Abstract
Zernike polynomials have been widely used in the description and shape retrieval of 3D objects. These orthonormal polynomials allow for efficient description and reconstruction of objects that can be scaled to fit within the unit ball. However, maps defined within box-shaped regions ¶ for example, rectangular prisms or cubes ¶ are not well suited to representation by Zernike polynomials, because these functions are not orthogonal over such regions. In particular, the representations require many expansion terms to describe object features along the edges and corners of the region. We overcome this problem by applying a Gram-Schmidt process to re-orthogonalize the Zernike polynomials so that they recover the orthonormality property over a specified box-shaped domain. We compare the shape retrieval performance of these new polynomial bases to that of the classical Zernike unit-ball polynomials.
A. Björk. Solving linear least-squares problems by Gram-Schmidt orthogonalization. B.I.T., 7:1–21, 1967. 10.1007/BF01934122Search in Google Scholar
M. Bray. Orthogonal polynomials: a set for square areas. Proc. SPIE, 5252:314-321, 2004. Search in Google Scholar
N. Canterakis. 3D Zernike moments and Zernike affine invariants for 3D image analysis and recognition. 11th Scandinavian Conf. on Image Analysis, 85–93, 1999. Search in Google Scholar
G. M. Dai and V. N. Mahajan. Nonrecursive determination of orthonormal polynomials with matrix formulation. Optics Letters, 32(1):74-76, 2007. 10.1364/OL.32.000074Search in Google Scholar PubMed
EMDB map distribution format description, version 1.0. Document created and published by the EMDataBank.org team, collaboration between PDBe, RCSB-PDB, NCMI, 2010. Search in Google Scholar
S. D. Fuller. Depositing electron microscopy maps. Structure, 11(1):11–12, 2003. 10.1016/S0969-2126(02)00942-5Search in Google Scholar
J. B. Heymann, M. Chagoyen, and D. M. Belnap. Common conventions for interchange and archiving of threedimensional electron microscopy information in structural biology. J Struct Biol, 151(2):196–207, 2005. 10.1016/j.jsb.2005.06.001Search in Google Scholar PubMed
K. M. Hosny. Exact and fast computation of geometric moments for gray level images. Applied Mathematics and Computation, 189:1214–1222, 2007. 10.1016/j.amc.2006.12.025Search in Google Scholar
K. M. Hosny. A novel symmetry-based method for exact computation of 2D and 3D geometric moments. International Journal of Innovative Computing, Information and Control, 8(9):6123–6140, 2012. Search in Google Scholar
K. M. Hosny and M. Hafez. An algorithm for fast computation of 3D Zernike moments for volumetric images. Mathematical Problems in Engineering, Article ID 353406, 17 pp, 2012. 10.1155/2012/353406Search in Google Scholar
V. N. Mahajan and G. M. Dai. Orthonormal polynomials in wavefront analysis: analytical solution. J. Opt. Soc. Am. A, 24(9):2994–3016, 2007. 10.1364/JOSAA.24.002994Search in Google Scholar
L. Mak, S. Grandison, and R. J. Morris. An extension of spherical harmonics to region-based rotationally invariant descriptors for molecular shape description and comparison. J Molecular Graphics and Modeling, 26(7):1035–1045, 2008. 10.1016/j.jmgm.2007.08.009Search in Google Scholar PubMed
M. Novotni and R. Klein. 3D Zernike descriptors for content based shape retrieval. Proceedings of the 8th ACM symposium on Solid modeling and Applications, 216–225, 2003. 10.1145/781606.781639Search in Google Scholar
M. Novotni and R. Klein. Shape retrieval using 3D Zernike descriptors. Computer-Aided Design, 36:1047–1062, 2004. 10.1016/j.cad.2004.01.005Search in Google Scholar
L. Sael, B. Li, D. La, Y. Fang, K. Ramani, R. Rustamov, and D. Kihara. Fast protein tertiary structure retrieval based on global surface shape similarity. Proteins, 72(4):1259–1273, 2008. 10.1002/prot.22030Search in Google Scholar
M. Tagari, R. Newman, M. Chagoyen, J. M. Carazo, and K. Henrick. New electron microscopy database and deposition system. Trends Biochem Sci, 27(11):589, 2002. 10.1016/S0968-0004(02)02176-XSearch in Google Scholar
V. Venkatraman, P. R. Chakravarthy and D. Kihara. Application of 3D Zernike descriptors to shape-based ligand similarity searching. J of Cheminformatics, 1:19, 2009. 10.1186/1758-2946-1-19Search in Google Scholar PubMed PubMed Central
V. Venkatraman, L. Sael, and D. Kihara. Potential for protein surface shape analysis using spherical harmonics and 3D Zernike descriptors. Cell Biochem Biophys, 54:23–32, 2009. 10.1007/s12013-009-9051-xSearch in Google Scholar PubMed
V. Venkatraman, Y. D. Yang, L. Sael and D. Kihara. Protein-protein docking using region-based 3D Zernike descriptors. BMC Bioinformatics, 10:407, 2009. 10.1186/1471-2105-10-407Search in Google Scholar PubMed PubMed Central
F. Zernike. Diffraction theory of knife-edge test and its improved form, the phase contrast method. Mon. Not. R. Astron. Soc., 94:377–384, 1934.10.1093/mnras/94.5.377Search in Google Scholar
©2013 Versita Sp. z o.o.
This content is open access.
