International Journal of Computer Vision
High-quality computational imaging through simple lenses
ACM Transactions on Graphics (TOG)
Convex Relaxation of a Class of Vertex Penalizing Functionals
Journal of Mathematical Imaging and Vision
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We introduce the novel continuous regularizer total curvature (TC) for images$\math$. It is defined as the Menger-Melnikov curvature of the Radon measure |Du|, which can be understood as a measure theoretic formulation of curvature mathematically related to mean curvature. The functional is not convex, therefore we define a convex relaxation which yields a close approximation. Similar to the total variation, the relaxation can be written as the support functional of a convex set, which means that there are stable and efficient minimization algorithms available when it is used as a regularizer in image processing problems. Our current implementation can handle general inverse problems, inpainting and segmentation. We demonstrate in experiments and comparisons how the regularizer performs in practice.