Morphological systems are a broad class of nonlinear systems that can provide rigorous and efficient solutions to many applications of image and signal processing that can benefit from and most often require the use of nonlinear analysis. Examples include multiscale filtering, feature extraction, image segmentation, and other geometry-based problems in computer vision. This research is investigating a range of theoretical and applied problems including: (1) the modeling of multiscale processing in image analysis, image distance transforms, and signal envelope detection using morphological systems; (2) the development of analytic tools for these nonlinear systems both in the time/space domain and in a new transform domain - the slope domain; and, (3) the exploration of exciting connections of the above to physics. The unifying theme is a collection of max-min differential/difference equations modeling the scale or time/space dynamics of morphological systems, and some novel nonlinear signal transforms, called slope transforms. Several nonlinear PDEs that model the evolution of multiscale morphological filters are being studied. These PDEs are related to the well-known eikonal equation of optics. Solutions based on max-min difference equations that can compute distance transforms are being examined. The analysis of the multiscale nonlinear PDEs and the solution of the eikonal equation through max-min equations constitutes a new area in nonlinear image analysis that we call differential morphology; many applications are envisioned. Specific research problems include the development of fast discrete slope transforms; the study of recursive morphological systems resulting from discretizing the nonlinear differential equations; the design of discrete slope-selective filters for signal envelope detection and image distance transforms; the design of analog slope filters for D/A conversion; multiscale image analysis using morphological PDEs; and image segmentation and gr idless image halftoning by solving the eikonal PDE using morphological systems.
