A finite hyperplane traversal algorithm for 1-dimensional \(L^1pTV\) minimization, for \(0<p\leq 1\)
type: Item
instance of: scholarly article
title: A finite hyperplane traversal algorithm for 1-dimensional \(L^1pTV\) minimization, for \(0<p\leq 1\)
zbMATH Open document ID: 1330.90108
author: Heather A. Moon
author: Thomas J. Asaki
published in: Computational Optimization and Applications
publication date: 2015-09-04T00:00:00Z
review text: This paper presents a finite hyperplane transversal (ht) algorithm for solving the 1D discrete \(L^{1}pTv\) problem, for all parameters \(\lambda\geq0\) \[ \min_{u\in \mathbb{R}^{m+1}}\left\{ G_{p}(u)\equiv\sum_{i=0}^{m-1}\left| u_{i+1} -u_{i}\right| ^{p}+\lambda\sum_{i=0}^{m}\left| f_{i}-u_{i}\right| \right\} , \] where \(0<p\leq1,\) \(f\in \mathbb{R}^{m+1}\) is some given data, with either fixed (\(u_{0}=f_{0}\) and \(u_{m}=f_{m} \)) or free boundary conditions. The ht algorithm requires only finitely many iterations to obtain a complete set of exact minimizers for all \(\lambda \geq0\). For \(p=1,\) these minimizers are global minimizers, while for \(p<1,\) the minimizers are local. Computational implementations of the ht algorithm are presented for both general and binary data. The paper shows results of time trials and considers an example of using ht for extracting scale information from daily sunspot number data.
Mathematics Subject Classification ID: 90C30
zbMATH DE Number: 6478556
zbMATH Keywords: signal processing
zbMATH Keywords: nonsmooth optimization
zbMATH Keywords: scales
reviewed by: Francisco Guerra Vazquez
MaRDI profile type: Publication
full work available at URL: s10589-015-9738-4
OpenAlex ID: W188373354
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DOI: 10.1007/S10589-015-9738-4
description: scientific article; zbMATH DE number 6478556