Piecewise polynomial Reconstruction of Scalar Fields from Simplified Morse-Smale Complexes

Léo Allemand-Giorgis 1 Georges-Pierre Bonneau 1 Stefanie Hahmann 2
1 MAVERICK - Models and Algorithms for Visualization and Rendering
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
2 IMAGINE - Intuitive Modeling and Animation for Interactive Graphics & Narrative Environments
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : Morse-Smale (MS) complexes have been proposed to visualize topological features of scalar fields defined on manifold domains. Herein, three main problems have been addressed in the past: (a) efficient computation of the initial combinatorial structure connecting the critical points; (b) simplification of these combinatorial structures; (c) reconstruction of a scalar field in accordance to the simplified Morse-Smale complex. The present paper faces the third problem by proposing a novel approach for computing a scalar field coherent with a given simplified MS complex that privileges the use of piecewise polynomial functions. Based on techniques borrowed from shape preserving design in Computer Aided Geometric Design, our method constructs the surface cell by cell using piecewise polynomial curves and surfaces. The benefit and limitations of using polynomials for reconstruction surfaces from topological data are presented in this paper.
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Léo Allemand-Giorgis, Georges-Pierre Bonneau, Stefanie Hahmann. Piecewise polynomial Reconstruction of Scalar Fields from Simplified Morse-Smale Complexes. Hamish Carr; Christoph Garth; Tino Weinkauf. Topological Methods in Data Analysis and Visualization IV - Theory, Algorithms, and Applications, Springer, pp.151-168, 2017, Mathematics and Visualization, 978-3-319-44684-4. ⟨10.1007/978-3-319-44684-4_9⟩. ⟨hal-01252477⟩

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