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Skeletons of 3D Surfaces Based on the Laplace-Beltrami Operator Eigenfunctions

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In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.
 
Cite this paper
Escalona-Buendia, A. , Hernández-Martínez, L. , Martínez-Vega, R. , Murillo-Torres, J. and Nieto-Crisóstomo, O. (2015) Skeletons of 3D Surfaces Based on the Laplace-Beltrami Operator Eigenfunctions. Applied Mathematics, 6, 414-420. doi: 10.4236/am.2015.62038.
 
References
[1]Reuter, M., Wolter, F.E. and Peinecke, N. (2006) Laplace-Beltrami Spectra as “Shape-DNA” of Surfaces and Solids. Computer-Aided Design, 38, 342-366.

http://dx.doi.org/10.1016/j.cad.2005.10.011
 
[2]Reuter, M., Biasotii, S., Patanè, G. and Spagnulo, M. (2009) Discrete Laplace-Beltrami Operators for Shape Analysis and Segmentation. Computer & Graphics, 33, 381-390.

http://dx.doi.org/10.1016/j.cag.2009.03.005
 
[3]Sadleir, R.J.T. and Whelan, P.F. (2005) Fast Colon Centerline Calculation Using Optimized 3D Topological Thinning. Computerized Medical Imaging and Graphics, 29, 251-318.

http://dx.doi.org/10.1016/j.compmedimag.2004.10.002
 
[4]Cornea, H.D., Siver, D. and Min, P. (2005) Curve-Skeleton Applications. IEEE Visualization, 23-28 October 2005, 95-102.
 
[5]Seo, S., Chung, M.K., Whyms, B.J. and Vorperian, H.K. (2011) Mandible Shape Modeling Using the Second Eigenfunction of the Laplace-Beltrami Operator. Proceedings of SPIE, Medical Imaging, 2011: Image Processing, 7962, 79620Z.

http://dx.doi.org/10.1117/12.877537
 
[6]Object File Format (2005) Princeton Shape Benchmark.

http://shape.cs.princeton.edu/benchmark/documentation/off_format.html
 
[7]Thompson, P.M., Hayashi, K.M., et al. (2004) Mapping Hippocampal and Ventricular Change in Alzheimer Disease. Neuroimage, 22, 1754-1766.

http://dx.doi.org/10.1016/j.neuroimage.2004.03.040                                                       eww150226lx
 
[8]Aho, A.V., Hopcfort, J.E. and Ullman, J.D. (1983) Data Structures and Algorithms. Addison-Welsey, Boston.    
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