Scalable Rigid-Invariant Distance for Shape Matching and Alignment

Zakk Heile, Peilin He, Jayson Tran, Ruiling Wang, Shrikant Chand

Submission Track: Short papers presenting ongoing research or work submitted to other venues (up to 5 pages, excluding references)

Keywords: shape alignment, matching, tissue, wasserstein, distance, unsupervised learning

TL;DR: We define a novel mathematically grounded distance function to compare shapes, point clouds, or distributions, and demonstrate superior accuracy and computability in downstream biological tasks using tissue slices and human meshes.

Abstract: Comparing probability distributions from biological images requires metrics that are geometrically grounded and invariant to orientation. Classical optimal transport (OT) distances are sensitive to rotations, while Gromov–Wasserstein (GW) offers invariance but is computationally prohibitive. We introduce **Rigid-Invariant Sliced Wasserstein via Independent Embeddings (RISWIE)**, a scalable pseudometric that achieves rigid invariance by aligning data-adaptive embeddings through optimal signed permutations, at negligible cost. Evaluated on 2D HuBMAP tissue slices and 3D MPI-FAUST meshes, RISWIE attains 95.8\% accuracy with over $10^4\times$ speedup over GW and an AUC of 0.94 for human pose matching. Its optimization also yields explicit axis alignments usable for downstream analysis, making RISWIE a practical and interpretable distance for large-scale geometric data.

Submission Number: 40