Discrete Curvature

Theory and Applications


A colloquium on discrete curvature

A Fast Algorithm for Euler's Elastica Model Using Augmented Lagrangian Method

Xue-Cheng Tai

Slides can be found by following the link.

Minimization of functionals related to Euler's elastica energy has a wide range of applications in computer vision and image processing. A high order nonlinear partial differential equation (PDE) needs to be solved, and the gradient descent method usually takes high computational cost. In this paper, we propose a fast and efficient numerical algorithm to solve minimization problems related to Euler's elastica energy and show applications to variational image denoising, image inpainting, and image zooming. We reformulate the minimization problem as a constrained minimization problem, followed by an operator splitting method and relaxation. The proposed constrained minimization problem is solved by using an augmented Lagrangian approach. Numerical tests on real and synthetic cases are supplied to demonstrate the efficiency of our method.

The technique can be extended to a number of other applications related to curvature minimization. In the end, we shall also show the essential ideas to use these techniques for applications that have replace the "length" regularization" by "curvature" regularizations.

Joint work with T. Chan, J. Hahn and G. Chung and W. Zhu.

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