Geometric Moments and Their Invariants -...

J Math Imaging Vis (2012) 44:223–235DOI 10.1007/s10851-011-0323-x

Geometric Moments and Their Invariants

Mark S. Hickman

Published online: 30 December 2011© Springer Science+Business Media, LLC 2011

Abstract Moments and their invariants have been exten-sively used in computer vision and pattern recognition.There is an extensive and sometimes confusing literature onthe computation of a basis of functionally independent mo-ments up to a given order. Many approaches have been usedto solve this problem albeit not entirely successfully. In thispaper we present a (purely) matrix algebra approach to com-pute both orthogonal and affine invariants for planar objectsthat is ideally suited to both symbolic and numerical com-putation of the invariants. Furthermore we generate basesfor both systems of invariants and, in addition, our approachgeneralises to higher dimensional cases.

Keywords Orthogonal and affine transformations ·Moments · Invariants · Covariants · Image recognition

1 Introduction

Moments are quantities that characterise the “shape” of anobject. That object may be a physical shape, an image orsomething more abstract like a statistical distribution. Letthe object of interest by described by ρ : R

n → Rm; that is a

vector-valued function defined on Rn. We define a geometric

moment of ρ by

ma =∫

xaρ(x) dΩ (1)

M.S. Hickman (�)Department of Mathematics & Statistics, University ofCanterbury, Private Bag 4800, Christchurch 8140, New Zealande-mail: [email protected]

where a = (a1, a2, . . . , an) is a multi-index (with non-negative components),

xa ≡ xa11 x

a22 · · ·xan

n

and

dΩ = dx1 ∧ dx2 ∧ · · · ∧ dxn

is the volume form on Rn. The order of the moment is given

by

|a| ≡ a1 + a2 + · · · + an.

We assume that ρ has compact support D and is bounded sothat the moments (1) are finite.

In the application of shape recognition of a planar objectthen n = 2 and ρ is usually a scalar valued function (greyscale image) or takes values in R

3 (RGB image). Thus

ma,b =∫∫

xaybρ(x, y) dx ∧ dy

is a moment of order a + b. It has a single component inthe case of a grey scale image. For RGB images we willconsider that there are three separate moments. Thus, forexample, the red component is given by

mRa,b =

∫∫xaybρR(x, y) dx ∧ dy

and so we will always consider a moment to be a scalarquantity.

These moments were introduce to pattern recognition bythe seminal paper of Hu [7]. In this paper he consider planarimages and introduced the a set of 7 rotational invariantsthat were constructed from moments up to order 3. This set

mailto:[email protected]

224 J Math Imaging Vis (2012) 44:223–235

has had a rather checkered history (see [5] for a brief accountof the history of Hu’s invariants).

The motivation for Hu’s paper was to construct combina-tions of moments that are unchanged under a rotation. Wewill consider any group G with a linear action on R

n; that isthe group acts on R

n via a matrix A

x = Ax. (2)

Clearly the standard actions of both the orthogonal O(2) andaffine GL(2) groups on R

2 are linear (they are also linearin higher dimensions). By contrast, the projective action ofGL(3) on R

2 is not. The choice of a linear action may seemto be excessively restrictive in that it does not allow transla-tions. However as we will see in Sect. 5, translations may beinvariantly normalised.

There has been a number of approaches in the literatureto the construction of orthogonal (rotational) and affine mo-ment invariants. The methods employed have ranged fromthe use of Fourier-Mellin transforms [11, 22] to algebraicinvariants [9, 19, 20, 23], orthogonal polynomials [2, 12, 16,17, 21], complex moments [1, 3, 4] and graphical methods[14]. In this paper, inspired by the theory of binary invari-ants [6, 13], we develop a purely matrix algebraic approachto obtain both orthogonal and affine invariants.

In Sect. 2 we will examine the transformation of themonomials xa under the action (2). In the following section,the action on moments is discussed. In Sect. 4 moment co-variants and invariants are introduced. Invariants for the or-thogonal group (and its conformal generalisation) are con-structed in Sect. 6. The affine case is examined in Sect. 7. Inthe final section we make some concluding remarks.

2 Monomials

Given a group G with a linear action on R2; that is the group

acts via a matrix A we wish to compute the induced actionon the monomial xayb . Our analysis is valid in any dimen-sion and so for this and the next section we work in R

n.We begin with a definition of the Kronecker product [18,p. 1867] (see [10, Chap. 13] for a recent discussion).

Definition 1 Let R be m × n matrix and S be an arbitrarymatrix. The Kronecker (or direct or tensor) product of R andS is the matrix given by

R ⊗ S ≡

⎡⎢⎢⎢⎣

R11S R12S · · · R1nS

R21S R22S · · · R2nS...

.... . .

...

Rm1S Rm2S · · · RmnS

⎤⎥⎥⎥⎦ .

The importance of this construction lies in the followingobservation.

Proposition 1 Let A,B,X and Y be matrices such that AX

and BY are defined. Then

(A ⊗ B)(X ⊗ Y) = AX ⊗ BY.

In addition

AT ⊗ BT = (A ⊗ B)T

and

A−1 ⊗ B−1 = (A ⊗ B)−1

whenever the inverses exist.

Proof We have

(A ⊗ B)(X ⊗ Y)

=⎡⎢⎣

A11B A12B · · ·A21B A22B · · ·

......

. . .

⎤⎥⎦⎡⎢⎣

X11Y X12Y · · ·X21Y X22Y · · ·

......

. . .

⎤⎥⎦

=⎡⎢⎣

(AX)11BY (AX)12BY · · ·(AX)21BY (AX)22BY · · ·

......

. . .

⎤⎥⎦

= AX ⊗ BY.

Next we observe

AT ⊗ BT =⎡⎢⎣

A11BT A21B

T · · ·A12B

T A22BT · · ·

......

. . .

⎤⎥⎦= (A ⊗ B)T .

Finally note that

(A−1 ⊗ B−1)(A ⊗ B) = (A−1A) ⊗ (B−1B)

= I ⊗ I = I. �

Thus there is an induced action on X ⊗ Y given by A ⊗ B .Let x(1) = x be the monomials of degree 1 in R

n. Define

x(d) =⊗

d

X(1).

For example, in R2,

x(2) = x(1) ⊗ x(1) =

⎡⎢⎢⎣

x2

xy

yx

y2

⎤⎥⎥⎦ , x(3) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

x3

x2y

xyx

xy2

...

y3

⎤⎥⎥⎥⎥⎥⎥⎥⎦

are the (with some redundancy) monomials of degree 2and 3.

J Math Imaging Vis (2012) 44:223–235 225

Proposition 2 Let G act linearly on Rn. Then the action of

A ∈ G on the monomials of degree d is given by

Ad =⊗

d

A;

that is

x(d) = Adx(d).

Proof The action of A on the monomials of degree d isgiven by

Ax(1) ⊗ Ax(1) ⊗ · · · ⊗ Ax(1)︸︷︷︸d factors

=(⊗

d

A)(⊗

d

x(1)

)

by Proposition 1. �

In [15] higher order monomials are constructed as matri-ces from outer products of lower order monomials; that isorder c + d monomials are initially given by x(c)xT

(d). How-ever these authors rely on a specific variable ordering andnormalisation of the monomials in order to obtain a trans-formation matrix that will be orthogonal if A is orthogonal.

The Kronecker product is implemented in both symbolicand numerical mathematical packages. For example, it isgiven by KroneckerProduct in both the LinearAl-gebra package of MAPLE and in MATHEMATICA, and bykron in MATLAB.

3 Transformations of Moments

Under a linear transformation A, the zeroth order momentwill transform

m0,0 = detA∫∫

ρ(Ax) dx ∧ dy.

Ideally we wish that moments transform under a so-calledmultiplier representation [13]; that is

m0,0 = (detA)qm0,0 (3)

for some q . This will not happen unless ρ has a very spe-cific form. In particular, if ρ is the characteristic functionthen we get the desired transformation law (with q = 1). Ofcourse ρ = 1 is not a particularly interesting case for patternrecognition. However we obtain a moment that transformsunder a multiplier representation if we consider the parts ofthe image where the value of ρ is in a chosen range.

Definition 2 For α > 0, let

ρ(k)(x) ={

1 if k ≤ ρ(x) < k + α,

0 otherwise.

The threshold moments based on ρ(k) are given by

m(k)a =

∫xaρ(k)(x)dΩ =

∫Dk

xadΩ

where Dk is the support of ρ(k).

In practice, ρ will take only a discrete range of values.For each of these values (fixing α) we have a complete setof threshold moments. For example, if the image of interestis a 8-bit grey scale image and we choose α = 1 then eachof the 256 threshold moments

m(k)0,0 =

∫∫Dk

dx ∧ dy

amounts to counting the number of pixels whose densityis k. All of these moments will transform under (3) withq = 1. Therefore any weighted sum of these threshold in-variants will also transform under (3) with q = 1.

Remark The level sets of ρ may not be closed curves. Forexample, consider the unit square with ρ(x, y) = x. Thelevel sets of ρ are line segments x = k, 0 ≤ y ≤ 1. This isthe reason why, from a theoretical point of view, we do notchoose threshold moments based on ρ(x) = k.

We define a “duality”

∗ : xa ↔ ma

between monomials of degree d and moments of order d

where d = |a|. Let

M(d) = ∗x(d).

For example, in R2,

M(2) =

⎡⎢⎢⎣

m2,0

m1,1

m1,1

m0,2

⎤⎥⎥⎦ .

For convenience we define

M(0) = [m0], A0 = [1].

Proposition 3 Under a linear transformation A, we have

M(d) = (detA)Ad M(d)

where the components of M(d) are threshold moments.

The proof of this proposition follows immediately fromProposition 2 and the discussion above. In the light of thisresult, all moments discussed in the rest of this paper will beunderstood to be threshold moments.


Remark In our definition of moments we used the volumeform dΩ rather than the volume “element” dx1dx2 · · ·dxn;that is, we used the signed volume rather than the unsignedvolume. If the unsigned volume is used then Proposition 3becomes

M(d) = |detA|Ad M(d).

While this choice makes no difference for the special or-thogonal group SO(n) (that is, the group of rotations) or thespecial affine group SL(n), it does make a difference for thefull orthogonal group O(n) (rotations plus reflections) andthe general affine group GL(n).

4 Moment Invariants and Covariants

A moment invariant is a combination of moments that is un-changed under a group transformation. Formally we have

Definition 3 A (rational) function ι of moments up to orderd is said to be an (rational) invariant of order d under agroup action G if

ι(M(0), . . . , M(d)) = (detA)qι(M(0), . . . , M(d)) (4)

for all A ∈ G. q is called the weight of the invariant ι. Inparticular if q = 0 then ι is called an absolute invariant. Ifq = 0 then ι is called a relative invariant.

Clearly product of invariants are also invariants and sumsof invariants of the same weight are invariants. Thereforeabsolute invariant may be construct from two relative invari-ants ι1, ι2 of weights q1 and q2 respectively by

ιq21 ι

−q12 .

For this reason, we restrict our attention to rational invari-ants.

The “volume” moment m0 is trivially an invariant ofweight 1. For the orthogonal group, a less trivial exampleis given by MT

(d)M(d) since

MT

(d)M(d) = MT(d)A

Td Ad M(d) = MT

(d)M(d)

(it follows from Proposition 1 that, if A is orthogonal thenAd is also orthogonal). This absolute invariant will be aquadratic polynomial in order d moments.

We extend this definition to allow dependency of the vari-able x (see, for example, [6, 13]).

Definition 4 A function η of x and moments up to order d

is said to be a covariant of order d and weight q under agroup G if

η(x, M(0), . . . , M(d)) = (detA)qη(x, M(0), . . . , M(d))

for all A ∈ G.

For the orthogonal group, the simplest example of a co-variant is xT x (in other words the Euclidean length). Sinceboth x(d) and M(d) transform under Ad then xT

(d)M(d) willbe a covariant of order d .

Remark If the unsigned volume element is used then|detA|d replaces (detA)d in Definitions 3 and 4.

Whilst invariants are of primary interest in pattern recog-nition, covariants allow us to generate the invariants. Thecovariants will be the building blocks from which we obtainthe invariants.

Let

∂(1) = ∂ =

⎡⎢⎢⎢⎣

∂x1

∂x2...

∂xn

⎤⎥⎥⎥⎦

and

∂(d) =⊗

d

∂(d)

with

∂(0) = [1](that is, ∂(0)P = P ). ∂(d) is a vector of d th order partialderivatives. Under a linear transformation A, ∂ will trans-form

∂ = A−T ∂

and so, by Proposition 1,

∂(d) = A−Td ∂(d). (5)

Our strategy will be to construct a system of covariantsand then use the derivative operator ∂(d) to generate the in-variants. It is clear that invariants that are not “independent”will yield no new information. Specifically

Definition 5 A set of invariants {ιk : k = 1, . . . ,m} is saidto be (functionally) dependent if there exists a non-trivialfunction F such that

F(ι1, ι2, . . . , ιm) = 0. (6)

Conversely, if no such function exists then the ιk are said tobe independent.

If the invariants ιk are dependent then, in principle atleast, (6) may be solved to find one of the invariants in termsof the remaining m − 1 invariants. Following [5], we define


Definition 6 A basis for invariants of order d or less is a setof independent invariants {ιk : k = 1, . . . ,m} such that forany invariant ι of order d or less, there exists a non-trivialfunction F such that

F(ι, ι1, ι2, . . . , ιm) = 0.

Thus a basis is a complete set of functionally independentinvariants.

This definition does not require that a basis separates theorbits; that is, the intersection of the level sets of all the basiselements is a single orbit of the group action. For example,an element in a separating basis ι could be replaced by ι2.The elements would remain functionally independent andthus form a basis. However new basis may not separate theorbits since ι2 = c2 only implies ι = ±c. The issue here isthat there is no restriction on the function F . For the separa-tion of the orbits, a basis must generate all rational invariantsrationally. Thus

Definition 7 A rational basis for invariants of order d is abasis such that for any rational invariant ι of order d or less,there exists a non-trivial rational function F such that

F(ι, ι1, ι2, . . . , ιm) = 0.

The number of elements in a basis (rational or otherwise)will depend on d and the group G.

5 Translations

If the group G includes translations (for example, the Eu-clidean group SE(2) of rotations and translation on theplane) then the linear action (2) is replaced by an affine ac-tion

x = Ax + ξ .

This action may be “converted” into a linear action on Rn+1

via[

x1

]=[

A ξ

0T 1

][x1

]. (7)

The analysis of the previous two sections may now be ap-plied to this action. However x(d) will be a vector of mixeddegree. For example, for an affine planar action

x(1) =⎡⎣x

y

1

⎤⎦ , x(2) =

⎡⎢⎢⎢⎢⎢⎣

x2

xy

x...

1

⎤⎥⎥⎥⎥⎥⎦

.

Similarly M(d) will be a vector of moments of mixed or-der. Therefore the transformation laws will no longer be ho-mogeneous in the degree of the monomial or order of themoment.

The zeroth order moment, m0, will still be an invariant ofweight 1 (the volume form is invariant under translations).For a pure translation (that is A = I ) the first order momentstransform

mei= mei

+ ξim0

where ei is the multi-index which is 1 in the ith position and0 everywhere else and ξi is the ith component of ξ . Choos-ing mei

= 0; that is,

ξi = ei

m0(8)

will (invariantly) normalise the translation component of thegroup. The remaining part of the group will yield a linearaction on R

n. However, as we shall see, there is a cost inchoosing this normalisation. In the literature the use of thisnormalisation is widespread and the resultant moments arecalled central moments [5].

6 Moment Invariants of the (Conformal) OrthogonalGroup

For the orthogonal group, invariants have either weight 0 or1 (the so-called pseudoinvariants [5] that change sign undera reflection). If we restrict the group to the special orthog-onal group (that is, rotations only) then all invariants areabsolute. However, if we choose the unsigned volume ele-ment then all orthogonal invariants will be absolute. In thiscase the orthogonal case is identical to the special orthogo-nal case. On the other hand, if we add dilations, that is (uni-form) scaling, to the orthogonal group we obtain the confor-mal orthogonal group also known as the similarity group. Inthis case invariants may have any weight.

A conformal orthogonal matrix has the form A = λB

where B is orthogonal. Thus

AT A = λ2I = (detA)2/nI

in n dimensions.

Lemma 1 Let A = λB be a conformal orthogonal matrixin R

n. Then Ad = λdBd and

ATd Ad = λ2dI = (detA)2d/nI.

Let

η0 = xT(1)x(1).


η0 is a covariant of weight 2/n for the conformal orthogonalgroup (and of weight 0 for the orthogonal group). For d > 0,let

ηd = xT(d)M(d).

We have

ηd = (detA)xT(d)A

Td Ad M(d)

and so, by Lemma 1, ηd is a covariant of the conformal or-thogonal group of weight 1 + 2d/n (and 1 in the orthogonalcase).

For n = 2, ηd has conformal weight d + 1 and

η0 = x2 + y2,

ηd =d∑

i=0

(d

i

)xd−iyimd−i,i .

Inspired by the theory of binary forms [6, 13] we have

Definition 8 The r th transvectant of two (conformal) or-thogonal covariants P and Q is given by

Λr(P,Q) = (∂(r)P )T ∂(r)Q.

Note that Λ0(P,Q) = PQ and Λr(P,Q) = Λr(Q,P ).

Proposition 4 Let P and Q be conformal orthogonal co-variants of weight p and q respectively. Then Λr(P,Q) is acovariant of weight p+q −2r/n. For the orthogonal group,its weight is (p + q) mod 2.

Proof Under a linear transformation A, Λr(P,Q) willtransform

(detA)p+q(∂(r)P )T A−1r A−T

r ∂(r)Q.

For the orthogonal case, Proposition 1 implies that A−1r is

orthogonal and so the result follows. For the conformal case,Lemma 1 gives

A−1r A−T

r = (detA)−2r/nI

and so the transvectant will be a covariant of weight p+q −2r/n. �

Lemma 2 Let ζ be a moment. Then

∂

∂ζΛr(P,Q) = Λr

(∂P

∂ζ,Q

)+ Λr

(P,

∂Q

∂ζ

).

Proof Since the transvectant is a matrix product and deriva-tives commute the result follows immediately. �

The transvectant allows invariants to be constructed fromcovariants. If P and Q have degree r (in the variables x(1))then Λr(P,Q) will be an invariant. For example, for n = 3,

1

4Λ2(η0, η2) = m2,0,0 + m0,2,0 + m0,0,2,

1

36Λ3(η3, η3) = m2

0,0,3 + 3m20,1,2 + 3m2

1,0,2 + 3m20,2,1

+ 6m21,1,1 + 3m2

2,0,1 + m20,3,0

+ 3m21,2,0 + 3m2

2,1,0 + m23,0,0

are invariants. Our strategy will be the choose invariants thatdepend on order d moments and, if possible, on only oneother order. This choice will facilitate the proof of indepen-dence. Moreover we wish to construct a complete set of in-variants in the n = 2 case.

There is considerable freedom in making this choice. Forexample one could consider

Λqd(ηpi , η

qd ) with

p = d

gcd(i, d), q = i

gcd(i, d). (9)

However these invariants grow in degree (with respect to themoments) alarmingly particularly if i and d are relativelyprime. In that case the invariant with be a polynomial ofdegree 2d − i. The following choice has the advantage thatthe degree of the elements does not grow and is, in fact,bounded by 3. This choice is also motivated by the strategyof using order 1 invariants to normalise translations. Insteadwe make the following choice.

Proposition 5 Let G be the (conformal) orthogonal groupand let

ηd = xT(d)M(d).

For d = 2q , let

σ0d = Λd(ηq

0 , ηd),

σid =

⎧⎪⎨⎪⎩

Λd(ηq−i/20 ηi, ηd), i = 2,4, . . . , d,

Λd(ηq−i

0 η2i , ηd), i = 3,5, . . . , s,

Λd(Λi−q(ηi, ηi), ηd), i = s + 2, s + 4, . . . , d − 1

with

s = 2

⌊q + 1

2

⌋− 1

and

σ1d ={

Λ2(η21, η2), d = 2,

Λd(ηq−20 η2

2, ηd), d > 2.


For d = 2q + 1, let

σ0d = Λ2d(ηd0 , η2

d),

σid =

⎧⎪⎨⎪⎩

Λd(η(d−i)/20 ηi, ηd), i = 3,5, . . . , d,

Λd(η(d−3−i)/20 η3ηi, ηd), i = 2,4, . . . , d − 3,

Λd+2(η3ηd−1, η0ηd), i = d − 1

and

σ1d ={

Λ2(η2,Λ2(η0, η

23)), d = 3,

Λd(η(d−3)/20 η3,Λ

1(η2, ηd)), d > 3.

Finally let

Σd = {σ0d, σ1d , σ2d, . . . , σdd

}.

For d > 1, Σd is a functionally independent set of invari-ants.

Proof Direct computation using MAPLE shows that Σd isfunctionally independent for d ≤ 3. Let d > 3 and supposeΣd is not functionally independent. Thus there exists F suchthat

F(σ0d , σ1d, σ2d , . . . , σdd) = 0. (10)

Let ζ be an order i > 3 with i = d moment. Note that

∂F

∂ζ= ∂F

∂σid

∂σid

∂ζ= 0.

Therefore

∂F

∂σid

= 0.

Thus F = F(σ0d , σ1d , σ2d, σ3d , σdd).Note that

∂ηd

∂md,0= xd,

∂ηd

∂m0,d

= yd

and so

∂(d)

∂ηd

∂md,0= d!e1, ∂(d)

∂ηd

∂m0,d

= d!e−1

where e1 is the unit vector with 1 in the first position ande−1 is the unit vector with 1 is the last position.

For d even, since only σ3d depends on order 3 moments,F cannot depend on σ3d . By Lemma 2, we have

∂σ0d

∂md,0= d!(∂(d)η

d/20

)T e1 = (d!)2 = ∂σ0d

∂m0,d

.

Similarly we have

∂σdd

∂md,0= 2(d!)2md,0,

∂σdd

∂m0,d

= 2(d!)2m0,d

and

∂σkd

∂md,0= (d!)2mk

2,0,∂σkd

∂m0,d

= (d!)2mk0,2

for k = 1,2. Also note that

∂σ2d

∂m2,0= (

∂(d)ηd/2−10 x2)T (∂(d)ηd) = (d!)2(md,0 + · · · )

where · · · indicate terms independent of md,0. Similarly

∂σ2d

∂m0,2= (d!)2(m0,d + · · · ),

∂σ1d

∂m2,0= (d!)2(2m2,0md,0 + · · · ),

∂σ1d

∂m0,2= (d!)2(2m0,2m0,d + · · · ).

Therefore the block in the Jacobian generated by thesederivatives has the form (after removing the numerical factorof d!2)

⎡⎢⎢⎣

0 0 1 12m2,0md,0 + · · · 2m0,2m0,d + · · · m2

2,0 m20,2

md,0 + · · · m0,d + · · · m2,0 m0,2

0 0 2md,0 2m0,d

⎤⎥⎥⎦ .

This matrix has rank 4 and so σ0d , σ1d, σ2d , σdd are func-tionally independent. Therefore F = 0 and Σd is function-ally independent for d even.

For d odd, differentiating (10) with respect to M2,0 andm0,2 yields the system

⎡⎣

∂σ1d

∂m2,0

∂σ2d

∂m2,0

∂σ1d

∂m0,2

∂σ2d

∂m0,2

⎤⎦[

∂F∂σ1d

∂F∂σ2d

]=[

00

]. (11)

Observe that

∂σ2d

∂m2,0= Λd(x2η

(d−5)/20 η3, ηd)

and so will not have any term that depends on m1,d−1

whereas

∂σ1d

∂m2,0= Λd(η

(d−3)/20 η3,Λ

1(x2, ηd)

will have a term that depends on m1,d−1. Similarly ∂σ1d

∂m0,2

will have a term that depends on md−1,1 whereas ∂σ2d

∂m0,2will

not. Therefore the coefficient matrix is non-singular and so

∂F

∂σ1d

= ∂F

∂σ2d

= 0


is the only solution to (11). This clearly implies that F can-not depend on σ3d . Finally a similar argument shows that

∂F

∂σ0d

= ∂F

∂σdd

= 0

and so F = 0. Therefore Σd is functionally independent. �

As we shall see shortly,

Ed =⋃j

Σj

is also functionally independent. Moreover, in the casen = 2, it is a complete set. For convenience, let

Σ0 = {m0}, Σ1 = {σ01}

(remember that m0 is an invariant of weight 1).

Lemma 3 Let

σ = G(σ0d , σ1d, . . . , σdd)

and suppose that

∂σ

∂ζ= 0

for all order d moments ζ . Then σ = 0.

Proof Let ξ be a order i moment with i = 2,3, d . Note that

0 = ∂2σ

∂ξ∂ζ= ∂G

∂σid

∂2σid

∂ξ∂ζ

and so

∂G

∂σid

= 0.

Next observe that, differentiating with respect to m2,0, md,0

and with respect to m0,2, m0,d yields the set of equations

⎡⎣

∂2σ1d

∂m2,0∂md,0

∂2σ2d

∂m2,0∂md,0

∂2σ1d

∂m0,2∂m0,d

∂2σ2d

∂m0,2∂m0,d

⎤⎦[

∂G∂σ1d

∂G∂σ2d

]=[

00

].

For d even, the coefficient matrix is

(d!)2[

2m2,0 12m0,2 1

]

which is non-singular. For d odd, note that

∂2σ

∂m2,0∂m1,d−1= ∂G

∂σ1d

∂2σ1d

∂m2,0∂m1,d−1= 0.

The proof of Proposition 5 shows that the second factor isnon-zero. Thus, in both cases, we obtain

∂G

∂σ1d

= ∂G

∂σ2d

= 0.

This immediately implies that

∂G

∂σ3d

= 0.

A similar argument shows that

∂G

∂σ0d

= ∂G

∂σdd

= 0.

Thus σ = 0 as required. �

Theorem 1 The set

Ed =d⋃

j=1

Σj

is functionally independent.

Proof As noted before E3 is functionally independent. As-sume that Ed−1 is functionally independent. Suppose thatEd is not functionally independent. Thus there exist a non-trivial invariant

σ = G(σ0d , σ1d, σ2d , . . . , σdd)

such that

E ′ = Ed−1 ∪ {σ }is functionally dependent. Consequently a relationshipF = 0 exists between the elements of E ′. Now for everyorder d moment ζ

0 = ∂F

∂σ

∂σ

∂ζ.

Therefore either F does not depend on σ or, by Lemma 3,σ = 0. Either case contradicts the assumption that Ed−1 isfunctionally independent. Thus Ed is functionally indepen-dent. �

For the special orthogonal group, σij is an absolute in-variant. For the orthogonal group, let

σ ′ij =

{σij if the weight of σij is 0,

σ 2ij if the weight of σij is 1.

For the conformal group, let

σ ∗ij = m

−wij

0 σij


where wij is the weight of σij . Clearly σ ′ij and σ ∗

ij are abso-lute invariants with respect to their group actions. Moreover,let

E ′d = {σ ′

ij : σij ∈ Ed}, E ∗d = {σ ∗

ij : σij ∈ Ed}.These normalisation do not destroy the independence of theinvariants and so E ′

d and E ∗d is an independent set of absolute

invariants.We now restrict our attention to the n = 2 case. The di-

mension of O(2) is 1 (and its conformal extension has di-mension 2). There are d + 1 moments of order d . Thus thereare 1

2 (d +1)(d +2) moments of order up to d . Hence the or-bits of the action of the orthogonal group on these momentshave codimension 1

2 (d + 1)(d + 2) − 1. In the conformalcase, the orbits have codimension 1

2 (d + 1)(d + 2) − 2. Theorbits lie in the level sets of the absolute invariants and sotheir codimension gives the number of independent abso-lute invariants. In particular this shows that the set given inProposition 5 is a complete set.

Theorem 2 Let G act on moments of order up to d .

1. For G = SO(2), {m0,0} ∪ Ed is a basis for the absoluteinvariants.

2. For G = O(2), {m20,0} ∪ E ′

d is a basis for the absoluteinvariants.

3. For the conformal orthogonal group, E ∗d is a basis for the

absolute invariants.

Proof The independence of these sets has already been es-tablished. Ed, E ′

d and E ∗d each have

1 + 3 + 4 + · · · + (d + 1) = 1

2(d + 1)(d + 2) − 2

elements. Thus in each case, the given set has the same num-ber of elements as the codimension of the orbits. The resulttherefore follows. �

If translations are normalised via (8) then the order 1moments are zero and σ01 = σ12 = 0. Ed loses 2 elements.However orbit codimension is also reduced by 2. ThereforeEd (with σ01 and σ12 removed) remains a basis.

These invariants are easily computed (it only requiresmatrix algebra). MAPLE code for this computation is givenin the Appendix. The elements of E4 are (with common nu-merical factors removed)

m21,0 + m2

0,1,

m2,0 + m0,2,

m21,0m2,0 + 2m0,1m1,0m1,1 + m2

0,1m0,2,

m22,0 + 2m2

1,1 + m20,2,

6m0,3m2,1 + 6m1,2m3,0 + 5m20,3 + 9m2

1,2 + 9m22,1 + 5m2

3,0,

10m2,0m23,0 + 10m0,2m

20,3 + 3m0,2m3,0m1,2 + m0,2m

23,0

+ 36m1,1m1,2m2,1 + 18m1,1m3,0m2,1 + 9m2,0m21,2

+ 3m2,0m3,0m1,2 + 3m0,2m2,1m0,3 + 9m0,2m22,1

+ 18m1,1m1,2m0,3 + m2,0m20,3 + 3m2,0m2,1m0,3

+ 18m2,0m22,1 + 18m0,2m

21,2,

m2,0m23,0 + 2m2,0m

22,1 + m2,0m

21,2 + 2m1,1m3,0m2,1

+ 4m1,1m1,2m2,1 + 2m1,1m1,2m0,3 + m0,2m22,1

+ 2m0,2m21,2 + m0,2m

20,3,

m23,0 + 3m2

2,1 + 3m21,2 + m2

0,3,

m4,0 + 2m2,2 + m0,4,

m2,0m4,0 + 2m1,1m3,1 + m2,2m2,0 + m2,2m0,2

+ 2m1,1m1,3 + m0,2m0,4,

m24,0 + 4m2

3,1 + 6m22,2 + 4m2

1,3 + m20,4,

m4,0m23,0 + m4,0m

22,1 + 4m3,1m3,0m2,1 + 4m3,1m1,2m2,1

+ 4m2,2m22,1 + 2m2,2m3,0m1,2 + 4m2,2m

21,2

+ 2m2,2m2,1m0,3 + 4m1,3m1,2m2,1 + 4m1,3m1,2m0,3

+ m0,4m21,2 + m0,4m

20,3,

m22,0m4,0 + 4m1,1m2,0m3,1 + 4m2,2m

21,1 + 2m2,2m2,0m0,2

+ 4m0,2m1,1m1,3 + m20,2m0,4.

By contrast, the i = 3, d = 4 case of (9) has 119 terms (theequivalent basis element in this choice has 12 terms).

Computations using the MAPLE package implementingthe Hubert-Kogan algorithm [8] shows that this is not a ra-tional basis. For d = 3 (with translations normalised) thereare two elements in the rational basis which cannot be ex-pressed rationally in terms of the above basis. Their squareshave rational expressions in this basis. However the rationalbasis generated by the Hubert-Kogan algorithm has a linearelement, a quadratic element and two elements each of de-gree 4 and 5 with up to 41 terms in each element. In contrast,the above basis E3 (with translations normalised) has a lin-ear element, 3 quadratic elements and two cubic elementsand up to 15 terms in each element. By comparison, the in-variants derived by Hu [7] have degree at most 4 and haveup to 29 terms. It is well known that the Hu invariants arenot functionally independent. The basis given by Flusser etal. [5] has degree at most 4 and each element has up to 16terms.

An advantage of the basis Ed is the relatively slow growthof the maximum number of terms in the invariants. The de-gree of the basis elements in the moments always remains


bounded by 3. The maximum number of terms in a basiselement of order d is

Order 5 6 7 8 9 10

Max # of terms 34 30 50 60 66 105

7 Affine Actions on the Plane

In this section we consider the linear actions of the affinegroup GL(2) and the special affine group SL(2) on R

2. Let

J =[

0 1−1 0

]

a so-called symplectic matrix, J 2 = −I . A ∈ GL(2) is char-acterised [18, p. 2925] by

AT JA = (detA)J. (12)

As before, let

Jd =⊗

d

J.

It follows from Proposition 1 that

J Td = (−1)dJd, AT

d JdAd = (detA)dJd

for any A ∈ GL(2). We immediately see that

νd = xT(d)Jd M(d) =

d∑j=0

(−1)i(

d

i

)xd−iyimi,d−i

are covariants of the GL(2) action. Since J is skew-symmetric, xT Jx = 0 and so we have d covariants in themoments up to order d .

The definition of the orthogonal transvectant is modifiedto give

Definition 9 The d th transvectant of two GL(2) covariantsP and Q is given by

Λd(P,Q) = (∂(d)P )T Jd∂(d)Q.

Proposition 6 Let P and Q be GL(2) covariants of weightp and q respectively. Then Λd(P,Q) is a covariant ofweight p + q − d . Furthermore

1. Λd(P,Q) = (−1)dΛd(Q,P ),2. Λ2d+1(P,P ) = 0.

The proof of the first part of this proposition is the sameargument as used in the proof of Proposition 4. The secondpart follows immediately from the skew-symmetry of Jd for

d odd. The skew symmetry of odd order transvectants im-plies that σdd = 0, for d odd, if we choose to define theinvariants in the same manner as the orthogonal case. Forconvenience, let

Dr (P ) = Λr(P,P ).

Clearly Dr (νi) will be non-zero and quadratic in order i mo-ments (and degree 2(i − r) in x and y) if r ≤ i is even.

Let d = 2q ≥ 4 be even. Define

θid =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Λ4(ν4, Dd−2(νd)), i = 0,

Λ4(ν22 , Dd−2(νd)), i = 1,

Λd(νiνd−i , νd), i = 2,3, . . . , q,

Λd(ν2 D2(νi), νd), i = q + 1,

Λd(Di−q(νi, )νd),

i = q + 2, q + 4, . . . , d − 1,

Λd(Λ1(νi, νd−i+2), νd),

i = q + 3, q + 5, . . . , d − 1,

Dd(νd), i = d.

For d = 2q + 1 ≥ 5, define

θid =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Λ2(ν2, Dd−1(νd)), i = 0,

Λ2(D2(ν3), Dd−1(νd)), i = 1,

Λd(νiνd−i , νd), i = 2,3, . . . , q,

Λd(Λ2(ν2νq, νq+1), νd), i = q + 1,

Λd(Λ1(νi, νd−i+2), νd), i = q + 2, . . . , d − 1,

D2(Dd−1(νd)), i = d.

Finally, let

θ12 = Λ2(ν21 , ν2),

θ22 = D2(ν2),

θ03 = Λ2(ν2, D2(ν3)),

θ13 = Λ3(ν31 , ν3),

θ23 = Λ6(ν32 , ν2

3),

θ33 = D2(D2(ν3)).

Lemma 4 For d > 2, let

Θd = {θ0d, θ1d , . . . , θdd}with

Θ2 = {θ12, θ22}.Θd is a functionally independent set of GL(2) invariants.

Proof We first must show that θid are not constants. Thereis a potential that cancellation may occur. Note that, since


J2s is symmetric and non-degenerate, Λ2s(P ,P ) is not zerofor any covariant, P of degree ≥ 2s. Furthermore Λj(P,Q)

will be non-zero if P depends on order i invariants butQ does not (and both P and Q have degrees ≥ j ). Thusθid = 0.

Direct calculation using MAPLE shows that⋃6

i=2 Θi isfunctionally independent. Consider the case d = 2q > 6,even. Assume, on the contrary, that there exists a non-trivialF such that

F(θ0d , θid , . . . , θdd) = 0.

Since order d − 1 moments only occur in θd−1,d , F cannotdepend on θd−1,d . Next consider the terms depending on or-der moments d − 2; that is θ2d and θd−2,d . Differentiatingwith respect to md−2,0 and m0,d−2 we obtain⎡⎣

∂θ2d

∂md−2,0

∂θd−2,d

∂md−2,0

∂θ2d

∂m0,d−2

∂θd−2,d

∂m0,d−2]

⎤⎦[ ∂F

∂θ2d

∂F∂θd−2,d

]=[

00

].

The coefficient matrix is either[Λd(xd−2ν2, νd) 2Λd(Λ2(xd−2, νd−2), νd)

Λd(yd−2ν2, νd) 2Λd(Λ2(yd−2, νd−2), νd)

]

or[Λd(xd−2ν2, νd) Λd(Λ1(xd−2, ν4), νd)

Λd(yd−2ν2, νd) Λd(Λ1(yd−2, ν4), νd)

].

In both cases this matrix is non-singular. Thus F cannot de-pend on θ2d or θd−2,d . In a similar manner, we see that F

cannot depend on θi,d or θd−i,d for i = 3,4, . . . , q with theexception of the case i = 4, d = 8. Thus, apart from thisexceptional case, we have

F = F(θ0d , θ1d , θdd).

However θ0d , θ1d and θdd are clearly independent and soF = 0. In the exceptional case (d = 8), we have

F = F(θ08, θ18, θ48, θ88).

A direct computation using MAPLE, shows that F = 0. ThusΘd is functionally independent for d even. A similar set ofarguments may be used in the case for d odd. �

Theorem 3 The set

Td =d⋃

j=2

Θj

is functionally independent.

The proof of this result follows exactly the same argu-ments as used in Theorem 1.

In the case of SL(2) all the elements of Td absolute in-variants. In the case of GL(2) we normalise these relativeinvariants using m0,0 to obtain absolute invariants T ∗

d . Thedimension of SL(2) is 3 while that of GL(2) is 4. Thereforethe codimension of orbits of their action on moments up toorder d is 1

2 (d + 1)(d + 2) − 3 and 12 (d + 1)(d + 2) − 4

respectively. We see that Td has

2 + 4 + 5 + · · · + (d + 1) = 1

2(d + 1)(d + 2) − 4

elements. Thus we obtain

Theorem 4 Let G act on the moments of order up to d .

1. For G = SL(2), {m0,0} ∪ Td is a basis for the absoluteinvariants

2. For G = GL(2), T ∗d is a basis for the absolute invariants.

Again, if translations are normalised via (8) then θ12 =θ13 = 0 and the dimension of Td reduces by 2. However theorbit codimension also reduces by 2 and so we still have amaximal functionally independent set. In this case remain-ing the elements of T3 are

m2,0m0,2 − m21,1,

m2,0m2,1m0,3 − m2,0m21,2 + m1,1m2,1m1,2

− m1,1m3,0m0,3 + m0,2m3,0m1,2 − m0,2m22,1,

36m2,0m21,1m

21,2 + 24m2,0m

21,1m2,1m0,3 + 9m2

2,0m0,2m21,2

+ 6m22,0m0,2m2,1m0,3 − 36m3

1,1m2,1m1,2

− 4m31,1m3,0m0,3 − 54m2,0m1,1m0,2m2,1m1,2

− 6m2,0m1,1m0,2m3,0m0,3 + 36m21,1m0,2m

22,1

+ 24m21,1m0,2m3,0m1,2 + 9m2,0m

20,2m

22,1

+ 6m2,0m20,2m3,0m1,2 + 5m3

2,0m20,3 + 5m3

0,2m23,0

− 30m22,0m1,1m1,2m0,3 − 30m1,1m

20,2m3,0m2,1,

6m3,0m1,2m2,1m0,3 − 4m3,0m31,2 − 4m3

2,1m0,3

+ 3m22,1m

21,2 − m2

3,0m20,3.

8 Conclusions

The construction of orthogonal invariants using the transvec-tant (8) and the covariants ηi may be applied in any dimen-sion. In higher dimensions the issue is whether one can gen-erate enough functionally independent invariants to matchthe codimension of the group orbits. For example the orbitsof the group SO(3) have codimension

1

6(d + 1)(d + 2)(d + 3) − 3


in the space of moments of order d or less. Thus the numberof independent invariants grows as d3 rather than as d2 inthe planar case. Whether a maximal independent set of in-variants can be constructed from the covariants ηi remainsopen.

A more subtle question is whether a rational basis; thatis, a set of invariants that separate the orbits of the groupaction, may be constructed from the covariants ηi . As notedabove, the basis constructed in Sect. 6 does not separate theorbits. The computation work for the d = 3 case suggest thata rational basis has significantly greater complexity that thebasis given here. From a operational viewpoint, this addedcomplexity may outweigh any benefits that a basis whichseparates the orbits may give.

In the case of the affine invariants, the move to higher di-mensions will only work in even dimensions since there isno matrix J such that J 2 = −I in odd dimensions. The iden-tification of matrices that satisfy (12) with the general affinegroup is only valid for n = 2. In general, matrices that sat-isfy (12) form a subgroup of GL(n) known as the symplecticgroup. Thus the invariants generated in the even dimensionalcase will be invariants of the symplectic group.

Acknowledgements I would like to thank the Galaad group at IN-RIA Sophia Antipolis for hosting me while this work was completed.In particular, I would like to thank Evelyne Hubert for fruitful dis-cussions on this material and for running her code and comparing itsoutput with the basis given in this paper.

Appendix: MAPLE code

The following MAPLE code allows one to construct orthog-onal invariants of any order and in any number of variables.

w i th ( L i n e a r A l g e b r a ) :

Moments := p roc ( d , { v a r s :=_VARS} )o p t i o n remember ;g l o b a l _VARS ;i f _VARS <> v a r s t h e n

f o r g e t ( Moments ) ;_VARS:= v a r s

e l i f _VARS= e v a l n (_VARS) t h e n_VARS : = [ x , y ]

f i ;i f d = 1 t h e n

<<op (_VARS)>>e l s e

K r o n e c k e r P r o d u c t (<<op (_VARS) > > ,Moments ( d−1,_VARS ) [ 1 ] )

f i ;[% , map (w−>m[ seq ( d e g r e e (w, i ) ,

i i n _VARS) ] , % ) ]end ;

e t a := d−> i f d = 0 t h e n ‘+ ‘ ( seq ( i ^2 ,i i n _VARS ) )

e l s e ( T r a n s p o s e ( Moments ( d ) [ 1 ] ) .Moments ( d ) [ 2 ] ) [ 1 , 1 ] f i ;

pd := r −>Q−>map (w−> d i f f (Q, seq ( i $ d e g r e e (w, i ) ,i i n _VARS ) ) , Moments ( r ) [ 1 ] ) ;

Lambda := r −>(P ,Q)−> i f r =0 t h e n P∗Q e l s e( T r a n s p o s e ( pd ( r ) ( P ) ) . ( pd ( r ) (Q ) ) ) [ 1 , 1 ]

f i ;

The default choice of variables is x, y. For other choicesthe command Moments must be executed first with theoptional argument vars. For example, execution of thecommand Moments(1,vars=[x,y,z]); will alloworthogonal invariants in R

3 to be computed. This code maybe easily modified to compute the affine invariants.

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