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，该框架整合了加权各向异性与各向同性总变分（AITV）之差。该分割框架通常包含两个阶段：平滑与阈值处理，因此简称为SaT。在第一阶段，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，其中涉及$\ell_1 -\alpha \ell_2$正则化子的邻近算子闭式解。本文分析了ADMM算法的收敛性。在第二阶段，通过$K$-均值聚类对平滑图像进行阈值处理，以获得最终分割结果。数值实验表明，该分割框架既能适用于灰度图像，也能适用于彩色图像，具有高效性（可在数秒内生成高质量分割结果），并且对受噪声、模糊或两者共同退化的输入图像具有鲁棒性。我们将AITV方法与其原始凸总变分（TV）和非凸总变分TV$^p (0<p<1)$方法进行了对比，展示了所提方法在定性与定量方面的优势。