An Efficient Smoothing and Thresholding Image Segmentation Framework with Weighted Anisotropic-Isotropic Total Variation

In this paper, we design an efficient, multi-stage image segmentation framework that incorporates a weighted difference of anisotropic and isotropic total variation (AITV). The segmentation framework generally consists of two stages: smoothing and thresholding, thus referred to as smoothing-and-thresholding (SaT). In the first stage, a smoothed image is obtained by an AITV-regularized Mumford-Shah (MS) model, which can be solved efficiently by the alternating direction method of multipliers (ADMMs) with a closed-form solution of a proximal operator of the \(\ell _1 -\alpha \ell _2\) ${\ell }_1-\alpha {\ell }_2$ regularizer. The convergence of the ADMM algorithm is analyzed. In the second stage, we threshold the smoothed image by \(K\) $K$ -means clustering to obtain the final segmentation result. Numerical experiments demonstrate that the proposed segmentation framework is versatile for both grayscale and color images, efficient in producing high-quality segmentation results within a few seconds, and robust to input images that are corrupted with noise, blur, or both. We compare the AITV method with its original convex TV and nonconvex TV \(^p (0<p<1)\) ${}^p(0<p<1)$ counterparts, showcasing the qualitative and quantitative advantages of our proposed method.