A convex approximation of Euler's elastica functional
Benedikt Wirth, CIMS

Euler's elastica energy has often been proposed for curvature regularization in image processing, however, global minimizers are difficult to obtain due to the high nonlinearity and nonconvexity.
We introduce a convex, lower semi-continuous, coercive approximation, which is thus very well-suited as a regularizer in image processing. The approximation is closely related to the exact convex relaxation (which seems very difficult to understand). Interestingly, the exact convex relaxation of the elastica energy reduces to constantly zero if the total variation part of the elastica energy is neglected. The convex approximation arises via functional lifting of the image gradient into a Radon measure on the four-dimensional space $\Omega \times S^1 \times \mathbb{R}$, $\Omega \subset \mathbb{R}^2$, of which the first two coordinates represent the image domain and the last two the normal and curvature of the image level lines. It can be expressed as a linear program that can robustly be solved on the GPU after discretizing the involved
Radon measures via a linear combination of line measures concentrated on curve segments. (joint work with Kristian Bredies and Thomas Pock)