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Principal moments for efficient representation of 2D shape

The analytic signature is a recently proposed 2D shape representation scheme. It is tailored to the representation of shapes described by arbitrary sets of unlabeled points, or landmarks, because its most distinctive feature is the maximal invariance to a permutation of those points. The shape similarity of two point clouds can then be obtained from a direct comparison of their representations. However, since the analytic signature is a continuous function, performing the comparison of their densely sampled versions may result excessively time-consuming, e.g., when dealing with large databases, even of simple shapes. In this paper we address the problem of efficiently storing and comparing such powerful representations. We start by showing that their frequency spectrum is related to particular complex moments of the shape. From this relation, we derive the bandwidth of the representation in terms of the shape complexity. Using this result, we show that the analytic signature can be described by a small set of complex moments. We call this compact description the Principal Moments (PMs) of a shape and show how to efficiently compare shapes using PMs. Our experiments illustrate that the gain in efficiency comes at no cost in performance.