Exact Algorithms for $ L ^{ 1 } $ -TV Regularization of Real-Valued or Circle-Valued Signals

We consider $L ^{ 1 }$ -TV regularization of univariate signals with values on the real line or on the unit circle. While the real data space leads to a convex optimization problem, the problem is nonconvex for circle-valued data. In this paper, we derive exact algorithms for both data spaces. A key ingredient is the reduction of the infinite search spaces to a finite set of configurations, which can be scanned by the Viterbi algorithm. To reduce the computational complexity of the involved tabulations, we extend the technique of distance transforms to nonuniform grids and to the circular data space. In total, the proposed algorithms have complexity $\mathcal{O}(KN)$, where $N$ is the length of the signal and $K$ is the number of different values in the data set. In particular, the complexity is $\mathcal{O}(N)$ for quantized data. It is the first exact algorithm for total variation regularization with circle-valued data, and it is competitive with the state-of-the-art methods for scalar data, assuming that the latter are quantized.