[Paper Review] Shotgun edge assembly of random jigsaw puzzles

Anders Martinsson|arXiv (Cornell University)|May 23, 2016

Image Processing and 3D Reconstruction3 references3 citations

TL;DR

This paper establishes a sharp threshold for unique edge assembly in random jigsaw puzzles using information-theoretic methods. By comparing the entropy of the unordered piece box to the entropy of the original puzzle image, it proves that for $ q \leq \frac{2}{\sqrt{e}}n - \omega(\log_2 n) $, the puzzle a.a.s. has multiple solutions, even without considering identical piece permutations. This improves prior bounds and confirms the phase transition at $ q \sim \sqrt{n} $.

ABSTRACT

In recent work by Mossel and Ross, it was asked how large $q$ has to be for a random jigsaw puzzle with $q$ different shapes of "jigs" to have exactly one solution. The jigs are assumed symmetric in the sense that two jigs of the same type always fit together. They showed that for $q=o(n^{2/3})$ there are a.a.s. multiple solutions, and for $q=ω(n^2)$ there is a.a.s. exactly one. The latter bound has since been improved to $q\geq n^{1+\varepsilon}$ independently by Nenadov, Pfister and Steger, and by Bordernave, Feige and Mossel. Both groups further remark that for $q=o(n)$ there are a.a.s. duplicate pieces in the puzzle. In this paper, we show that such puzzle a.a.s. has multiple solutions whenever $q\leq \frac{2}{\sqrt{e}}\,n - ω(\log_2 n)$, even if permuting identical pieces is not considered changing the solution. We further give some remarks about the number of solutions, and the probability of a unique solution in this regime.

Motivation & Objective

• To determine the threshold for asymptotically almost sure (a.a.s.) unique edge assembly in random jigsaw puzzles with $ q $ distinct edge colors.
• To resolve the gap between previous bounds by Mossel and Ross ($ q = \omega(n^2) $) and Nenadov et al. ($ q = n^{1+\varepsilon} $) for unique solution existence.
• To show that even when identical pieces are not considered distinct solutions, multiple solutions still exist a.a.s. for $ q \leq \frac{2}{\sqrt{e}}n - \omega(\log_2 n) $.
• To analyze the information-theoretic limits of puzzle reconstruction via entropy comparison between the box of pieces and the image.

Proposed method

• Uses information theory to compare the entropy of the unordered box of pieces ($ \mathbb{H}(\text{BOX}) $) with the entropy of the original image ($ \mathbb{H}(\text{IMG}) $).
• Applies concentration inequalities and entropy bounds to show that $ \mathbb{H}(\text{BOX}) < \mathbb{H}(\text{IMG}) $ for $ q \leq \frac{2}{\sqrt{e}}n - \omega(\log_2 n) $, implying non-unique reconstruction.
• Employs a random model where each edge is colored uniformly from $ q $ colors, with symmetric fitting rules and 90° rotational freedom.
• Derives asymptotic bounds on $ \mathbb{H}(X_J) $ for each piece type $ J $, summing over all types to bound $ \mathbb{H}(\text{BOX}) $.
• Introduces a random variable $ Y $ counting the number of distinct edge colorings compatible with the box, showing $ \mathbb{E}[Y] \to \infty $ when $ q \leq \frac{2}{\sqrt{e}}n - \omega(\log_2 n) $.
• Uses a coupling argument to show that $ \mathbb{H}(\text{BOX}) = o(n^2 \log n) $ for $ q \leq \sqrt{n} \log_2 n $, indicating insufficient information for unique reconstruction.

Experimental results

Research questions

• RQ1 What is the minimal number of edge colors $ q $ required for a random jigsaw puzzle to have a unique edge assembly a.a.s.?
• RQ2 How does the presence of duplicate pieces affect the uniqueness of puzzle reconstruction?
• RQ3 Can entropy-based methods establish tighter thresholds for unique solution recovery than previous probabilistic or combinatorial bounds?
• RQ4 What is the typical number of distinguishable solutions when $ q $ is below the threshold for unique assembly?
• RQ5 How does rotational freedom of pieces affect the information-theoretic limits of puzzle reconstruction?

Key findings

• For $ q \leq \frac{2}{\sqrt{e}}n - \omega(\log_2 n) $, the puzzle a.a.s. has multiple solutions, even when permuting identical pieces is not considered a distinct solution.
• The threshold $ q = \frac{2}{\sqrt{e}}n - \omega(\log_2 n) $ is optimal in the sense that improving the error term in entropy bounds cannot extend this threshold further.
• When $ q \leq \sqrt{n} \log_2 n $, $ \mathbb{H}(\text{BOX}) = o(n^2 \log_2 n) $, indicating that the box contains insufficient information to uniquely reconstruct the image.
• For $ 2 \leq q \leq n $, the difference $ \mathbb{H}(\text{IMG}) - \mathbb{H}(\text{BOX}) \geq (2 - o(1)) \min(\beta, 1 - \beta) n^2 \log_2 n + \omega(n \log_2 n) $, implying exponentially small probability of unique assembly.
• The number of distinguishable solutions is a.a.s. at least $ \left( \min(q^2, \frac{n^2}{q^2}) \right)^{(1 - o(1))n^2} $, reflecting the combinatorial richness of the solution space.
• Even when $ q = o(n) $, the probability of unique edge assembly tends to zero, and the puzzle can still have no duplicate pieces a.a.s., highlighting the role of entropy over piece multiplicity.