Efficient minimization of the non-local potts model

The Potts model is a well established approach to solve different multi-label problems. The classical Potts prior penalizes the total interface length to obtain regular boundaries. Although the Potts prior works well for many problems, it does not preserve fine details of the boundaries. In recent years, non-local regularizers have been proposed to improve different variational models. The basic idea is to consider pixel interactions within a larger neighborhood. This can for example be used to incorporate low-level segmentation into the regularizer which leads to improved boundaries. In this work we study such an extension for the multi-label Potts model. Due to the increased model complexity, the main challenge is the development of an efficient minimization algorithm. We show that an accelerated first-order algorithm of Nesterov is well suited for this problem, due to its low memory requirements and its potential for massive parallelism. Our algorithm allows us to minimize the non-local Potts model with several hundred labels within a few minutes. This makes the non-local Potts model applicable for computer vision problems with many labels, such as multi-label image segmentation and stereo.