A Levenberg--Marquardt scheme for nonlinear image registration
Two images are represented by compactly supported functionals \(T,R:\Omega\to \mathbb{R}\) where \(\Omega\subset\mathbb{R}^2\). A displacement vector field \(u:\Omega\to\mathbb{R}^2\) is considered which matches the two images, \(R(x)\approx T(x- u(x))\). This leads to the optimization problem \[ D[u]:= \int_\Omega (T(x- u(x))- R(x))^2 \,dx\to\min!.\tag{1} \] The images are encoded as two-dimensional arrays of pixel values, and the problem (1) is replaced by its discrete approximation. The Levenberg-Marquardt (LM) method is applied which uses the linearization of the functional \(D\) around a current approximation. The resulting quadratic minimization problem is ill-posed. To regularize the problem, an auxiliary Lagrange term is used which is borrowed from linear elasticity theory. Numerical experiments show that: a) the LM scheme usually converges in 5 to 7 iterations, b) the method outperforms the steepest descent methods, currently used for image registration, see \textit{M. Bro-Nielsen} and \textit{C. Gramkow} [Fast fluid registrationof medical images. Lecture Notes Comput. Sci. 1131, 267--276 (1996)], and c) the approach works well when the images are contaminated by small errors.
scientific article; zbMATH DE number 2057840