Invariants of a pair of tonics revisited

Long Quan, Patrick Gros and Roger Mohr

The invariants of a pair of quadratic forms and a pair of coplanar conks are revisited following Forsyth et al.‘,’ and Semple and Kneebone3. For a given pair of tonics, with their associated matrices C, and C2, we show that Trace (Cl’ C,), Trace (C:’ C2) and 1 C, ) I / C, 1 are only invariants of associated quadratic forms, but not invariants of the tonics. Two of true invariants of the conies are

Trace (C2’ C,) 1 C, /

(Trace(C!’ C,))’ 1 Cl 1

und

‘I‘race(~Y,‘C2) 1 C, 1 -___

(Trace(C_~‘C,))’ / C! )

Then the true invariants of the tonics are geometrically interpreted. in terms of cross ratios, through the common self-polar triarzgle of the two tonics.

Keywords: invariant, cross-ratio, conic

Following ;I series of publications of Forsyth c’t al.‘.‘.“, invariants of ;I pair of coplanar tonics provoke much interest. Wc first review some general aspects of invariants of ;I pair of tonics, with reference to Scmple and Kneehone-‘. WC emphasize the difference between invariants of quadratics and those of tonics. For a given pair of tonics, with their associated matrices C, and C2. WC show that Truce(C2’CI), Trace(C;‘C2), and 1 Cl )/ 1 Cl 1 arc only invariants of associated quadratic forms, but not inva;iants of the tonics. Two of true invariants of the tonics are:

Trace (C 1 ’ C , ) 1 C, 1 - -

and:

Trace(Cj’CI) 1 C, /

(Trace(Cj’C,))’ I C3 /

Secondly, geometric interpretation of conic invariants based on cross ratios is presented using the common self-polar triangle. Note that Kapur and Mundy’ have tried to make a geometrical interpretation using polarity transformation for their proposed invariants. but it is not complete. The results presented in this paper can be considered as a complementary part of the work by Forsyth et al.‘.‘. Zisserman et ~1.’ and Kapur and Mundy’ to better understand conic invariants, and to make better practical use of them.

The general aspect on the theory of invariants is developed by Semple and Kneebone”. Most demon- strations of fundamental projective properties used in this paper are not explicit: readers can find them in Semple and Kneebonej, particularly in Chapter 5 dealing with tonics. For more detailed discussion on algebraic projective geometry, interested readers. are referred to that reference.

REVIEW OF THEORY OF INVARIANTS

Invariants of a pair of quadratic forms

Consider a quadratic form: 9 = s’C.r = I&,, N!,x,.x,, i, j = I, 2, 3; where C is B symmetric 3 x 3 matrix and .r = (X ,, X1,. Q) is the vector of homogeneous coordin- ates in the plane. We suppose j C I #O. where j C / is the determinant of C; this insures the associated tonics are not degenerate. We change the coordinates by the substitution x = Px’; P can be any non-singular trans- formation ( I PI #(I). We look for-invariants which arc independent of any P.

It follows that C’ = P’CP. The asxociated determi- nants have the important property 1 C“ 1 = / P 1’ 1 C /.

Let 9, = _r’C,x and 3, =.r’C, v be two quadratic forms in .Y = (A-,. .t-,. r?)‘. The gcncral linear combina- tion of 3, and I, m,ry bc written as 9A,p = A 3 I + /IL ‘3, = .I’Ch t

‘I‘hereforir- / I A,p I := /,h’ + I~h‘,.u I- I,h/uu’-c /,/u3.

02~~~-XXS6/93-/OOS~l~~OS @) I092 Dutterworth-Heinern~~n~~ l.tcl

where:

Uii and bjj being elements of Ci and C2; and A,, and B, being the cofactors of aii and b,.

Take any non-singular linear transformation P, $t, $ and 9h,p are respectively transformed by x = Px ’ into the forms:

9.‘, =x’~C’,X’, 9.5 =x”C$x’, and

%,, = ha; +/&z; =x’fC~,&

Write / Ci,, j = Z;h3 + ZkA2p + Z$hp2 + Z&p’, since / Ci,, 1 = j P /’ / C,+ 1, which holds for any A and p, we obtain:

Z\=IP12Z,, z;=IP12z2,

G=lP12~3, and Zb= IPI’Z,

As a consequence, any ratios of Zi/Zj, for i #j are independent of any transformation P, and thus are invariants for the discussed pair of quadratic forms.

Invariants of a pair of tonics

Let us consider the tonics associated with these quadratic forms. It should be noted that the tonics are associated in such a way, not with uniquely defined quadratic forms 9.r and 9z, but with families of forms A9i and ,&&, where A and p are arbitrary scalar. The ratios Zi/Zj are no more invariant for the associated con&, since invariants of tonics also have to be dependent of any multiplicative scalar A amd fi for their matrices. Therefore, the powers of Zi, i = 1, 2, 3, 4, with respect to h and ,u, have to be taken into account. Two invariants for a pair of conies can be obtained (cf. Semple and Kneebone’) by:

IlZ3 12 I4

ff=x and /3=-

Z5

which are independent not only of any transformation P, but also of any A and ,u. In addition, any other invariant which can be derived in this way is a rational function of LY and Z3. It is evident that all invariants of tonics are those of associate quadratic forms.

GEOMETRIC INTERPRETATION OF INVARIANTS OF A PAIR OF CONICS

Basic concepts

The cross ratio of four points Pi, i = 1, 2, 3, 4, with

Figure I. Q and R are conjugate points with respect to the conic

their homogeneous coordinates (hi, pi) on a projec- tive line is defined by:

By convention, {PI, P2; P3, co} = (t?, - &)/(c?~- 0,). For a given conic %, if Q and R are two points of the

plane, Q and R are said to be conjugate points with respect to the conic % if they are harmonic with respect to the points P1, P2 which are the intersection points of QR on the conic (cc Figure l), i.e. {Q, R; P,, Pz} = -1.

The locus of points conjugated to a fixed point P is a line. This line is called the polar of P. For a given line 1, the is a unique point which is conjugate to every points of I with respect to %. This point is calied the pole of 1. The conic % thus defines a transformation of the points (poles) of the plane into the lines (polars) of the plane and lines (polars) into points (poles). Recall that lines and points are dual entities and have the same (formal) homogeneous coordinate representation in the projec- tive plane. In matrix notation, this transformation

5 nown as the polarity defined by %) can be written as p0[ar = C P and p Pt,“te = C-‘I, where A and ,u are

any non-zero scalars. If the polar of P passes through a point Q, then the

polar of Q passes through P. The three points Q,, Q2, Q3 are said to form a serf-

polar triangle for %, if the polar of each is the line joining the other two. There is an infinity of self-polar triangles for a given conic.

Where is the common self-polar triangle?

If A, B, C, D are four points of the conic %, the diagonal triangle XYZ of the quadrangle ABCD is self-polar for ‘f: (c$ Figure 2).

Consider two coplanar tonics (e, and (ez, they always intersect in four points, say A, B, C, D. A, B, C, D form a quadrangle whose diagonal triangle X, Y, Z is self-polar both with respect to %r and Ce2. That is, it is the common self-polar triangle of ‘c: , and (e2 (cf- Figure 3). This common self-polar triangle is unique:

The common self-polar triangle of two tonics is the diagonal triangle of the quadrangle defined by the intersection points of the two tonics.

320 image and vision computing

Figure 2. Self-polar triangle XYZ for a conic deter- mined by four points of the conic A, B, C, D

Y

Figure 3. Common .relf-polar triangle XY Z for a pair of coplanar tonics

What are the vertices of the common self-polar triangle for?

The polar of X with respect to se, can be obtained by ,ul, = C, X, and the polar of X with respect to (e2 by Yl;(. As XYZ is the common self-polar triangle, lx and I ;( are the same polar YZ of X. Homogeneous coordinates for points and lines are determined apart from an arbitrary scalar factor. Two lines are the same if fx=hf$. Therefore, C,X=AC2X, leading to (CT’ C ,)X= AX. This is the standard definition for eigenvalue and eigenvector, meaning X is the eigen- vector of CT’ C,. Similarly, we can prove that the other two vertices Y and Z are also eigenvectors of CT’ C,:

The three vertrices of the common self-polar triangle are the three eigenvectors of C ;’ C , .

What are the eigenvalues of C;’ C, related to?

Since the three vertices of the common self-polar

triangle are the three eigenvectors of CT’ C,. so what are the eigenvalues of CT’ C , related to?

Let now see how the equation of a conic may be put in a simple form by taking a special triangle of reference.

When the triangle of reference is self-polar, the equation of a conic reduces to the diagonal form aIlx:+azzxf+a33x:=().

When the common self-polar triangle XYZ is selected as the triangle of reference, i.e. X= (1, 0, O)‘, Y = (0, 1, 0)’ and Z = (0, 0, l)‘, (4, and %j2 reduce at the same time to the diagonal form. These diagonal forms are obtained by the substitution x = Px’, where P = (X/Y/Z’):

The equations of the two tonics can be written as:

x’C,x= (Px’)‘C,(Px’) =xff(PfC,P)x

xtczx = (Px’)‘C2(Px’) =xfl(P’c~P)x

Thus, C;-‘C; = (P’&P)-‘(P’C,P) = P-‘(C;‘C,)P. Recall that the matrix P is formed by the eigen-

vectors X, Y, Z of C;’ C,, so we prove that C;’ C, and C$-‘C’, have the same eigenvalues, noted as A,, AZ and A3. They can be directly obtained by:

The polar XY meets each of two tonics in two points: M,, N, for %,, and M2, Nz for 32. The equation of XY reduces to x; = 0. These intersection points can be obtained by solving the following equations:

a;1x;2+a;2x;2+a)37_r3 ‘2=() and

x;=o

12 _ b’,,x;‘+b;2x;2+b;3xj -0

x;=o

If we denote the projective coordinates of Ml, N,, M2 and N2 on XY as 13,) O’, , O2 and 0;. They are respectively the roots of:

As homogeneous coordinates are unchanged by an arbitrary scalar, so only ratios of homogeneous coor- dinates are significant. Let us consider one of the ratios hi/hi, say AllA2.

Obviously, we have

Al a’,,lb’l, -= A2 a ‘22/b )22

On the other hand, the cross-ratio of M,, M2 with respect to X, Y is

{M,, Mz; X, Y} = {e,, e2; 00, 0} =” I

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It becomes:

{M,, M,; X, Y>‘= @2= $$ I1 22

So, A,lA2= {M,, M,; X, Y}2 Note that:

{M,, M2; X, Y}“= {M,, /V,; X, Y}‘=

{N,,Mz;X, Y>‘= (N,,%;X, Y>”

That means that the ratio of two eigenvalues is the square of the cross ratio of M,, M2 (either M,, N2 or N,, M2 or N,, N2) with respect to X and Y. Similarly, other ratios of eigenvalues have the same geometric interpretation in terms of cross ratios.

Invariants of a pair of tonics and their interpretation

Now we give a geometric interpretation of the invariants of tonics presented earlier.

Taking the reduced form of the tonics through the common self-polar triangle, we have:

1, = a ‘, I B ‘, I + a ‘22 B ‘22 + a ;3 B ‘.m

14=b’,,b;2b;3

Three invariants of the associated quadratic forms are:

13 12 1 1 1 -=A, +hz+h3, -=

I,

14 II -+-+- andr =A,A2A3 AI A2 A3 4

All other ratios Z,ll,, for i fj (invariants of quadratic forms) can be expressed from these three, and are symmetric functions of A,, AZ, A3.

Obviously:

Truce(c$C,) = +, 4

Truce(C~‘C2) =$ and ICI1 II -=-

IG 14

so that Truce(C;‘C,), Truce(Ci’Cz) and IC, I/ICZl are only invariants of a pair of quadratic forms, but not invariants of the tonics. We cannot express them in terms of ratios of A, which are some known cross ratios.

Two of true invariants of the tonics are:

IIIR (A, +A~+A&IA~Ax) (y=-=

1: (W2+Wx+A,Ad2 = 1 +A,lA, +A,/A,

AJAz (1 +AJA, +A3/A2)2

12 14

‘=I:=

A,A2+A2A3+A,b

(A,+A~+A~)~ =

AZ/A, 1 +A,lA, +A,lA,

(~+A~/A, +A31A,)’

Every ratio A,/A, is the square of a known cross ratio, so that the invariants of a pair of tonics are rational functions of some known cross ratios.

Note that Forsyth et al.,.2 propose Truce(C;‘C ,) and Truce(Ci’C2) as the two invariants of a pair of tonics. They are actually only invariants for the associated quadratics, not for the tonics. They become the true invariants of tonics when the associated matrices are normalized. As normalization makes no scaling possible, a quadratic form is uniquely associated with a conic. On the other hand, with I C, / = I, = 1 and I C2 1 = I4 = I, Truce(C;‘C ,) and Truce(CT’C2) be expressed in terms of a and /3:

113

and

Truce(Ci’C2) = & i 1

113

CONCLUSION

The invariants of a pair of tonics are first revisited, and then geometrically interpreted based on cross ratios with the help of the common self-polar triangle of the tonics. This work is a continuation of the invariant investigation of Forsyth et al. We hope to provide a better understanding and use of conic invariants in computer vision (see also Gross and Quan6). Our conviction is that cross ratio is the simplest fundamental projective invariant. Other invariants, at least those concerning point, line and conic entities, should be expressible by cross ratios. Therefore, the invariants for a set of geometric entities can often be directly constructed. For example, to get the two invariants of a pair of tonics, the most direct ones are the two cross ratios of the four intersection points defined for each conic’. However, the stability of invariants obtained with respect to the other invariants in this way should be verified through experimentation.

ACKNOWLEDGEMENTS

We would like to thank Professor Michel Brion and Oscar Burlet for many interesting discussions.

REFERENCES

I Forsyth, D, Mundy, J and Zisserman, A ‘Transfor- mational invariance -a primer’, Proc. 1st Br. Much. Vision Conf., Oxford, UK (September 1990) pp l-6

2 Forsyth, D, Mundy, J I,, Zisserman, A and Rothwell, C ‘Invariant descriptors for 3D object recognition and pose’, Proc. DAKPA-ESPRIT Work- shop on Applic. Invuriunts in Comput. Vision, Reykjavik, Iceland (March 1991) pp 171-208

322 imuge and vision computing

3 Semple, J G and Kneebone, G T Algebraic Projec- tive Geometry, Oxford Science Publication, UK (1952)

4 Zisserman, A, Marinos, C, Forsyth, D A, Mundy, J L and Rothwell, C A ‘Relative motion and pose from invariants’, Proc. 1st Br. Mach. Vision Conf., Oxford, UK (September 1990) pp 7-12

5 Kapur, D and Mundy, J L ‘Fitting affine invariant tonics to curves’, Proc. DARPA-ESPRIT Workshop on Applic. Invariants in Comput. Vision, Reykjavik,

Iceland (March 1991) pp 209-233 6 Gros, P and Quan, L PrPsentation de la thPorie des

invariants sous une forme utilisable en vision par ordinateur, Technical Report RT69 IMAG - 7 LIFIA, IMAG - LIFIA. Grenoble, France (July 1991)

7 Maybank, S J ‘The projection of two non-coplanar tonics’, Proc. DARPA-ESPRIT Workshop on Applic. Invariants in Comput. Vision, Reykjavik. Iceland (March 1991) pp 47-53

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