Maximal iV-Ghosts and Minimal Information Recovery from N Projected Views of an Array

摘要

Digital data is now frequently stored privately and securely in the "cloud". One repository stores several different sets of projections of the original data. Each set is kept on a separate, remote server. The information residing on any local server, purposefully, is insufficient to exactly reconstruct the full data. Here we ask: how much useful information can be gleaned from one local projection set? We answer that question by examining projection ghosts. A ghost is an assembly of signed pixels positioned to have zero sums along chosen discrete directions. The shape of each ghost is defined uniquely by its distinct set of directions. An TV-ghost with a shape that fits snuggly inside the boundary of an array defines precisely all of the array locations that cannot be exactly reconstructed from those N projected views. Minimal N-ghosts contain 2N elements: one (-1/+1) pair is needed for zero-sums along each of the N directions. Maximal N-ghosts contain 2~N elements: the number of (-1/+1) elements can double N times, once for each of the N directions. Here we construct maximal N-ghosts that cover a large area of their bounding array. By maximising the number of unrecoverable ghosted pixels, we minimise the information that can be reconstructed from N projected views. We show that at least 60% of the data in an m × m array, for m ≈ N~2/4, can be masked or made "unreadable", for a maximal set of N noise-free projections of the original m × m data.