Invariant Shape Analysis of Surfaces with Spherical Topology

T. Shaska; M. -R. Siadat

cs.CVSep 30, 2026

Invariant Shape Analysis of Surfaces with Spherical Topology

Authors: T. Shaska, M. -R. Siadat

Abstract

Spherical harmonic descriptors of closed 3D shapes depend on the parameterization, the pose and the scale of the surface, and the standard rotation-invariant reductions, the power spectrum and the bispectrum, discard the relative orientation of the harmonic bands and cannot distinguish a shape from its mirror image. We construct a descriptor that removes all three dependencies exactly and loses nothing else: a conformal parameterization normalized by its conformal barycenter, followed by polynomial invariants of the rotation group. Identifying each harmonic band with a binary form turns the rotation quotient into classical invariant theory and makes reflections visible as the sign of an invariant, so chirality is recorded. The descriptor is complete for the truncated expansion, stable in the orbit distance, and comes with numerical diagnostics. Benchmarks confirm the guarantees, and on bilateral anatomical structures the descriptor separates mirror-image pairs from asymmetric pairs, which parity-blind descriptors cannot.

Figures & tables

Panel of Figure 1 — Figure 1. The conformal barycenter on $\mathbb{S}^{2}$ . Left: a probability measure $\mu$ consisting of $42$ point masses of weight $1/42$ on $\mathbb{S}^{2}$ (points on the far hemisphere are drawn lighter; each atom has mass less than $1/2$ , see Remark 2.2 ), its Euclidean center of mass $c$ (cross), and its conformal barycenter $x=B(\mu)\in\mathbb{B}^{3}$ (circle). The dashed segment is the geodesic of $\mathbb{B}^{3}$ through $0$ and $x$ . Right: the pushforward $(\tau_{x})_{*}\mu$ under the hyperbolic translation $\tau_{x}$ ; its Euclidean center of mass is the origin. The points were computed from 1 .

harmonic side	binary-form side
$\mathcal{H}_{l}^{\mathbb{C}}$ with basis $\mathcal{Y}_{lm}$	$\mathrm{Sym}^{2l}$ with basis $z_{0}^{l-m}z_{1}^{l+m}$	(i), (iii)
real harmonics $\mathcal{H}_{l}$	$\Fix(\sigma_{l})$ : $a_{2l-k}=(-1)^{l+k}\bar{a}_{k}$	(v)
harmonic part $\pi(fg)$ of a product	product $F_{f}F_{g}$	(i)
Clebsch–Gordan map $\mathcal{H}_{l}\otimes\mathcal{H}_{m}\to\mathcal{H}_{l+m-r}$	transvectant $(F_{f},F_{g})_{r}$	Proposition 3.12
rotation $\rho(g)\in\mathrm{SO}(3)$	substitution by $g\in\mathrm{SU}(2)$	(ii)
reflection $-1\in\mathrm{O}(3)$	$F\mapsto(-1)^{l}F$	Corollary 3.10

Table 1. The dictionary of Theorem 3.8 . Roman numerals refer to its parts.

Panel of Figure 2 — Figure 2. The image of $\mathcal{H}_{2}$ in the $(I,J)$ -plane is the shaded region $\{I\geq 0,\ I^{3}\geq 27J^{2}\}$ , bounded by the curves $J=\pm I^{3/2}/\sqrt{27}$ of axially symmetric quadrupoles. It is not convex: the points $(1,c)$ and $(4,8c)$ with $c=27^{-1/2}$ lie on the boundary, and their midpoint lies outside. Linear interpolation of invariant coordinates therefore leaves the image.

Panel of Figure 3 — Figure 3. Maxwell axes of a nonzero real harmonic of degree three, drawn as lines through the center of $\mathbb{S}^{2}$ . Left: coplanar axes; the half-turn about the normal $n$ of their plane reverses $f$ . Middle: bisector type; $n$ bisects the angle of $\mathbb{R}a$ and $\mathbb{R}b$ , and $\mathbb{R}c$ is orthogonal to $n$ , so the half-turn about $n$ exchanges $\mathbb{R}a$ and $\mathbb{R}b$ and reverses $\mathbb{R}c$ . In both cases $f$ is achiral and $R(F_{f})=0$ . Right: a generic configuration, for which $f$ is chiral.

diagnostic	value (copy A)
(i) maximal weight $\max_{i}w_{i}$ ; centering residual $\lvert\sum_{i}w_{i}u_{i}\rvert$	$1.9\cdot 10^{-4}$ ; $6\cdot 10^{-16}$
(i) quasi-conformal distortion of the discrete map, max / mean (additional)	$1.030$ / $1.009$
(ii) $\lVert\mathsf{Y}^{\top}\Omega\mathsf{Y}-1\rVert_{2}$ ; $\lVert C_{0}\rVert$	$1.4\cdot 10^{-3}$ ; $2.3\cdot 10^{-5}$
(iii) truncation residual at $L=6$ ; bound of Proposition 2.4 ; ratio	$2.5\cdot 10^{-5}$ ; $4.5\cdot 10^{-2}$ ; $5.6\cdot 10^{-4}$
(iii) $\sup e^{2\rho}$	$3.10$
(iv) quadrupole eigenvalues	$-0.2186,\ -0.0843,\ 0.3029$

Table 2. Diagnostics of Section 8.4 for two independently processed copies of one synthetic mesh. Copy B agrees with copy A to the digits shown. The last row is a first-order sensitivity ratio; see the text.

Figure 4. Benchmarks of Section 9 . (a) Nearest-neighbour accuracy on the four classes

A,A^{*},B,B^{*}

of Section 9.1 as a function of coefficient noise, mean and standard deviation over five repetitions; chance is

1/4

. (b) Local Lipschitz ratio

\lvert\Delta D\rvert/\delta

of the frame coordinates of Theorem 6.2 and of the polynomial invariants of Theorem 4.9 , as the gap between two eigenvalues of the quadrupole tends to zero. (c) Change of the invariants of Theorem 4.9 against the exact orbit distance at a principal point, at a point with stabilizer

\mathrm{SO}(2)

(perturbed in the directions of its slice moved by the stabilizer), and at the origin, with fitted slopes.

structure	flips	$K_{\max}$	$\sup e^{2\rho}$	$\delta(L,L^{\prime})$	$\delta(L,R)$	$\delta(L,R^{*})$	$\chi_{L}$	$\chi_{R}$	$\sgn\varepsilon$	$\sgn(-i\tilde{R})$
hippocampus	0	2.19	$8.2\cdot 10^{7}$	$5.1\cdot 10^{-14}$	0.1225	0.0088	0.119	0.126	+ / -	- / +
amygdala	0	1.42	140	$2.8\cdot 10^{-13}$	0.3411	0.0039	0.341	0.342	- / +	- / +
putamen	0	1.32	18	$1.1\cdot 10^{-6}$	0.1729	0.0049	0.173	0.173	+ / -	+ / -
globus pallidus	0	1.23	26	$6.2\cdot 10^{-14}$	0.3161	0.0009	0.316	0.317	+ / -	- / +
thalamus	0	1.06	5.6	$4.9\cdot 10^{-14}$	0.1940	0.0015	0.194	0.194	- / +	+ / -
parahippocampal gyrus	0	1.23	$1.3\cdot 10^{3}$	$1.8\cdot 10^{-4}$	0.1890	0.0049	0.188	0.190	+ / -	- / +

Table 3. Eleven bilateral pairs of BodyParts3D through the pipeline of Section 8 , radial model,

L=6

. flips: faces with reversed orientation on the sphere;

K_{\max}

: largest quasi-conformal distortion of a face;

\sup e^{2\rho}

: largest ratio of a face area to the area of its spherical image (diagnostic (iii)).

L^{\prime}

is a rigidly moved, relabeled, Möbius-perturbed copy of

L

, and

R^{*}

the reflected right structure; all orbit distances are divided by

\lVert c\rVert

\chi

is the chirality index. The last two columns give the signs of

\varepsilon

and of

-i\tilde{R}

on the left and right side; a sign in parentheses belongs to a value whose gauge coordinate 17 is below

0.1

, that is, within numerical distance of the divisor.

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