平滑算子；另一个部分沿边缘法向流动，因此作为增强算子。
