Rotation recovery from spherical images without correspondences

This paper addresses the problem of rotation estimation directly from images defined on the sphere and without correspondence. The method is particularly useful for the alignment of large rotations and has potential impact on 3D shape alignment. The foundation of the method lies in the fact that the spherical harmonic coefficients undergo a unitary mapping when the original image is rotated. The correlation between two images is a function of rotations and we show that it has an SO(3)-Fourier transform equal to the pointwise product of spherical harmonic coefficients of the original images. The resolution of the rotation space depends on the bandwidth we choose for the harmonic expansion and the rotation estimate is found through a direct search in this 3D discretized space. A refinement of the rotation estimate can be obtained from the conservation of harmonic coefficients in the rotational shift theorem. A novel decoupling of the shift theorem with respect to the Euler angles is presented and exploited in an iterative scheme to refine the initial rotation estimates. Experiments show the suitability of the method for large rotations and the dependence of the method on bandwidth and the choice of the spherical harmonic coefficients

[1]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[2]  Berthold K. P. Horn.  Extended Gaussian images , 1984, Proceedings of the IEEE.

[3]  Shahriar Negahdaripour,et al.  IEEE Transactions on Pattern Analysis and Machine Intelligence , 2004 .

[4]  Reiner Lenz,et al.  Rotation-invariant operators and scale-space filtering , 1987, Pattern Recognit. Lett..

[5]  P. Anandan,et al.  Hierarchical Model-Based Motion Estimation , 1992, ECCV.

[6]  Yehoshua Y. Zeevi,et al.  The Canonical Coordinates Method for Pattern Deformation: Theoretical and Computational Considerations , 1992, IEEE Trans. Pattern Anal. Mach. Intell..

[7]  Katsushi Ikeuchi,et al.  The Complex EGI: A New Representation for 3-D Pose Determination , 1993, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  D. Healy,et al.  Computing Fourier Transforms and Convolutions on the 2-Sphere , 1994 .

[9]  W. Kiel,et al.  Direct Perception of Three-Dimensional Motion from Patterns of Visual Motion , 1995 .

[10]  Y. Aloimonos,et al.  Direct Perception of Three-Dimensional Motion from Patterns of Visual Motion , 1995, Science.

[11]  Gilles Burel,et al.  Determination of the Orientation of 3D Objects Using Spherical Harmonics , 1995, CVGIP Graph. Model. Image Process..

[12]  Harpreet S. Sawhney,et al.  True multi-image alignment and its application to mosaicing and lens distortion correction , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[13]  Shree K. Nayar,et al.  Catadioptric omnidirectional camera , 1997, Proceedings of IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[14]  Harpreet S. Sawhney,et al.  True Multi-Image Alignment and Its Application to Mosaicing and Lens Distortion Correction , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[15]  Václav Hlavác,et al.  Zero Phase Representation of Panoramic Images for Image Vased Localization , 1999, CAIP.

[16]  Roberto Cipolla,et al.  Extracting Group Transformations from Image Moments , 1998 .

[17]  A. Leonardis,et al.  Robust localization using eigenspace of spinning-images , 2000, Proceedings IEEE Workshop on Omnidirectional Vision (Cat. No.PR00704).

[18]  G. Chirikjian,et al.  Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups , 2000 .

[19]  W. Wriggers,et al.  Fast rotational matching. , 2002, Acta crystallographica. Section D, Biological crystallography.

[20]  Szymon Rusinkiewicz,et al.  Rotation Invariant Spherical Harmonic Representation of 3D Shape Descriptors , 2003, Symposium on Geometry Processing.

[21]  Kostas Daniilidis,et al.  Direct 3D-rotation estimation from spherical images via a generalized shift theorem , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[22]  Szymon Rusinkiewicz,et al.  Symmetry descriptors and 3D shape matching , 2004, SGP '04.

[23]  Kostas Daniilidis,et al.  Rotation estimation from spherical images , 2004, Proceedings of the 17th International Conference on Pattern Recognition, 2004. ICPR 2004..

[24]  Kostas Daniilidis,et al.  Catadioptric Projective Geometry , 2001, International Journal of Computer Vision.

[25]  Szymon Rusinkiewicz,et al.  Modeling by example , 2004, ACM Trans. Graph..

[26]  Richard Szeliski,et al.  Systems and Experiment Paper: Construction of Panoramic Image Mosaics with Global and Local Alignment , 2000, International Journal of Computer Vision.

[27]  S. Shankar Sastry,et al.  Radon-based structure from motion without correspondences , 2005, 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05).

[28]  D. Rockmore,et al.  FFTs on the Rotation Group , 2008 .