Particle filter with state permutations for solving image jigsaw puzzles

We deal with an image jigsaw puzzle problem, which is defined as reconstructing an image from a set of square and non-overlapping image patches. It is known that a general instance of this problem is NP-complete, and it is also challenging for humans, since in the considered setting the original image is not given. Recently a graphical model has been proposed to solve this and related problems. The target label probability function is then maximized using loopy belief propagation. We also formulate the problem as maximizing a label probability function and use exactly the same pairwise potentials. Our main contribution is a novel inference approach in the sampling framework of Particle Filter (PF). Usually in the PF framework it is assumed that the observations arrive sequentially, e.g., the observations are naturally ordered by their time stamps in the tracking scenario. Based on this assumption, the posterior density over the corresponding hidden states is estimated. In the jigsaw puzzle problem all observations (puzzle pieces) are given at once without any particular order. Therefore, we relax the assumption of having ordered observations and extend the PF framework to estimate the posterior density by exploring different orders of observations and selecting the most informative permutations of observations. This significantly broadens the scope of applications of the PF inference. Our experimental results demonstrate that the proposed inference framework significantly outperforms the loopy belief propagation in solving the image jigsaw puzzle problem. In particular, the extended PF inference triples the accuracy of the label assignment compared to that using loopy belief propagation.