Abstract
We propose different criteria for classifying algorithms for morphology computation (Sect. 2.1). Such criteria are based on the dimension of the input scalar field (2D, 3D, or dimension-independent), on the input format (simplicial models, regular grids), on the output information (ascending or descending Morse complex, Morse-Smale complex), on the format of the output information, and on the algorithmic approach applied. This last criterion leads to a classification into boundary-based and region-growing algorithms (coming from Banchoff’s piecewise linear Morse theory), algorithms based on the watershed transform, and based on Forman’s discrete Morse theory. We will use this classification to organize the survey provided in the remainder of the book. We discuss methods to compute the critical points of the scalar field, which is a basic subcomponent of most morphology computation algorithms (Sect. 2.2), and solutions for dealing with the domain boundary (Sect. 2.3), and with plateaus (Sect. 2.4), which are common issues when applying such algorithms to real-world data.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
L. Arge, J. Chase, P. Halpin, L. Toma, D. Urban, J.S. Vitter, and R. Wickremesinghe. Flow computation on massive grid terrains. Geoinformatica, 7(4):283–313, 2003.
C. L. Bajaj, V. Pascucci, and D. R. Shikore. Visualization of scalar topology for structural enhancement. In Proc. IEEE Visualization’98, pages 51–58. IEEE Computer Society, 1998.
C. L. Bajaj and D. R. Shikore. Topology preserving data simplification with error bounds. Computers and Graphics, 22(1):3–12, 1998.
T. Banchoff. Critical points and curvature for embedded polyhedral surfaces. American Mathematical Monthly, 77(5):475–485, 1970.
S. Biasotti, L. De Floriani, B. Falcidieno, P. Frosini, D. Giorgi, C. Landi, L. Papaleo, and M. Spagnuolo. Describing shapes by geometrical-topological properties of real functions. ACM Computing Surveys, 40(4):Article 12, 2008.
Y.-J. Chiang, T. Lenz ans X. Lua, and G. Rote. Simple and optimal output-sensitive construction of contour trees using monotone paths. Computational Geometry: Theory and Applications, 30(2):165–195, 2005.
L. Čomić, L. De Floriani, and F. Iuricich. Building morphological representations for 2D and 3D scalar fields. In E. Puppo, A. Brogni, and L. De Floriani, editors, Eurographics Italian Chapter Conference, pages 103–110. Eurographics, 2010.
L. Čomić, L. De Floriani, and L. Papaleo. Morse-Smale decompositions for modeling terrain knowledge. In Proc. International Conference on Spatial Information Theory (COSIT), volume 3693 of Lecture Notes in Computer Science, pages 426–444. Springer, 2005.
H. Edelsbrunner. Geometry and Topology for Mesh Generation. Cambridge University Press, England, 2001.
H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci. Morse-Smale complexes for piecewise linear 3-manifolds. In Proc. 19th ACM Symposium on Computational Geometry, pages 361–370, 2003.
H. Edelsbrunner, J. Harer, and A. Zomorodian. Hierarchical Morse complexes for piecewise linear 2-manifolds. In Proc. 17th ACM Symposium on Computational Geometry, pages 70–79, 2001.
H. Edelsbrunner and E. P. Mücke. Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics, 9(1):66–104, 1990.
T. Gerstner and R. Pajarola. Topology preserving and controlled topology simplifying multi-resolution isosurface extraction. In Proc. IEEE Visualization’00, pages 259–266, 2000.
A. Gyulassy, P.-T. Bremer, B. Hamann, and V. Pascucci. A practical approach to Morse-Smale complex computation: Scalability and generality. IEEE Transactions on Visualization and Computer Graphics, 14(6):1619–1626, Nov-Dec 2008.
A. Gyulassy, V. Natarajan, V. Pascucci, and B. Hamann. Efficient computation of Morse-Smale complexes for three-dimensional scalar functions. IEEE Transactions on Visualization and Computer Graphics, 13(6):1440–1447, Nov-Dec 2007.
R. Klette and A. Rosenfeld. Digital Geometry - Geometric Methods for Digital Picture Analysis. Computer Graphics and Geometric Modeling. Morgan Kaufmann, San Francisco, 2004.
P. Magillo, L. De Floriani, and F. Iuricich. Morphologically-aware elimination of flat edges from a tin. In Proc. 21st ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM SIGSPATIAL GIS 2013), November 5-8 2013.
A. Mangan and R. Whitaker. Partitioning 3D surface meshes using watershed segmentation. Transactions on Visualization and Computer Graphics, 5(4):308–321, 1999.
F. Meyer. Topographic distance and watershed lines. Signal Processing, 38:113–125, 1994.
X. Ni, M. Garland, and J. C. Hart. Fair Morse functions for extracting the topological structure of a surface mesh. In International Conference on Computer Graphics and Interactive Techniques ACM SIGGRAPH, pages 613–622, 2004.
L. Papaleo. Surface Reconstruction: Online Mosaicing and Modelling with Uncertainty. PhD thesis, University of Genova – Department of Computer Science, 2004.
T. K. Peucker and D. H. Douglas. Detection of surface-specific points by local parallel processing of discrete terrain elevation data. Computer Graphics and Image Processing, 4:375–387, 1975.
J. Roerdink and A. Meijster. The watershed transform: Definitions, algorithms, and parallelization strategies. Fundamenta Informaticae, 41:187–228, 2000.
B. Schneider. Extraction of hierarchical surface networks from bilinear surface patches. Geographical Analysis, 37(2):244–263, 2005.
B. Schneider and J. Wood. Construction of metric surface networks from raster-based DEMs. In S. Rana, editor, Topological Data Structures for Surfaces, pages 53–70. John Wiley & Sons Ltd, 2004.
P. Soille. Morphological Image Analysis: Principles and Applications. Springer-Verlag, Berlin and New York, 2004.
S. Takahashi, T. Ikeda, T. L. Kunii, and M. Ueda. Algorithms for extracting correct critical points and constructing topological graphs from discrete geographic elevation data. In Computer Graphics Forum, volume 14, pages 181–192, 1995.
S. Takahashi, Y. Takeshima, and I. Fujishiro. Topological volume skeletonization and its application to transfer function design. Graphical Models, 66(1):24–49, 2004.
J. Toriwaki and T. Fukumura. Extraction of structural information from gray pictures. Computer Graphics and Image Processing, 7:30–51, 1978.
L. T. Watson, T. J. Laffey, and R. M. Haralick. Topographic classification of digital image intensity surfaces using generalized splines and the discrete cosine transformation. Computer Vision, Graphics, and Image Processing, 29:143–167, 1985.
G. Weber and G. Scheuermann. Automating transfer function design based on topology analysis. In G. Brunnett, B. Hamann, H. Müller, and L. Linsen, editors, Geometric Modeling for Scientific Visualization, Mathematics and Visualization. Springer Verlag, Heidelberg, 2004.
G. H. Weber, G. Scheuermann, H. Hagen, and B. Hamann. Exploring scalar fields using critical isovalues. In Proc. IEEE Visualization’02, pages 171–178. IEEE Computer Society, 2002.
G. H. Weber, G. Scheuermann, and B. Hamann. Detecting critical regions in scalar fields. In G.-P. Bonneau, S. Hahmann, and C. D. Hansen, editors, Proc. Data Visualization Symposium, pages 85–94. ACM Press, New York, 2003.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 The Author(s)
About this chapter
Cite this chapter
Čomić, L., De Floriani, L., Magillo, P., Iuricich, F. (2014). Morphology Computation Algorithms: Generalities. In: Morphological Modeling of Terrains and Volume Data. SpringerBriefs in Computer Science. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2149-2_2
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2149-2_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-2148-5
Online ISBN: 978-1-4939-2149-2
eBook Packages: Computer ScienceComputer Science (R0)