Augmented Lagrangian method for Euler's elastica based variational models

Euler's elastica is widely applied in digital image processing. It is very challenging to minimize the Euler's elastica energy functional due to the high-order derivative of the curvature term. The computational cost is high when using traditional time-marching methods. Hence developments of fast methods are necessary. In the literature, the augmented Lagrangian method (ALM) is used to solve the minimization problem of the Euler's elastica functional by Tai, Hahn and Chung and is proven to be more efficient than the gradient descent method. However, several auxiliary variables are introduced as relaxations, which means people need to deal with more penalty parameters and much effort should be made to choose optimal parameters. In this dissertation, we employ a novel technique by Bae, Tai, and Zhu, which treats curvature dependent functionals using ALM with fewer Lagrange multipliers, and apply it for a wide range of imaging tasks, including image denoising, image inpainting, image zooming, and image deblurring. Numerical experiments demonstrate the efficiency of the proposed algorithm. Besides this, numerical experiments also show that our algorithm gives better results with higher SNR/PSNR, and is more convenient for people to choose optimal parameters.