Behavioral analysis of anisotropic diffusion in image processing

Abstract

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.

Forward-and-backward diffusion for hyperspectral remote sensing image smoothing and enhancement

“…(c) In edge feature regions, we need to define ( ) s sφ or sublinear to avoid too many minima for the function in Eq. (2) [10]. Using the above discussion, we proposed an alternative class of regularization function for the minimization of ( ) ( )dx…”

“…(1) At homogeneous regions (low gradients), smoothing is the same in all directions: (10) (2) Near the edges where the magnitude of the gradient is very large, we need to diffusion along the ξ -direction and not across it: …”

“…(10)(11)(12) and the relation between regularization and diffusion process (7), we characterize a general image restoration-enhancement in the following way: (a) In homogeneous regions, isotropic diffusion is preferred, i.e., ( ) 2 s s φ . (b) In mid-gradient regions, we apply forward diffusion to smooth the image and simultaneously inverse diffusion for enhancing the edges back.…”

“…They developed an adaptive smoothing and edge detection scheme in which the linear heat diffusion equation is replaced by a selective diffusion that preserves edges. This development led some researchers to focus on the development of various anisotropic diffusion models and diverse numerical schemes to obtain the steady-state solutions [4][5][6][7][8][9][10][11][12][13]. While a great deal of effort has been dedicated to the behavioral analysis of anisotropic diffusion in the last decades, many researchers have discussed its extension to remote sensing multispectral imagery [14,15], after the pioneering works of Di Zenzo [16] and Cumani [17] on vector-valued images, popularized by Sapiro and Ringach [18].…”

A new framework of quantifying differences between images by matching gradient fields and its application to image blending

“…The PDE that governs anisotropic diffusion can be written in the following form: div( ( ) ) I t c I I ∂ ∂ = ∇ ∇ where I is an image and c is the diffusion coefficient. It has been noted in [19] that this PDE could also be derived by minimizing the following energy:…”