Article

Topological analysis and characterization of discrete scalar fields

Authors:

Emanuele Danovaro,

Leila De Floriani,

Mohammed Mostefa MesmoudiAuthors Info & Claims

Proceedings of the 11th international conference on Theoretical foundations of computer vision

Pages 386 - 402

Published: 07 April 2002 Publication History

Abstract

In this paper, we address the problem of analyzing the topology of discrete scalar fields defined on triangulated domains. To this aim, we introduce the notions of discrete gradient vector field and of Smale-like decomposition for the domain of a d-dimensional scalar field. We use such notions to extract the most relevant features representing the topology of the field.We describe a decomposition algorithm, which is independent of the dimension of the scalar field, and, based on it, methods for extracting the critical net of a scalar field. A complete classification of the critical points of a 2-dimensional field that corresponds to a piecewise differentiable field is also presented.

References

[1]

M.K. Agoston. Algebraic Topology, A First Course, Pure and Applied Mathematics, Marcel Dekker, 1976.

[2]

C. L. Bajaj, V. Pascucci, and D. R. Shikore. Visualization of scalar topology for structural enhacement. In Proceedings of the IEEE Conference on Visualization' 98 1998, pages 51-58, 1998.

Digital Library

[3]

C. L. Bajaj and D. R. Shikore. Topology preserving data simplification with error bounds. Journal on Computers and Graphics, 22(1):3-12, 1998.

[4]

T. F. Banchoff. Critical Points and Curvature for Embedded Polyhedral Surfaces. Amer. Math. Monthly, 77(1):475-485, 1977.

[5]

S. Biasotti, B. Falcidieno, and M. Spagnuolo. Extended reeb graphs for surface understanding and description. In Proc. 9th DGCI'2000, Upsala, LNCS 1953, Springer-Verlag, pages 185-197, 2000.

Digital Library

[6]

L. DeFloriani, M. M. Mesmoudi, F. Morando, and Enrico Puppo. Nonmanifold decomposition in arbitrary dimension. In Proc. DGCI'2002, LNCS 2301, pages 96-80, 2002.

Digital Library

[7]

H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse complexes for piecewise linear 2-manifolds. In Proc 17th Sympos. Comput. Geom., pages 70-79, 2001.

Digital Library

[8]

R. Forman. Morse theory for cell complexes. Advances in Mathematics, 134:90- 145, 1998.

[9]

T. Gerstner and R. Pajarola. Topology preserving and controlled topology simplifying multiresolution isosurface extraction. In Proceedings IEEE Visualization 2000, pages 259-266. IEEE Computer Society, 2000.

Digital Library

[10]

J.C Hart. Morse theory for implicit surface modeling. In H.C. Hege and K. Poltihier (Eds), Mathematical Visualization, Springer-Verlag, pages 256-268, 1998.

[11]

J.C Hart. Using the CW-complex to represent topological structure of implicit surfaces and solids. In Proc. Implicit Surfaces 1999, Eurographics/SIGGRAPH, pages 107-112, 1999.

Digital Library

[12]

C Johnson, M. Burnett, and W. Dumbar. Crystallographic topology and its applications. In Crystallographic Computing 7: Macromolecular Crystallography data, P. E. Bourne, K. D. Watenpaugh, eds., IUCr Crystallographic Symposia, Oxford University Press, 2001.

[13]

J. Toriwaki and T. Fukumura. Extraction of structural information from grey pictures. Computer Graphics and Image Processing, 7:30-51, 1975.

[14]

F. Meyer. Topographic distance and watershed lines. Signal Processing, 38(1):113-125, 1994.

Digital Library

[15]

J. Milnor. Morse Theory. Princeton University Press, 1963.

[16]

L. R. Nackman. Two-dimensional critical point configuration graph. IEEE Transactions on Pattern Analysisand Machine Intelligence, PAMI-6(4):442-450, 1984.

[17]

T. K. Peucker and E. G. Douglas. Detection of surface-specific points by local pa-prallel processing of discrete terrain elevation data. Graphics Image Processing, 4:475-387, 1975.

[18]

S. Smale. Morse inequalities for a dynamical system. Bulletin of American Mathematical Society, 66:43-49, 1960.

[19]

R. Thom. Sur une partition en cellule associées a une fonction sur une variété. C.R.A.S., 228:973-975, 1949.

[20]

L. T. Watson, T. J. Laffey, and R.M. Haralick. Topographic classification of digital image intensity surfaces using generalised splines and the discrete cosine transformation. Computer Vision, Graphics and Image Processing, 29:143-167, 1985.

[21]

G.H. Weber, G Scheuermann, H. Hagen, and B. Hamann. Exploring scalar fields usign criticla isovalues. In Proceedings IEEE Visualization 2002, pages 171-178. IEEE Computer Society Press, 2002.

Digital Library

Cited By

De Floriani LFugacci UIuricich FMagillo P(2015)Morse complexes for shape segmentation and homological analysisComputer Graphics Forum10.1111/cgf.1259634:2(761-785)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1111/cgf.12596
Magillo PDe Floriani LIuricich FKnoblock CSchneider MKröger PKrumm JWidmayer P(2013)Morphologically-aware elimination of flat edges from a TINProceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems10.1145/2525314.2525341(244-253)Online publication date: 5-Nov-2013
https://dl.acm.org/doi/10.1145/2525314.2525341
Vitali MDe Floriani LMagillo P(2011)Computing morse decompositions for triangulated terrainsProceedings of the 16th international conference on Image analysis and processing: Part I10.5555/2042620.2042689(565-574)Online publication date: 14-Sep-2011
https://dl.acm.org/doi/10.5555/2042620.2042689
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Published In

cover image Guide Proceedings

Proceedings of the 11th international conference on Theoretical foundations of computer vision

April 2002

437 pages

ISBN:3540009167

Editors:
Tetsuo Asano
JAIST, School of Information Science, Tatsunokuchi, Ishikawa, Japan
,
Reinhard Klette
University of Auckland, Computer Science Dept. and CITR, Glen Innes, Auckland, New Zealand
,
Chrisitan Ronse
LSIIT UMR CNRS-ULP, Illkirch, France

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 07 April 2002

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Cited By

De Floriani LFugacci UIuricich FMagillo P(2015)Morse complexes for shape segmentation and homological analysisComputer Graphics Forum10.1111/cgf.1259634:2(761-785)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1111/cgf.12596
Magillo PDe Floriani LIuricich FKnoblock CSchneider MKröger PKrumm JWidmayer P(2013)Morphologically-aware elimination of flat edges from a TINProceedings of the 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems10.1145/2525314.2525341(244-253)Online publication date: 5-Nov-2013
https://dl.acm.org/doi/10.1145/2525314.2525341
Vitali MDe Floriani LMagillo P(2011)Computing morse decompositions for triangulated terrainsProceedings of the 16th international conference on Image analysis and processing: Part I10.5555/2042620.2042689(565-574)Online publication date: 14-Sep-2011
https://dl.acm.org/doi/10.5555/2042620.2042689
Čomí LMesmoudi MDe Floriani L(2011)Smale-like decomposition and forman theory for discrete scalar fieldsProceedings of the 16th IAPR international conference on Discrete geometry for computer imagery10.5555/1987119.1987167(477-488)Online publication date: 6-Apr-2011
https://dl.acm.org/doi/10.5555/1987119.1987167
Čomić LDe Floriani L(2008)Cancellation of critical points in 2D and 3D Morse and Morse-Smale complexesProceedings of the 14th IAPR international conference on Discrete geometry for computer imagery10.5555/1792454.1792469(117-128)Online publication date: 16-Apr-2008
https://dl.acm.org/doi/10.5555/1792454.1792469
Mesmoudi MDe Floriani LPort UAlliez PRusinkiewicz S(2008)Discrete distortion in triangulated 3-manifoldsProceedings of the Symposium on Geometry Processing10.5555/1731309.1731312(1333-1340)Online publication date: 2-Jul-2008
https://dl.acm.org/doi/10.5555/1731309.1731312
Biasotti SDe Floriani LFalcidieno BFrosini PGiorgi DLandi CPapaleo LSpagnuolo M(2008)Describing shapes by geometrical-topological properties of real functionsACM Computing Surveys10.1145/1391729.139173140:4(1-87)Online publication date: 15-Oct-2008
https://dl.acm.org/doi/10.1145/1391729.1391731
Danovaro EDe Floriani LVitali MMagillo PSamet HShahabi CSchneider M(2007)Multi-scale dual morse complexes for representing terrain morphologyProceedings of the 15th annual ACM international symposium on Advances in geographic information systems10.1145/1341012.1341050(1-8)Online publication date: 7-Nov-2007
https://dl.acm.org/doi/10.1145/1341012.1341050
Čomić LDe Floriani LPapaleo L(2005)Morse-Smale decompositions for modeling terrain knowledgeProceedings of the 2005 international conference on Spatial Information Theory10.1007/11556114_27(426-444)Online publication date: 14-Sep-2005
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Saux EThibaud RLi KKim MCruz IPfoser D(2004)A new approach for a topographic feature-based characterization of digital elevation dataProceedings of the 12th annual ACM international workshop on Geographic information systems10.1145/1032222.1032235(73-81)Online publication date: 12-Nov-2004
https://dl.acm.org/doi/10.1145/1032222.1032235

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