Learning an Integral Equation Approximation to Nonlinear

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 diicult to distinguish from noise, while larger scale structure requires more computational eeort. In both cases good localization can be problematic. Errors can also arise when connicting results at diierent 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 diierent scale Marr and Hildreth, The kernels used for the blurring are predeened 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 diiusion) equation. He thus transformed the integral equation representing the convolution process into the solution of a partial diierential equation (PDE). Structure sensitive multi-scale techniques attempt to analyze an image at a variety of scales within a single image 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 nal 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 coeecient based on the magnitude of the intensity gradient in the image, giving rise to a nonlinear PDE. Like the quadtree, the intent is to produce 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 speciic time, via the solution of an integral equation which is the nonlinear analog of convolution. The kernel function …