Learning an Integral Equation Approximation to Nonlinear Anisotropic Diffusion in Image Processing

Abstract

Multi-scale image enhancement and representation is an important part of biological and machine early vision systems. The process of constructing this representation must be both rapid and insensitive to noise, while retaining image structure at all scales. This is a complex task as small scale structure is difficult to distinguish from noise, while larger scale structure requires more computational effort. In both cases good localization can be problematic. Errors can also arise when conflicting results at different scales require cross-scale arbitration.

Broadly speaking, multi-scale image analysis has historically been accomplished using two types of techniques: those which are sensitive to image structure and those which are not. Algorithms in the latter category typically use a set of variously sized blurring kernels to produce images each of which retain structure at a different scale (Marr and Hildreth, 1980; Burt and Adelson, 1983; Koenderink, 1984; Hummel, 1986). The kernels used for the blurring are predefined and independent of the content of the image. Koenderink showed that if the kernels are Gaussian, then this process is equivalent to the evolution of the linear heat (or diffusion) equation. He thus transformed the integral equation representing the convolution process into the solution of a partial differential equation (PDE).

Structure sensitive multi-scale techniques attempt to analyze an image at a variety of scales within a single image (Klinger, 1971; Perona and Malik, 1987; Nitzberg and Shiota, 1992). Klinger (Klinger, 1971) proposed the quad tree, one of the earliest structure-sensitive multi-scale image representations. In this approach, a tree structure is built by recursively subdividing an image based on pixel variance in subregions. The final tree contains leaves representing image regions whose variance is small according to some measure. Recently (Perona and Malik, 1987; Perona and Malik, 1990), the PDE formalism introduced by Koenderink has been extended to allow structure-sensitive multi-scale analysis. Instead of the uniform blurring of the linear heat equation which destroys small scale structure as time evolves, Perona and Malik use a space-variant conductance coefficient based on the magnitude of the intensity gradient in the image, giving rise to a nonlinear PDE. Like the quadtree, the end result is a single image representation which contains information at all scales of interest.

The Perona and Malik approach produces impressive results, but the numerical integration of a nonlinear PDE is a costly and inherently serial process. In this paper we present a technique which obtains an approximate solution to the PDE for a specific time, via the solution of an integral equation which is the nonlinear analog of convolution. The kernel function of the integral equation plays the same role that a Green's function does for a linear PDE, allowing the direct solution of the nonlinear PDE for a specific time without requiring integration through intermediate times. We then use a learning technique to approximate the kernel function for arbitrary input images. The result is an improvement in speed and noise-sensitivity, as well as providing a means to parallelize an otherwise serial algorithm.