Calculus of Variations and Geometric Measure Theory
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P. Liu - X. Y. Lu

Real order (an)-isotropic total variation in image processing - Part I: analytical analysis and functional properties

created by lu on 01 May 2019


Submitted Paper

Inserted: 1 may 2019
Last Updated: 1 may 2019

Year: 2018

ArXiv: 1805.06761 PDF


In this paper, a variational, multi-dimensional model for image reconstruction is proposed, in which the regularization term consists of the $r$-order (an)-isotropic total variation seminorms $TV^r$, with $r\in \mathbb R^+$, defined via the Riemann-Liouville fractional derivative. Key properties, such as the lower semi-continuity and compactness with respect to both the function $u$ and the order of derivative $r$, are studied. This paper, the first of our series works on analytical and numerical aspects of the model, as well as the learning of optimal order $r$ for particular imaging tasks, provides a comprehensive analysis of the behavior of $TV^r$ in the space of functions with bounded (fractional order) total variation.

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