A Levenberg–Marquardt Scheme for Nonlinear Image Registration

Abstract

This paper presents a Levenberg—Marquardt scheme to obtain a displacement vector field u(x)=(u 1(x),u 2(x)) t , which matches two images recorded with the same imaging machinery. The displacement vector should transform the image location x=(x 1,x 2) t of an image T, such that the grey level are equal to another image R. The so-called mono-modal image registration problem leads to minimize the nonlinear least squares functional D(u(x))=‖R(x)−T(x−u(x))‖2. To apply the Levenberg—Marquardt method, we replace the nonlinear functional D by its linearization around a current approximation. The resulting quadratic minimization problem is ill-posed, due to the fact that determining the unknown components of the displacements merely from the images is an underdetermined problem. We use an auxiliary Lagrange term borrowed from linear elasticity theory, which incorporates smoothness constraints to the displacement field. Finally, numerical experiments demonstrate the robustness and effectiveness of the proposed approach.