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

TL;DR

This work develops polylogarithmic-space streaming algorithms for exact two-dimensional geometric congruence testing and geometric hashing with finite-precision rational inputs. It introduces a 3-pass CongIden algorithm that leverages complex moments and randomized finite-field embeddings to recover a rotation and translation, and a 6-pass GeomHash that outputs compact congruence hashes of length $O(\log n+\log U+\log m)$. A central technical contribution is the non-vanishing moment guarantee in the rational setting, which ensures rotation recovery from $O(\log n)$ moments, together with fingerprinting to certify equality. The authors extend these ideas to three dimensions with a six-pass CongIden in $O(n^{5/6}\log^4 U\log n)$ space and provide matching lower bounds that highlight the inherent trade-offs between input precision and streaming space. Collectively, the results bridge RAM-based congruence algorithms and space-bounded streaming models, enabling efficient, exact congruence queries on large, precision-limited datasets.

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.

Theorems & Definitions (50)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Definition 1.4: Complex Moments
• Theorem 1.5
• Corollary 2.2
• Corollary 2.3
• Lemma 2.4: MR80123
• Theorem 3.1
• Claim 3.2