Abstract
Simplification of Morse and Morse-Smale complexes is an important issue to eliminate noise and reduce over-segmentation. Moreover, different users may have different requirements in terms of degree of simplification, which usually vary over time and location within the field domain. Thus, a multi-resolution representation of morphology is critical for interactive analysis and exploration of data. In this chapter, we first describe and compare simplification operators on Morse functions and Morse and Morse-Smale complexes (Sect. 6.1). We then present multi-resolution models for the morphology of scalar fields (Sect. 6.2), and we specify two models in more detail: the first one providing a multi-resolution description of the combinatorial structure of Morse and Morse Smale complexes in arbitrary dimensions, and the second one addressing the problem of coupling a multi-resolution representation of geometry and of morphology in 2D.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
S. Beucher. Watershed, hierarchical segmentation and waterfall algorithm. In J. Serra and P. Soille, editors, Mathematical Morphology and its Applications to Image Processing, volume 2 of Computational Imaging and Vision, pages 69–76. Springer, 1994.
P.-T. Bremer, H. Edelsbrunner, B. Hamann, and V. Pascucci. A topological hierarchy for functions on triangulated surfaces. Transactions on Visualization and Computer Graphics, 10(4):385–396, July/August 2004.
P.-T. Bremer, V. Pascucci, and B. Hamann. Maximizing adaptivity in hierarchical topological models. In A.G. Belyaev, A.A. Pasko, and M. Spagnuolo, editors, Proc. International Conference on Shape Modeling and Applications 2005 (SMI ’05), pages 300–309, Los Alamitos, California, 2005. IEEE Computer Society Press.
P.-T. Bremer, V. Pascucci, and B. Hamann. Maximizing adaptivity in hierarchical topological models using cancellation trees. In Moeller T., Hamann B., and Russell R. D., editors, Mathematical Foundations of Scientific Visualization, Computer Graphics, and Massive Data Exploration, pages 1–18. Springer, 2009.
L. Čomić. Operators for Multi-Resolution Morse and Cell Complexes. PhD thesis, University of Novi Sad – Faculty of Technical Sciences, Serbia, 2014.
L. Čomić and L. De Floriani. Dimension-independent simplification and refinement of Morse complexes. Graphical Models, 73(5):261–285, September 2011.
L. Čomić, L. De Floriani, and F. Iuricich. Dimension-independent multi-resolution Morse complexes. Computers & Graphics, 36(5):541–547, 2012.
L. Čomić, L. De Floriani, and F. Iuricich. Simplification operators on a dimension-independent graph-based representation of Morse complexes. In C. L. Luengo Hendriks, G. Borgefors, and R. Strand, editors, ISMM, volume 7883 of Lecture Notes in Computer Science, pages 13–24. Springer, 2013.
E. Danovaro, L. De Floriani, P. Magillo, M. M. Mesmoudi, and E. Puppo. Morphology-driven simplification and multiresolution modeling of terrains. In E.Hoel and P.Rigaux, editors, Proc. ACM GIS 2003 - The 11th International Symposium on Advances in Geographic Information Systems, pages 63–70. ACM Press, 2003.
E. Danovaro, L. De Floriani, P. Magillo, and M. Vitali. Multiresolution Morse triangulations. In G. Elber, A. Fischer, J. Keyser, and M.-S. Kim, editors, Symposium on Solid and Physical Modeling, pages 183–188. ACM, 2010.
E. Danovaro, L. De Floriani, L. Papaleo, and M. Vitali. A multi-resolution representation for terrain morphology. In Proc. fourth International Conference on Geographic Information Science - GIScience 2006, Münster, Germany, September 2006.
E. Danovaro, L. De Floriani, and M. Vitali. Multi-resolution Morse-Smale complexes for terrain modeling. In 14th International Conference on Image Analysis and Processing, Modena, September 10–14 2007.
L. De Floriani, P. Magillo, and E. Puppo. VARIANT - processing and visualizing terrains at variable resolution. In Proc. 5th ACM Workshop on Advances in Geographic Information Systems, Las Vegas, Nevada, 1997.
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, D. Morozov, and V. Pascucci. Persistence-Sensitive Simplification of Functions on 2-Manifolds. In SCG’06: Proc. of the 22nd Annual Symposium on Computational Geometry 2006, 2006.
M. Garland and P. S. Heckbert. Surface simplification using quadric error metrics. In Computer Graphics Proceedings, Annual Conference Series (SIGGRAPH ’97), ACM Press, pages 209–216, 1997.
A. Gyulassy, P.-T. Bremer, B. Hamann, and V. Pascucci. Practical considerations in Morse-Smale complex computation. In V. Pascucci, X. Tricoche, H. Hagen, and J. Tierny, editors, Topological Methods in Data Analysis and Visualization: Theory, Algorithms, and Applications, Mathematics and Visualization, pages 67–78. Springer Verlag, Heidelberg, 2011.
A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann. Topology-based simplification for feature extraction from 3D scalar fields. In Proc. IEEE Visualization’05, pages 275–280. ACM Press, 2005.
F. Iuricich. Multi-resolution shape analysis based on discrete Morse decompositions. PhD thesis, University of Genova – DIBRIS, Italy, 2014.
Y. Matsumoto. An Introduction to Morse Theory, volume 208 of Translations of Mathematical Monographs. American Mathematical Society, 2002.
T. Weinkauf, Y. I. Gingold, and O. Sorkine. Topology-based smoothing of 2D scalar fields with c 1-continuity. Computer Graphics Forum, 29(3):1221–1230, 2010.
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). Simplification and Multi-Resolution Representations. 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_6
Download citation
DOI: https://doi.org/10.1007/978-1-4939-2149-2_6
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)