Total Cyclic Variation and Generalizations

We introduce a general framework for regularization of signals with values in a cyclic structure, such as angles, phases or hue values. These include the total cyclic variation ${TV_{{S}^{1}}}$, as well as cyclic versions of quadratic regularization, Huber-TV and Mumford-Shah regularity. The key idea is to introduce a convex relaxation of the original non-convex optimization problem. The method handles the periodicity of values in a simple way, is invariant to cyclical shifts and has a number of other useful properties such as lower-semicontinuity. The framework allows general, possibly non-convex data terms. Experimental results are superior to those obtained without special care about wrapping interval end points. Moreover, we propose an equivalent formulation of the total cyclic variation which can be minimized with the same time and memory efficiency as the standard total variation. We show that discretized versions of these regularizers amount to NP-hard optimization problems. Nevertheless, the proposed framework provides optimal or near-optimal solutions in most practical applications.