I‑AIMS Institute for Artificial Intelligence
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Exact Euclidean-Invariant Surface Descriptors for Data-Efficient Learning from Segmented Anatomy

Tanush Shaska

I-AIMS preprint sh-182 · 2026

Three-dimensional anatomical shape is commonly represented by hand-crafted radiomic features or by learned voxel representations. Radiomic features are compact but incomplete. Voxel representations require substantial data and acquire rotational stability only through augmentation or architectural constraints. We develop, implement, and test a third approach: polynomial invariants of spherical-harmonic surface representations. A conformal parameterization of a genus-zero surface is normalized by its conformal barycenter, which reduces the parameterization ambiguity to a rotation of the sphere. Each harmonic band is identified with a binary form through the Veronese correspondence, and the residual rotation is removed by classical invariants: band powers, bispectra, the Igusa invariants J₂,J₄,J₆,J₁₀ of the sextic attached to band three, and two-sided Gram invariants of the coordinate expansion. A weighted-projective normalization separates scale-free shape from size. In the continuous model the descriptor is exactly invariant under translations, orthogonal transformations, and positive scalings.

We give a complete discrete pipeline from a binary mask to the descriptor and measure how much of the continuous invariance survives discretization. On 26 real anatomical surfaces from the BodyParts3D atlas, voxelized at 1 mm after random rotations, reflections, translations, and scalings, the median within-structure descriptor displacement is 0.061 of the median between-structure distance; the error is governed by the discrete conformal map, and every displacement exceeding half the between-structure distance occurs on two structures with thin protrusions, whose conformal map concentrates area beyond what the mesh samples. Increasing the mesh budget reduces these failures to within-to-between ratios near 0.1, although the area concentration of the conformal map is unchanged. In synthetic populations matched in canonical coordinates on the band powers, bispectra add +0.168 in AUC, the Igusa invariants J₄,J₆,J₁₀ are statistically equivalent to no gain, and linear, MLP, and polynomial-kernel learners on raw coefficients remain at chance even with exact rotation augmentation. In a positive control in which the classes differ only in the configuration of the three Maxwell poles of band three, powers, bispectra, and Gram invariants remain at chance while the Igusa invariants raise the AUC by 0.15 through the complete pipeline, which identifies what that layer measures. On the 335 glioma cases of BraTS 2019, with the pipeline and hypotheses of the synthetic studies unchanged, the pipeline extracts 274 tumor cores and 116 whole tumors, with exclusion unrelated to grade. The invariant models change their predicted probability under rotation, reflection, rescaling, and regridding of the tumor by a median of 0.009, against 0.127 for raw coefficients, against 0.033 for an augmented 3D CNN on the mask, and are learned from 32 examples where raw-coefficient learners stay at chance and the CNN trails them. Grade is only weakly predictable from tumor-core shape (AUC about 0.64), and the invariant layers add nothing detectable to shape radiomics; on this endpoint the representation is stable but redundant.

MSC classes: Primary 13A50; Secondary 68T07, 68U05, 92C55

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