A finite hyperplane traversal Algorithm for 1-dimensional $$L^1pTV$$L1pTV minimization, for $$0

In this paper, we consider a discrete formulation of the one-dimensional $$L^1pTV$$L1pTV functional and introduce a finite algorithm that finds exact minimizers of this functional for $$0<p\le 1$$0<p≤1. Our algorithm for the special case for $$L^1TV$$L1TV returns globally optimal solutions for all regularization parameters $$\lambda \ge 0$$λ≥0 at the same computational cost of determining a single optimal solution associated with a particular value of $$\lambda $$λ. This finite set of minimizers contains the scale signature of the known initial data. A variation on this algorithm returns locally optimal solutions for all $$\lambda \ge 0$$λ≥0 for the case when $$0<p<1$$0<p<1. The algorithm utilizes the geometric structure of the set of hyperplanes defined by the nonsmooth points of the $$L^1pTV$$L1pTV functional. We discuss efficient implementations of the algorithm for both general and binary data.