MRF Energy Minimization and Beyond via Dual Decomposition

Abstract

This paper introduces a new rigorous theoretical framework to address discrete MRF-based optimization in computer vision. Such a framework exploits the powerful technique of Dual Decomposition. It is based on a projected subgradient scheme that attempts to solve an MRF optimization problem by first decomposing it into a set of appropriately chosen subproblems and then combining their solutions in a principled way. In order to determine the limits of this method, we analyze the conditions that these subproblems have to satisfy and we demonstrate the extreme generality and flexibility of such an approach. We thus show that, by appropriately choosing what subproblems to use, one can design novel and very powerful MRF optimization algorithms. For instance, in this manner we are able to derive algorithms that: 1) generalize and extend state-of-the-art message-passing methods, 2) optimize very tight LP-relaxations to MRF optimization, 3) and take full advantage of the special structure that may exist in particular MRFs, allowing the use of efficient inference techniques such as, e.g, graph-cut based methods. Theoretical analysis on the bounds related with the different algorithms derived from our framework and experimental results/comparisons using synthetic and real data for a variety of tasks in computer vision demonstrate the extreme potentials of our approach.

Modeling Shapes with Higher-Order Graphs: Methodology and Applications

“…Then the shape matching and inference problem boils down to determining the correspondence from the target shape (or the observed image data) for each point on the source shape (model). Recent significant development in graph-based methods and inference techniques (e.g., Markov Random Field (MRF) inference algorithms [10,25,27] and graph matching [37,28,38]) have demonstrated their potential in solving such a correspondence problem. In particular, the newly developed techniques for higher-order models [24,27,23,17] enhance significantly the applicable extent and the performance of graph-based methods.…”

“…Recent significant development in graph-based methods and inference techniques (e.g., Markov Random Field (MRF) inference algorithms [10,25,27] and graph matching [37,28,38]) have demonstrated their potential in solving such a correspondence problem. In particular, the newly developed techniques for higher-order models [24,27,23,17] enhance significantly the applicable extent and the performance of graph-based methods. In such a context, we employ higher-order potentials to characterize measures/statistics that are g-invariant (e.g., similarity-invariant and isometry-invariant) and optimize the energy function using discrete optimization methods to address 3D shape matching and inference (e.g., [42,39,40,41]).…”

“…Inspired by [38], the dual-decomposition optimization framework [4,27] and the order-reduction technique proposed in [23] are adopted to deal with the problem in Eq. 3.…”

“…3. After solving the subproblems, the dual variables are updated using a projected subgradient method [27,38] to maximize the lower bound.…”

“…More specifically, we decompose the original problem into a set of subproblems, each corresponding to a factor-tree [6] and perform the exact inference efficiently in each subproblem in polynomial time using the max-product belief propagation algorithm [6], with complexity O(NL K ), where N, L and K denote the number of nodes, the number of candidates for each node, and the maximum order of the factors, respectively. The solutions of the subproblems are combined using projected subgradient method to solve the Lagrangian dual so as to obtain the solution of the original problem [27].…”