Reconstructing random pictures

TL;DR

We study reconstructing a random $n\\times n$ binary picture from its $k\\times k$ subgrid deck, identifying a sharp reconstruction threshold with high-probability recoverability above it and non-recoverability below it. The main result shows two-point concentration around $k_c(n)\\sim\\sqrt{2\\log_2 n}$: reconstructibility transitions from unlikely to likely as $k$ crosses $k_c(n)$. The proofs combine entropy-based $0$-statement arguments with interface-exploration techniques, and the work situates the problem within shotgun-reconstruction and related jigsaw/algorithmic reconstruction literature, including extensions to colors and higher dimensions. The findings establish fine-grained asymptotics for the threshold and motivate future directions in higher dimensions and correlated models.

Abstract

Given a random binary picture $P_n$ of size $n$, i.e., an $n\times n$ grid filled with zeros and ones uniformly at random, when is it possible to reconstruct $P_n$ from its $k$-deck, i.e., the multiset of all its $k\times k$ subgrids? We demonstrate ``two-point concentration'' for the reconstruction threshold by showing that there is an integer $k_c(n) \sim (2 \log n)^{1/2}$ such that if $k > k_c$, then $P_n$ is reconstructible from its $k$-deck with high probability, and if $k < k_c$, then with high probability, it is impossible to reconstruct $P_n$ from its $k$-deck. The proof of this result uses a combination of interface-exploration arguments and entropic arguments.

Figures (1)

• Figure 1: A picture of size $3$ and its $2$-deck.

Theorems & Definitions (1)

• Theorem 1.1