In this paper, we propose an algorithm based on augmented Lagrangian method and give a performance comparison for two segmentation models that use the $L$1- and $L$2-Euler's elastica energy respectively as the regularization for image segmentation. To capture contour curvature more reliably, we develop novel augmented Lagrangian functionals that ensure the segmentation level set function to be signed distance functions, which avoids the reinitialization of segmentation function during the iterative process. With the proposed algorithm and with the same initial contours, we compare the performance of these two high-order segmentation models and numerically verify the different properties of the two models.