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.

翻译：几何全等问题是计算机视觉与图像识别任务中的基础构建模块。该问题旨在判定两个点集在平移和旋转变换下是否全等。一个相关且更一般化的问题是几何哈希，其任务是对多个点集进行紧凑编码以支持高效的全等查询。尽管应用广泛，这两个问题在空间敏感场景中先前鲜有研究。本文研究流式模型下的二维全等测试与几何哈希问题，其中数据以流形式到达，主要目标是最小化空间占用。为合理分析空间复杂度，我们通过有限精度有理数设定处理了长期被忽视的输入精度问题：输入点坐标为形如$p/q$的有理数，其中$|p|, |q| \le U$。我们的结果考虑了一种更强的全等测试变体——全等识别，为此我们提出了一种使用$O(\log n(\log U+\log n))$空间的三轮随机化流式算法。以全等识别算法为基础，我们进一步给出一个六轮$O(m\log n (\log n + \log U + \log m))$空间的随机化流式算法，可输出长度为$O(\log n+\log U+\log m)$的哈希函数。实现空间效率的关键技术工具是复矩方法的应用。虽然复矩方法在物体识别中常被用作启发式方法，但其有效性常受限于矩消失问题（Flusser与Suk [IEEE Trans. Image Process 2006]）。我们证明，在有理数设定下，仅需追踪$O(\log n)$个复矩即可保证非零矩的存在，从而为在正例中恢复有效旋转提供了可靠的理论保证。