Image alignment with rotation manifolds built on sparse geometric expansions

In this paper we discuss the problem of alignment of patterns under arbitrary rotation. When a generic image pattern is geometrically transformed, it typically spans a (possibly nonlinear) manifold in a high dimensional space. When the pattern of interest is given by a sparse approximation over a structured dictionary of geometric atoms, we show that the rotation manifold can be expressed analytically as a function of the transformation parameters. At the same time, its high order derivatives are also given in a closed form when the pattern is represented as a sparse linear combination of a few differentiable basis functions. In this framework, the alignment problem is formulated as the minimization of the distance between the reference pattern and the manifold, which boils down to a nonlinear least squares optimization problem. We propose to solve this problem by a Newton-type method, whose solution is facilitated by the analytical expressions of the manifold derivatives. We further derive a global optimization heuristic algorithm based on Newton, and provide sufficient conditions for computing the global minimizer. Experimental results demonstrate the effectiveness of the proposed methodology for image alignment and rotation invariant pattern recognition.