Inverse problems in imaging and computer vision are typically addressed as data-fidelity optimization problems, where data-regularizers such as H1 or TV (total variation) are included to render the problem well-posed. However, while H1 regularization is known to produce overly smooth reconstructions, the TV (or ROF) model is feature-preserving but introduces staircasing artifacts.
The geometrically derived Beltrami framework, introduced by Sochen, Kimmel and Malladi (1998) offers an ideal compromise between feature preservation and avoidance of staircasing artifacts. Until now, one of the main limiting factors of the Beltrami regularizers was the lack of really efficient optimization schemes.
Here, we start from one of the most efficient TV-optimization methods, primal-dual projected gradients, and apply it to the Beltrami functional. Doing so, we achieve better performance than ROF denoising for the basic grey-scale denoising problem, then extend the method to more involved problems such as inpainting, deconvolution, and the color case, all in a straightforward fashion.
With the proposed primal-dual projected gradients optimization algorithm, the benefits of the geometric Beltrami regularizer become available at no extra computational cost, compared to state-of-the-art TV/ROF regularizers.
When using this code, please do cite our paper:
D. Zosso, A. Bustin, "A Primal-Dual Projected Gradient Algorithm for
Efficient Beltrami Regularization," submitted to Computer Vision and
Image Understanding (check for updates at http://www.math.ucla.edu/~zosso).
Preprint available at: ftp://ftp.math.ucla.edu/pub/camreport/cam14-52.pdf
link to author webpage/preprint added to description