A Convex, Lower Semicontinuous Approximation of Euler's Elastica Energy

We propose a convex, lower semi-continuous, coercive approximation of Euler’s elastica energy for images, which is thus very well-suited as a regularizer in image processing. The approximation is not quite the convex relaxation, and we discuss its close relation to the exact convex relaxation as well as the difficulties associated with computing the latter. Interestingly, the convex relaxation of the elastica energy reduces to constantly zero if the total variation part of the elastica energy is neglected. Our convex approximation arises via functional lifting of the image gradient into a Radon measure on the four-dimensional space Ω × S1 × R, Ω ⊂ R2, 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 which also admits a predual representation. Combined with a tailored discretization of measures via linear combinations of short line measures, the proposed functional becomes an efficient regularizer for image processing tasks, whose feasibility and effectiveness is verified in a series of exemplary applications.