An algebra for local histograms

In this article, we consider local overlapping histograms of functions between discrete domains and codomains. We develop a simple algebra for local histograms. Based on a separation of overlapping domains into non-overlapping domains, we (1) show how these can be used to enumerate the size of the set of possible histograms given the local histogram domains, and (2) enumerate the number of functions, which share a specific choice of a set of local histograms. Finally, we present a decoding algorithm, which given a set of overlapping histograms, and calculate the set of functions, which share these histograms.