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.
We describe linear-time algorithms for solving a class of problems that involve transforming a cost function on a grid using spatial information. These problems can be viewed as a generalization of classical distance transforms of binary images, where the binary image is replaced by an arbitrary function on a grid. Alternatively they can be viewed in terms of the minimum convolution of two functions, which is an important operation in grayscale morphology. A consequence of our techniques is a simple and fast method for computing the Euclidean distance transform of a binary image. Our algorithms are also applicable to Viterbi decoding, belief propagation, and optimal control.