Sampling conditions for anisotropic diffusion

Multi-resolution image analysis utilizes subsampled image representations for applications such as image coding, hierarchical image segmentation and fast image smoothing. An anti-aliasing filter may be used to insure that the sampled signals adequately represent the frequency components/features of the higher resolution signal. Sampling theories associated with linear anti-aliasing filtering are well-defined and conditions for nonlinear filters are emerging. This paper analyzes sampling conditions associated with anisotropic diffusion, an adaptive nonlinear filter implemented by partial differential equations (PDEs). Sampling criteria will be defined within the context of edge causality, and conditions will be prescribed that guarantee removal of all features unsupported in the sample domain. Initially, sampling definitions will utilize a simple, piecewise linear approximation of the anisotropic diffusion mechanism. Results will then demonstrate the viability of the sampling approach through the computation of reconstruction errors. Extension to more practical diffusion operators will also be considered.