Piecewise Polynomial Reconstruction of Functions from Simplified Morse-Smale complex

Simulation of phenomena like climate often deals with large datasets. A process to extract the most salient features is needed to assist in the understanding of the dataset. The Morse-Smale (MS) complex is a topological structure defined on scalar functions which extracts critical points of the function and the links between them. Furthermore, it encodes a hierarchy between critical points, and less important critical points can be deleted in order to simplify the structure. Starting from an initial function f : R2 → R, the Morse-Smale complex of this function is computed, then simplified. From this simplified structure, we aim to construct a new function, which corresponds to this structure, closed to the initial function, thus approximating the initial data set by preserving the most salient features. The main difficulty we face, lies in the fact that both, the boundary curves (corresponding to the 1-cells of the MS complex) and the surface patches inside each 2-cell have to be monotonic functions. Furthermore, the geometry of the 2-cells may be very complex, see Fig. 1 (middle and right).