 
Summary: Rotational Invariance Based on Fourier
Analysis in Polar and Spherical Coordinates
Qing Wang, Olaf Ronneberger, and
Hans Burkhardt, Member, IEEE
AbstractIn 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, rotationinvariant descriptors based on
polar and spherical Fourier coefficients are tested on pattern classification
problems.
Index TermsInvariants, Fourier analysis, radial transform, multidimensional.
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1 INTRODUCTION
NOTHING needs to be said about the importance of Fourier
transform in image processing and pattern recognition. Usually,
Fourier transform is formulated in Cartesian coordinates, where a
