Unlocking the Potential of Operations Research for Multi-Graph Matching

TL;DR

This work revisits respective algorithms for the MDAP, adapt them to incomplete multi-graph matching, and propose their extended and parallelized versions, and shows that the new method substantially outperforms the previous state of the art in terms of objective and runtime.

Abstract

We consider the incomplete multi-graph matching problem, which is a generalization of the NP-hard quadratic assignment problem for matching multiple finite sets. Multi-graph matching plays a central role in computer vision, e.g., for matching images or shapes, so that a number of dedicated optimization techniques have been proposed. While the closely related NP-hard multi-dimensional assignment problem (MDAP) has been studied for decades in the operations research community, it only considers complete matchings and has a different cost structure. We bridge this gap and transfer well-known approximation algorithms for the MDAP to incomplete multi-graph matching. To this end, we revisit respective algorithms, adapt them to incomplete multi-graph matching, and propose their extended and parallelized versions. Our experimental validation shows that our new method substantially outperforms the previous state of the art in terms of objective and runtime. Our algorithm matches, for example, 29 images with more than 500 keypoints each in less than two minutes, whereas the fastest considered competitor requires at least half an hour while producing far worse results.

Figures (8)

• Figure 1: Multi-Graph Matching and Cycle Consistency.
• Figure 2: Conceptual Diagram of Our Method
• Figure 3: Feasible solution contruction.
• Figure 4: Comparison of the construction and GM local search algorithm variants. Results averaged over 10 instances of worms-10. For reference, the best and the worst result of Our over 10 runs is depicted as a baseline. (\ref{['fig:results-cost-distribution']}) Cost distributions over 100 runs. Result after construction with either seq-c, par-c or inc-c (initial) and after improving with either seq-ls or par-ls ($+$ LS). (\ref{['fig:results-ablation-local-search']}) Convergence of seq-ls, par-ls and best-ls variants. Two plots are shown for each algorithm, for seq-c and par-c results as a starting point respectively.
• Figure 5: Objective over time comparison. Objective plotted in log-scale and offset by the inconsistent solution's objective $C^\mathrm{inc}$ obtained from independently solving $d(d-1)/2$ pairwise GM problems. The value $C^\mathrm{inc}$ approximates, since hutschenreiter_fusionmoves_2021 is approximative, a lower bound to Eq. \ref{['eq:MGM']} provided by the MGM relaxation that ignores cycle consistency. Results are averaged over all instances of the dataset. For Our, the best and worst result over 10 runs is given. \ref{['subfig:results-compare-b']} The first 60 minutes of \ref{['subfig:results-compare-a']}, illustrating the considerable difference in construction time between Our and MP-T for large problems. \ref{['subfig:results-compare-c']} and \ref{['subfig:results-compare-d']} provide averaged results over all 12 object synthetic density and all 8 object hotel datasets respectively. \ref{['subfig:results-compare-a']}, \ref{['subfig:results-compare-b']} compare only methods able to find allowed solutions of sparse problems. DS* results are shown for \ref{['subfig:results-compare-c']} only, as the other datasets contain incomplete problems. Note, that even 10 sequential runs of Our to attain Our(best)'s results would still render the fastest algorithm.

Theorems & Definitions (7)

• Theorem 1
• Definition 1
• Definition 2
• Definition 3
• Theorem 2
• proof
• proof : Proof of \ref{['thm:reduction']}