As one of the most important image segmentation models, the Mumford-Shah functional was developed to pursue a piecewise smooth approximation of a given image based on the regularization on the total length of curves. In this paper, we modify the Mumford-Shah model using Euler's elastic a as the regularization. A two-stage segmentation method is applied the Euler's elastic a regularized Mumford-Shah model. The first stage is to find a smooth solution of the variant Mumford-Shah functional based on augmented Lagrangian method while a thresholding is performed in the second stage to obtain different phases for the segmentation. The K-means clustering method is used as the technique to find the thresholds for the segmentation. For intensity inhomogeneous images, we eliminate the effect of the bias field by bias-corrected fuzzy c-means method. Experimental results show that as the regularization, Euler's elastic a makes the Mumford-Shah model perform better for many kinds of images, including tubular and irregular shaped, CT Angiography (CTA) and MRI images in different noise level.