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

Multiscale 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. Structure sensitive multiscale techniques attempt to analyze an image at a variety of scales within a single image. Various techniques are compared. In this paper, we present a technique which obtains an approximate solution to the partial differential equation (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.