This paper studies  image decomposition models which involve functional related to total variation and Euler's elastica energy. Such kind of variational models with first order and higher order derivatives have been widely used in image processing to accomplish advanced tasks. However, these non-linear partial differential equations usually take high computational cost by the gradient descent method.  In this paper, we propose a proximal alternating direction method of multipliers (ADMM) for total variation (TV) based Vese-Osher's decomposition model [L. A. Vese and S. J. Osher, J. Sci.  Comput., 19.1 (2003), pp. 553-572]  and its extension with Euler's elastica regularization. We demonstrate that efficient and effective solutions to these minimization problems can be obtained by proximal based numerical algorithms. In numerical experiments, we present numerous results on image decomposition and image denoising, which conforms significant improvement of the proposed models over standard models.