Computing Morse Decompositions for Triangulated Terrains: An Analysis and an Experimental Evaluation

We consider the problem of extracting the morphology of a terrain discretized as a triangle mesh. We discuss first how to transpose Morse theory to the discrete case in order to describe the morphology of triangulated terrains. We review algorithms for computing Morse decompositions, that we have adapted and implemented for triangulated terrains. We compare the the Morse decompositions produced by them, by considering two different metrics.

[1]  Serge Beucher,et al.  Use of watersheds in contour detection , 1979 .

[2]  Qian Huang,et al.  Quantitative methods of evaluating image segmentation , 1995, Proceedings., International Conference on Image Processing.

[3]  Luc Vincent,et al.  Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  Jos B. T. M. Roerdink,et al.  The Watershed Transform: Definitions, Algorithms and Parallelization Strategies , 2000, Fundam. Informaticae.

[5]  Leila De Floriani,et al.  A Geometric Approach to Curvature Estimation on Triangulated 3D Shapes , 2010, GRAPP.

[6]  Thomas Funkhouser,et al.  A benchmark for 3D mesh segmentation , 2009, SIGGRAPH 2009.

[7]  Wolfgang Strasser,et al.  Extracting regions of interest applying a local watershed transformation , 2000 .

[8]  Jos B. T. M. Roerdink,et al.  Computation of Watersheds Based on Parallel Graph Algorithms , 1996, ISMM.

[9]  Emanuele Danovaro,et al.  Topological Analysis and Characterization of Discrete Scalar Fields , 2002, Theoretical Foundations of Computer Vision.

[10]  Herbert Edelsbrunner,et al.  Hierarchical morse complexes for piecewise linear 2-manifolds , 2001, SCG '01.

[11]  Emanuele Danovaro,et al.  A Discrete Approach to Compute Terrain Morphology , 2007, VISIGRAPP.

[12]  Chandrajit L. Bajaj,et al.  Topology preserving data simplification with error bounds , 1998, Comput. Graph..

[13]  Emanuele Danovaro,et al.  Multiresolution morse triangulations , 2010, SPM '10.

[14]  William M. Rand,et al.  Objective Criteria for the Evaluation of Clustering Methods , 1971 .

[15]  Frédéric Chazal,et al.  Molecular shape analysis based upon the morse-smale complex and the connolly function , 2002, SCG '03.

[16]  Emanuele Danovaro,et al.  Morphology-driven simplification and multiresolution modeling of terrains , 2003, GIS '03.

[17]  Fernand Meyer,et al.  Topographic distance and watershed lines , 1994, Signal Process..

[18]  Leila De Floriani,et al.  Modeling and Generalization of Discrete Morse Terrain Decompositions , 2010, 2010 20th International Conference on Pattern Recognition.

[19]  Valerio Pascucci,et al.  Morse-smale complexes for piecewise linear 3-manifolds , 2003, SCG '03.

[20]  Daniela Giorgi,et al.  Describing shapes by geometrical-topological properties of real functions , 2008, CSUR.

[21]  T. Banchoff Critical Points and Curvature for Embedded Polyhedral Surfaces , 1970 .

[22]  Tosiyasu L. Kunii,et al.  Algorithms for Extracting Correct Critical Points and Constructing Topological Graphs from Discrete Geographical Elevation Data , 1995, Comput. Graph. Forum.

[23]  Emanuele Danovaro,et al.  Multi-scale dual morse complexes for representing terrain morphology , 2007, GIS.

[24]  Leila De Floriani,et al.  Discrete Distortion for Surface Meshes , 2009, ICIAP.

[25]  R. Forman Morse Theory for Cell Complexes , 1998 .

[26]  Claudio Gutierrez,et al.  Survey of graph database models , 2008, CSUR.

[27]  Alina N. Moga,et al.  A connected component approach to the watershed segmentation , 1998 .

[28]  Emanuele Danovaro,et al.  Surface Segmentation through Concentrated Curvature , 2007, 14th International Conference on Image Analysis and Processing (ICIAP 2007).

[29]  Reinhard Klette,et al.  Geometry, Morphology, and Computational Imaging , 2003 .

[30]  Bernd Hamann,et al.  A topological hierarchy for functions on triangulated surfaces , 2004, IEEE Transactions on Visualization and Computer Graphics.

[31]  Thomas S. Huang,et al.  Image processing , 1971 .

[32]  Leila De Floriani,et al.  Sparse terrain pyramids , 2008, GIS '08.