Bridging nonlinear diffusion and multiscale analyses

We revisit the nonlinear diffusion problem of signals/images which has been and remains of great research interest in image analysis and computer vision. We build on a previously reported reformulation of the linear diffusion in a probabilistic setting to address a limitation inherent to nonlinear diffusion, namely a loss of texture particularly noticeable in natural images. In the course of our probabilistic reinterpretation of nonlinear diffusion we have, in addition to gaining much insight into Perona-Malik (1988) equation and its properties, proposed a robust technique which achieved a remarkable noise removal while preserving sharp features like edges. While this has resolved a long standing problem of a required prior knowledge of a stopping time of the evolution, the associated cost was a limitation similar to that of all existing nonlinear diffusion techniques, namely a loss of texture information. Using a method based on frames, we bridge the world of multiscale analysis and that of partial differential equation-based diffusion and effectively address this problem. We give substantiating examples of our approach, particles (pixels or sixels), a reformulation of the linear diffusion in a probabilistic setting and interpret, as a result, the filtering as random motions of particles.