Invariant Shape Analysis of Surfaces with Spherical Topology

Motivation

• Conventional power-spectrum reductions discard relative band orientation, and even-degree rotation invariants cannot distinguish a shape from its mirror image.The paper frames this distinction as consequential for bilateral anatomy, where handedness differentiates left from right.Source: §1 · p. 1

Construction

• The descriptor removes parameterization, position, and scale dependencies exactly, and is complete for the truncated expansion.Conformal parameterization and conformal-barycenter normalization reduce parameterization freedom to rotations; polynomial invariants remove that residual rotation.Source: §1 · pp. 1–2

Harmonics as binary forms

• The rotation quotient uses binary-form invariants whose real or imaginary phase distinguishes ordinary invariants from pseudo-invariants, making reflection sign visible.For a multihomogeneous invariant, the parity of the weighted degrees determines whether it is real or purely imaginary on real harmonics.Source: §1 · p. 2

Separation and chirality

• Seven invariants of degrees 1, 2, 2, 3, 3, 4, and 6 separate SO(3)-orbits through degree two, while a degree-three skew invariant characterizes chirality.The skew invariant vanishes exactly when the harmonic is congruent to its mirror image.Source: §1 · p. 2

Frames, metrics, stability

• The orbit distance is a quotient metric; invariants are Lipschitz on bounded sets, and compact-set orbit distance is bounded by a Hölder power of invariant discrepancy.Frame coordinates can be ill-conditioned near degenerate strata, but the stated stability result does not depend on the frame.Source: §1 · p. 2

Benchmarks

• On shared-spectrum and bispectrum classes, those descriptors classify at chance, O(3)-invariants reach one half, and seven invariants classify all classes; anatomical tests distinguish handedness pairs from different shapes.The benchmarks also reject two structures whose conformal maps fold.Source: §1 · pp. 2–3

2.1. Spherical harmonics.

• Spherical-harmonic bands are rotation-invariant as norms, but individual coefficients change with placement on the sphere; rotations mix coefficient orders.The paper uses the degree-l harmonic space H_l, of dimension 2l+1, as the band representation.Source: §2.1 · pp. 3–4

2.2. Conformal parameterizations and the conformal barycenter.

• A conformal parameterization is centered by choosing the Möbius transformation that moves its conformal barycenter to the origin; centered choices then differ only by a rotation.For triangulated surfaces, the discrete barycenter is unique provided no atom has mass at least 1/2; uniqueness does not guarantee conditioning.Source: §2.2 · pp. 5–6

2.3. The conformal center and the harmonic coefficients.

• The conformal center is similarity-equivariant, and the coefficient tuple of a centered parameterization is determined up to one common rotation; the full coordinate expansion reconstructs the centered surface.The radial expansion retains only distance from the center, and the paper asserts no reconstruction theorem from radial data.Source: §2.3 · pp. 6–7

2.4. Truncation.

• At degree L, the squared L2 truncation error is at most |S|/(2π(L + 1)(L + 2)), independently of the conformal factor.The corresponding area-weighted geometric error also depends on sup e2ρ; elongated or convoluted surfaces can have large area distortion, so that factor should be reported.Source: §2.4 · p. 8

3. Spherical Harmonics as Binary Forms

• A real degree-l harmonic determines l Maxwell axes, whose antipodal pairs on S2 are the roots of its associated binary form.The axes are determined up to order and multiplicity.Source: §3 · p. 8, §3.3 · p. 13

3.1. Harmonic polynomials and the null cone.

• Every degree-l polynomial has a unique harmonic part modulo qP_{l−2}(C3), and this decomposition is SO3(C)-invariant and equivariant.The harmonic and q-multiple summands are orthogonal under a positive-definite Hermitian form.Source: §3.1 · p. 9

3.2. The spin covering and the map Ψl.

• The map ρ is a surjective double cover from SU(2) to SO(3), and the binary-form correspondence is equivariant under this action.Its kernel is {±1}; the corresponding SL2(C) map onto SO3(C) has the same kernel.Source: §3.2 · p. 10, §3.4 · p. 14

3.3. The real form and reflection parity.

• Real harmonics map to a specific real form of the binary forms, and odd-parity invariants change sign under reflections while taking purely imaginary values on real harmonics.The real-form coefficient condition is a_{2l−k} = (−1)^{l+k} conjugate(a_k), not ordinary real coefficients.Source: §3.3 · pp. 11, 13
• Every nonzero real harmonic is the harmonic part of a product of l real linear forms, with the corresponding l Maxwell axes determined up to order and multiplicity.On the sphere, these axes are the antipodal root pairs of the associated binary form.Source: §3.3 · p. 13
• For degree-three real harmonics, exact arithmetic verifies that the Igusa invariants J2, J4, J6, and J10 are rational, while the degree-15 skew invariant R is a rational multiple of i.The passage relates this behavior to the different real form arising from degree-three real harmonics and to reflection parity.Source: §3.3 · p. 13

3.4. The harmonic–binary-form correspondence.

• The map f ↦ f ◦ v identifies the degree-l harmonic space with binary forms of degree 2l and intertwines the corresponding group actions.The substitution from homogeneous polynomials onto binary forms has kernel qP_{l−2}(C3); its restriction to harmonic polynomials is an isomorphism.Source: §3.4 · p. 14
• For l=1, ∥f1∥2 = disc(Qa)/12; for l=2, ∥f2∥2 = I(Ff2)/15; for l=3, J2(Ff3) = 2100 ∥f3∥2.J2 is positive on every nonzero real band-three harmonic; band norms are phased quadratic invariants, while other invariants refine the power spectrum.Source: §3.4 · p. 15
• Every nonzero real degree-l harmonic corresponds to a binary form factoring into l real quadratic forms, whose roots map to l antipodal pairs on S2.Conversely, every effective antipodally invariant divisor of degree 2l arises from a nonzero real harmonic, unique up to a nonzero real scalar.Source: §3.4 · p. 15
• Rotations carry the roots of the binary form to the rotated roots on S2, while inversion multiplies the form by (−1)^l.Thus the correspondence tracks the rotation action and the parity of each band.Source: §3.4 · p. 16
• Composing with Ψ transfers the invariant ring and rotation-orbit equivalence; invariants of multidegree (dl) change by (−1)^p under reflection and t^w under dilation.Here p = Σl dl and w(I) is the invariant's weight.Source: §3.4 · p. 16
• The shape quotient removes one positive scale common to all bands; independently rescaling binary forms can lose amplitude, relative band normalization, and the sign of an even band.A form in the SL2(C) orbit of Ff need not be σl-fixed; rotations correspond to SU(2) substitutions.Source: §3.4 · p. 17
• Joint invariants of (Ff0, . . . , FfL) are needed to record relative band orientations, which invariants of separate forms miss.The tuple consists of binary forms of degrees 0, 2, . . . , 2L.Source: §3.4 · p. 17

4. Invariants and Separation

• A generating set of phased real invariants separates SO(3)-orbits, and its Hilbert map identifies the quotient with a closed semialgebraic image; separation, not generation, is the key requirement.The invariant map vanishes only at the zero descriptor, and any separating subset also suffices for orbit distinction.Source: §4 · p. 18

4.1. The coordinate model.

• The coordinate model represents bands as double binary forms; its invariants separate two-sided SO(3)×SO(3) orbits, while real invariants are mirror-invariant and imaginary ones are mirror pseudo-invariants.Agreement of these invariants at every truncation implies that the surfaces differ by an orientation-preserving rigid motion; higher-dimensional quotient structure can require more invariants than its dimension suggests.Source: §4.1 · pp. 19–20

4.2. Scale.

• Positive scaling acts on each invariant according to its degree, so weighted projectivization removes scale while retaining a separate size variable and preserving pseudo-invariant signs.The weighted sup-gauge gives a canonical normalized representative, and bounded normalized coordinates diagnose approach to an invariant’s zero set.Source: §4.2 · p. 20

4.3. Counting.

• Band-wise invariants leave a dimension deficit of 3L−4 for L≥2, which joint invariants must capture through the relative orientations of bands.For the coordinate model, the quotient dimension is 3(L+1)^2−9; at L=6 the stated counts are 46 for the radial quotient and 138 for the coordinate quotient.Source: §4.3 · p. 21

4.4. Band-by-band invariants.

• For V2, seven weighted homogeneous invariants of degrees 1, 2, 2, 3, 3, 4, and 6 separate SO(3)-orbits, with the degree-six pseudo-invariant’s sign encoding chirality.The invariants form a complete scale-free descriptor and realize the quotient as a closed semialgebraic subset of the hypersurface defined by the squared pseudo-invariant relation.Source: Degree two with the lower bands: a… · pp. 22–23
• For a band-three shape, the weighted Igusa invariants [J2 : J4 : J6 : J10] classify up to rotation, scale, and reflection; adjoining the skew pseudo-invariant restores chirality-sensitive completeness.The even invariants are real and the skew invariant is purely imaginary; when J10 is nonzero, the even-invariant point is also the genus-two curve’s moduli point.Source: Degree three · p. 24, The locus L2 and the square of the… · p. 25

4.4. Band-by-band invariants.

• Bispectral invariants are transvectants; they are real when l1 + l2 + l3 is even and pseudo-invariants when it is odd.Source: Joint invariants · p. 25
• A cubic pseudo-invariant requires three distinct bands with odd sum, first occurring as B234; degree-four pseudo-invariants already occur at L = 3.When two bands coincide, Bl1ll vanishes for odd l1; an example at L = 3 is P123 = ((F1, F2)1, (F3, F3)4)4.Source: Joint invariants · p. 25
• Invariants of degree at most three do not separate in general; for a single band Hl with l ≥ 3, invariant quadratics are multiples of the norm and cubics have dimension at most one.Source: Joint invariants · p. 25

Degree-three harmonics

• Figure 3 contrasts coplanar and bisector-type achiral axes, both with R(Ff) = 0, against a generic chiral configuration.In the bisector case, the half-turn exchanges two axes and reverses the third.Source: Figure 3 · p. 26

Chirality criterion

• A reflected radial descriptor lies in the original SO(3)-orbit exactly when every pseudo-invariant vanishes.These pseudo-invariants detect whether a shape is congruent to its mirror image.Source: §5 · pp. 25–26

Other band combinations

• Every descriptor supported on even bands, and every band-one descriptor, is achiral; mixed bands one and two are achiral exactly when det(a, Qa, Q2a) = 0.For distinct eigenvalues of Q, this means a lies in a plane spanned by two eigenvectors of Q.Source: §5 · p. 26

Degree-three harmonics

• For a nonzero degree-three harmonic, achirality is equivalent to R(F) = 0 and to a half-turn reversing the harmonic.The equivalent Maxwell-axis configurations are coplanar axes or one axis orthogonal to a bisector of the other two.Source: §5 · pp. 26–27
• When J10(F) ≠ 0, the degree-three achirality criterion is also equivalent to the associated genus-two curve having an elliptic involution.The paper gives a practical alternative test using the degree-30 real polynomial F2 in O(3)-invariants, avoiding evaluation of R and complex arithmetic.Source: §5 · pp. 27–28

Anatomical comparisons

• For anatomical comparisons, homologous left and right structures are only approximately related by reflection, so one side must first be reflected into a common handedness before comparing pseudo-invariants.Otherwise, raw pseudo-invariants confound pathological asymmetry with laterality.Source: §5 · p. 29

Radial model

• For radial descriptors with a quadrupole having distinct eigenvalues, an ordered eigenframe leaves only the Klein four-group K, and K-invariant monomials of degree at most three separate orbits.The aligned coordinates number dim VL − 3, the dimension of the quotient.Source: §6 · p. 29

Coordinate model

• For coordinate descriptors with distinct singular values in C1, the signed SVD frame leaves a diagonal K-action, and K-invariant monomials of degree at most three separate orbits.The aligned coordinates number dim WL − 6, the dimension of the quotient.Source: §6 · pp. 29–30

Strata and stability

• On the open stratum, aligned tuples are real-analytic, but the frame class has no continuous extension to the complement.Near symmetry strata, frame conditioning is governed by the reciprocal eigenvalue or singular-value gap.Source: §6 · pp. 30–31

Relation to SPHARM-PDM

• The coordinate-model frame formalizes first-order-ellipsoid alignment in SPHARM-PDM, while identifying and quotienting the residual finite group instead of fixing it by convention.The paper also makes explicit that the conformal parameterization makes the frame a function of the surface, not optimizer initialization.Source: §6 · p. 31

Global alternative

• Global polynomial invariants can separate orbits continuously using 2 dim V + 1 invariants, at the cost of roughly doubling dimension and using generic combinations of high-degree generators.The frame approach instead supplies dim(V/G) coordinates on the principal stratum, but is discontinuous at symmetry strata.Source: §6 · p. 31

7.1. The orbit distance.

• The orbit distance is the smallest coefficient-space distance after nuisance rotations, and for band-one coordinate descriptors it equals the Euclidean distance between signed singular-value vectors.For band one in the radial model, it reduces to the difference of coefficient norms.Source: §7.1 · pp. 32–33

7.2. Stability in both directions.

• Invariant coordinates vary Lipschitz-continuously with orbit distance, and separating invariants have a Hölder-continuous inverse on compact sets; locally, a full-rank slice Jacobian gives a bi-Lipschitz embedding.The Hölder exponent need not be one; near zero, quadratic invariants can make orbit distance scale as the square root of invariant discrepancy.Source: §7.2 · pp. 33–34
• The coefficient-to-invariant stability results do not quantify stability under mesh perturbations, which remains open for the surface-to-coefficient stage.The uniformizing map and centered parameterization depend continuously on the metric, but a quantitative mesh-perturbation statement is left open.Source: §7 · p. 31, §7.2 · p. 35

7.3. The certified descriptor.

• The certified descriptor is parameterization-independent, complete for the truncated expansion modulo rotations, and records mirror handedness through pseudo-invariants.Its invariants are Lipschitz on bounded sets, while on compact sets the orbit distance is bounded by a Hölder power of invariant discrepancy.Source: §7.3 · pp. 35–36

8.1. From a mesh to a centered parameterization

• The discrete conformal map must be a flipped-triangle-free map of a connected, closed, oriented genus-zero mesh; its residual freedom is Möbius up to discretization error.The mesh is assumed embedded without self-intersections and free of degenerate faces.Source: §8.1 · p. 36
• Centering uses the unique discrete conformal barycenter when every vertex weight is below 1/2, then leaves only a sphere rotation for the invariants to remove.The implementation checks the weight condition, computes the barycenter, and records the residual after recentering.Source: §8.1 · p. 36

8.2. Harmonic coefficients

• Harmonic coefficients use round-sphere quadrature weights rather than the mesh area measure, with truncation degree guided by the residual and the stated error bound.The quadrature matrix’s deviation from identity, the constant coefficient, and the maximum face-to-image area ratio are also reported.Source: §8.2 · pp. 36–37

8.3. Binary forms and invariants

• Binary-form coefficients and invariants are computed with independent polynomial checks, while frame coordinates are used only when spectral gaps exceed a threshold.Floating-point imaginary or real residuals provide error checks for the invariants; the transvectant sequence is verified in exact rational arithmetic.Source: §8.3 · p. 37

8.4. Normalization and diagnostics

• The diagnostic suite checks centering, quadrature, truncation, spectral gaps, normalization, and an end-to-end orbit-distance test on independently processed equivalent meshes.The end-to-end distance should be zero exactly; if its computed value fails, the earlier diagnostics help locate the error.Source: §8.4 · p. 37

8.5. Verified identities

• Independent verified identities cover conventions, the harmonic-to-binary-form correspondence, parity, and invariants in degrees two and three.The checks use exact rational arithmetic where possible and machine precision otherwise, with fixed-seed random inputs.Source: §8.5 · pp. 37–38

8.6. A synthetic end-to-end test

• The synthetic mesh copies have exact orbit distance zero and computed distance 2·10−8; discretization error exceeds quotient error by four orders of magnitude.The invariants agree to the same order, while the coarser mesh has normalized orbit distance 1.1·10−4 and invariant differences up to 2·10−2 in the dipole invariants.Source: §8.6 · pp. 38–39, Table 2 · p. 39

9.1. Discrimination

• The seven invariants and frame coordinates classify all four dipole–quadrupole classes without error at σ = 0 and above 0.97 accuracy at σ = 0.02, while power spectrum and bispectrum remain at chance.O(3)-invariants reach one half, separating A from B but not either class from its mirror; Figure 4(a) plots accuracy against coefficient noise.Source: §9.1 · p. 40, Figure 4 · p. 40

9.2. Frames near the degenerate strata

• Near an axial stratum, the frame ratio grows like 0.65/g, whereas the polynomial-invariant ratio stays between 4.2 and 5.7 over four decades of g.The frame’s fitted slope is −1.00 over 10−4 ≤ g ≤ 10−2; the authors prefer polynomial invariants when spectral gaps are uncontrolled.Source: §9.2 · p. 41

9.3. Stability exponents

• Invariant discrepancy has fitted slopes 0.96 at a principal point, 1.97 at an SO(2)-stabilized point, and 2.00 at the origin.The invariants are locally bi-Lipschitz at the principal point; at the other points the quadratic discrepancy implies inverse Hölder exponent at most 1/2 nearby.Source: §9.3 · p. 41

9.4. Paired anatomical structures

• For seven valid brain and gonadal pairs, reflected-right distances are 0.001 to 0.009 versus left–right distances of 0.12 to 0.34; kidney and adrenal pairs instead remain different shapes after reflection.The experiment uses one atlas subject and makes no anatomical claim; two structures were rejected by diagnostics, and two others had initialization-sensitive optimization error.Source: §9.4 · pp. 41–43

9. Benchmarks on Harmonic Coefficients

• The experiments find polynomial invariants complete and chirality-recording, with uniform Lipschitz stability; their inverse is Lipschitz on the principal stratum and Hölder of exponent 1/2 where stabilizers jump.On real anatomy the invariants separate mirror-image pairs from asymmetric pairs, while diagnostics identify surfaces whose conformal parameterization cannot be trusted.Source: §9.4 · p. 43

• The construction formulates each nuisance freedom as a group action, proving completeness through degree two, characterizing degree-three chirality, and establishing metric and stability results.It also constructs separating frame coordinates on the principal strata and makes the residual Klein four-group explicit.Source: §10 · p. 43
• Unlike SPHARM-COM’s area-preserving coefficients, the descriptor quotients the remaining freedoms; its radial model is complete through degree two with seven invariants.The sextic invariants and ˆR complete band three as an isolated band, while the coordinate model retains direction information and is faithful in the all-band limit.Source: §10 · p. 43
• A cohort-level quantitative comparison is left to the SPHARM-COM manuscript and future joint work.Source: §10 · p. 44
• Three problems remain open: a global polynomial separating set for WL, the sharp Hölder exponent and its stratum dependence, and the geometry of the real band-three locus.The authors caution against assigning anatomical meaning to such loci before empirical validation.Source: §10 · p. 44
• The pipeline is validated on synthetic surfaces and a single atlas subject; learning with its fixed input layers and mesh-level anatomical validation are next steps.The computed class is independent of parameterization, position, and scale.Source: §10 · p. 44

Funding.

• The authors report receiving no organizational support for the submitted work.Source: Funding · p. 44

Competing interests.

• The authors declare no competing interests relevant to the article; ethics approval is not applicable.Source: Competing interests · p. 44

Data availability.

• The synthetic surface is defined in the text, and the anatomical meshes are publicly available BodyParts3D version 3.0 files.Source: Data availability · p. 44

Code availability.

• Supplementary materials include scripts for verification, synthetic tests, benchmarks, and the mesh pipeline.Source: Code availability · p. 44