Partial differential equations in image analysis: Continuous modeling, discrete processing

This paper presents an overview of selected topics from an emerging new image analysis methodology that starts from continuous models provided by partial differential equations (PDEs) and proceeds with discrete processing of the image data via the numerical implementation of these PDEs on some discrete grid. We briefly discuss basic ideas, examples, algorithms, and applications for PDEs modeling nonlinear multiscale analysis, geometric evolution of curves and signals, nonlinear image/signal restoration via shock filtering, and the eikonal PDE of optics. Wherever possible, we compare the PDE approach with the corresponding all-discrete method. The PDE approach is very promising for solving (or improving previous all-discrete solutions of) many problems in image processing and computer vision because it provides new and more intuitive mathematical models, has connections with physics, gives better approximations to the Euclidean geometry of the problem, and is supported by efficient discrete numerical algorithms based on difference approximations.