Multi-resolution Morse-Smale Complexes for Terrain Modeling

Abstract

We propose a hierarchical representation for the morphology of a terrain. The basis of our morphological model is a decomposition of the terrain model, composed of the stable and unstable manifolds defined by its critical points, called a Morse-Smale complex. We propose a compact dual representation of the Morse-Smale complex and we define new simplification operators of the terrain morphology, which act on such representation. Based on these operators, we define a hierarchical morphology-based representation for a terrain, that we call a Multi-resolution Morse-Smale Complex (MMSC). Results from our implementation of the MMSC are shown.

Multi-Scale 3D Morse Complexes

“…Hierarchical watershed approaches have been developed to cope with this problem both in 2D and 3D images [3]. Other approaches proposed for terrains are based on a hierarchical representation of the 1-skeleton of a Morse-Smale complex, generated through the simplification operations mentioned above [5,10]. All these techniques consider the 1-skeleton at full resolution and generate a sequence of simplified representations of the complex by repeatedly applying a simplification operation.…”

“…Current multiresolution models are just mesh-based models, defined through a progressive simplification process applied to a triangle or a tetrahedral mesh discretizing the domain of the field at full resolution [11]. Hierarchical structural models based on the Morse-Smale complexes have been proposed for terrains [5,6,15,10]. Simplification operators for the MorseSmale complex of a 3D scalar field have been proposed in [17].…”

Multi-scale dual morse complexes for representing terrain morphology

“…Current multi-scale terrain models are just based on a progressive simplification process applied to a TIN describing a terrain at full resolution. Only a few hierarchical representations are based on a simplification of the network formed by the critical points and the integral lines joining them [5,6,9,13,10]. Such techniques provide a structural terrain description at different degrees of granularity, but they rely on strong assumptions on the height function defining the terrain.…”

Critical Nets and Beta-Stable Features for Image Matching

“…Thus, they encode image content that is rich, local, and stable in a formally well-defined sense. Our critical net is a close relative of the two dimensional Morse-Smale (M-S) graph [11,12], but is both simpler in concept and more reliable in computation. The following three aspects distinguish the critical net from the M-S graph and underscore the computational advantages of the former: (1) Critical nets are well defined for any discrete or continuous function, while M-S graphs requires the extra assumptions that all critical points are non-degenerate and there is no saddle-saddle connection.…”

“…Repeatability is a consequence of the fact that the critical net is invariant to affine deformations of the image domain and a certain wide class of changes in the function values. Our critical nets are a close relative of the Morse-Smale graph [11,12], but can be computed much more reliably and very efficiently on images defined on the integer grid.…”