3-D Symmetry Detection and Analysis Using the Pseudo-polar Fourier Transform

Symmetry detection and analysis in 3D images is a fundamental task in a gamut of scientific fields such as computer vision, medical imaging and pattern recognition to name a few. In this work, we present a computational approach to 3D symmetry detection and analysis. Our analysis is conducted in the Fourier domain using the pseudo-polar Fourier transform. The pseudo-polar representation enables to efficiently and accurately analyze angular volumetric properties such as rotational symmetries. Our algorithm is based on the analysis of the angular correspondence rate of the given volume and its rotated and rotated-inverted replicas in their pseudo-polar representations. We also derive a novel rigorous analysis of the inherent constraints of 3D symmetries via groups-theory based analysis. Thus, our algorithm starts by detecting the rotational symmetry group of a given volume, and the rigorous analysis results pave the way to detect the rest of the symmetries. The complexity of the algorithm is O(N ³log (N)), where N×N×N is the volumetric size in each direction. This complexity is independent of the number of the detected symmetries. We experimentally verified our approach by applying it to synthetic as well as real 3D objects.