Indexing and querying color sets of images

TL;DR

This paper develops algorithms and data structures for efficiently indexing and querying the set of color fingerprints of all rectangular regions in an image, enabling fast retrieval of maximal regions with specific color sets.

Contribution

It introduces methods to compute, index, and query maximal rectangular regions by their color sets with improved time and space complexities, including Monte Carlo and deterministic algorithms.

Findings

Expected time for maximal location determination: O(nm^2σ)

Data structure query time: O(|f|+loglog n+k)

Space complexity for indexing: O(nm log n + |L|)

Abstract

We aim to study the set of color sets of continuous regions of an image given as a matrix of $m$ rows over $n\geq m$ columns where each element in the matrix is an integer from $[1,\sigma]$ named a {\em color}. The set of distinct colors in a region is called fingerprint. We aim to compute, index and query the fingerprints of all rectangular regions named rectangles. The set of all such fingerprints is denoted by ${\cal F}$. A rectangle is {\em maximal} if it is not contained in a greater rectangle with the same fingerprint. The set of all locations of maximal rectangles is denoted by $\mathcal{L}.$ We first explain how to determine all the $|\mathcal{L}|$ maximal locations with their fingerprints in expected time $O(nm^2\sigma)$ using a Monte Carlo algorithm (with polynomially small probability of error) or within deterministic $O(nm^2\sigma\log(\frac{|\mathcal{L}|}{nm^2}+2))$ time. We then show how to build a data structure which occupies $O(nm\log n+\mathcal{|L|})$ space such that a query which asks for all the maximal locations with a given fingerprint $f$ can be answered in time $O(|f|+\log\log n+k)$, where $k$ is the number of maximal locations with fingerprint $f$. If the query asks only for the presence of the fingerprint, then the space usage becomes $O(nm\log n+|{\cal F}|)$ while the query time becomes $O(|f|+\log\log n)$. We eventually consider the special case of squared regions (squares).