# A Weighted Difference of Anisotropic and Isotropic Total Variation for Relaxed Mumford-Shah Color and Multiphase Image Segmentation

arXiv: https://arxiv.org/abs/2005.04401 · HTML: https://mlsanity.com/paper/2005.04401/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 May 2020.

## Built on by (most cited first, 3 of 3)

- [An Efficient Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation](https://mlsanity.com/paper/2202.10115/) (2022)
- [Difference of Anisotropic and Isotropic TV for Segmentation under Blur and Poisson Noise](https://mlsanity.com/paper/2301.03393/) (2023)
- [An Image Segmentation Model with Transformed Total Variation](https://mlsanity.com/paper/2406.00571/) (2024)

## Abstract (arXiv)

In a class of piecewise-constant image segmentation models, we propose to incorporate a weighted difference of anisotropic and isotropic total variation (AITV) to regularize the partition boundaries in an image. In particular, we replace the total variation regularization in the Chan-Vese segmentation model and a fuzzy region competition model by the proposed AITV. To deal with the nonconvex nature of AITV, we apply the difference-of-convex algorithm (DCA), in which the subproblems can be minimized by the primal-dual hybrid gradient method with linesearch. The convergence of the DCA scheme is analyzed. In addition, a generalization to color image segmentation is discussed. In the numerical experiments, we compare the proposed models with the classic convex approaches and the two-stage segmentation methods (smoothing and then thresholding) on various images, showing that our models are effective in image segmentation and robust with respect to impulsive noises.