A convex representation for the vectorial Mumford-Shah functional

We propose the first tractable convex formulation of the vectorial Mumford-Shah functional which allows to compute high-quality solutions independent of the initialization. To this end, we generalize recently introduced convex formulations for scalar functionals to the vector-valued scenario in such a way that discontinuities in the different color channels preferably coincide. Furthermore, we propose an efficient solution which makes the overall optimization problem as tractable as in the scalar-valued case. Numerous experimental comparisons with the naive channel-wise approach, with the well-known Ambrosio-Tortorelli approximation, and with the classical total variation confirm the advantages of the proposed relaxation for contrast-preserving and edge-enhancing regularization.

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