A Scale-space Approach to the Morphological Simplification of Scalar Fields

We present a multi-scale morphological model of scalar fields based on the analysis of the spatial frequencies of the underlying function. Morphological models partition the domain of a function into homogeneous regions. The most popular tool in this field is the Morse-Smale complex, where each region is spanned by all integral lines that join a minimum to a maximum, with the integral lines departing from saddles as region boundaries. Morphological features usually occur at very different scales, from noise and high frequency details up to large trends at the lowest frequencies. Without some form of multi-scale analysis, only the morphology at the finest scale is visible and explicit in such a model. The most popular approach in the literature is the filtration provided by persistent homology, a method that combines the amplitude values of critical points with the topology of the sublevel sets of the function. We propose the adoption of an alternative filtration method, based on the analysis of the deep structure of the linear scale-space of the function. To retrieve an adequately fine-grained ranked sequence of pairs of critical points that vanish through the scales, we adopt a continuous representation of the scale-space that overcomes the limits of discrete scale-space approaches. This sequence provides a progressive simplification of the Morse-Smale complex, resulting in a progressive multi-scale model of the morphology that always refers to the geometry of the original function, which is not changed by our model. We apply our method to digital elevation models, with results providing a multi-scale representation of the network of ridges and valley lines that joins peaks, pits and passes and divide the land into mountains and basins.