LAUSR

In this article, we propose a class of piecewise-smooth image segmentation models in a variational framework. The models involve $$L^1$$L1 data fidelity measures and assume that an image can be approximated by the sum of a piecewise-constant function and a smooth function. The smooth function models intensity inhomogeneity, and the $$L^1$$L1 data-fitting terms enable to segment images with low contrast or outliers such as impulsive noise. The regions to be segmented are represented as smooth functions, almost binary functions, instead of the Heaviside expression of level set functions. The existence of minimizers of our main model is shown. Furthermore, we design fast and efficient optimization algorithms based on the augmented Lagrangian method and present a partial convergence result. Numerical results validate the effectiveness of the proposed models compared with other state-of-the-art methods.