Reconstructing random pictures

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.

v3: 9 figures, 23 pages, substantial additions made to the presentation of main proofs; final version appearing in RSA