Sobolev gradients and joint variational image segmentation,denoising and deblurring

摘要

We consider several variants of the active contour model without edges,~4extended here to the case of noisy andblurry images, in a multiphase and a multilayer level set approach. Thus, the models jointly perform denoising,deblurring and segmentation of images, in a variational formulation. To minimize in practice the proposedfunctionals, one of the most standard ways is to use gradient descent processes, in a time dependent approach.Usually, the L~2gradient descent of the functional is computed and discretized in practice, based on the L~2inner product. However, this computation often requires theoretically additional smoothness of the unknown,or stronger conditions. One way to overcome this is to use the idea of Sobolev gradients.~(8,13,19)We comparein several experiments the L~2andH~1gradient descents for image segmentation using curve evolution, withapplications to denoising and deblurring. The Sobolev gradient descent is preferable in many situations andgives smaller computational cost.