  We consider the problem of minimizing the continuous valued total variation
subject to different unary terms on trees and propose fast direct algorithms
based on dynamic programming to solve these problems. We treat both the convex
and the non-convex case and derive worst case complexities that are equal or
better than existing methods. We show applications to total variation based 2D
image processing and computer vision problems based on a Lagrangian
decomposition approach. The resulting algorithms are very efficient, offer a
high degree of parallelism and come along with memory requirements which are
only in the order of the number of image pixels.
