ML optimality of PDE-based segmentation algorithms

Summary form only given, as follows. Building on our image restoration and segmentation algorithms developed in Pollak et al. (2000), we present a very simple nonlinear diffusion equation and show its utility both for image segmentation and for the detection of abrupt changes in 1-D. We show that it may be interpreted as a variant of the Perona-Malik (1990) equation, as the steepest descent equation for the total variation (Bouman and Sauer 1991, Rudin et al. 1992), and-in 1-D-as a solver of a simple version of the Mumford-Shah (1985) problem. The analysis of our equation in 1-D reveals it to be an exact solver of certain maximum likelihood detection/estimation problems. The major advantage over other methods to solve these problems is O(N log N) computational complexity in one spatial dimension. Finally, we show our method to be a robust estimator (in the spirit of H-infinity estimation (Nagpal and Khargonekar 1991)) for a restricted class of 1-D problems. Experiments suggest that the 2-D version of our algorithm retains robustness properties of the 1-D version. Moreover, if only a binary segmentation is required, the computational complexity of our 2-D algorithm is still O(N log N). A remaining challenge is to extend our probabilistic analysis of the 1-D algorithm to 2-D, designing fast and optimal image segmentation algorithms