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2D moment invariants from the point of view of the classical invariant theory

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Pith's one-line read Image moment invariants are exactly joint SO(2)-invariants of binary forms, and solving one linear derivation gives the complete minimal generating sets: 14, 65, and 562 invariants for orders 3, 4, and 5.

desk verdict Useful so2/Kravchuk machinery and explicit polynomial generators, but Theorem 7's rational-invariant set has a fixable index typo and the generation proof is incomplete. read the letter →

arxiv 1908.08927 v2 pith:QDQT6Q45 submitted 2019-08-20 cs.CV

Leonid Bedratyuk This is my paper · ORCID
classification cs.CV MSC 13A5015A72
keywords momentinvariantsbinaryformsSO(2)-invariantsLiealgebraactionKravchukpolynomialsHilbertbasisclassicalinvarianttheorypatternrecognition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the search for 2D rotation moment invariants can be recast exactly as a classical invariant theory problem: the algebra of polynomial (or rational) moment invariants of order up to d is isomorphic to the algebra of joint SO(2)-invariants of binary forms of degrees 2 through d. Because SO(2) is one-dimensional, the group action can be replaced by a single derivation D, and all invariants are just the kernel of this derivation. This yields minimal polynomial generating sets: 14 invariants for d=3, 65 for d=4, and 562 for d=5. It also yields an explicit minimal rational invariant set of size dim W_d - 1 for every d. If correct, this unifies moment-based image classification with the classical theory of binary forms.

What carries the argument

The central object is the direct sum W_d = V_2 oplus V_3 oplus ... oplus V_d of binary-form spaces and the algebra C[W_d]^so2 of joint invariants under the one-dimensional Lie algebra action generated by the derivation D. The construction diagonalizes D: its eigenvalues are {-di, -(d-2)i, ..., di}, and its eigenvectors are e_d(+-si) = sum_j i^j K_j((d -/+ s)/2, d) a_{d-j,j}, where K_j are binary Kravchuk polynomials. This turns the search for invariants into finding monomials with zero total eigenvalue, and Hilbert bases of the resulting monoid give minimal polynomial generating sets. For rational invariants, a transcendence-basis argument produces the smaller explicit set $G^{{(d)}}$_{p,q}.

What would settle it

Compute the derivative with respect to $\theta$ at $\theta$=0 of the rotated normalized central moment for eta_{1,1}; if the result is not eta_{2,0} - eta_{0,2}, then the isomorphism in Theorem 1 fails. This can be checked symbolically from eta_{p,q}=mu_{p,q}/mu_{0,0}^{(p+q)/2+1} or numerically on a simple compact image.

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Extended reading notes

Core claim

Theorem 1 asserts that C[eta]^SO(2)_d is isomorphic to C[W_d]^so2 and C(eta)^SO(2)_d is isomorphic to C(W_d)^so2, where W_d is the direct sum V_2 oplus V_3 oplus ... oplus V_d of binary-form spaces. The proof passes from the rotation group to its Lie algebra, so that invariant construction becomes solving D(f)=0 for the diagonalizable derivation D(eta_{p,q}) = q eta_{p+1,q-1} - p eta_{p-1,q+1}. The paper proves that D has spectrum {-di, -(d-2)i, ..., di} and eigenvectors expressible through binary Kravchuk polynomials, and then uses Hilbert bases of a monoid to obtain minimal generating sets of polynomial invariants. For rational invariants, it proves that a minimal generating set has exactly dim W_d - 1 elements and gives the explicit set $G^{{(d)}}$_{p,q}. The final result confirms and generalizes the known minimal rational invariant set for rotation moment invariants.

Load-bearing premise

The load-bearing premise is that normalized central moments rotate according to the quoted formula D(eta_{p,q}) = q eta_{p+1,q-1} - p eta_{p-1,q+1}, which the paper takes from the literature rather than deriving from the definition.

Editorial extensions

If this is right

  • For every order d there is a finite set of polynomial invariants whose equality on two images forces the images into the same orbit of the translation-rotation-scale group.
  • Minimal polynomial generating sets can be computed mechanically as Hilbert bases of a monoid; the paper carries this out for d=3, 4, and 5, giving 14, 65, and 562 generators.
  • The rational invariants are even simpler: a minimal generating set of size dim W_d - 1 exists for every d, and the paper writes it down explicitly.
  • The Cayley-Sylvester analogue gives the number of independent homogeneous invariants of each degree as the count of non-negative integer solutions of a linear Diophantine system, with a closed Poincare-series integral.
  • The eigenvectors of the derivation are Kravchuk-polynomial combinations, and complex moments are exactly these eigenvectors.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Lie-algebra eigenvector method could be applied to other one-parameter subgroups of GL(2), such as shear or anisotropic scale, producing analogous Hilbert-basis generator problems.
  • Representing an invariant in the Hilbert basis gives its monomial coefficients, so sensitivity of moment invariants to image noise could be studied directly in this monomial model.
  • Because the rational invariant set is much smaller than the polynomial minimal sets, a practical large-d classifier could use rational invariants for the actual features and polynomial invariants as a completeness certificate.
  • A direct symbolic derivation of the rotation formula D(eta_{p,q}) = q eta_{p+1,q-1} - p eta_{p-1,q+1} from the definition of normalized central moments would make the whole isomorphism self-contained.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a bridge between the classical theory of 2D moment invariants and the classical invariant theory of binary forms. The author defines the algebras C[η]_d^{SO(2)} and C(η)_d^{SO(2)} of simultaneous polynomial and rational rotation moment invariants of order up to d, and claims in Theorem 1 that these are isomorphic to the joint SO(2)-invariant algebras C[W_d]^{so_2} and C(W_d)^{so_2}, where W_d = V_2 ⊕ ⋯ ⊕ V_d. The proof proceeds by passing to the Lie algebra action, diagonalizing the resulting derivation D with the help of Kravchuk polynomials, and then solving the invariant condition as a monomial-weight equation. The paper gives explicit minimal generating sets of the polynomial invariant algebra for d=3,4,5 (14, 65, and 562 generators, respectively), proves a Cayley–Sylvester-type counting formula and a Poincaré-series formula, and claims in Theorem 7 a minimal generating set of the rational invariant field C(W_d)^{so_2} of size dim W_d − 1, confirming Flusser's earlier result.

Significance. If the technical gaps are repaired, the paper would be a useful contribution: it gives a conceptually clean reduction of rotation moment invariants to classical invariant theory, produces explicit and reproducible Hilbert-basis computations for low orders, and provides a parameter-free derivation with no fitted constants. The use of Kravchuk polynomials to diagonalize the rotation action is elegant, and the small cases d=2,3,4 are checked in detail. The claimed rational-invariant generating set, however, is the central output of the paper, and the stated form of Theorem 7 is internally inconsistent with Example 6.2; this must be fixed before the main claim can be accepted.

major comments (3)
  1. [Section 6, Theorem 7] The displayed generating set G(d)_{p,q} imposes the condition q ≠ s in the third family e_n(si)^q e_p(−qi)^s. For d=5, p=3, q=1, this excludes the invariant e_5(i)e_3(−i) = x_{51}y_{31}, whose weight is i − i = 0. That invariant is nevertheless listed as β_{11} in Example 6.2, and the example's set has 17 elements, the required dim W_5 − 1, whereas the theorem's stated set has only 16 elements. The correct exclusion is not 'q ≠ s' but '(n,s) = (p,q)': the product e_n(si)e_p(−qi) with n ≠ p and s = q is an independent invariant, while e_p(qi)^q e_p(−qi)^q is already generated by e_p(qi)e_p(−qi). Since Theorem 7 is the statement that supplies the minimal generating set of the rational invariant field, the paper's central claim is not supported by the theorem as written.
  2. [Section 6, proof of Theorem 7] In the auxiliary derivation proof, the first integrals are written as x_k^{λ_q} y_q^{λ_k}. But for the actual derivation D one has λ_j = i s_j, so these expressions involve complex powers such as x_k^{i q} y_q^{i s}, which are not elements of the rational function field C(W_d) and are not rational invariants. The proof can be repaired by applying the scalar multiple D' = −iD, which has the same kernel and integer weights, but as written the algebraic-independence and first-integral arguments do not apply to the stated rational monomials. The proof should also state explicitly that the constructed first integrals form a complete set of first integrals, not merely an algebraically independent set, since algebraic independence plus the transcendence-degree count alone does not by itself show that the listed monomials generate the whole invariant field.
  3. [Section 2, Theorem 1] The proof of Theorem 1 concludes the isomorphism of invariant algebras from the fact that the moment action and the binary-form action satisfy 'the same partial differential equation'. This is too terse for a foundational theorem: the author should explicitly define the algebra isomorphism φ: C[η]_d → C[W_d] by η_{p,q} ↦ a_{p,q}, verify that φ intertwines the two derivations, and then justify the equality of the fixed subrings with the kernels for both the polynomial and the rational cases. The intended argument is straightforward, but the current wording leaves the isomorphism asserted rather than proved.
minor comments (5)
  1. [Section 2, Theorem 1 proof] The summation index in the formula 'd∑_{p+q=2} (qη_{p+1,q−1} − pη_{p−1,q+1}) ∂/∂η_{p,q}' should be over all pairs with 2 ≤ p+q ≤ d, not only the pair p+q=2.
  2. [Section 6, Theorem 7] The sentence 'The the set of dim Wd−1 invariants' contains a duplicated article and should read 'The set of dim W_d − 1 invariants'.
  3. [Abstract] The phrase 'This allow us' is ungrammatical and should be 'This allows us'.
  4. [Section 4, Example 4.4] The text contains the Cyrillic-looking string 'СоСоA'; this should be the ASCII name 'CoCoA' for consistency with reference [28].
  5. [References] References [14], [25], and [31] are the same Flusser paper and should be merged to avoid duplicate citations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and anchored in standard invariant theory and external references.

full rationale

The paper's central claim is an isomorphism between algebras of moment invariants and algebras of SO(2)-invariants of binary forms, with explicit generating sets for small orders. The load-bearing input is the rotation action on normalized moments, quoted from Teague [6] and Prokop/Reeves [10], which is an external anchor rather than a consequence of the paper's own target claims. The eigenvectors of the derivation D are derived in the paper from the Sylvester matrix and Kravchuk polynomials, not assumed from the later invariant lists. The polynomial generating sets for d = 3, 4, 5 are obtained by computing Hilbert bases of monoids with CoCoA, a computation independent of the announced counts; the rational invariant count dim W_d - 1 follows from the standard transcendence-degree theorem for rational invariants of a one-dimensional group, not from the proposed generating set. There are no fitted constants, no self-citations, and no author-imported uniqueness theorem; the comparison with Flusser's result is post-hoc confirmation, not an input. The internal inconsistency in Theorem 7's condition q ≠ s versus Example 6.2 is a correctness or completeness concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters enter the derivation, so the free parameters list is empty. The main imported inputs are standard theorems of invariant theory plus the SO(2) action law on normalized central moments, which is taken from the image-moments literature rather than derived here. No new particles, forces, dimensions, or ad hoc entities are postulated.

assumptions (5)
  • standard math For a connected Lie group G acting on polynomial functions, the invariant algebra equals the kernel of the Lie algebra derivations: C[V]^G = C[V]^g.
    Invoked in Section 2 to replace SO(2) by its Lie algebra so2; standard for connected Lie groups with rational actions.
  • standard math The invariant monomials of a diagonalizable derivation form a monomial algebra whose minimal generating set is the Hilbert basis of its exponent monoid, and Gordan's lemma ensures finite generation.
    Used in Theorem 3 and the examples to compute polynomial invariant algebras via CoCoA.
  • standard math For a one-dimensional group acting rationally, the transcendence degree of the rational invariant field equals dim W_d - dim SO(2), so C(W_d)^SO(2) has transcendence degree dim W_d - 1.
    Used in Theorem 6 to fix the size of the minimal rational generating set; standard invariant theory.
  • domain assumption Normalized central moments transform under rotation according to the action formula quoted in the proof of Theorem 1 from references [6] and [10].
    This is the central bridge from image moments to binary forms; it is accepted from prior literature without derivation in this paper.
  • domain assumption Images are real, piecewise continuous, compactly supported functions, so geometric moments and their normalizations are well-defined.
    Assumed in the Section 1 setup; this is the standard image model for moment methods.

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Pith. "Pith review of 2D moment invariants from the point of view of the classical invariant theory." pith.science (2026). https://pith.science/paper/QDQT6Q45

@misc{pith2026190808927,
  author       = {Pith},
  title        = {Pith review of: 2D moment invariants from the point of view of the classical invariant theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QDQT6Q45}},
  note         = {Machine review of arXiv:1908.08927}
}
abstract

Invariants allow to classify images up to the action of a group of transformations. In this paper we introduce notions of the algebras of simultaneous polynomial and rational 2D moment invariants and prove that they are isomorphic to the algebras of joint polynomial and rational $SO(2)$-invariants of binary forms. Also, to simplify the calculating of invariants we pass from an action of Lie group $SO(2)$ to an action of its Lie algebra $\mathfrak{so}_2$. This allow us to reduce the problem to standard problems of the classical invariant theory.

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