Robust registration and learning using multi-radii spherical polar Fourier transform

This paper presents effective methods using spherical polar Fourier transform data for two different appli-cations, with active areas of research, one as a conventional volumetric registration algorithm and other as machine learning classification network. For registration purposes the proposed method has the following advantageous features: (i) it is a unique and effective technique for estimating up to 7 degrees of freedom for 3D volumetric registration, that has a closed-form solution for 3D rotation estimation, and which does not resort to recurrence relations or search for point correspondences between two objects/volumes, (ii) it allows for robust rotation estimation determined simultaneously on multiple spectral spheres, therefore complete stack of such spherical layers can be processed concurrently to obtain accurate and optimal all three angles, and (iii) it has the ability to handle arbitrary large rotation angles, is shown to be robust against the presence of noise, holes/missing data, and partial overlaps. We demonstrate the effectiveness of our solution with extensive experimentation, including a set of scanned MRI images, a crashed car parking dataset, and the Princeton shape benchmark dataset with hundreds of 3D objects. For the classification solution we modify and adapt an existing network in the literature, a type of spherical convolutional network, that is suitable for processing multi-radii spectral spherical data, and showcase the resulting robustness achieved in classification of objects from the ModelNet40 dataset, especially in the presence of outliers, additive noise, and missing data.