Fast Minimization of Region-Based Active Contours Using the Shape Hessian of the Energy.

We propose a novel shape optimization algorithm for region-based active contour models. Region-based active contours are preferred for many segmentation problems, because they incorporate more global information by aggregating cues or statistics over the distinct regions defined by the contour configuration. This makes them effective in a diverse array of segmentation scenarios, also more robust to contour initializations, Unfortunately they are also more expensive computationally, because a significant part of the optimization involves repeated integrations of the image features over the regions through the many iterations of the contour updates. Accordingly, we aim to decrease the overall computational cost of region-based active contours by reducing the cost of an individual iteration, and the total number of iterations. To this end, we first develop a Lagrangian curve representation that is spatially adaptive and economical in terms of the number of nodes used. Then we perform the shape sensitivity analysis of the general form of the region-based segmentation energy. In particular, we compute the second variation or the shape Hessian of the energy, and we use this to compute fast descent directions for the contours to significantly reduce the computational cost. Our implementation builds on a finite element discretization of the whole framework, including the contours. This results in efficient velocity computations in linear time with respect to the number of contour nodes.