Euclidean Invariants of Linear Scale-Spaces - Semantic Scholar

Euclidean Invariants of Linear Scale-Spaces

Alfons Salden? INRIA, 2004 route des Lucioles, BP 93, F-06902 Sophia-Antipolis Cedex, France

Abstract. The similarity jet of a linear scale-space is described in its most concise set of local and multi-local Euclidean invariants. The stability and (partial) equivalence of topologies on these invariants regardless additive uniform Gaussian noise is demonstrated.

1 Introduction In this paper Hilbert's method [2] is applied to nd a solution to the equivalence problem for the similarity jet of a linear scale-space of a two-dimensional greyvalued image [3]. Furthermore, a partially well-ordered topology induced on the found set of local and multi-local invariants is shown to be stable and (partially) equivalent above an eective scale related to that of additive uniform Gaussian noise [3]. The found set of invariants becomes important in describing magnetic resonance images or setting up of ane invariant frames in case of weak perspective imaging [3]. Its solution is also relevant as input to modern geometry and topology to quantify image formation and to construct dynamic scale-space theories [3]. The paper is organised as follows. In section 2 the equivalence problem is dened, in section 3 Hilbert's method is briey treated and in section 4 a partially well-ordered topology of the invariants is shown to be stable and (partial) equivalent for two input images that are slightly perturbed versions of each other.

2 The Equivalence Problem On the basis of a conservation law for the exchange of energy between a region and its surrounding on a two-dimensional Euclidean space E and application of the divergence theorem a linear scale-space of a two-dimensional grey-valued input image satises an isotropic linear diusion equation with suitable initialboundary value conditions [3]. The solution space to this initial-boundary value problem is a linear scale-space which has a particular jet-structure: 2

?

This work is supported by the Netherlands Organisation of Scientic Research, grant nr. 910-408-09-1, and by the European Communities, H.C.M. grant nr. ERBCHBGCT940511

Denition 1. The jet of a linear scale-spacen is dened by: j 1(L ) = fx; s; Lnjn 2 G i ; G0 = G, where x are the spaZZ  ZZ g, with Ln = L  Gn ; Gn = @xi1@:::@x n 0

+

+

0

tial coordinates, s is the scale parameter, L the input image and G the Green's function corresponding to the initial-boundary value problem. Requiring invariance under the similarity group [3] yields a similarity jet: Denition 2. The similarity jet of (Denition 1) is dened by: j 1 ( ) = n +2 f; nj = pxs ; n = s 2 L  Gn ; n 2 ZZ  ZZ g, where  labels Euclidean distances and n have the same physical dimension and are dierential energies. As our image domain coincides with Euclidean space E the fundamental properties of (Denition 2) are invariants under the group of spatially homogeneously Euclidean movements E (2; IR). Denition 3. A function I on (2) is called a Euclidean invariant or simply an invariant, if and only if, I (g(j 1 ) ) = detw (J (g))I (j 1  ) ; 8g 2 E (2; IR), where J is the Jacobian of the Euclidean group action and detw (J (g)) = 1; 8g 2 E (2; IR) is the weight of the invariant I . Now (Denition 2) and (Denition 3) provide a natural basis for stating the equivalence problem. Denition 4. The equivalence problem is the problem of nding a complete and irreducible set of invariants necessary and sucient to describe any property I of (Denition 2) irrespective any action of E (2; IR). 0

0

0

0

0

+

+

0

0

2

0

0

3 Hilbert's Solution Method The lenght measures  i and the dierential energies r can locally coordinatise so-called binary forms. Denition 5. A binary form Q of nP-th order is a homogeneous polynomial in  with coecients i1 :::ir : Q( ) = n ik i1 :::in  i1 : : :  in : In the context of (Denition 4) energy can be approximated by a truncated Taylor series of binary forms up to r-th order. The above mentioned complete and irreducible set of invariants will primarily be generated by considering the resultant of two binary forms of equal order (see section (3.1) and (3.2)). In order to dene such a resultant rst of all the transvection of two binary forms P and Q of arbitrary orders, the polar of a binary form and the Cayley-form of two binary forms of equal order have to be introduced [1]: Denition 6. The k-th order transvection [ : ; : ]k ofQtwo binary forms P and Q of arbitrary orders is dened by: [P; Q]k ( ) = lim! kl il jl @@il @@jl P ( )Q(); in which ij is the parity of the ordered pair (ij ). 1

!

2

=1

=1

Denition7. The -th order polar Q of a binary form Q of order n is dened

? by: Q = n

1

Q

p

=1

 ip @Q @ip : ( )

Denition 8. The Cayley-from F of two binary forms Q and Q both of order 1 2 1 2 n is dened by: F = Q Qn?Qn Q ; () =   ?   : 1

1

(



2

2

2

1

)





n n   P j i Using Q Qn = k  2nk? k +k 1  Qi ; Qj kn?k ()k it appears that the Cayleyk P P form F can be written as: F = in? jn? cij ( )i ( )n? ?i ( )j ( )n? ?j with cij a symmetric matrix function c on the coecients Qi1 :::in and Qj1 :::jn of the binary forms Q and Q , respectively. On the basis of these denitions the resultant can be dened as the determinant of the matrix c [1]: Denition 9. The resultant R of two binary forms Q and Q , respectively, both of order n is dened by: R(Q ; Q ) = det(c). 1

=0

1

1

2

1

1

2

2

1

1

1

=0

2

1

1

2

2

Because our solution method hinges on that of Hilbert it is necessary to know which systems of binary forms to consider. Hilbert searches for the invariants of forms under actions of the general linear group Gl(n; IR), whereas we are interested in the invariants under E (2; IR). Felix Klein, however, pointed out that equivalence problems for forms under special transformation groups can be conceived as particular projective ones [5]: Theorem 10. A complete set of integral and rational invariants of the system of (binary) forms Q ; : : : ; Qr is equivalent to a complete set 1of projective invariants of the system of equations: qi = Qi (~; 1) = 0; ~ = 2 2 IRIP adjoined to equation: (~) = (~ + 1) = 0 under the fractional action ~ ! pr sq on the projective line IRIP induced by the linear group GL(2; IR) on IR . Therefore, we rst solve the ane equivalence problem by means of Hilbert's method before tackling the Euclidean one. 1

1

1

~+

2

~+

2

1

2

3.1 The Solution of the Ane Equivalence Problem A local binary form On the basis of (Theorem 10) the ane equivalence problem is a special case of a projective problem for binary forms. For this projective problem there always exists a complete system of a nite number of invariants [5]. If such a system exists, then the next problem is nding such a system. In this context Hilbert addressed the problem of nding a complete set of integrally and algebraically independent invariants for a system of (binary) forms that is invariant under the general linear group GL(n; IR). His construction method for such a set is based on the concept of null forms [2].

Denition 11. A null form is a (binary) form all of which invariants vanish.

In the case of a single binary form the question arises how such a null form looks like and which invariants have to vanish in order it to be a null form. To this end Hilbert proves the following [2]: Theorem 12. If all the invariants of a binary form of order n = 2h + 1, respectively n = 2h are zero, then the binary form possesses an (h + 1)-fold linear factor-and conversely, if it possesses an (h + 1)-fold linear factor, then all invariants are equal to zero. Thus a binary form of order n = 2h + 1 or n = 2h is a (canonical) null form if it has a (h +1)-fold linear factor and therefore is parametrised by n ? h coecients. Denition 13. A canonical null form Nn of order n h= 2h + 1 or n = 2h is dened as a product of a (h + 1)-fold linear factor  and a binary form qn? h of order (n ? (h + 1)): Nn ( ) =  h qn? h ( ): Now the proof of (Theorem 12) in [2] supplies us with a method for constructing a set of invariants of a single binary form, through which all others can be expressed as an integral algebraic function of that set. Let's follow closely and elucidate the rst part of the proof presented by Hilbert [2]. In order to explicitly state which invariants of a single binary form Qn of order n = 2h + 1, respectively n = 2h, have to vanish such that it is a null form Hilbert starts o with the construction of the set Zn of transvectants [Qn ; Qn ]k : = 2h + 1 . Next Zn  fQn; [Qn ; Qn] ; : : : ; [Qn ; Qn ] g g where g = h ?h 1 ifif nn = 2h he uses the following auxiliary theorem to retrieve his invariants [2,1,4]: Theorem 14. The necessary and sucient condition for a binary form Qn of order n = 2h + 1, respectively n = 2h, to have a root of multiplicity h + 1 is equivalent to requiring the set of transvectants, Zn , to have one linear factor in common. The latter condition on the set of transvectants, Zn , is now formulated in terms of the vanishing of the resultant of two linear independent combinations U and V of powers of the transvectants that have the same order, namely M . Here M is dened to be the least common multiple of the numbers n; : : : ; 2(n ? 2g), such that the powers m ; m ; : : : ; mg of the transvectants are related to this least common multiple M and the order n of the binary form Qn as follows: M = m n = 2m (n ? 2) = : : : = 2mg (n ? 2g), where g is h and h ? 1, respectively, depending on whether n = 2h + 1 and n = 2h, respectively. On the basis of these numbers the two forms U and V with indeterminatePparameters u = u ; u ; : : :P ; ug and v = v ; v ; : : : ; vg , are given by: U = gk uk ([Qn ; Qn ]k )mk ; V = gk vk ([Qn ; Qn ]k )mk . Now (Theorem 14) holds, if and only if, the resultant R of the forms U and V vanishes: R(U; V ) = P  J P = 0, where each P is a product of powers of the parameters u and v above and J are the sought invariants for odd order n and J and [Qn ; Qn ]n those for even order n. +1

1

+1

(

+1)

1

2

1

1

0

=0

+1)

2

0

0

(

1

0

=0

1

A local system of binary forms Now the search for invariants boils down to deriving the conditions for a system of binary forms (Q ; : : : ; Qr ) to be a system of canonical null forms (N ; : : : ; Nr ). The set of simultaneous invariants come subsequently about upon application of the following theorem [3]: 1

1

Theorem 15. A complete and irreducible set of simultaneous integrally and al-

gebraically independent invariants of a system of two binary forms (Qp ; Qq ) of orders say p and q, respectively, is generated rstly by determining the least common multiple Mp;q of the orders p and q of the binary forms and secondly by computing the simultaneous invariants of the new system of two binary forms  Mp;q  Mp;q p Qp ; Qq q through calculation of the integrally and algebraically independent invariants of a linear combination of this new pair of binary forms of order Mp;q .

Calculating the invariant of each binary form Qn and the simultaneous invariants of pairs of binary forms (Qp ; Qq ) of system (Q ; : : : Qr ) is necessary and sucient to solve the ane equivalence problem for (Denition 2). 1

Multi-local systems of binary forms Let's conclude by solving the ane equivalence problem for multi-local systems of binary forms f(Q ; : : : ; Qr ) kQ g. The solution is an extension of the solution for the local problem for a system of binary forms (Q ; : : : ; Qr ) to a bi-local problem concerning two systems of binary forms f(Q ; : : : ; Qr ); (P ; : : : ; Pr )g, to a tri-local, etc. The solution is obtained by consecutively adjoining forms P to the system (Q ; : : : ; Qr ) and solving the problem for the extended problem analogous that for one local system of binary forms treated above. 1

1

1

1

1

Fig. 1. To the left level contours of

of a surface with a cusp at the origin together with its parabolic curve projected onto the ( )-plane. To the right level contours on which invariant 1 1 is approximately zero. The disconnected curve segments are obviously close to the projected parabolic curve. z (x; y )

=

x

2

+ xy

2

x; y

R ;

Example The systems of linear forms f(Q ); (P )g can be put in the system of linear forms fN (Q); N (P )g, if and only if, the following invariant vanishes: R ; = [Q ; P ]. This image property is just the resultant of both linear forms of the systems of the corresponding linear forms. Such an invariant can be useful in establishing an ane invariant frame. Here, however, this invariant is used to nd the parabolic curves on a surface S . Let us consider the invariant R ; on the basis of the image gradient elds rI and rI 0 of two orthographic images I and I 0 , respectively, of a Lambertian surface illuminated from two dierent source directions s^ and s^0 at the same point in both orthographic images. It's shown [3] that the above image gradient elds have to be parallel along the projected parabolic curves. Consequently, the invariant R ; has to vanish. Thus one nds as photometric invariant objects of the pair of images the projected parabolic curves of the surface (see gure 1). 1

1

1 1

1

1

1

1

1 1

1 1

3.2 The Solution of the Equivalence Problem The solution of (Denition 4) using the projective formulation in (Theorem 10) comes about by adjoining the quadratic binary  =  i  i to the systems of the previous section (3.1) and extend Hilbert's method to the extended systems. 1

2

4 Stability and Partial Equivalence As the energies n satisfy a maximum principle [3] numerical stability under the linear ltering is denitively guaranteed. The stability of (Denition 2) or its invariants I under addition of uniform Gaussian noise  with zero mean and xed variance comes about by conceiving j 1 () of L = L +  as a by scale s parametrised subset of the Sobolev-space H1 (E ) and xing tolerances on the basis of the resolution properties of the camera system [3]. 0

0

2

Denition 16. The Sobolev-space H1(E ) is the set of innitely many times continuous R dierentiable functions u on a subset  E such that u 2 L (E ): jjPujj =P ju(Rx)j dx 0 and scale sn such that: jjI ? I jj12 s < ; 8 s  sn . 0

( )

Besides stability in the above sense one also expects some (partial) equivalence of the topology induced by the invariants I of (Denition 2). Partial in the sense that certain subsets have to be omitted or added to realise equivalence. In order to demonstrate stability and (partial) equivalence we rst create the full topology O of the eld of invariants I derived in the previous section.

Denition 18. The topology O on a eld of invariants I is a set of open subsets S of this eld satisfying ; 2 O ^ S 2 O, (A 2 O ^ B 2 O) ) A \ B 2 O and Ai 2 O ) [i Ai 2 O. This topology O can be endowed by means of s and the attained levels (I ) of the invariants I with an ordering relation  yielding an ordered topology (O; ).

Denition 19. An ordering relation  on the topology (18) is a relation that satises transitivity v  w ^ w  x if v  x; v; w; x 2 S; and trichotomy, 8v; w 2 S jj v = w _ v  w _ w  v, respectively. Denition 20. An ordered topology (O; ) is the topology (18) endowed with and ordering relation (19). On the basis of s and (I ) one can subsequently endow (20) with a partial wellordering yielding a partial well-ordered topology. Denition 21. An ordered topology (20) is partially well-ordered if and only if 8T  S j9w 2 T j w  w0 ; 8w0 2 S . Denition 22. A partial well-ordered topology (O; w ) is the ordered topology (20) satisfying the well-ordering relation (21). Let's make explicit the partially well-ordered topology (22) induced by s and (I ) attained on the set of critical points and the set of critical zero-crossings. Denition 23. The set @Kx(I ) of critical points of I is dened by: @Kx(I ) = f(x; s)jrI (x; s) = 0g. Denition 24. The set @Ks(I ) of zero-crossings of I is dened by: @Ks(I ) = f(x; s)j@s I (x; s) = 0g. Denition 25. The set @Kx;s(I ) of critical zero-crossings of I is dened by: @Kx;s(I ) = @Kx(I ) \ @Ks (I ). The partially well-ordering by s comes about by extending the scale ranges such that they include the previous ones. The additional partial well-ordering of the eld of invariants on each subset of the critical sets (23) and (25) at one scale s comes about by extending analogously the ranges of attained levels (I ). These considerations lead us to inict the following topology on the eld of invariants I in order to assess stability and (partial) equivalence of L and L . Proposition 26. The partially well-ordered topology (22) of the eld of invariants I on the critical sets (23) and (25) is proposed to be generated by wellordering all possible subsets of those critical sets rstly on the basis of the scale parameter s and secondly on the basis of the levels (I ) at one scale s. Note that metrical relations on the subsets of the critical sets can be added to rene the well-ordering [3]. In gure 2 the stability and (partial) equivalence are visually demonstrated. Above a certain scale-width the d -levels at the critical sets and these sets themselves of the two input images become partially indistinguishable. Realise that the set-theoretic dierence operator has to be used in order to eventually establish (partial) equivalence. 0

3

0

Fig. 2. First row from left to right: the 256  256 input image of a capital letter "A" of dynamic range of 200 for the background and 0 pfor the letter itself and its third order p discriminant 3 at scale-width of 2 = 15 and 2 = 20 pixel-distances, respectively. Second row from left to right: the 256  256 input image of the capital letter "A" from the rst row with additive uncorrelated Gaussian noise with mean zero and variance of 5  105 and its third order discriminant 3 at the same scale-widths as the gures in the two other gures on the rst row. d

s

s

d

References 1. Gordan, P.: Ueber die Bildung der Resultante zweier Gleichungen. Math. Annalen, 50 (1871) 355414 2. Hilbert, D.: Ueber die vollen Invariantensystemen. Math. Annalen, 42 (1893) 313 373 3. Salden, A.H.: Dynamic Scale-Space Paradigms. PhD thesis, Utrecht University, The Netherlands, 1996 4. van der Waerden, B.L.: Moderne Algebra. Springer-Verlag, Berlin, 1940. 5. Weitzenböck, R.: Invariantentheorie. P. Noordho, Groningen, 1923.