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 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 (ADMM) with a closed-form solution of a proximal operator of the $\ell_1 -\alpha \ell_2$ regularizer. Convergence of the ADMM algorithm is analyzed. In the second stage, we threshold the smoothed image by $K$-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$^p (0<p<1)$ counterparts, showcasing the qualitative and quantitative advantages of our proposed method.

翻译：本文设计了一种高效的多阶段图像分割框架，该框架融合了加权各向异性与各向同性全变分之差（AITV）。该分割框架通常包含两个阶段：平滑与阈值，因此简称SaT。第一阶段通过AITV正则化的Mumford-Shah（MS）模型获得平滑图像，该模型可利用交替方向乘子法（ADMM）高效求解，其中$\ell_1 -\alpha \ell_2$正则化子的近端算子具有闭式解。文中分析了ADMM算法的收敛性。第二阶段对平滑图像进行$K$-均值聚类阈值处理，以获取最终分割结果。数值实验表明，所提分割框架既适用于灰度图像也适用于彩色图像，具有通用性；能在数秒内高效生成高质量分割结果；且对受噪声、模糊或两者共同污染的输入图像具有鲁棒性。我们将AITV方法与其原始凸TV及非凸TV$^p (0<p<1)$方法进行对比，展示了本方法在定性与定量上的优势。