Efficient Streaming Algorithms for Two-Dimensional Congruence Testing and Geometric Hashing

TL;DR

This paper introduces space-efficient streaming algorithms for 2D congruence testing and geometric hashing, addressing input precision and using complex moments to enable practical, low-memory solutions in computer vision tasks.

Contribution

It develops the first space-efficient streaming algorithms for 2D congruence testing and geometric hashing, leveraging complex moments in finite-precision rational settings.

Findings

Achieves a 3-pass algorithm with $O( ext{log } n ( ext{log } U + ext{log } n))$ space for congruence identification.

Designs a 6-pass algorithm with $O(m ext{log } n ( ext{log } n + ext{log } U + ext{log } m))$ space for geometric hashing.

Shows that $O( ext{log } n)$ complex moments suffice to prevent vanishing moments in rational input.

Abstract

The geometric congruence problem is a fundamental building block in many computer vision and image recognition tasks. This problem considers the decision task of whether two point sets are congruent under translation and rotation. A related and more general problem, geometric hashing, considers the task of compactly encoding multiple point sets for efficient congruence queries. Despite its wide applications, both problems have received little prior attention in space-aware settings. In this work, we study the two-dimensional congruence testing and geometric hashing problem in the streaming model, where data arrive as a stream and the primary goal is to minimize the space usage. To meaningfully analyze space complexity, we address the underaddressed issue of input precision by working in the finite-precision rational setting: the input point coordinates are rational numbers of the form $p/q$ with $|p|, |q| \le U$. Our result considers a stronger variant of congruence testing called congruence identification, for which we obtain a 3-pass randomized streaming algorithm using $O(\log n(\log U+\log n))$ space. Using the congruence identification algorithm as a building block, we give a 6-pass $O(m\log n (\log n + \log U + \log m))$-space randomized streaming algorithm that outputs a hash function of length $O(\log n+\log U+\log m)$. Our key technical tool for achieving space efficiency is the use of complex moments. While complex moment methods are widely employed as heuristics in object recognition, their effectiveness is often limited by vanishing moment issues (Flusser and Suk [IEEE Trans. Image Process 2006]). We show that, in the rational setting, it suffices to track only $O(\log n)$ complex moments to ensure a non-vanishing moment, thus providing a sound theoretical guarantee for recovering a valid rotation in positive instances.