A finite hyperplane traversal algorithm for 1-dimensional \(L^1pTV\) minimization, for \(0<p\leq 1\)
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.
scientific article; zbMATH DE number 6478556