An Extension of 3D Zernike Moments for Shape Description and Retrieval of Maps Defined in Rectangular Solids

Atilla Sit; Julie C Mitchell; George N Phillips; Stephen J Wright

Molecular Based Mathematical Biology (2013)

  • Volume: 1, page 75-89
  • ISSN: 2299-3266

Abstract

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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.

How to cite

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Atilla Sit, et al. "An Extension of 3D Zernike Moments for Shape Description and Retrieval of Maps Defined in Rectangular Solids." Molecular Based Mathematical Biology 1 (2013): 75-89. <http://eudml.org/doc/267024>.

@article{AtillaSit2013,
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.},
author = {Atilla Sit, Julie C Mitchell, George N Phillips, Stephen J Wright},
journal = {Molecular Based Mathematical Biology},
keywords = {Zernike polynomials; 3D shape retrieval; reconstruction; Gram-Schmidt orthogonalization; Electron Microscopy Data Bank; electron microscopy data bank},
language = {eng},
pages = {75-89},
title = {An Extension of 3D Zernike Moments for Shape Description and Retrieval of Maps Defined in Rectangular Solids},
url = {http://eudml.org/doc/267024},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Atilla Sit
AU - Julie C Mitchell
AU - George N Phillips
AU - Stephen J Wright
TI - An Extension of 3D Zernike Moments for Shape Description and Retrieval of Maps Defined in Rectangular Solids
JO - Molecular Based Mathematical Biology
PY - 2013
VL - 1
SP - 75
EP - 89
AB - 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.
LA - eng
KW - Zernike polynomials; 3D shape retrieval; reconstruction; Gram-Schmidt orthogonalization; Electron Microscopy Data Bank; electron microscopy data bank
UR - http://eudml.org/doc/267024
ER -

References

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  1. A. Björk. Solving linear least-squares problems by Gram-Schmidt orthogonalization. B.I.T., 7:1–21, 1967. Zbl0183.17802
  2. M. Bray. Orthogonal polynomials: a set for square areas. Proc. SPIE, 5252:314-321, 2004. 
  3. N. Canterakis. 3D Zernike moments and Zernike affine invariants for 3D image analysis and recognition. 11th Scandinavian Conf. on Image Analysis, 85–93, 1999. 
  4. G. M. Dai and V. N. Mahajan. Nonrecursive determination of orthonormal polynomials with matrix formulation. Optics Letters, 32(1):74-76, 2007. [WoS][PubMed][Crossref] 
  5. EMDB map distribution format description, version 1.0. Document created and published by the EMDataBank.org team, collaboration between PDBe, RCSB-PDB, NCMI, 2010. 
  6. S. D. Fuller. Depositing electron microscopy maps. Structure, 11(1):11–12, 2003. [Crossref][PubMed] 
  7. 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. 
  8. K. M. Hosny. Exact and fast computation of geometric moments for gray level images. Applied Mathematics and Computation, 189:1214–1222, 2007. [WoS] Zbl1124.94003
  9. 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. 
  10. 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. Zbl1264.94012
  11. V. N. Mahajan and G. M. Dai. Orthonormal polynomials in wavefront analysis: analytical solution. J. Opt. Soc. Am. A, 24(9):2994–3016, 2007. [Crossref] 
  12. 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. 
  13. 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. 
  14. M. Novotni and R. Klein. Shape retrieval using 3D Zernike descriptors. Computer-Aided Design, 36:1047–1062, 2004. [Crossref] 
  15. 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. [Crossref][PubMed][WoS] 
  16. 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. [PubMed][Crossref] 
  17. 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. 
  18. 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. [PubMed][Crossref][WoS] 
  19. V. Venkatraman, Y. D. Yang, L. Sael and D. Kihara. Protein-protein docking using region-based 3D Zernike descriptors. BMC Bioinformatics, 10:407, 2009. [Crossref][PubMed] 
  20. 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. 

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