Reconstructing random pictures
Bhargav Narayanan, Corrine Yap

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 of size , i.e., an grid filled with zeros and ones uniformly at random, when is it possible to reconstruct from its -deck, i.e., the multiset of all its subgrids? We demonstrate ``two-point concentration'' for the reconstruction threshold by showing that there is an integer such that if , then is reconstructible from its -deck with high probability, and if , then with high probability, it is impossible to reconstruct from its -deck. The proof of this result uses a combination of interface-exploration arguments and entropic arguments.
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Taxonomy
TopicsDigital Image Processing Techniques · Medical Image Segmentation Techniques · Image Processing and 3D Reconstruction
