Second order total variation [feature request]

[aizvorski]

I'd like to use second-order total variation, as described in for example these couple of papers:
https://www.ipol.im/pub/art/2013/40/article.pdf
http://www.franklenzen.de/pdf/lenzen_et_al_ssvm2013.pdf

In other words, instead of trying to minimize sum of first derivative |x_i - x_i-1|, this would be minimizing second derivative |x_i - 2*x_i-1 + x_i-2| (in the simple 1d case)

This has really important applications to image inpainting and filtering; the second-order TV denoising tends to produce better results for natural gradients by not introducing steps into them as first-order TV does.

Is it possible to add this? Where to start? I like this library a lot but I'm not familiar enough with the implementation to know what needs to be modified to add this.

[albarji]

Hi Alex! Thanks for your interest in proxTV. I must admit I have not been giving maintenance to this library for more than a year now. Since then I have moved on from Total Variation to other research topics. I did some research into second-order TV some years ago, but did not come down to write an implementation for proxTV.

I'm afraid I won't have time to add this feature, but if you want to try developing it yourself I would suggest to start by looking at the C implementations of the other methods, which you can find in the src folder. I must admit the code is not easy to follow, as efficiency is a priority over clarity.

If you are able to implement a C function for second order taut-string, the following steps should be adding a Python interface (here) and some tests to make sure the implementation works (here). If you prefer Matlab instead, the interface functions can be found in the matlab folder.

[Vldchlms]

Hi Alex,

I would recommend you the check a recent paper

Multivariate Trend Filtering for Lattice Data