Rotation-Invariant Vectorized Shape Representations

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 $\mathbb{S}<sup>1$, 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.

• The paper introduces a vector-valued shape sketch that is rotation-invariant by construction and supports direct nearest-neighbor search, clustering, and machine learning without alignment optimization.
• The authors prove injectivity up to rotation, constant shifts, and reverse-of-complement transformations, making the induced Euclidean distance a valid metric on the resulting shape equivalence classes.
• For Lipschitz star-shaped objects, discretization and FFT computation achieve dimension and runtime of O(1/ε), while SQUID fish experiments reach over 0.99 clustering accuracy at sufficient angular resolution.

Overview

This paper introduces a rotation-invariant vector representation of planar shapes, developed by Shafieasl and Phillips at the University of Utah (2605.27498). The central object is a "sketch" of a shape: a Euclidean vector whose coordinates aggregate pairwise differences of a radial function over all angular offsets. Because the sketch depends only on index-offset pairs of function values, it is invariant to rotation by construction, and the Euclidean distance between two sketches serves as a valid distance between the underlying shapes—up to rotation and one additional transformation the authors call reverse-of-complement (RoC). The practical payoff is that shapes become ordinary vectors, so nearest neighbor search, $k$-means clustering, and standard machine learning pipelines apply directly in shape space without any alignment optimization.

The work sits between three established approaches to shape comparison: alignment-minimizing distances (e.g., Hausdorff under optimal rigid motion), intrinsic characteristic-based distances (turning functions, Gromov–Hausdorff, Persistent Homology Transform), and heuristic learned or hand-crafted rotation-invariant descriptors (spherical harmonics, PointNet). The distinguishing claim is that this method combines formal metric guarantees with vector-valued output: unlike heuristic embeddings, the authors prove injectivity (modulo rotation, constant offset, and RoC), and unlike alignment-based distances, no optimization over rotations is required at query time.

The sketch construction

Shapes are first standardized (mean-centered, scaled to unit maximum radius) and restricted to star-shaped objects, so each ray from the origin intersects the shape in an interval $[0, a_\theta]$. Discretizing angles into $m$ wedges yields a function $f : [m] \to \mathbb{R}$ with $f(j) = a_j$. The sketch is defined coordinate-wise as

$$V_f(k) = \frac{1}{m}\sum_{j \in [m]} \Phi\big(f(j) - f(j+k)\big),$$

with $\Phi(z) = e^{-z}$ as the primary choice (Laplace and Gaussian kernels are also discussed). Each coordinate $V_f(k)$ summarizes the multiset $\Delta_f(k)$ of all pairwise value differences at offset $k$.

Two structural facts anchor the analysis. First, the signature is invariant to rotations and constant shifts of $[0, a_\theta]$0, and also to the RoC operation $[0, a_\theta]$1, which acts as an involution $[0, a_\theta]$2. Second, for $[0, a_\theta]$3 the continuous signature $[0, a_\theta]$4 factors as a periodic convolution of $[0, a_\theta]$5 and $[0, a_\theta]$6. This convolution structure both explains the RoC invariance and enables fast computation via the FFT.

Injectivity and the metric guarantee

The core technical contribution is proving that the sketch is strictly injective up to rotation, shift, and RoC—not merely invariant to them. Two functions with equal sketches are called lag-homometric ($[0, a_\theta]$7 for all $[0, a_\theta]$8). Recovering $[0, a_\theta]$9 from its difference multisets uses the Lindemann–Weierstrass theorem: since sums of exponentials of distinct algebraic numbers are linearly independent over $m$0, equality of sketched values forces equality of the underlying difference multisets. This is a notable use of transcendental number theory to establish injectivity of a shape descriptor.

The remaining argument proceeds via canonical forms. A canonical form fixes the minimum value at zero and rotates it to index zero; an RoC-canonical form additionally resolves the ambiguity introduced by the mirror-flip operation using the second-largest difference. An inductive lemma then shows that, given lag-homometric functions in general position (all values and all pairwise differences distinct), once the locations of the $m$1 smallest and $m$2 largest values are resolved, the next extreme can be located; iterating until $m$3 proves that RoC-canonical lag-homometric functions must be identical. A parallel, more intricate case analysis handles permutation-valued functions $m$4, where tied differences require tracking position functions modulo $m$5.

The consequence is a clean statement: for functions in $m$6, $m$7 if and only if $m$8 or $m$9 for some discrete rotation $f : [m] \to \mathbb{R}$0 and constant $f : [m] \to \mathbb{R}$1. Hence the induced distance $f : [m] \to \mathbb{R}$2 is a genuine metric on equivalence classes modulo rotation and RoC, and the same holds for standardized star shapes in general position.

Approximation guarantees and computation

For $f : [m] \to \mathbb{R}$3-Lipschitz star-shaped shapes, discretizing into $f : [m] \to \mathbb{R}$4 wedges incurs star-distance error at most $f : [m] \to \mathbb{R}$5. Combined with the fact that the sketch map is $f : [m] \to \mathbb{R}$6-Lipschitz with respect to star distance (and hence Hausdorff distance, since Hausdorff distortion bounds star distance), choosing $f : [m] \to \mathbb{R}$7 yields an $f : [m] \to \mathbb{R}$8-accurate distance. The full pipeline—discretization plus FFT-based sketching—runs deterministically in $f : [m] \to \mathbb{R}$9 time per pair, producing vectors of dimension $f(j) = a_j$0. This deterministic near-linear guarantee is stronger than what randomized feature maps typically offer, though the authors also give a Random-Fourier-Features-style Monte Carlo estimator of the associated positive semi-definite kernel $f(j) = a_j$1, requiring $f(j) = a_j$2 samples.

Appendix results quantify discretization error further: for $f(j) = a_j$3-Lipschitz $f(j) = a_j$4, trapezoidal-rule analysis gives $f(j) = a_j$5 when $f(j) = a_j$6, with spectral convergence for analytic $f(j) = a_j$7—so doubling angular resolution roughly quadruples accuracy in the smooth regime.

Experimental validation

Experiments use the SQUID dataset of 1100 fish outlines, converted to star-shaped form by taking the maximal radial extent along rays within each wedge (default $f(j) = a_j$8).

Clustering robustness: on 100 objects (10 shapes × 9 random rotations plus originals), $f(j) = a_j$9-means with $$V_f(k) = \frac{1}{m}\sum_{j \in [m]} \Phi\big(f(j) - f(j+k)\big),$$0 achieves mean accuracy around 0.95 once $$V_f(k) = \frac{1}{m}\sum_{j \in [m]} \Phi\big(f(j) - f(j+k)\big),$$1 and consistently above 0.99 for $$V_f(k) = \frac{1}{m}\sum_{j \in [m]} \Phi\big(f(j) - f(j+k)\big),$$2, never exactly 1.00 only because some fish shapes are nearly identical. When discretization precedes the random rotation, accuracy is exactly 1.00, isolating residual error to the interaction of discretization with rotation rather than the sketch itself.

Retrieval: 5-nearest-neighbor search over all 1100 vectorized shapes returns visually similar fish at arbitrary orientations, confirming rotation invariance empirically.

These experiments are deliberately simple; they demonstrate stability and plug-and-play usability rather than benchmark superiority against competing descriptors, which the paper does not attempt.

Limitations and open questions

Several restrictions bound the applicability of the results. The injectivity theorem requires functions in general position—all values and all pairwise differences distinct—and the approximation theorem applies only to $$V_f(k) = \frac{1}{m}\sum_{j \in [m]} \Phi\big(f(j) - f(j+k)\big),$$3-Lipschitz star-shaped standardized shapes; non-star-shaped objects must be preprocessed into star-shaped form (as done for SQUID by taking maximal radial extents), which discards interior structure. The RoC ambiguity is inherent to the representation: two shapes related by this mirror-flip-type operation are indistinguishable, and while the authors show it can be controlled canonically, it means the descriptor cannot distinguish chiral variants under reflection combined with reversal. The Lindemann–Weierstrass-based injectivity argument relies on algebraic function values, and the analysis is specific to $$V_f(k) = \frac{1}{m}\sum_{j \in [m]} \Phi\big(f(j) - f(j+k)\big),$$4 for the exactness results, with Laplace and Gaussian variants discussed but not given matching guarantees. Finally, the paper leaves open whether the approach extends beyond planar star-shaped objects—for instance to higher dimensions or to functions without the Lipschitz regularity assumption—and does not compare retrieval or clustering quality against established descriptors such as shape contexts or harmonic-based embeddings.

Conclusion

This paper provides a formally grounded rotation-invariant shape embedding: a deterministic, FFT-computable vector sketch of dimension $$V_f(k) = \frac{1}{m}\sum_{j \in [m]} \Phi\big(f(j) - f(j+k)\big),$$5 that provably approximates a metric on star-shaped planar forms up to rotation and a characterized RoC operation. Its main contribution is closing the gap between heuristic invariant descriptors and alignment-based shape distances, delivering both metric guarantees and direct compatibility with vector-space infrastructure. The restriction to star-shaped, general-position inputs and the inherent RoC ambiguity remain the principal constraints on broader deployment.