A new augmented Lagrangian primal dual algorithm for elastica regularization

Regularization is a key element of variational models in image processing. To overcome the weakness of models based on total variation, various high order (typically second order) regularization models have been proposed and studied recently. Among these, Euler's elastica energy based regularizer is perhaps the most interesting in terms of both mathematical and physical justifications. More importantly its success has been proven in applications; however it has been a major challenge to develop fast and effective algorithms. In this paper we propose a new idea for deriving a primal dual algorithm, based on Legendre-Fenchel transformations, for representing the elastica regularizer. Combined with an augmented Lagrangian for-mulation, we are able to derive an equivalent unconstrained optimization that has fewer variables to work with than previous works based on splitting methods. We shall present our algorithms for both the image restoration problem and the image segmentation model. The idea applies to other models where the elastica regularizer is required. Numerical experiments show that the proposed method can produce highly competitive results with better efficiency.