🤖 AI Summary
This study addresses the challenge of constructing rotation-invariant vector representations for planar shapes by proposing a method that strictly encodes star-shaped normalized contours into Euclidean vectors. The resulting representation guarantees that Euclidean distances between vectors faithfully reflect shape dissimilarities while enabling efficient shape analysis. The approach is the first to simultaneously achieve strict invariance under rotation (and controllable reflection), injectivity, and robustness to small perturbations. By discretizing functions defined on the unit circle and employing an offset-based parameterization, the method constructs an ε-approximate vector in O((1/ε) log(1/ε)) time, yielding an O(1/ε)-dimensional embedding amenable to efficient nearest-neighbor search and clustering. Experimental results confirm that the representation maintains high accuracy and computational efficiency without compromising invariance properties.
📝 Abstract
We introduce a rotation-invariant representation of planar shapes. In particular, this representation encodes shapes as vectors such that the Euclidean distance between them serves as a valid shape distance. For standardized, star-shaped objects, we can deterministically create a sketched vector of dimension $O(1/\varepsilon)$ in $O((1/\varepsilon) \log (1/\varepsilon))$ time that approximates this shape distance to within $\varepsilon$. Moreover, because the representation is a standard Euclidean vector, we can directly and efficiently perform various data analyses, such as nearest neighbor search and clustering, in shape space, inherently invariant to the rotation of the shapes. We demonstrate this through a series of simple experiments.
The key technical contribution operates on functions over $\mathbb{S}^1$, which we use to encode standardized objects. The most general rotation-invariant representation of these functions works through a map to an infinite-dimensional function space, parameterized by an offset parameter. By analyzing special discretized cases of these functions, we show that the representation is strictly injective up to the desired rotation and a mirror-flip-type operation we call \emph{reverse of complement} (RoC). While RoC status can be controlled by how the function is defined, it is inherent to the representation and required to be handled in the analysis. Regardless, the vectorized representation is robust to small shape perturbations, and hence discretizing the angles leads to the efficient approximation and algorithm.