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A finite hyperplane traversal Algorithm for 1-dimensional \(L^1pTV\) minimization, for \(0<p\le 1\)

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Abstract

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

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Notes

  1. The sunspot number is commonly refered to as the Wolf Number in honor of Rudolf Wolf who is credited with the concept in 1848.

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Acknowledgments

We would like to acknowledge Jamie O’Brien for finding a mistake in our algorithm description. We would also like to thank the reviewers for taking the time to give valuable feedback on our original manuscript.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, St. Mary’s College of Maryland, St. Mary’s City, MD, 20686, USA

    Heather A. Moon

  2. Department of Mathematics, Washington State University, Pullman, WA, 99164-3113, USA

    Thomas J. Asaki

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  1. Heather A. Moon
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  2. Thomas J. Asaki
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Correspondence to Heather A. Moon.

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Moon, H.A., Asaki, T.J. A finite hyperplane traversal Algorithm for 1-dimensional \(L^1pTV\) minimization, for \(0<p\le 1\) . Comput Optim Appl 61, 783–818 (2015). https://doi.org/10.1007/s10589-015-9738-4

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  • DOI: https://doi.org/10.1007/s10589-015-9738-4

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