Non-local Unsupervised Variational Image Segmentation Models

New image denoising models based on non-local image information have been recently introduced in the literature. These so-called ”non-local” denoising models provide excellent results because these models can denoise smooth regions or/and textured regions simultaneously, unlike standard denoising models. Standard variational models s.a. Total Variation-based models are defined to work in a small local neighborhood, which is enough to denoise smooth regions. However, textures are not local in nature and requires semi-local/non-local information to be denoised efficiently. Several papers have introduced non-local filters and non-local variational models for image denoising. Yet, few studies have been done to develop unsupervised image segmentation models based on non-local information. This will be the goal of this paper. We define and study three unsupervised non-local segmentation models. These models will be based on the continuous global minimization approach for image segmentation recently introduced in [10, 6]. The energy of [10, 6] is a first order energy composed of the weighted Total Variation norm and a linear term. The first proposed non-local segmentation model will extend the Total Variation regularization term of [10, 6] to the non-local Total Variation energy. We will see that the non-local energy can segment fine and small structures better than the standard Total Variation energy. The second model will extend the data-based term of [10, 6] to a non-local term using the Chan-Vese model. The proposed non-local Chan-Vese model will overcome the main limitation of the original model, that does not work with local intensity inhomogeneities. Finally, the third model will also extend the data-based term of [10, 6] to a non-local term using the Mumford-Shah energy. The original Mumford-Shah energy is designed to work for piecewise smooth images only. We suggest to extend it to textures, defining a non-local Mumford-Shah model that works with real-world images. Numerical minimization schemes presented in this paper are based on continuous and discrete (graph cut) approaches. Experimental results will illustrate the improvements provided by the three proposed non-local unsupervised segmentation models.