Rotation-Invariant Vectorized Shape Representations

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.