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 nonconvex case and derive worst-case complexities that are equal to or better than existing methods. We show applications to total variation based two dimensional image processing and computer vision problems based on a Lagrangian decomposition approach. The resulting algorithms are very effcient, offer a high degree of parallelism, and come along with memory requirements which are only in the order of the number of image pixels.
SIAM Journal on Imaging Sciences
605 - 636
Kolmogorov V, Pock T, Rolinek M. Total variation on a tree. SIAM Journal on Imaging Sciences. 2016;9(2):605-636. doi:10.1137/15M1010257
Kolmogorov, V., Pock, T., & Rolinek, M. (2016). Total variation on a tree. SIAM Journal on Imaging Sciences, 9(2), 605–636. https://doi.org/10.1137/15M1010257
Kolmogorov, Vladimir, Thomas Pock, and Michal Rolinek. “Total Variation on a Tree.” SIAM Journal on Imaging Sciences 9, no. 2 (2016): 605–36. https://doi.org/10.1137/15M1010257.
V. Kolmogorov, T. Pock, and M. Rolinek, “Total variation on a tree,” SIAM Journal on Imaging Sciences, vol. 9, no. 2, pp. 605–636, 2016.
Kolmogorov V, Pock T, Rolinek M. 2016. Total variation on a tree. SIAM Journal on Imaging Sciences. 9(2), 605–636.
Kolmogorov, Vladimir, et al. “Total Variation on a Tree.” SIAM Journal on Imaging Sciences, vol. 9, no. 2, Society for Industrial and Applied Mathematics , 2016, pp. 605–36, doi:10.1137/15M1010257.