Higher dimensional affine registration and vision applications

Affine registration has a long and venerable history in computer vision literature, and extensive work have been done for affine registrations in ℝ² and ℝ³. In this paper, we study affine registrations in ℝ ^m for m > 3, and to justify breaking this dimension barrier, we show two interesting types of matching problems that can be formulated and solved as affine registration problems in dimensions higher than three: stereo correspondence under motion and image set matching. More specifically, for an object undergoing non-rigid motion that can be linearly modelled using a small number of shape basis vectors, the stereo correspondence problem can be solved by affine registering points in ℝ³ⁿ . And given two collections of images related by an unknown linear transformation of the image space, the correspondences between images in the two collections can be recovered by solving an affine registration problem in ℝ ^m , where m is the dimension of a PCA subspace. The algorithm proposed in this paper estimates the affine transformation between two point sets in ℝ ^m . It does not require continuous optimization, and our analysis shows that, in the absence of data noise, the algorithm will recover the exact affine transformation for almost all point sets with the worst-case time complexity of O(mk ²), k the size of the point set. We validate the proposed algorithm on a variety of synthetic point sets in different dimensions with varying degrees of deformation and noise, and we also show experimentally that the two types of matching problems can indeed be solved satisfactorily using the proposed affine registration algorithm.