The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.09 - September (2009 vol.31)
pp: 1715-1722
Qing Wang , University of Freiburg, Freiburg
Olaf Ronneberger , University of Freiburg, Freiburg
Hans Burkhardt , University of Freiburg, Freiburg
ABSTRACT
In this paper, polar and spherical Fourier analysis are defined as the decomposition of a function in terms of eigenfunctions of the Laplacian with the eigenfunctions being separable in the corresponding coordinates. The proposed transforms provide effective decompositions of an image into basic patterns with simple radial and angular structures. The theory is compactly presented with an emphasis on the analogy to the normal Fourier transform. The relation between the polar or spherical Fourier transform and the normal Fourier transform is explored. As examples of applications, rotation-invariant descriptors based on polar and spherical Fourier coefficients are tested on pattern classification problems.
INDEX TERMS
Invariants, Fourier analysis, radial transform, multidimensional.
CITATION
Qing Wang, Olaf Ronneberger, Hans Burkhardt, "Rotational Invariance Based on Fourier Analysis in Polar and Spherical Coordinates", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.31, no. 9, pp. 1715-1722, September 2009, doi:10.1109/TPAMI.2009.29
REFERENCES
[1] N.N. Lebedev, Special Functions and Their Applications (translated from Russian by R.A. Silverman), chapters 5, 6, 8. Dover, 1972.
[2] W. Kaplan, Advanced Calculus, fourth ed., pp. 696-698. Addison-Wesley, 1991.
[3] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, seventh ed., pp. 523-525. Cambridge, 1989.
[4] E.W. Weisstein, “Jacobi-Anger Expansion,” From MathWorld—A Wolfram Web Resource, http://mathworld.wolfram.comJacobi-AngerExpansion. html , 2009.
[5] K.E. Schmidt, “The Expansion of a Plane Wave,” http://fermi.la.asu.edu/PHY577/notesplane.pdf , 2009.
[6] S. Erturk and T.J. Dennis, “3D Model Representation Using Spherical Harmonics,” Electronics Letters, vol. 33, no. 11, pp. 951-952, 1997.
[7] M. Kazhdan, T. Funkhouser, and S. Rusinkiewicz, “Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors,” Proc. Symp. Geometry Processing, pp. 167-175, 2003.
[8] H. Huang, L. Shen, R. Zhang, F. Makedon, A. Saykin, and J. Pearlman, “A Novel Surface Registration Algorithm with Medical Modeling Applications,” IEEE Trans. Information Technology in Biomedicine, vol. 11, no. 4, pp.474-482, July 2007.
[9] C.-H. Teh and R.T. Chin, “On Image Analysis by the Methods of Moments,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 10, no. 4, pp. 496-513, July 1988.
[10] Y. Zana and R.M. CesarJr., “Face Recognition Based on Polar Frequency Features,” ACM Trans. Applied Perception, vol. 3, no. 1, pp. 62-82, 2006.
[11] 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.
[12] W.-Y. Kim and Y.-S. Kim, “Robust Rotation Angle Estimator,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 21, no. 8, pp. 768-773, Aug. 1999.
[13] E.M. Arvacheh and H.R. Tizhoosh, “Pattern Analysis Using Zernike Moments,” Proc. IEEE Instrumentation and Measurement Technology Conf., vol. 2, pp. 1574-1578, 2005.
[14] A. Averbuch, R.R. Coifman, D.L. Donoho, M. Elad, and M. Israeli, “Accurate and Fast Discrete Polar Fourier Transform,” Proc. Conf. Record 37th Asilomar Conf. Signals, Systems, and Computers, vol. 2, pp. 1933-1937, 2003.
[15] A. Makadia, L. Sorgi, and K. Daniilidis, “Rotation Estimation from Spherical Images,” Proc. 17th Int'l Conf. Pattern Recognition, vol. 3, pp.590-593, 2004.
[16] R. Skomski, J.P. Liu, and D.J. Sellmyer, “Quasicoherent Nucleation Mode in Two-Phase Nanomagnets,” Physical Rev. B, vol. 60, pp. 7359-7365, 1999.
[17] B. Pons, “Ability of Monocentric Close-Coupling Expansions to Describe Ionization in Atomic Collisions,” Physical Rev. A, vol. 63, p. 012704, 2000.
[18] R. Bisseling and R. Kosloff, “The Fast Hankel Transform as a Tool in the Solution of the Time Dependent Schrödinger Equation,” J. Computational Physics, vol. 59, no. 1, pp. 136-151, 1985.
[19] D. Lemoine, “The Discrete Bessel Transform Algorithm,” J. Chemical Physics, vol. 101, no. 5, pp. 3936-3944, 1994.
[20] O. Ronneberger, E. Schultz, and H. Burkhardt, “Automated Pollen Recognition Using 3D Volume Images from Fluorescence Microscopy,” Aerobiologia, vol. 18, pp. 107-115, 2002.
[21] H. Burkhardt, Transformationen zur lageinvarianten Merkmalgewinnung, Ersch. als Fortschrittbericht (Reihe 10, Nr. 7) der VDI-Zeitschriften. VDI-Verlag, 1979.
[22] O. Ronneberger, J. Fehr, and H. Burkhardt, “Voxel-Wise Gray Scale Invariants for Simultaneous Segmentation and Classification,” Proc. 27th DAGM Symp., pp. 85-92, 2005.
[23] Q. Wang, O. Ronneberger, and H. Burkhardt, “Fourier Analysis in Polar and Spherical Coordinates,” internal report, http://lmb.informatik.uni-freiburg.depapers /, 2008.
[24] C.C. Chang and C.J. Lin, “LIBSVM—A Library for Support Vector Machines,” http://www.csie.ntu.edu.tw/cjlinlibsvm/, 2009.
[25] FFTW Home Page, http:/www.fftw.org/, 2009.
[26] Fast Spherical Harmonic Transforms, http://www.cs.dartmouth.edu/ geelongsphere /, 2009.
[27] The Gnu Scientific Library, http://www.gnu.org/softwaregsl/, 2009.
7 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool