Reconstructing random pictures

TL;DR

This paper determines the threshold size of subgrids needed to reconstruct a random binary picture with high probability, revealing a sharp phase transition at a critical subgrid size related to the logarithm of the image dimension.

Contribution

It establishes a precise asymptotic threshold for reconstructing random binary images from their subgrid multisets, using novel interface and entropy-based methods.

Findings

Reconstruction threshold at k_c(n) ~ (2 log n)^{1/2}

High probability of reconstructibility for k > k_c

Impossibility of reconstruction for k < k_c

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.