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We consider $L^1$-TV regularization of univariate signals with values on thereal line or on the unit circle. While the real data space leads to a convexoptimization problem, the problem is non-convex for circle-valued data. In thispaper, we derive exact algorithms for both data spaces. A key ingredient is thereduction of the infinite search spaces to a finite set of configurations,which can be scanned by the Viterbi algorithm. To reduce the computationalcomplexity of the involved tabulations, we extend the technique of distancetransforms to non-uniform grids and to the circular data space. In total, theproposed algorithms have complexity $\mathscr{O}(KN)$ where $N$ is the lengthof the signal and $K$ is the number of different values in the data set. Inparticular, the complexity is $\mathscr{O}(N)$ for quantized data. It is thefirst exact algorithm for TV regularization with circle-valued data, and it iscompetitive with the state-of-the-art methods for scalar data, assuming thatthe latter are quantized.

Exact algorithms for $L^1$-TV regularization of real-valued or circle-valued signals