A Translation and Rotation Invariant Gauss–Newton Like Scheme for Image Registration

A Gauss–Newton like method is considered to obtain a d–dimensional displacement vector field ${\bf\vec{u}}$, which minimizes a suitable distance measure D between two images. The key to find a minimizer is to substitute the Hessian of D with the Sobolev-H2(Ω)d norm for ${\bf\vec{u}}$. Since the kernel of the associated semi-norm consists only of the affine linear functions we can show in this way, that the solution of each Newton step is a linear combination of an affine linear transformation and an affine-free nonlinear deformation. Our approach is based on the solution of a sequence of quadratic subproblems with linear constraints. We show that the resulting Karush–Kuhn–Tucker system, with a 3×3 block structure, can be solved uniquely and the Gauss–Newton like scheme can be separated into two separated iterations. Finally, we report on synthetic as well as on real-life data test runs.