Sobolev gradients and joint variational image segmentation, denoising and deblurring

We consider several variants of the active contour model without edges,⁴ extended here to the case of noisy and blurry 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 proposed functionals, one of the most standard ways is to use gradient descent processes, in a time dependent approach. Usually, the L² gradient descent of the functional is computed and discretized in practice, based on the L2 inner 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 compare in several experiments the L² and H¹ gradient descents for image segmentation using curve evolution, with applications to denoising and deblurring. The Sobolev gradient descent is preferable in many situations and gives smaller computational cost.