Abstract
We investigate a morphological approach to the analysis and understanding of three-dimensional scalar fields, and we consider applications to 3D medical and molecular images as examples.We consider a discrete model of the scalar field obtained by discretizing its 3D domain into a tetrahedral mesh. In particular, our meshes correspond to approximations at uniform or variable resolution extracted from a multi-resolution model of the 3D scalar field, that we call a hierarchy of diamonds. We analyze the images based on the concept of discrete distortion, that we have introduced in [26], and on segmentations based on Morse theory. Discrete distortion is defined by considering the graph of the discrete 3D field, which is a tetrahedral hypersurface in R 4, and measuring the distortion of the transformation which maps the tetrahedral mesh discretizing the scalar field domain into the mesh representing its graph in R 4. We describe a segmentation algorithm to produce Morse decompositions of a 3D scalar field which uses a watershed approach and we apply it to 3D images by using as scalar field both intensity and discrete distortion. We present experimental results by considering the influence of resolution on distortion computation. In particular, we show that the salient features of the distortion field appear prominently in lower resolution approximations to the dataset.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
P. Aleksandrov. Combinatorial Topology. Dover Publications Inc., 1998.
M. T. Anderson. G éom étrisation des Vari ét és de Dimension 3 via le Flot de Ricci. Soci ét é Math ématique de France, Gazette, 103:25-40, 2005.
E. J. Ansari, N. andE Delp. On detecting dominant points. Pattern Recognition, 24(5):441-451,1991.
W. Beil, K. Rohr, and H. S. Stiehl. Investigation of approaches for the localization of anatomical landmarks in 3d medical images. Computer Assisted Radiology and Surgery, CARS, Berlin, Germany, 265-270, 1997.
J. Bey. Tetrahedral mesh refinement. Computing, 55:355-378, 1995.
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 Comput. Surv., 4(40), 2008.
L. Comic and L. De Floriani. Cancellation of critical points in 2d and 3d Morse and Morse-Smale complexes. In 14th IAPR International Conference on Discrete Geometry for Computer Imagery, volume 4992 of Lecture Notes in Computer Science, 117-128, Lyon, France, 16-18 April 2008.
L. De Floriani and P. Magillo. Multiresolution mesh representation: models and data structures. In M. Floater, A. Iske, and E. Quak, editors, Principles of Multi-resolution Geometric Modeling, Lecture Notes in Mathematics, 364-418, Berlin, 2002. Springer Verlag.
N. Dyn, K. Hormann, K. Sun-Jeong, and D. Levin. Optimizing 3d triangulations using discrete curvature analysis. In T. Lyche and L. Schumaker, editors, Mathematical Methods for Curves and Surfaces: Oslo 2000, 135-146, 2000.
H. Edelsbrunner, J. Harer, V. Natarajan, and V. Pascucci. Morse-Smale complexes for piece-wise linear 3-manifolds. In Proceedings 19th ACM Symposium on Computational Geometry, 361-370, 2003.
E. Fatemizadeh, C. Lucas, and H. Soltanian-Zadeh. Automatic landmark extraction from image data using modified growing neural gas network. IEEE Transactions on Information Technology in Biomedicine, 7(2):77-85, 2003.
T. Gatzke and C. Grimm. Estimating curvature on triangular meshes. International Journal on shape Modeling, 12:1-29, 2006.
T. Gerstner and M. Rumpf. Multiresolutional parallel isosurface extraction based on tetrahedral bisection. In Proceedings Symposium on Volume Visualization, 267-278. ACM Press, 1999.
B. Gregorski, M. Duchaineau, P. Lindstrom, V. Pascucci, and K. Joy. Interactive view-dependent rendering of large isosurfaces. In Proceedings IEEE Visualization, 475-484, 2002.
A. Gyulassy, P. 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, 2008.
A. Gyulassy, V. Natarajan, V. Pascucci, P.-T. Bremer, and B. Hamann. Topology-based simplification for feature extraction from 3d scalar fields. In Proceedings, IEEE Visualization 2005,275-280, October 2005.
A. Gyulassy, V. Natarajan, V. Pascucci, and B. Hamann. Efficient computation of Morse-Smale complexes for three-dimensional scalar functions. IEEE Trans. Vis. Comput. Graph. (IEEE Visualization 2007), 6(13):1440-1447, 2007.
S. Hahmann, A. Belayev, L. Busé, G. Elber, B. Mourrain, and C. Rössl. Shape Interrogation. L. De Floriani, M. Spagnuolo (Eds.), Shape Analysis and Structuring (Mathematics+Visualization), 2009.
T. Hartkens, K. Rohr, and H. Stiehl. Evaluation of 3d operators for the detection of anatomical point landmarks in MR and CT images. ComputerVision and Image Understanding, 86(2):118-136,2002.
D. Hebert. Symbolic local refinement of tetrahedral grids. Journal of Symbolic Computation, 17(5):457-472, May 1994.
A. Kimura, Y. Takama, Y. Yamazoe, S. Tanaka, and H. Tanaka. Parallel volume segmentation with tetrahedral adaptive grid. International Conference on Pattern Recognition, 2:281-286,2004.
M. Lee, L. De Floriani, and H. Samet. Constant-time neighbor finding in hierarchical tetrahedral meshes. In Proceedings International Conference on Shape Modeling, 286-295,Genova, Italy, May 2001. IEEE Computer Society.
J. M. Maubach. Local bisection refinement for n-simplicial grids generated by reflection. SIAM Journal on Scientific Computing, 16(1):210-227, January 1995.
V. Mello, L. Velho, and G. Taubin. Estimating the in/out function of a surface represented by points. In Symposium on Solid Modeling and Applications, 108-114, 2003.
M. M. Mesmoudi, L. De Floriani, and P. Magillo. Morphological analysis of terrains based on discrete curvature and distortion. In Proceedings of International Conference on Advances in Geographic Information Systems (ACMGIS 2008), Irvine, California, USA, 2008.
M. M. Mesmoudi, L. De Floriani, and U. Port. Discrete distortion in triangulated 3-manifolds. Computer Graphics Forum (Proceedings SGP 2008), 27(5):1333-1340, 2008.
J. Milnor. Morse Theory. Princeton University Press, 1963.
V. Natarajan, Y. Wang, P. Bremer, V. Pascucci, and B. Hamann. Segmenting molecular surfaces. Computer Aided Geometric Design, 23(6):495-509, 2006.
M. Ohlberger and M. Rumpf. Hierarchical and adaptive visualization on nested grids. Computing, 56(4):365-385, 1997.
V. Pascucci. Slow Growing Subdivisions (SGS) in any dimension: towards removing the curse of dimensionality. Computer Graphics Forum, 21(3):451-460, 2002.
S. C. Pei and C. Lin. The detection of dominant points on digital curves byscale-space filtering. Pattern Recognition, 25(11):1307-1314, 1992.
T. Regge. General relativity without coordinates. Nuovo Cimento, 19(3):558-571, 1961.
M. Reuter, F.-E. Wolter, and N. Peinecke. Laplace-Beltrami spectra as “Shape-DNA” of surfaces and solids. Computer-Aided Design, 38:342-366, 2006.
M. Rivara and C. Levin. A 3D refinement algorithm suitable for adaptive and multigrid techniques. Communications in Applied Numerical Methods, 8(5):281-290, 1992.
J. Roerdink and A. Meijster. The watershed transform: definitions, algorithms, and parallelization strategies. Fundamenta Informaticae, 41:187-228, 2000.
H. Samet. Foundations of Multidimensional and Metric Data Structures. The Morgan Kaufmann series in computer graphics and geometric modeling. Morgan Kaufmann, 2006.
A. Shamir. Segmentation and shape extraction of 3d boundary meshes. In EG 2006 - State of the Art Reports, Vienna, 2006.
T. Surazhsky, E. Magid, O. Soldea, G. Elber, and E. Rivlin. A comparison of gaussian and mean curvatures estimation methods on triangular meshes. In Proceedings of Conference on Robotics and Automation, Proceedings. ICRA ’03. IEEE International, volume 1, 739-743, 2003.
C. Teh and R. T. Chin. On the detection of dominant points on digital curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(8):859-872, 1989.
M. Troyanov. Les surfaces Euclidiennes à singularités coniques. L’enseignement Mathématique, 32:79-94, 1986.
L. Vincent and P. Soille. Watershed in digital spaces: an efficient algorithm based on immersion simulation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(6):583-598, 1991.
K. Weiss and L. De Floriani. Diamond hierarchies of arbitrary dimension. Computer Graphics Forum (Proceedings SGP 2009), 28(5):1289-1300, 2009.
K. Weiss and L. De Floriani. Supercubes: A high-level primitive for diamond hierarchies. IEEE Transactions on Visualization and Computer Graphics (Proceedings IEEE Visualization 2009), 15 (6):1603-1610, 2009.
K. Weiss and L. De Floriani. Simplex and diamond hierarchies: Models and applications. In H. Hauser and E. Reinhard, editors, EG 2010 - State of the Art Reports, 113-136, Norrköping, Sweden, 2010. Eurographics Association.
K. Weiss, M. Mesmoudi, and L. De Floriani. Multiresolution analysis of 3D images based on discrete distortion. In International Conference on Pattern Recognition (ICPR), 4093-4096, Istanbul, Turkey, 2010.
Y. Zhou, B. Chen, and A. Kaufman. Multi-resolution tetrahedral framework for visualizing regular volume data. In R. Yagel and H. Hagen, editors, Proceedings IEEE Visualization, 135-142, 1997.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
De Floriani, L., Iuricich, F., Magillo, P., Mesmoudi, M.M., Weiss, K. (2012). Discrete Distortion for 3D Data Analysis. In: Linsen, L., Hagen, H., Hamann, B., Hege, HC. (eds) Visualization in Medicine and Life Sciences II. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21608-4_1
Download citation
DOI: https://doi.org/10.1007/978-3-642-21608-4_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21607-7
Online ISBN: 978-3-642-21608-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)