Nonlocal linear image regularization and supervised segmentation

A nonlocal quadratic functional of weighted di. erences is examined. The weights are based on image features and represent the a. nity between di. erent pixels in the image. By prescribing di. erent formulas for the weights, one can generalize many local and nonlocal linear denoising algorithms, including the nonlocal means. lter and the bilateral. lter. In this framework one can easily show that continuous iterations of the generalized. lter obey certain global characteristics and converge to a constant solution. The linear operator associated with the Euler-Lagrange equation of the functional is closely related to the graph Laplacian. We can thus interpret the steepest descent for minimizing the functional as a nonlocal di. usion process. This formulation allows a convenient framework for nonlocal variational minimizations, including variational denoising, Bregman iterations, and the recently proposed inverse scale space. It is also demonstrated how the steepest descent. ow can be used for segmentation. Following kernel based methods in machine learning, the generalized di. usion process is used to propagate sporadic initial user's information to the entire image. Unlike classical variational segmentation methods, the process is not explicitly based on a curve length energy and thus can cope well with highly nonconvex shapes and corners. Reasonable robustness to noise is still achieved.