Image segmentation is a fundamental problem in both image processing and computer vision with numerous applications. In this paper, we propose a two-stage image segmentation scheme based on inexact alternating direction method. Specifically, we first solve the convex variant of the Mumford-Shah model to get the smooth solution, the segmentation are then obtained by apply the K-means clustering method to the solution. Some numerical comparisons are arranged to show the effectiveness of our proposed schemes by segmenting many kinds of images such as artificial images, natural images, and brain MRI images.

We present a multi-phase image segmentation method based on the histogram of the Gabor feature space, which consists of a set of Gabor-filter responses with various orientations, scales and frequencies. Our model replaces the error function term in the original fuzzy region competition model with squared 2-Wasserstein distance function, which is a metric to measure the distance of two histograms. The energy functional is minimized by alternative minimization method and the existence of closed-form solutions is guaranteed when the exponent of the fuzzy membership term being 1 or 2. We test our model on both simple synthetic texture images and complex natural images with two or more phases. Experimental results are shown and compared to other recent results.

This paper studies the gradient flow of a regularized Mumford-Shah functionalproposed by Ambrosio and Tortorelli (1990, 1992) for image segmentation, and adopted by Esedoglu and Shen (2002) for image inpainting. It is shown that the gradient flow with L2 x L∞ initial data possesses a global weak solution, and it has a unique global in timestrong solution, which has at most finite number of point singularities in the space-time, when the initial data are in H1 x H1 ∩ L∞ . A family of fully discrete approximation schemes using low order finite elements is proposed for the gradient flow. Convergence of a subsequence (resp. the whole sequence)of the numerical solutions to a weak solution (resp. the strong solution) of the gradient flow is established as the mesh sizes tend to zero, and optimal and suboptimal order error estimates, which depend on $\frac{1}{{\varepsilon}}$ and $\frac{1}{k_{\varepsilon}}$ only in low polynomial order, are derived for the proposed fully discrete schemes under the mesh relation $k=o(h^{\frac12})$ . Numerical experiments are also presented to show effectiveness of the proposed numerical methods and to validate the theoretical analysis.