An Approach to Vectorial Total Variation Based on Geometric Measure Theory
Abstract
We analyze a previously unexplored generalization of the scalar total variation to vector-valued functions, which is motivated by geometric measure theory. A complete mathematical characterization is given, which proves important invariance properties as well as existence of solutions of the vectorial ROF model. As an important feature, there exists a dual formulation for the proposed vectorial total variation, which leads to a fast and stable minimization algorithm. The main difference to previous approaches with similar properties is that we penalize across a common edge direction for all channels, which is a major theoretical advantage. Experiments show that this leads to a significantly better restoration of color edges in practice.
Cite
Text
Goldlücke and Cremers. "An Approach to Vectorial Total Variation Based on Geometric Measure Theory." IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2010. doi:10.1109/CVPR.2010.5540194Markdown
[Goldlücke and Cremers. "An Approach to Vectorial Total Variation Based on Geometric Measure Theory." IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2010.](https://mlanthology.org/cvpr/2010/goldlucke2010cvpr-approach/) doi:10.1109/CVPR.2010.5540194BibTeX
@inproceedings{goldlucke2010cvpr-approach,
title = {{An Approach to Vectorial Total Variation Based on Geometric Measure Theory}},
author = {Goldlücke, Bastian and Cremers, Daniel},
booktitle = {IEEE/CVF Conference on Computer Vision and Pattern Recognition},
year = {2010},
pages = {327-333},
doi = {10.1109/CVPR.2010.5540194},
url = {https://mlanthology.org/cvpr/2010/goldlucke2010cvpr-approach/}
}