5. Convex Analysis

The main motivation for the following discussion is that the variational problem (4.3) for total variation denoising is not differentiable and we formally cannot compute a solution to this problem directly with the Euler-Lagrange formalism. However, since the energy functional is convex, we can apply methods from convex analysis to perform nonsmooth optimization.

For most of the results stated in the following we follow the standard textbook by [].