Total variation regularization superior to classical quadratic choice

Question

Where can I find literature on why the choice of regularization $$\int_{\Omega} |\nabla u| \,{\rm d} x\,{\rm d}y$$ is more effective than the classical choice $$\int_{\Omega} |\nabla u|^2 \,{\rm d} x\,{\rm d}y$$ when it comes to inverse problems occuring in the context of images?

Answer 1

It has been discussed it pretty much every numerical inverse problems talk that uses TV regularization. Presumably following the references from any of the associated papers, or any paper in image processing that uses the technique, would lead you to good sources.

And indeed: The first one I found cites Rudin-Osher-Fatemi (also cited in the comments by Eric Towers)

The main heuristic seems to be: If you want smooth results, you should use $L^2$ regularization. If you want sharp edges, and artifical staircasing artefacts are not a huge issue, you should consider TV.