A combined segmentation and registration framework with a nonlinear elasticity smoother

In this paper, we present a new non-parametric combined segmentation and registration method. The shapes to be registered are implicitly modeled with level set functions and the problem is cast as an optimization one, combining a matching criterion based on the active contours without edges for segmentation (Chan and Vese, 2001) [8] and a nonlinear-elasticity-based smoother on the displacement vector field. This modeling is twofold: first, registration is jointly performed with segmentation since guided by the segmentation process; it means that the algorithm produces both a smooth mapping between the two shapes and the segmentation of the object contained in the reference image. Secondly, the use of a nonlinear-elasticity-type regularizer allows large deformations to occur, which makes the model comparable in this point with the viscous fluid registration method. In the theoretical minimization problem we introduce, the shapes to be matched are viewed as Ciarlet–Geymonat materials. We prove the existence of minimizers of the introduced functional and derive an approximated problem based on the Saint Venant–Kirchhoff stored energy for the numerical implementation and solved by an augmented Lagrangian technique. Several applications are proposed to demonstrate the potential of this method to both segmentation of one single image and to registration between two images.

► Binary image segmentation and registration using a nonlinear elasticity smoother. ► Numerical method based on an auxiliary variable and the augmented Lagrangian method. ► Theoretical results of existence of minimizers. ► Applications to image segmentation with topology preservation and to registration of gene data to mouse atlas.