CSDL Home IEEE Transactions on Pattern Analysis & Machine Intelligence 2011 vol.33 Issue No.03 - March

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Issue No.03 - March (2011 vol.33)

pp: 471-484

José María Pozo , Universitat Pompeu Fabra, Barcelona and CIBER-BBN

Maria-Cruz Villa-Uriol , Universitat Pompeu Fabra, Barcelona and CIBER-BBN

Alejandro F. Frangi , Universitat Pompeu Fabra, Barcelona, CIBER-BBN, and Institució de Recerca i Estudis Avançats, Barcelona

ABSTRACT

This paper introduces and evaluates a fast exact algorithm and a series of faster approximate algorithms for the computation of 3D geometric moments from an unstructured surface mesh of triangles. Being based on the object surface reduces the computational complexity of these algorithms with respect to volumetric grid-based algorithms. In contrast, it can only be applied for the computation of geometric moments of homogeneous objects. This advantage and restriction is shared with other proposed algorithms based on the object boundary. The proposed exact algorithm reduces the computational complexity for computing geometric moments up to order N with respect to previously proposed exact algorithms, from N^9 to N^6. The approximate series algorithm appears as a power series on the rate between triangle size and object size, which can be truncated at any desired degree. The higher the number and quality of the triangles, the better the approximation. This approximate algorithm reduces the computational complexity to N^3. In addition, the paper introduces a fast algorithm for the computation of 3D Zernike moments from the computed geometric moments, with a computational complexity N^4, while the previously proposed algorithm is of order N^6. The error introduced by the proposed approximate algorithms is evaluated in different shapes and the cost-benefit ratio in terms of error, and computational time is analyzed for different moment orders.

INDEX TERMS

Image analysis, geometric moments, 3D Zernike moments, shape characterization, object characterization.

CITATION

José María Pozo, Maria-Cruz Villa-Uriol, Alejandro F. Frangi, "Efficient 3D Geometric and Zernike Moments Computation from Unstructured Surface Meshes",

*IEEE Transactions on Pattern Analysis & Machine Intelligence*, vol.33, no. 3, pp. 471-484, March 2011, doi:10.1109/TPAMI.2010.139REFERENCES

- [1] R. Mukundan and K.R. Ramakrishnan,
Moment Functions in Image Analysis—Theory and Applications. World Scientific, 1998.- [2] J. Flusser, "Moment Invariants in Image Analysis,"
Proc. World Academy of Science, Eng., and Technology, vol. 11, pp. 196-201, Feb. 2006.- [3] L. Yang, F. Albregtsen, and T. Taxt, "Fast Computation of Three-Dimensional Geometric Moments Using a Discrete Divergence Theorem and a Generalization to Higher Dimensions,"
Graphical Models and Image Processing, vol. 59, no. 2, pp. 97-108, 1997.- [4] S.X. Liao and M. Pawlak, "On Image Analysis by Moments,"
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 18, no. 3, pp. 254-266, Mar. 1996.- [5] S.X. Liao and M. Pawlak, "On the Accuracy of Zernike Moments for Image Analysis,"
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 20, no. 12, pp. 1358-1364, Dec. 1998.- [6] S. Rodtook and S.S. Makhanov, "Numerical Experiments on the Accuracy of Rotation Moments Invariants,"
Image and Vision Computing, vol. 23, pp. 577-586, 2005.- [7] S. Lien and J.T. Kajiya, "Symbolic Method for Calculating the Integral Properties of Arbitrary Nonconvex Polyhedra,"
IEEE Computer Graphics and Applications, vol. 4, no. 10, pp. 35-41, Oct. 1984.- [8] C. Cattani and A. Paoluzzi, "Boundary Integration over Linear Polyhedra,"
Computer-Aided Design, vol. 22, no. 2, pp. 130-135, 1990.- [9] F. Bernardini, "Integration of Polynomials over n-Dimensional Polyhedra,"
Computer-Aided Design, vol. 23, no. 1, pp. 51-58, 1991.- [10] S.A. Sheynin and A.V. Tuzikov, "Explicit Formulae for Polyhedra Moments,"
Pattern Recognition Letters, vol. 22, pp. 1103-1109, 2001.- [11] A.V. Tuzikov, S.A. Sheynin, and P.V. Vasiliev, "Computation of Volume and Surface Body Moments,"
Pattern Recognition, vol. 36, pp. 2521-2529, 2003.- [12] G.C. Best, "Helpful Formulas for Integrating Polynomials in Three Dimensions,"
Math. of Computation, vol. 18, no. 86, pp. 310-312, http://www.jstor.org/stable2003308, 1964.- [13] A. DiCarlo and A. Paoluzzi, "Fast Computation of Inertia through Affinely Extended Euler Tensor,"
Computer-Aided Design, vol. 38, no. 11, pp. 1145-1153, 2006.- [14] H.G. Timmer and J.M. Stern, "Computation of Global Geometric Properties of Solid Objects,"
Computer-Aided Design, vol. 12, no. 6, pp. 301-304, 1980.- [15] J. Liggett, "Exact Formulae for Areas, Volumes and Moments of Polygons and Polyhedra,"
Comm. in Applied Numerical Methods, vol. 4, no. 6, pp. 815-820, 1988.- [16] B. Li, "The Moment Calculation of Polyhedra,"
Pattern Recognition, vol. 26, no. 8, pp. 1229-1233, 1993.- [17] B. Mirtich, "Fast and Accurate Computation of Polyhedral Mass Properties,"
J. Graphics Tools, vol. 1, no. 2, pp. 31-50, 1996.- [18] H.T. Rathod and H.S.G. Rao, "Integration of Polynomials over Linear Polyhedra in Euclidean Three-Dimensional Space,"
Computer Methods in Applied Mechanics and Eng., vol. 126, nos. 3/4, pp. 373-392, 1995.- [19] R. Millan, L. Dempere-Marco, J. Pozo, J. Cebral, and A. Frangi, "Morphological Characterization of Intracranial Aneurysms Using 3-D Moment Invariants,"
IEEE Trans. Medical Imaging, vol. 26, no. 9, pp. 1270-1282, Sept. 2007.- [20] P.C. Hammer, O.P. Marlowe, and A.H. Stroud, "Numerical Integration over Simplexes and Cones,"
Math. Tables and Other Aids to Computation, vol. 10, no. 55, pp. 130-137, 1956.- [21] A.H. Stroud, "Approximate Integration Formulas of Degree 3 for Simplexes,"
Math. of Computation, vol. 18, no. 88, pp. 590-597, 1964.- [22] P. Hillion, "Numerical Integration on a Tetrahedron,"
Calcolo, vol. 18, no. 2, pp. 117-130, 1981.- [23] M. Gellert and R. Harbord, "Moderate Degree Cubature Formulas for 3-D Tetrahedral Finite-Element Approximations,"
Comm. in Applied Numerical Methods, vol. 7, no. 6, pp. 487-495, 1991.- [24] H. Rathod, K. Nagaraja, and B. Venkatesudu, "Numerical Integration of Some Functions over an Arbitrary Linear Tetrahedra in Euclidean Three-Dimensional Space,"
Applied Math. and Computation, vol. 191, no. 2, pp. 397-409, 2007.- [25] C. Gonzalez-Ochoa, S. McCammon, and J. Peters, "Computing Moments of Objects Enclosed by Piecewise Polynomial Surfaces,"
ACM Trans. Graphics, vol. 17, no. 3, pp. 143-157, 1998.- [26] C.-H. Teh and R.T. Chin, "On Digital Approximation of Moment Invariants,"
Computer Vision, Graphics, and Image Processing, vol. 33, pp. 318-326, 1986.- [27] A. Khotanzad and Y.H. Hong, "Invariant Image Recognition by Zernike Moments,"
IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 12, no. 5, pp. 489-497, May 1990.- [28] N. Canterakis, "Fast 3D Zernike Moments and Invariants," Technical Report 5/97, Inst. für Informatik, Albert-Ludwigs-Univ. Freiburg, citeseer.ist.psu.educanterakis97fast.html , 1997.
- [29] N. Canterakis, "3D Zernike Moments and Zernike Affine Invariants for 3D Image Analysis and Recognition,"
Proc. 11th Scandinavian Conf. Image Analysis, pp. 85-93, 1999.- [30] M. Novotni and R. Klein, "Shape Retrieval Using 3D Zernike Descriptors,"
Computed-Aided Design, vol. 36, pp. 1047-1062, Jan. 2004.- [31] W. Qiuting and Y. Bing, "3D Terrain Matching Algorithm and Performance Analysis Based on 3D Zernike Moments,"
Proc. 2008 Int'l Conf. Computer Science and Software Eng., vol. 6, pp. 73-76, Dec. 2008.- [32] P. Shilane, P. Min, M. Kazhdan, and T. Funkhouser, "The Princeton Shape Benchmark,"
Proc. Shape Modeling Int'l, pp. 167-178, 2004.- [33] M. Attene and B. Falcidieno, "ReMESH: An Interactive Environment to Edit and Repair Triangle Meshes,"
Proc. Shape Modeling Int'l, pp. 271-276, 2006. |