Di erential and Integral Geometry of Linear Scale-Spaces - CiteSeerX

Journal of Applied Intelligence, 2, 528 (1992)

c 1992 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

Dierential and Integral Geometry of Linear Scale-Spaces ALFONS H. SALDEN, BART M. TER HAAR ROMENY AND MAX A. VIERGEVER Image Sciences Institute, Utrecht University Hospital, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands [email protected]

Received April 29, 1991; Revised October 11, 1991 Editors: S. Devadas and P. Michel

Abstract. Linear scale-space theory provides a useful framework to quantify the dierential and integral geometry of spatio-temporal input images. In this paper that geometry comes about by constructing connections on the basis of the similarity jets of the linear scale-spaces and by deriving related systems of Cartan structure equations. A linear scale-space is generated by convolving an input image with Green's functions that are consistent with an appropriate Cauchy problem. The similarity jet consists of those geometric objects of the linear scale-space that are invariant under the similarity group. The constructed connection is assumed to be invariant under the group of Euclidean movements as well as under the similarity group. This connection subsequently determines a system of Cartan structure equations specifying a torsion two-form, a curvature two-form and Bianchi identities. The connection and the covariant derivatives of the curvature and torsion tensor then completely describe a particular local dierential geometry of a similarity jet. The integral geometry obtained on the basis of the chosen connection is quantied by the ane translation vector and the ane rotation vectors, which are intimately related to the torsion two-form and the curvature two-form, respectively. Furthermore, conservation laws for these vectors form integral versions of the Bianchi identities. Close relations between these dierential geometric identities and integral geometric conservation laws encountered in defect theory and gauge eld theories are pointed out. Examples of dierential and integral geometries of similarity jets of spatio-temporal input images are treated extensively.

Keywords: linear scale-space theory, similarity jet, dierential geometry, integral geometry, ane connection, metric, structure equations, Bianchi identities, torsion, curvature, translation vector eld, ane rotation vector elds, superposition principles 1. Introduction Our aim is to quantify the dierential and integral geometry of spatio-temporal input images. In order to achieve our goal this geometric quanti-

cation is carried out for similarity jets of linear scale-spaces of those images [47, 43, 36]. A reason for considering these similarity jets instead of merely the initial input images is the desire for reproducibility of dierential and topological

6

Salden, ter Haar Romeny and Viergever

measurements despite perturbations of the input images due to small scale uncorrelated noise contributions, Euclidean movements and similarity transformations. In the past decades several attempts have been made to describe spatio-temporal images based on various mathematical and physical disciplines, namely:

Classical continuum mechanics [23, 14, 9, 10], Dierential and integral geometry [5, 41, 35], Invariant theory [40, 39, 42, 36], Singularity theory [15, 26, 29, 16, 6], Logical ltering methods [24, 3], Fingerprints of zero-crossings [48], Topological ltering methods [8, 13, 30, 31, 18, 35], Primal sketches [27].

The rst three approaches concern local and multi-local description methods, whereas the remaining methods are heading for a global description. All these approaches for describing spatiotemporal images have their advantages and disadvantages. Let's make these aspects explicit for the local and multi-local description methods. In the classical continuum mechanical approach [23, 14, 9, 10] the assumption of the existence of an invertible (time-dependent) transformation of a spatial conguration mapped bijectively onto another (dieomorphisms of spatial congurations over time) allows a quantication of optic ow including rigid motions. But this assumption and that of constancy of total density are not always met in real images due to the niteness of the resolution properties of the vision system or the dynamics of the scene that evidently involves morphisms of spatial congurations over time. For example, a change in topology over time represented by a jump in the Euler-number, indicating the number of holes of a surface, excludes the presence of spatial (point) correspondence. Another example is an image gradient eld that is initially constant over space, but over time becomes inhomogeneous and singular due to non-integrable deformations of the input image. More general the spatio-temporal dynamics over an ensemble of input images consists not only of a totally integrable deformational spatio-temporal part, but also of non-exact (non-integrable) parts due to the break-

ing over space and time of the (in)homogeneity of e.g. the Galilean transformation group (scenes in which the number of objects are not the same can be described by such a symmetry breaking). In the dierential and integral geometric approach [5, 41, 35] the breaking of spatio-temporal symmetries over input images is associated to socalled Volterra processes, i.e. deformation, insertion or removal of spatio-temporal congurations. These processes are intensively studied in defect theory [17] and gauge eld theories [19], and can be nicely quantied in terms of a Burgers vector eld and Frank vector elds related to the torsion and the curvature of the process. These vector elds can display so to speak the image formation over space-time and scale [41, 35]. In grey-valued input images normally there does not exist any order of contact, i.e. the left and right derivatives of the input image along any direction in the image domain are not the same. This lack of order of contact boils down to the fact that the image gradient eld is multi-valued and not as naively assumed in most applications of linear scale-space theories innitely many times continuously dierentiable. The latter structure is boldly inicted on the image data. In defect eld theories [19] such defects in the order of contact are identied as manifestations of dislocations and disclinations of a lattice. In order to be able to quantify these kind of defects in an input image an external observer using still a linear scale-space paradigm can take advantage of reecting boundary conditions along hyperplanes going through a point of interest [41, 43, 36, 35]. If one now computes connection one-forms and frame vector elds after linear scaling under reective boundary conditions each pair of subimages bordered by the initial image boundary and the hyperplane, then one can quantify defects through integral invariants indicating how much the image is curved and twisted. Although the computational complexity reduces considerably in the fully discretised linear case computation capacity can still become in practice a limiting factor. The question arises whether it is possible to derive and measure such image formation aspects by means of torsion and curvature machines without introducing blurring of the image dened on the intersection of a half space and the spatio-temporal image domain. Adopting the viewpoint of an internal observer [41, 35], who in-

Dierential and Integral Geometry of Linear Scale-Spaces troduces an ane connection, one can circumvent capacity problems and derive torsion and curvature aspects for the similarity jet of a linear scalespace despite single-valuedness of the dierential structure of the input image [41, 35]. Moreover, one can exploit this image induced ane connection to dene new scaling paradigms, which only locally assume e.g. a Minkovski space-time to exist with a Lorentz metric structure, but with a global structure determined by the (metric)-ane connection [41, 35]. Degeneracy of the connection can cause undesirable singularities in related geometric objects, such as the torsion tensor and curvature tensor. But choosing a proper ane connection essential singularities due to the vanishing of structures of the similarity jet of the spatiotemporal input images can be avoided. It should be reckoned that the machines related to a particular connection for quantifying the Volterra process are intrinsically multi-local operators. In the invariant theoretic approach [40, 39, 42, 36] essential singularities do not occur, unless rational properties of the similarity jet are used to read out the morphology over space-time and under linear scaling. The morphology of the image under linear scaling and small perturbations is described in terms of a complete and irreducible set of (multi)-local invariants. The importance of multi-local invariants cannot be underestimated, for they naturally pop up as components of vector elds in the slot-machines or in these machines themselves measuring some twist or curvature of the input image. Having indicated some of the advantages and drawbacks of the dierential and integral geometric method for describing spatio-temporal input images the similarity jet of a spatio-temporal input image is dened in section 2. In section 3 a summary of dierential and integral geometry is presented. Finally, in section 4 the geometric theory presented in section 3 is applied to geometries living on the similarity jet of spatio-temporal input images.

2. The Similarity Jet The linear scale-space theory is treated in order to obtain a similarity jet of a spatio-temporal greyvalued input image. For extensive expositions on

7

the continuous, semi-discrete and discrete linear scale-space theory the reader is referred to [21, 29, 47, 43, 36]. Assume that space-time (M; g) can be modeled by the product of Riemannian manifolds (Mx; gx ) and (Mt ; gt ), i.e. M = Mx Mt and g = gx gt . Here Mx is a D-dimensional manifold representing the space of positions, and Mt a 1-dimensional manifold representing the space of times. Furthermore, the spatial and temporal metric tensor elds gx and gt are bilinear forms on the product of corresponding tangent bundles and coincide with the standard inner product on RD and R, respectively. So the space-time manifold (M; g) can be represented by a (D; 1)-dimensional Euclidean product space. Now let L0 be a (not necessarily smooth) spatio-temporal input image of a density eld dened on a (D; 1)-dimensional (bounded) space-time domain D M onto a 1-dimensional manifold N representing the density values: L0 : D M ! N: In case of a bounded domain D this domain can be a product of a D-dimensional spatial domain Dx and an 1-dimensional temporal domain Dt , i.e. D = Dx Dt . The reason for explicitly stating that space-time has a product topology of Euclidean spaces is that space and time are in the Newtonian context physically not on an equal footing. However, in a vision system one may assume them to be commensurable, for the realisation of the observed eld over horizontal and vertical layers allows such an interpretation (of course, if so, it would be more than natural to assume space-time to be a Minkovski space-time with a Lorentzian metric structure. However, this would only aect the physical interpretation, but not the line of reasoning in the sequel). If the vision system performs a smoothing of a density eld observed over space-time, then the system may generate a 2-parameter scale-space of images L of the input image L0 [21, 43, 36]: L : D Rx Rt ! N; with initial condition lim L(x; t; sx ; st ) = L0 (x; t); (s ;s )#(0;0) x t

where positions x 2 Dx , times t 2 Dt , spatial scale sx 2 Rx and temporal scale st 2 Rt are

8


all independent physical observables having only meaning within the vision system. If the smoothing of the eld satises a law of conservation of total density, then, according to the divergence theorem, the smoothing by the vision system should be governed by a system of spatio-temporal diusion equations:

@L ? @ 2L = 0; @sx @xi @xi @L ? @ 2 L = 0: @st @t@t

Note that, whenever the space-time domain D is bounded the initial conditions have to be supplemented by boundary conditions (for examples see again [43, 36]). The solution of the above Cauchy problem, i.e. the diusion equation and the initialboundary value condition, can be found by deriving their Green's functions. These functions can be obtained by reducing the diusion equations to so-called Weber dierential equations in terms of coordinates and energies [43, 36]:

= pxs ; x t = ps ; t ? a = sx sbt L; a; b 2 R;

that are invariant under the 2-parameter (; )scaling group of point transformations, the socalled group of similarity transformations: (~x; t~; s~x; s~t ; L~ ) = (x; t; 2 sx; 2 st ; 2a 2b L); with ; 2 R+ . After this reduction procedure the fundamental geometric and physical object constructed on the basis of the spatio-temporal input image L0 is the continuous similarity jet j 1 0 : Denition 1. by:

with where

The similarity jet j 1 0 is dened

j 1 L0 = f; ; ~n;~mg ; n+D m+1

~n;~m = sx 2 st 2 L~n;~m ;

L~n;~m = (L0 G~xn ) Gtm~ ;

and n = j~nj 2 Z+0 and m = jm ~ j 2 Z+0 are the spatial order and temporal order of dierentiation, respectively, and Gx and Gt are the spatial and temporal Green's functions consistent with the above Cauchy problem. These Green's functions can be derived by using the normalised Gaussian as Green's function on Mx RD and Mt R, and by applying, subsequently, the method of images or Fourier transform techniques [43, 36]. The found similarity jet is a unique and complete physical object for quantifying the dynamics of a scene (see section 4), as it represents given an input mage the states of a vision system under linear scaling (latent in those of the vertical and horizontal layers) in terms of measurable physical entities, namely distances in lengths and energies, with which units of measure are associated, and a certain topology [35]. Let's conclude our discussion of linear scalespace theory by pointing out some of its drawbacks and virtues in describing the geometry of the similarity jets of spatio-temporal input images. In practice fully discretising the continuous theory becomes indispensable, for in the continuous and semi-discrete case the kernels are dened at innitely many points in the image domain, and consists of an innite sum of translated Gaussians in the case of boundary conditions [43, 36]. In the fully discretised case the support of the kernel is for any integer scale denitely bounded, such that, even in the case of diusion with boundary conditions, the number of "Gaussians" contributing, to the kernel is also nite. Furthermore, integration over the image domain can be simply replaced by summations over a nite number of measurements of the initial input image. Linear scale-space theory enables the application of modern dierential and integral geometry [41, 35]. An external observer can quantify the lack of order of contact, the multi-valuedness (non-dierentiability) of the input image or better the torsion and curvature in the initial image formation process over scale and on a hyper-tube in space-time. An internal observer only aware of the outcomes of linear scaling and reading out the similarity jet according to a rule determined by the input image under that scaling can also

Dierential and Integral Geometry of Linear Scale-Spaces detect the torsion and curvature of a geometry determined by the similarity jet. Note that an external observer can detect multi-valuedness of the similarity jet, whereas the internal observer does not. Instead the internal observer concludes that the induced ane connection, that is the rule for relating structures over the image domain, implies a non-trivial geometry of the image formation process under linear scaling. If the image domain is curved, twisted and has a dierent lattice symmetry in the fully discretised case of linear scale-space theory [43, 36], and if the scaling paradigm has to be also invariant under other symmetries than similarity and Euclidean invariance as postulated in linear scalespace theory [47, 41, 35], then the scaling has to be adjusted to these geometric aspects in the image formation. The most obvious dierence between the linear and nonlinear scaling theories is latent in the supposed connection living on the image domain, which is needed to relate image properties with each other. For example, in the linear case the connection is assumed to be globally Euclidean, whereas in the nonlinear case it is assumed to be locally Euclidean, but (hardly reckoned or made explicit) non-at and twisted according to properties of the input image. The geometry of the similarity jets obtained through the use of the nonlinear scaling mechanisms, however, does not yield any new insights in the dierential and integral geometric description of spatiotemporal input images, except for the imposed additional symmetries.

3. Dierential and Integral Geometry The dierential and integral geometry of manifolds is formally quantied by means of (metric or non-metric) connection one-forms, torsion tensors, curvature tensors, ane translation vectors and ane rotation vectors. For extensive expositions on all the above subjects the reader is referred to [46, 32, 44, 45, 4]. The relations between these geometric entities and physical objects encountered in defect theory, general relativity and gauge eld theories [17, 19] are pointed out. This relationship will encourage us to combine the above mathematical and physical disciplines with linear scale-space theory (see section 4).

9

3.1. Dierential Geometry

Firstly, the system of Cartan structure equations for a manifold with a general ane connection are stated. The dierential geometry is caught by the frame eld, the connection one-forms, the torsion two-form, the curvature two-form and the Bianchi identities over the manifold. Relations between these geometric (mathematical) objects and those physical objects encountered in defect theory and gauge eld theories are briey indicated. Secondly, manifolds with a metric connection are dened as manifolds, for which the metric tensor is compatible with the connection. After imposing a duality constraint between the connection one-forms, that may dene the metric tensor, and a frame vector eld, the implications of (in)compatibility for the remaining connectionone forms are made partly explicit together with those for the related torsion tensor and curvature tensor. Let M be a n-dimensional base manifold and consider the ane frame bundle B P (M; ; A(n; R)) where P is the total space consisting of all ane frames Vp at each point p 2 M , : P ! M is the projection and A(n; R) = GL(n; R) T (n; R) the full ane group, where Gl(n; R) is the general linear group and T (n; R) the translational group, acting on the right semidirectly on each local ane frame Vp . Denition 2. A local ane frame Vp at a point p of a n-dimensional manifold M is dened as Vp = (x; e1 ; : : : ; en ); x; ei 2 TpM; where Tp M is the tangent space to the manifold M at point p.

In order to compare and relate the local ane frames over the manifold M an ane connection ? is specied which denes in turn the covariant derivative operator. Denition 3. An ane connection ? on a ndimensional manifold M is dened by one-forms !i and !ji on the tangent bundle TM :

rx = !i ei ; rei = !ij ej ;

10


where denotes the tensor product, and r the covariant derivative operator. Note that, the connection one-forms and the frame vector eld need not be related through some set of constraints. This freedom will be crucial in describing the similarity jet (Denition 1) in section 4. Despite the seemingly immense freedom of choice of an ane connection ? the connection one-forms !i and !ij do satisfy so-called structure equations:

Theorem 1. The structure equations for an ane connection ? are given by:

D!i = d!i + !ki ^ !k = i ; D!ij = d!ij + !kj ^ !ik = i j ; where d denotes the ordinary exterior derivative, D the covariant exterior derivative operator, ^ the exterior product of tensors and , i the torsion two-forms and i j the curvature two-forms.

Proof: See [46]

Note that the torsion two-forms i and the curvature two-forms i j are related to the components Tjk i and Rikl j of the torsion tensor T and curvature tensor R as follows:

i = 12 Tjk i !j ^ !k ;

i j = 12 Rikl j !k ^ !l ;

with

T = Tjk i !j !k ei ; R = Rikl j !i !k !l ej : Applying again the covariant exterior derivative operator D to these structure equations yields the integrability conditions for the ane connection ?, i.e. the Bianchi identities:

Theorem 2. The Bianchi identities for the

ane connection ? of the manifold M are given by:

D i = j i ^ !j ; D i j = 0:

Proof: See [46, 32]

The above structure equations and the Bianchi identities constitute now a system of so-called Cartan structure equations. This system is closed, which implies that the torsion and curvature tensor together with the connection one-forms form the fundamental dierential geometric objects for assessing equivalence of two manifolds with an ane connection and for deriving higher order differential geometric objects. The latter geometric objects are obtained by taking covariant derivatives of the torsion tensor and the curvature tensor. For irreducible components of these derived geometric objects see [4]. In defect theory and gauge eld theories [12, 19, 17] the above geometric objects and identities have particular physical counterparts. For example, the Bianchi identities can be identied with conservation laws for dislocation densities and disclination densities, or conservation laws for the angular momentum density and the energy-momentum density, respectively. Now one can subdivide the above manifolds with an ane connection into those with an ane connection compatible with a metric tensor (metric-ane connection) and those with an ane connection incompatible with a metric tensor [19, 12]. In order to dene a metric-ane connection one assumes that the connection oneforms !i determine a metric tensor , that the connection one-forms !i and the frame vector elds ei are each duals, and that the covariant derivative of the metric tensor vanishes identically. Denition 4. A metric tensor on a ndimensional manifold M with ane connection (Denition 3) is dened by:

= !i !i : Denition 5. For a n-dimensional manifold with ane connection (Denition 3) the connection one-forms !i and the frame vector elds ej are dual, if and only if:

!i (ej ) = ji ; where ji is the Kronecker delta-function.

Dierential and Integral Geometry of Linear Scale-Spaces Denition 6. A n-dimensional manifold with a metric-ane connection is a manifold with an ane connection (Denition 3) and metric tensor (Denition 4), for which the following compatibility condition holds:

r = 0: This means that the angles between and lengths of vectors measured by the metric tensor under parallel transport associated with the ane connection ? are preserved. The question arises how to express the connection one-forms !j i in terms of the one-forms !i and the frame vector elds ej given the compatibility constraint (Denition 6) and the duality constraint (Denition 5). In order to establish this relationship quantify with respect to a local reference frame Vp0 = (x0 ; e0i ), the component functions epi of the frame vector elds ei as: ei = e0p epi ;

and the components ej q of the one-forms !j through the duality constraints: ep i ep j = g i j ; epi eq i = gpq ; where gpq are the components of the metric tensor living on the reference tangent bundle spanned locally by the reference frame vector elds e0p . On the basis of the denition of an ane connection (Denition 3) and the above representations of the frame vector elds ei it is easily veried that the connection one-forms !j i are directly related to so-called connection coecients ?jk i :

!j i = ?jk i !k ; ?jk i = ej (log E ); E = (epi ): Note that the connection coecients evidently directly relate to the well-known Weber-fractions often encountered in any eld of exact science. In this case the components of the torsion tensor T and the curvature tensor R can be expressed by means of the frame vector elds ei and the connection coecients ?jk i as follows: Tjk k = 21 (?jk i ? ?kj i ); Rjkl i = ek ?jl i ? ej ?kl i + ?jl m ?mki ? ?kl n ?nji :

11

Note that the torsion not necessarily has to be identically zero for a manifold with a metric-ane connection. If the torsion tensor vanishes one speaks of Riemannian manifolds, else of CartanEinstein manifolds. Furthermore, the reference frame may be twisted and curved due to the connection and possibly metric living on the reference frame bundle. An external observer is aware of these reference properties, but will consider them as unimportant for the essential dynamics of a scene. Similarly, the latter reference frame intricacies are eliminated by an internal observer only being sensible to twist and curvature through an image induced connection. Summarising, the ane connections on a manifold satisfy systems of Cartan structure equations. These equations determine through the torsion two-forms and the curvature two-forms and the connection one-forms the equivalence relations for a manifold. Choosing a metric on the basis of the connection one-forms !i and imposing the duality constraint between these forms and the frame vector elds ej , one can construct a metric connection, such that the components of the connection one-forms !ji are fully determined by the former geometric objects. Note that, the dierential geometry treated here mainly concerns that from the viewpoint of an internal observer [17]. However, in section 4 also examples are presented of dierential geometry (Riemannian geometry) from the viewpoint of an external observer. 3.2. Integral Geometry

In this subsection integral invariants are dened for n-dimensional manifolds M with an ane connection ?, and those with an ane connection ? and a metric tensor . Firstly, consider a n-dimensional manifold M with an ane connection ? (metric or non-metric) as dened in the previous subsection 3.1. Two fundamental integral invariants, based on the connection one-forms (!i ; !ji ) and the frame vector elds ek , are the ane translation vector and the ane rotation vectors belonging to an ane displacement around an innitesimally small contour [4]:

12


Denition 7. The ane translation vector b and the ane rotation vectors are dened by:

b = fi =

I

IC C

rx; rei ;

where C is a innitesimally small closed loop on a 2-dimensional submanifold S of M with the same induced ane connection ?. One may obtain the submanifold S by setting D ? 2 of the connection one-forms !i equal to zero. Using Stokes' theorem and the structure equations (Theorem 1) the ane translation and rotation vectors can be written as [4]:

b = fi =

Z

ZSc Sc

i ei ;

i j ej ;

where Sc S is an innitesimally small patch with boundary C . The nonvanishing of the translation vector b indicates the presence of torsion on the given manifold, whereas that of the ane rotation vectors fi indicates the presence of curvature. The above vectors constitute the integral versions of the structure equations (Theorem 1) and form measures for the inhomogeneity of the ane group action. The Bianchi identities (Theorem 2) imply that the vectors satisfy the following superposition principles [19]: X B = b; X Fi = fi : For integral invariants of higher degree and interesting dimensional restrictions the reader is again referred to Cartan's memoir [4]. These superposition principles play a crucial role in quantifying the geometry or dynamics at critical points and along ridges and ruts as will be shown in section 4. Furthermore, it is also worthwhile to mention that these integral invariants are very stable within the context of linear scale-space framework given a non-degenerate (metric) ane connection. Secondly, consider a n-dimensional manifold M with an ane connection ? and a metric tensor

as dened in the previous subsection. For such a manifold the denitions of the above integral

invariants remain the same. But because of the existence of the metric tensor it is possible to decompose e.g. the ane rotation vectors fi into "rotational" parts firot, "dilational" parts fdil and "shearing" parts fshear :

fi = firot + fidil + fishear ;

with

firot = fidil =

Z ZSc Sc

jijj ej ;

(kk) i j ej ;

where j j and ( ) denote the full antisymmetrisation and symmetrisation operator, respectively. Besides these local integral invariants there exist other integral invariants, e.g. the volume measure and the Euler characteristic [46, 32], that can be constructed on the basis of a metric connection, the covariant derivatives of the torsion tensor and those of the curvature tensor [4]. In defect theory [17] and gauge eld theory [19] one refers to the translation vector b and the rotation vectors fi as the Burgers vector eld and the Frank vector elds, respectively. They are conceived as manifestation of dislocations and disclinations along defect lines created by Volterra processes (see gures 1 and 2). Furthermore, the superposition principles for those vectors coincide with Kirchho's law for currents. Summarising, the dierential and integral geometry of manifolds with ane connection and possibly a metric structure is captured by means of the Cartan structure equations. The chosen frame elds, connection one-forms and metric structure fully determine through these equations the curvature tensor, the torsion tensor and their covariant derivatives, as well as integral invariants, such as the ane translation vector and the ane rotation vectors. Some of the physical counterparts for the frame elds, the connection one-forms and the Cartan structure equations are briey indicated. The latter observation encourages us to proceed in applying the just presented formalism to geometries determined by similarity jets in the next section. As pointed out in the introduction in general there does not exists any order of contact in structures related to an input image implying that the above analysis holds only

Dierential and Integral Geometry of Linear Scale-Spaces

a

according to a rule specied by a (metric)-ane connection such that the image formation under linear scaling can still be expressed in terms of non-trivial curvature and torsion aspects.

b

u x

b

x + u(x)

Fig. 1. Burgers vector eld due to a dislocation caused by a displacement eld . The underlying Volterra process breaks the square lattice symmetry (a) into (b) by removing lattice points and moving the horizontal lattice lines closer to each other. Traversing the circuit in (a) one returns to the initial starting point, whereas in (b) one ends up being Burgers vector eld away from the starting point. b

u

C

b

a

b

x C

O

13

f O

Fig. 2. Frank vector f due to a wedge disclination. The underlying Volterra process breaks the square lattice symmetry (a) into (b) by cutting the initial square lattice (a) open along the negative x-axis and inserting new lattice points (b) such that the lattice surfaces are smoothly continued along the cutting line except at the origin. Traversing the circuit C in (a) and measuring the change in a vector carried along one observes no change in that vector, whereas in (b) one experiences the Frank vector eld f .

locally. However, although the internal observer in a linear scale-space cannot detect any lack of order of contact for the similarity jet, because of the regularisation property of the linear scaling paradigm, one can read out the similarity jet

4. Applications Manifolds with a metric connection, and those with ane connection and metric tensor together with their geometries are constructed on the basis of the similarity jet (Denition 1) (see sections 4.1 and 4.2). First, however, some preliminary remarks should be made about restrictions imposed by the linear scaling paradigm, and some auxiliary geometric objects and machines should be dened. The rst remark concerns the value of imposing additional invariance conditions other than those underlying the linear scaling paradigm. Although the fundamental symmetries underlying the linear scaling paradigm concern Euclidean invariance and self-similarity, there is no reason to adopt an external observer point of view and to require additional invariance of the similarity jet such as that under the group of (monotonic) grey-value transformations L~ 0 = f (L0) 6= constant, unless one is really interested in the Euclidean geometry of isophotes and owlines [38, 11]. One should realise that the linear scaling paradigm is intrinsically not invariant under such a group of transformations nor is intended to be so. For scaling paradigms adjusted to the above invariance condition and heading for an analysis of other types of image structures the reader is referred to [47, 41, 35]. The second remark concerns the degeneracy of a metric connection and consequently the introduction of essential singularities of derived geometric objects such as torsion and curvature tensors [47, 41, 35]. These singularities occur especially in geometric scale-space theories in which the invariance under the group of grey-value transformations is taken care o. Then the pure metric relations, e.g. the ane curvatures of curves or nets, and thus the connection can become singular. However, restricting to pure classical ane invariance one can derive ane image ows that do not yield essentially singular scale-spaces [41, 35]. In order to describe objects such as branching points and defect lines in the similarity jet a local

14


reference frame is needed, that is a frame which at least is invariant under the group of similarity transformations (see section 2). The simplest local dual reference frame satisfying this invariance condition can already be constructed on the basis of only the spatio-temporal positions and scales (see Theorem 3).

Theorem 3. (; ; d ) constructed on the basis of the similarity jet and invariant under the similarity group is given by the following one-forms d :

A local dual reference frame V0 =

i di0 = pdxs ; x dD0 +1 = dss x ; x dt dD0 +2 = ps ; t ds t dD0 +3 = s : t

Proof: Show that d~0 = d0 under the simi-

larity group (Denition 1). The local reference frame V 0 , consisting of frame vectors e0 , can be derived from the following duality constraint:

d0 (e0 ) = : Note that the Greek indices denote spatiotemporal variables and scales, whereas the Latin indices denote only spatial variables. Furthermore, that the metric tensor g0 = d0 d0 constructed on the basis of the above one-forms d has components g = . The latter metric property is a consequence of the required invariance of physical measurements under a similarity group action and duality constraint. The latter metricity also considered by Eberly [7] plays however no role in describing the relevant geometry of the similarity jet as to be demonstrated shortly. However, if one accepts it as an additional structure on the similarity jet it can be used to read out particular Riemannian geometries as done by Eberly in [7]. Having constructed a local self-similar reference frame one can continue dening manifolds on the image domain with a geometry brought about by

endowing it with an image dependent connection (see sections 3, 4.1 and 4.2). In the next subsection most of the manifolds concern isophotes and owlines of grey-valued images represented by the similarity jet at a xed spatial and temporal resolution and a xed time. Denition 8. An isophote Ci of a grey-valued image at a xed spatial and temporal resolution, s0x and s0t , respectively, and a xed time t0 is dened by: Ci = f 0 ( 0 ) = constant g: Denition 9. A owline Ci of a grey-valued image at a xed spatial and temporal resolution, s0x and s0t , respectively, and a xed time t0 is dened by:

d0 0 0 Cf = dp = ( (p)) ;

where p is an arbitrary parameter.

In order to quantify their classical Euclidean differential geometries in terms of the similarity jet extensive use has been made of the method of implicit dierentiation [22, 11]. Another object studied in both the following subsections and determined by the similarity jet is a n-dimensional net N on a D-dimensional manifold M . Denition 10. A n-dimensional net N on a Ddimensional manifold M is dened by the set of integral curves for n D vector elds vi 2 TM dened on M through the similarity jet.

A reason for investigating a net is that it so to speak represents a slice of the space of observations represented by the similarity jet and inherits all the image formation (morphological) aspects brought about by linear scaling. In order to connect nets (Denition 10) over scales and space-time so-called projection oneforms are needed [20]. As the geometry of the similarity ? jet is contained in let's rst propose log u = , with u being the dynamic unit (dynamic inner scale) of the vision system, as a natural order of variation of the system activity as function of the dimensionless canonical

Dierential and Integral Geometry of Linear Scale-Spaces coordinates ; it can also be conceived as a connection one-form on the input image. The latter concept yields a solution to the commensurability (aspect ratio) problem between space-time and the image grey-values. The point of view of an internal observer is adopted, who's only aware of or adjusts to the rst order similarity jet (the internal observer has no other reference frame available than that induced by the rst order image structure). This observation xes the connection on the similarity jet without bothering about the reference frame on the system states as needed in Eberly's approach [7]. With our connections identically vanishing there is no topological or better physical information latent in the similarity jet other than the trivial one of a constant input image. Furthermore, labeling the space-time events attains only meaning in the context of the input image. The desired connection one-forms for isophotes on the image domain come about by considering the following variational principle: = 0; and decomposing it into three parts, the rst part due to the spatial scaling of the input image, ss , the second part due to the temporal scaling of the input image, ts and the third part due to the space-time behaviour of the input image, st : = ss + ts + st = 0; ss = i i + D+1 D+1 = 0; ts = D+2 D+2 + D+3 D+3 = 0; st = i i + D+2 D+2 = 0:

15

One could, of course, divide each one-form above by the component function of its last term yielding other projection one-forms: !~ D+1 = ? D+1i di + dD+1 ; i i D +3 D+2 dD+2 + dD+3 ; D +2 !~ = ? D+2 D+2 !~ D+3 = ? D+2i di + dD+2 : i i The second term of these one-forms are readily identied with translations along the spatial scale axis, the temporal scale axis and the time axis. The rst term in !~ D+3 corresponds to the translation in the spatial domain due to a velocity. Unfortunately, these alternative one-forms can become singular and physically not measurable as in the case of so-called drift velocities introduced by Lindeberg [28] as soon as one approaches extrema. These one-forms or velocities ask for adaptive sampling and further analysis in the neighbourhood of such points. The similarity jet, however, is not hampered by such a singular behaviour, neither are the projection one-forms (Denition 11) or related vector elds with the same component functions (that are not satisfying some duality constraint). The latter property of the mentioned geometric objects will play a crucial role in reading out the geometry of the similarity jet or better the morphology of the image formation over time and under linear scaling at critical points (see subsection 4.2.2). The non-degeneracy also occurs by denition for multi-local invariants of the similarity jet computed around a critical point between two patches bordered by ridges and ruts [35]; for example, measuring the jump in the average image gradient eld between the two patches will Solving the above variational problems under the certainly not yield essential singularities. assumption of proximity and equal luminance [20] However, one might object that the linear the desired projection one-forms are derived easily. scaling paradigm is not concerned with connecting points, but rather in connecting images Denition 11. The projection one-forms for and subimages, whereas the nonlinear scaling the similarity jet are dened by: paradigms, such as the shortening ows, are concerned with connecting points. A similar reason!D+1 = ? D+12 i di + i 2 i dD+1 ; ing can be applied for the space-time dynamics of the image. Realising this dierence in the ob !D+2 = ? D+32 D+2 dD+2 + D+22 D+2 dD+3 ; jectives of the scaling paradigms and the dynamics of a space-time eld one is immediately in!D+3 = ? D+22 i di + i 2 i dD+2 : clined to replace the above projection one-forms

16


by the following very simple one-forms, namely !D+i = dD+i where i = 1; 2; 3. It should be clear that the analysis of the above singularities on the basis of these one-forms becomes trivial and not hampered by all kinds of degeneracies in the similarity jet. As soon as space, time and the scale variables are viewed as incommensurable physical entities, then one is urged to come up with a machine for reading out the geometry other than a metric connection. This happens as soon as e.g. space-time is viewed as a product of two manifolds with their own metric structures as is the case in linear scalespace (see section 2). A geometric machine that is not troubled by such a compatibility constraint, but still is capable to compare geometric objects living on a product manifold is the Lie derivative with respect to some vector eld [46, 33]: Denition 12. The Lie derivative LX Y of a geometric object Y with respect to vector eld X is dened at each point p on a manifold M by: ? ? ?Y ; (L Y ) = lim 1 Y X

p

?t

(t;p)

p

t where t : Tp M ! T(t;p) an isomorphism. t!0

Note that the geometric object compared along a certain trajectory by the Lie derivative (Denition 12) can be e.g. a connection, a torsion tensor or a curvature tensor dened in the previous section 3 living on either submanifold. For expressions of Lie derivatives in terms of component functions of the vector eld and the geometric object, and their well-known properties the reader is referred to [33]. 4.1. Manifolds with Metric Connection

Spatio-temporal geometric objects are constructed on the basis of the similarity jet in order to derive for a related manifold with a metric connection the fundamental dierential and integral geometric properties. Firstly, at one level of spatial and temporal scale the Euclidean geometry of space curves, hypersurfaces and nets are quantied in terms of the similarity jet (see subsection 4.1.1). Secondly, a metric-ane geometry

of nets is formulated and quantied in terms of the similarity jet again at xed scales (see subsection 4.1.2). 4.1.1. Euclidean Geometry of Space Curves, Surfaces and Nets. Consider a space curve C in D-dimensional Euclidean space E D at xed levels of spatial and temporal resolutions and at a xed time. Denition 13. A space curve C in Ddimensional Euclidean space E D is dened as a collection of vectors x 2 E D that are twice differentiable with respect to an arbitrary parameter p: C fx 2 C 2 (E D )jx = x(p) ^ p 2 [0; 1]g:

If curve C is a non-degenerate curve parametrised by Euclidean arclength parameter s, then in the neighbourhood of regular points the frame V , i.c. a positively oriented orthonormal basis, is given by: V = (x; e1 ; : : : ; en );

dxi dxi ; ds = e1 = dx ds dp dp

12

dp;

where x is determined in the reference system and the remaining unit frame vector elds e2; : : : ; en are obtained by means of the Gramm-Schmidt orthonormalisation process applied to the vectors d1 x ; : : : ; dnnx . Now the Euclidean metric connecds ds tion one-forms !i and !ji for its metric connection (see previous section 3) are given by: !1 = ds; !2 = : : : = !n = 0; !ij = ?!ji ; !kk+1 = kk ds; !lk = 0; 8l > k + 1; where the latter one-forms !ij can be simply expressed in the following matrix form: (!j ) = ds d log E ; i

ds

where the matrix E has n columns dened by the component functions of the frame vector elds ei : E = (e1 : : : en ): Higher order Euclidean dierential geometric invariants are subsequently straightforwardly derived by taking derivatives of the found curvatures kk with respect to Euclidean arclength parameter s. Furthermore, space curves can be described in

Dierential and Integral Geometry of Linear Scale-Spaces terms of a Euclidean canonical expansion:

x(s) =

1 1 dk x X

k k s : k ! ds k=0

Do realise that the above curvatures kk are invariant zero-forms being connection coecients for the rotation group and have nothing to do with the curvature two-form encountered in section 3. Although the frame eld construction is impossible at e.g. inection points, the Euclidean curvatures might still be measurable. Whether these curvatures can be computed is set by the local Euclidean structure left and right from those points. In order to derive integral invariants for (segments of) a curve the frames and the curvatures form appropriate observables. For example, the Euclidean curvatures integrated over a curve segment are global invariants unaected by the associated group action. Let's give two examples of the above geometry for space curves dened on the basis of the spatial layers of the similarity jet, i.e. isophotes and owlines of a 2-dimensional grey-valued image, and owlines of a 3-dimensional grey-valued image. Example 1. For a planar curve the Euclidean frame vector elds are given by: i ei1 = dx ds ; 1 de1 ? 21 dei 1 ei2 = de ds ds ds ;

and the connection one-forms are given by: 2 0; !1 = ds; !0 =kds (!ij ) = ?kds 0 ; with the Euclidean curvature k given by: 1 k = de ds e2 ;

where denotes the standard inner product on a Euclidean space. Choose in case of a 2-dimensional grey-valued image at a point 0 of an isophote (Denition 8) image dependent local Cartesian coordinates v and w, in which the curve can be described by = (v; w(v)) such that wv = dw dv = 0. Applying the

17

method of implicit dierentiation introduced and exploited notably by [22, 11] yields the following variations up to second order: (v; w(v)) = 0 2 R; v = v v + w wv v = 0; vv = vv 2 v + 2wv wv 2 v + ww wv2 2 v + w wvv 2 v = 0; with e.g. = @ ; v

@v

and using the requirement, wv = 0, one obtains up to second order the following description of the 2 v vv isophote w(v) = ? 2w . Thus the w coordinate is a consequence of a projection of a vector onto the local unit normal image gradient dening the second frame vector eld e2 , and the coordinate v that onto the rst frame vector eld e1 . In terms of the similarity jet the Euclidean frame vector elds eij and the curvature eld ki of the isophotes (Denition 8) are given by: ! ki k i ; e1 = (j j ) 12 ! i i e2 = ; (j j ) 12 ki = i ij jp pq23 q : (j j ) Note that the derivatives with respect to the coordinates v and w are directional derivatives: @ @ i i @v = e1 j=0 r ; @w = e2 j=0 r ; with

r = @@ i : s=s0

Here the viewpoint of an external observer is adopted by assuming the curve to be embedded in a higher dimensional Euclidean space. For the owlines of a 2-dimensional grey-valued image the Euclidean frame vector elds efi and the owline curvature eld kf are given by: ef1 = p i ; i i

18

with


def1 ; ds f kf = ef2 deds1 ;

de1 de1 ? 21 de1 ds ds ds ; 1 de d

ef2 =

e2 =

d = p i @ : ds j j @ i

e3 = t ke1 + ds2 ; ds = p i @x@ i ; j j (1) (2) (1) (2) k = ((x x ) (x x )) 21 ; (1) (2) (3) t = (x xk2 ) x ;

Note that the latter derivative operator is not @ , and equivalent to the directional derivative @w that immediately the viewpoint of an internal observer is adopted. Integral R invariants, such as the total rotation index 21 C kds for the image of isophotes or owlines can be computed by considering fully discretised similarity jets [43, 36]. One may even consider a total rotation index on a submanifold consisting of a set of segments of e.g. isophotes that lie within a region bounded by ridges and ruts [25, 7, 35] (see also Example 4). Example 2. For a 3-dimensional space curve C the frame vector elds ei are given by:

e1 = dx ds ; de1 de1 ? 21 de1 e2 = ds ds ds ; e3 = e 1 e 2 ;

and the connection one-forms by: 2 3 !1 = ds; 0 !0 = !kds= 0;0 1 (!ji ) = @ ?kds 0 tds A ; 0 ?ds 0 with the Euclidean curvature k and torsion t:

1 e de1 k = e1 de 1 ds ds de t = ?e2 ds3 :

12

;

The frame vector elds ei , the curvature eld k and the torsion eld t for 3-dimensional owlines can be straightforwardly expressed in terms of a corresponding similarity jet as: i e1 = ; (n n ) 12

i where the xi = ddsxi are given in terms of image properties as:

x(1) = e1 ; x(2) = (s s )p (pj ? ()2p pq q )j ; n n

x(3)

= p pq qk3 + p q pqk3 ? (n n ) 2 (n n ) 2 3 (p pq q )5r rk ? 2 (p pq qr )5 r k ? (n n ) 2 (n n ) 2 (p q r pqr )k ? 4 (ppq q )2 k : (n n) 52 (n n) 72

Again realise that the owline coincides with the integral curve corresponding to the normalised gradient vector eld of the dierential invariant . Consider now hypersurfaces S in Euclidean space E D . The Euclidean frame eld V for such a surface S and its Euclidean connection one-forms are given by:

V = (x; e1 ; : : : ; eD ); !D = 0; !1 ; : : : ; !D?1 6= 0; !ij = 0; j = 1; : : : D ? 1; !iD = hij !j ; hij = hji : Here e1 ; : : : ; eD?1 form an orthonormal basis spanning the tangent bundle to the surface S and eD its unit normal bundle. The innitely many times dierentiable functions hij on S dene the elementary symmetric polynomials pk jk k j1 pk = (n ?n! k)! ji11 :::i :::jk hi1 : : : hik :


19

of the surface in terms of u- and v-coordinates: if i = i ; j = j ; hspace1ex 6= ; w(u; v) = 1 @ 2 w u2 + 2 @ 2w uv + @ 2 w v2 ; 2 @u2 @u@v @v2 fj1 ; : : : ; jk g 6= fi1; : : : ; ik g if j1 ; : : : ; jk even permutation with of i1 ; : : : ; ik @ 2 w = ? uu ; if j1 ; : : : ; jk odd permutation @u@u w of i1 ; : : : ; ik 2 @ w = ? uv ; that in turn determine the principal curvatures @u@v w [46]. @ 2 w = ? vv : @v@v w Example 3. For an isophote of a 3-dimensional Note that, again, the derivatives of the similarity grey-valued image an appropriate frame eld V solution concern directional derivatives with reto position the surface is readily obtained [37]: spect to the frame vector elds constituting frame V0 = (x; u^; v^; w^); V0 . The mean curvature H and the Gaussian curvature K of the surface are then given by: with uu + vv ; H = ? ijk j km m uî = 2w 1 2 (ijk j km m ipq p qr r ) ? 2uv : vî = ijk w^j u^k K = uu vv 2w i wî = These curvatures are in turn directly related to (j j ) 12 the principal curvatures i through the eigenvalue problem for the components of the second fundaIt is shown how the desired so-called Frenet frame mental form: eld V [4] can be realised by applying a suitable ! rotation around the w-axis to the unit tangent vec@2w ? @2w @u@v tors u^ and v^. This rotation must bring the direc= det @u@u @2w @2w tions of the new x1 - and x2 -axis in the tangent @u@v @v@v ? plane to the surface in line with the principal di2 ? 2H + K = 0: rections corresponding to the principal curvatures Thus the principal curvatures can be expressed as: 1 and 2 , respectively. Performing such an operp 2 ation yields a Euclidean description of the surface = H H ? K: 1 ; 2 1 2 3 with respect to the coordinate system (x ; x ; x ) in terms of derivatives of the principal curvatures In order to nd the rotation angle for which with respect to both the x1 - and the x2 -axis: the frame lines up with the principle curvature directions consider an arbitrary unit tangent vector 2 X 1 p i X 1 @ ^: x3 (x1 ; x2 ) = i1 : : : @xip ( p + 2)! @x i=1 p=0 ^ u^ cos + v^ sin ; xi xi xi1 : : : xip : and normal curvature ^ along that direction: The explicit quantication of the principal cur^ ^ vatures and directions in terms of the similarity ^ ? i ij j12 : (p p ) jet of a 3-dimensional grey-valued image follows straightforwardly through the application of the The principal directions are now obtained by submethod of implicit dierentiation [22, 11]. Using stituting ^ into the expression for ^ and solving the gauging conditions, u = 0 and v = 0, the for angle the following equation: solution of the variational principle up to second i = ^: order gives the following second order description

with

80 > > > < i :::i 1 k j1 :::jk = > 1 > > : ?1

20


This is consistent with the fact that the extrema of the normal curvature coincide with the principal curvatures. Using elementary properties of trigonometric functions and the expression for the mean curvature H the equation reduces to: H ? i = uu2? vv cos 2 + uv sin 2; w w or sin (2 + ) = i cos with = arctan uu? vv ; uv?1 i = (H ? i ) uv : w Solving this trigonometric equation for one of the principal curvatures, the rotation angle i for letting the u- and v-axis be parallel to the x1 - and x2 -axis is given by: i = ? 21 + arcsin (i cos ): Using the method of implicit dierentiation to nd a description of the surface in terms of w = w(u; v) and applying the derived rotation over angle i to go from the coordinates (u; v; w) to the new coordinates (x1 ; x2 ; x3 ) a canonical Euclidean invariant description is established: x3 (x1 ; x2 ) = w(x1 cos i ? x2 sin i ; x1 sin i + x2 cos i ): Finally, consider an arbitrary n-dimensional net embedded in a D-dimensional Euclidean space E D . The Euclidean frame eld V for such a net N and its Euclidean connection one-forms are given by:

V = (x; e1 ; : : : ; en); ei = pvvi v ; i i i i i i k ! = ds ; !j = ?kj ! ; where e1; : : : ; en form an orthonormal frame eld

spanning the tangent bundle to the local Euclidean space E D and to N simultaneously, dsi innitesimally small Euclidean displacements along the integral curves on N generated by vector elds ei (satisfying the duality constraint (Denition

5)), and the connection coecients ?jk i are given by:

log E i ; E = (e1 : : : en ) : ?kj i = @ @s k j

The torsion tensor T and the curvature tensor R of the net, endowed with the above Euclidean

connection, can easily be expressed in terms of the connection coecients and derivatives with respect to ei = dsdi . Similarly the integral invariants for these nets can be obtained (see section 3). Example 4. Consider a 2-dimensional net dened by isophotes and owlines of a 2-dimensional grey-valued image . Choose on this net as Euclidean frame vector elds ei :

e1 = ei1 ; e2 = ef1 ; where the frame vector elds e1 and e2 are unit tangent vector elds to the isophotes and owlines, respectively (see gure 3 and also Example 1). Next choose on the net the following Euclidean connection one-forms:

!1 = dsi ; !2 = dsf ; !ji = ?kj i !k ; with connection coecients: @ log E i i ?kj = @sk ; E = (e1 e2 ) ; sk = si ; sf ; j

where si is the Euclidean arclength along the isophote and sf the Euclidean arclength along the owline. These coecients are readily expressed in terms of the Euclidean isophote curvature ki and owline curvature kf (see Example 4):

0 ki ?ki 0 ; 0 ?k i (?2j ) = k 0 f : f One observes that the connection is not symmetric, and that the torsion tensor and the curvature tensor are not vanishing. It is easily shown that the component functions of these tensors can be expressed in terms of linear combinations (?1j i ) =


Fig. 3. From left to right input image and its Euclidean frame eld V with the rst frame vector eld tangent to the isophotes.

Fig. 4. From left to right the curves on which the isophote curvature eld i and the owline curvature eld f vanish together with the isophotes of the input image depicted in gure 3. k

k

of derivatives of the owline curvature and the isophote curvature as already noticed in [38]. In gure 4 the locations where the isophote and owline curvature eld are vanishing are computed on the basis of the Euclidean connection induced by the input image in gure 3. The question arises which are the essential physical objects of the above net invariant under diffeomorphisms of the image domain E 2 . It is clear that the set of (non)-isolated singularities of L0 is one of them, for the vanishing of the image gradient is not inuenced by such a dieomorphism. Not so obvious is that for the landscape of ridges and ruts of the input image L0 [25, 7, 35] (i.e. the singular curves of the steepest lines of descent and ascent in the input image)1 . The topological equivalence of these singularity sets can be explained by the fact that across them the integral curves of the image gradient have opposite con-

21

Fig. 5. Left frame: a 256 256 pixel-resolution discrete input image L0 (x; y) = 1 ? y2 ; x < 0 and L0 (x; y) = 1 ? 1 2 2 2 ((y ? x) + (y + x) )); x 0 with its center as origin. Right frame: the Euclidean length of the translation vector jbj for a linearly scaled version of that image. Note that the non-isolated set of singularities x 0 and y = 0 will instantaneously disappear upon continuous linear scaling.

vexity; the curvature vector to the owlines ips across ridges and ruts along the isophotes. Consequently the connection on the owlines at the ruts and ridges along the isophotes is completely degenerate implying that any order of derivative with respect to the Euclidean arclength parameter si of the owline curvature kf is vanishing. Because taking all orders into account and the fact that to a nite order there will always be non-ridge or non-rut points for which they are zero, it is impossible to distinguish on the basis of a pure local analysis between ridges, ruts and the borders of their inuencing zones consisting of e.g. inection points [35]. Thus it is impossible to tell on the basis of a pure local analysis whether all, a particular subset or none of the disjoint components of the zero-crossings of the owline curvature are parts of the landscape of ridges and ruts. But computing instead the torsion or equivalently the translation vector of the normalised owline curvature vector eld highlights the characteristics of ridge or rut points, i.e. the multivaluedness of image structures, in this case the normalised owline curvature vector eld, dened by the single-valued similarity jet. In gure 5 the length of the translation vector eld b corresponding to the frame vector elds ei in Example 4 for a discretised grey-valued input image L0 is computed by means of linear scale-space theory. It seems that the ridge (end)points on the nonisolated singularity set are detected (realise that ridges and ruts can be discerned on the basis of

22


the isophote curvature 1 [25, 7, 35]. If 1 > 0 and 2 = 0, then the points possibly belong to ridges; if 1 < 0 and 2 = 0, then to ruts). However, on

the basis of gure 3 and gure 4 one might object that the length of the translation vector eld corresponding to the above frame elds on the outer components of the zero-crossings of the owline curvature eld is non-vanishing says that one did not yet use the proper machine to point out ruts and ridges. In order to nd ridges and ruts one actually has to investigate, as mentioned in the beginning, the normalised owline curvature vector eld (preserving its orientation) and read out the change of this orientation eld along the integral curve belonging to the normalised owline curvature vector eld, i.e. an isophote. Applying the half-space method used by an external observer [35] one detects a reversal of the orientation eld across ridges and ruts along isophotes, but across e.g. inection points one observes no change in the orientation eld. Alternatively, a topological method introduced in [35] can be used. It wraps a strip around a (non)isolated critical point and locates the extrema of the length of the image gradient eld while covering by means of the encircling tip-point of the strip the whole e.g. spatial image domain. Applying one of both methods above one obtains a branching pattern (and thus connectivity relations) of the landscape of ridges and ruts, and other (non)isolated singularity sets for free and within a linear scale-space setting. Although in the above example the ane translation vector and the ane rotation vectors can be computed for also continuous scale-space theories other integral invariants such as the Euler characteristic can only be retrieved in the case of (semi)discretised linear scale-spaces. In the continuous case the computation of the latter image properties can become a mathematical nuisance and an undesirable action from a physical point of view given the nite resolution properties of a vision system and uncertainty relations underlying any measurement. 4.1.2. Metric-ane Geometry of Nets. Assuming space and time commensurable physical entities an (D + 1)-dimensional spatio-temporal net (Denition 10) on a (D + 1)-dimensional mani-

fold M representing space-time is endowed with a metric-ane connection (Denition 6). Note that the assumption of a separate Euclidean connection on the spatial and the temporal part of the image domain has been dropped. The geometry comes about through the connection one-forms that are not constructed on the basis of a Galilean frame work [4] (see Example 5). In this framework the coupling of space-time with the external source elds are only reminiscent in the connection oneforms !ji . Example 5. Choose a (2+1)-dimensional net on a (2 + 1)-dimensional grey-valued image at xed spatial scale and temporal scale with a metricane connection brought about by the following connection one-forms !i dened on the basis of the similarity jet (Denition 1): !1 = @ log di ;

@i log dj ; !2 = ij @ @ i @ log @ log i !3 = ? @3 i d + @ @ log @ log 3 @i @i d ;

where !i refers to the spatial part of the image, and !3 to the spatio-temporal part coinciding with space-time projection one-forms. Computing on the basis of the duality constraint the frame vector elds ei and subsequently the connection coecients as in Example 4, one immediately observes that the dynamics captured by the connection one-forms, the torsion tensor and the curvature tensor inherit typical image-dependent "accelerations", and other spatio-temporal image formation aspects from the imposed metricity. As stressed in the introduction of this section a metric connection can be inconsistent with the underlying physical postulates about the commensurability or topology of space and time. Another rather disturbing fact is that in practice the metric connections or derived geometric objects all tend to become essentially singular at certain locations. In particular points, e.g. critical points and top-points a geometric description can become unfeasible. A last problem concerns the fact that

Dierential and Integral Geometry of Linear Scale-Spaces in practice the dynamics of a scene represented by an input image needs to be spatio-temporally updated. Especially, if the capacity of the vision system is limited and the analysis should be performed real-time, then the consecutive images should be spatio-temporally bounded. The question arises how to connect the similarity jets of these bounded spatio-temporal input images. In the next subsection the above mentioned problems are tackled by constructing suitable ane connections (Denition 3). 4.2. Manifolds with Ane Connection

In subsection 4.2.1 two examples of ane geometry of nets are treated. In subsection 4.2.2 the similarity jet is provided with an ane connection in order to quantify a geometry at critical points. Finally, in subsection 4.2.3 the geometry of a set of similarity jets of spatio-temporal input images is captured by supplementing this set with additional connection one-forms. 4.2.1. Ane Geometry of Nets. Two examples of ane geometries on nets are considered, because space-time is modeled by the product of Euclidean spaces (see section 2). In Example 6 a metric-ane connection on space is permitted. The additional connection-one forms come about by applying Lie derivatives (Denition 12) with respect to a velocity vector eld of these spatial one-forms, and of a one-form with the same component functions as that vector eld. In Example 7 the duality constraint and metricity is completely dropped in order to avoid the generation of singular geometric objects determined by the connection. Example 6. Choose a (2; 1)-dimensional net on (2; 1)-dimensional Euclidean space-time M and endow the spatial part of the net with a metricane connection by taking the connection oneforms !i equal to those in Example 5. The frame vector elds ei living on the spatial subtangent space of space-time are determined by the duality constraint, but are, of course, not equal to those found in Example 5. The other connection one-form !3 is used to construct a corresponding frame vector eld e3 which can be imposed a sep-

23

arate duality constraint. Now in order to obtain the connection one-forms ! other than those related to the metric connections living on the separate submanifolds Lie derivatives are taken with respect to the frame vector elds e of the components ! ( @@ ) of the connection one-forms ! :

! =

Le !

@

@ d :

Evidently, the "connection coecients" ? = (Le ! ( @@ )) in this case reect an absolute change in the order of magnitude. Subsequently, the torsion and curvature tensor given this connection can be straightforwardly computed and can be identied with spatio-temporal creation and annihilation of image details due to external elds. The morphology of an input image jumping from isophote to isophote within the similarity jet is so to speak operationalised by an ane translation vector eld and an ane rotation vector eld, such as the Burgers and Frank vector eld found in defect theory [17]. These morphological entities satisfy the superposition principle at branching points as mentioned in section 3. Note that taking the alternative projection oneform !~ 3 (frame vector eld e~3 ) on space-time given in the introduction of this section one could construct a momentum like vector eld, for example p = log e3 where log could be interpreted as some (topological) mass for the external elds yielding the similarity solution . The commutator of the Lie derivatives of this vector eld p and the spatial frame vector eld ei can subsequently related to "uncertainty relations" within the similarity jet. These uncertainty relations in turn dene the structure functions of the nonlinear Lie-algebra underlying the image formation under the linear scale-space paradigm. In the next example a completely ane connection is set up for a similar net as in the previous example. This ane connection will not yield undesirable essential singularities in derived geometric objects as one would like to on the basis of the measurability of the similarity jet. Example 7. Choose the same connection oneforms ! as in example 6, but without imposing duality nor metricity on either submanifold of space-time. Next set the components of the

24


frame vector elds e equal to those of ! . Finally, dene the connection one-forms ! again as the Lie derivatives of the components of connection one-forms ! with respect to the vector elds e . Now the connection even if it is vanishing does ensure that the torsion and the curvature and their derivatives remain measurable. One actually quanties the torsion and curvature of the similarity jet [33, 35]. 4.2.2. Geometry at Singularities. Normally singularity or catastrophe theory is used to describe the similarity jet at critical points and top-points [20, 28, 16, 34]). But here the geometry is induced by a connection constructed on the basis of the similarity jet that has on critical sheets appropriate projection one-forms related to drift velocity vector elds as introduced by Lindeberg [28]. The reader not familiar with the denitions of critical points, top-points and drift velocity vector elds should consult the references listed above. Let's restrict ourselves to a (D; 1)-dimensional net on the similarity jet of a D-dimensional spatial input image. For other type of images the procedure to follow is the same. Let's study the geometry of the similarity jet along critical curves and in the neighbourhood of top-points. Now constrain the above net to one with an ane connection by specifying rst the following connection one-forms spanning the dual tangent space to the spatial translation group:

!k = (Ak )pq Bpqi di ; k = 1; : : : ; D; with the components of matrix A and matrix B given by:

2 A = @@log i @j ; 3 log B = @@i @ j @k :

In general these connection one-forms are not vanishing on critical curves. A supplementary connection one-form !D+1 , that is not hampered by singularities on critical curves but still relates to the drift velocity introduced by Lindeberg [28], is given by:

!D+1 = (det AA?1 G)i di + det AdD+1 ;

with

2 G = @@ Dlog +1 @j :

Use now the components of these connection oneforms to construct the frame vector elds e , and dene subsequently the additional connection onefroms ! as Lie derivatives of the components of the one-forms ! with respect to the vector elds e as done in subsection 4.2.1. Now the geometry and thus the morphology of the similarity jet along critical curves can nicely be expressed in terms of measurable torsion and curvature aspects (if the reader prefers to impose a Euclidean metric connection on the spatial domain, then he has only to proceed as in the previous subsection). In a similarity jet the critical curves merge upon increasing scale, for example two Gaussian blobs with three extrema merge into one blob with one extremum. In practice these critical curves can be viewed as defect lines in defect theory [19] and the bifurcations as branching points that can be characterised by their integral invariant properties, namely the ane translation vector and the ane rotation vectors (see section 3). It is interesting to attach to these critical curves the landscape of ridges and ruts over scale in order to obtain a particular decomposition of the rst order similarity jet into cells. The poset structure (the structure of partially ordered sets), connectivity relations and (local) topological dimensions of this jet can subsequently be nicely read out by logical or topological ltering methods [24, 35]. Note that it can happen that the second and higher order parts of the similarity jet are vanishing too at critical points and that these critical points form non-isolated singularity sets. Fortunately, one almost never encounters them in similarity jets, because continuous linear scaling blurs them away instantaneously and because they disappear abruptly under perturbations of the input image. In the discrete linear scale-spaces, however, one may nd them over a whole range of scales. But beyond half the outer scale all nonisolated singularities are blurred away. Furthermore, as mentioned in the introduction of this section other projection one-form !D+1 can be considered for reading out the similarity jet along the scale axis. Taking the connection one-form dictated by the linear scaling paradigm,

Dierential and Integral Geometry of Linear Scale-Spaces i.e. !D+1 = dD+1 , the morphology of the input image under the linear scaling paradigm can be followed by tracing the transitions in the dierential and integral geometry across scale in the neighbourhood (not at) of the critical points. In this case one can still rely on only the rst and second order information content of the similarity jet instead of investigating the third order similarity jet as done above. 4.2.3. Updating Similarity Jet. Normally a spatio-temporal input image is bounded, that is its support or domain is bounded. The latter boundedness is due to the nite resolving capacity of a vision system. Now the question arises how to update such an input image. First of all one should note that it is possible to vary the spatial domain Dx as well as the temporal domain Dt with the present moment T0 running as a function of the external eld 0 . Assume the spatio-temporal domains, for convenience, to be independent of the present moment T0 and of the external eld, and to be of xed extents. Thus the scene dynamics is studied at every present moment T0 on spatiotemporal domains Dx;t(T0 ) of xed lengths. Note that the present moment is the upper bound of the temporal domain. Next let the temporal domain Dt (T0 ) at present moment T0 overlap with the temporal domain Dt (T00 ) at the present moment T00 T0 . Now squeeze the time-dierence between the two present moments, but keep T00 xed. At the present moment T00 the transition between both scene dynamics can be quantied in terms of their similarity jets as follows. Conceive the present moment as an additional degree of freedom as Koenderink did [21] and assume the classical scaling paradigm also applicable to this set of similarity jets. Extending our ideas of section 4 leads to an updated (metric)-ane scene dynamics represented by the updated similarity jets. The dual frame of the reference system is supplemented by the one-form:

dD+4 = pdT ; ST T dD+5 = dS STu ;

where ST is the scale parameter with respect to the present moment T . In order to measure the updated scene dynamics two additional connec-

25

tion one-forms are needed. On the basis of a fourth variational principle allowing to connect similarity jets under present moment scaling: TS = D+4 D+4 + D+5 D+5 = 0; and the proximity and equal total luminance condition [20] and the assumption that the present moment should be connected through the vector eld D+4 , one has the following two additional connection one-forms: !D+4 = ? D+52 D+4 dD+4 + D+4 D+4 dD+5 ; 2 D +5 D +5 ! = d : Having constructed the necessary connection oneforms ! the related frame vector elds and the other connection one-forms ! can be found analogously as in the previous subsections 4.2.1 and 4.2.2. Of course, the connection between the updated similarity jets can also be supplemented by the natural ones !D+4 = dD+4 and !D+5 = dD+5 .

5. Conclusion and Discussion The (updated) dynamics of a scene observed by a vision system at various resolutions of time and space is quantied by imposing (metric)-ane connections on a set of similarity jets of the spatiotemporal input images of the scene. The similarity jets are obtained by applying linear scale-space theory to the spatio-temporal input images. The observed dynamics is expressed in terms of dierential and integral invariants of the images at different spatial and temporal scales. Alternatively, the whole similarity jet can be described by similar invariants. Especially, the torsion two-forms, the curvature two-forms, the ane translation vector and the ane rotation vector appear to be appropriate machines to read out the dynamics or the similarity jet structure. The importance of these machines is demonstrated at critical point and top-points, and for ridges and ruts. The close relation between our dierential and integral geometric theory of scene dynamics, defect theory and gauge theory is pointed out by showing the close relation between branching points and de-

26


fect lines, and critical points and non-isolated singularity sets, respectively. Up to now space-time is modeled globally as a (D; 1)-dimensional Euclideanly at and nontwisted space, whereas an internal observer can verify this space-time property only locally. It makes, therefore, sense to generalise linear scalespace theory in this context [47, 41, 35]. It is evident that our formalism is readily extended to nonlinear scale-space theories, and also permits the construction of other theories. If the internal observer, for example, experiences locally a Minkovski space-time structure it is reasonable to adjust the scale-space theories for such a physical fact. Finally, it may be crucial to exploit the loss of order of contact or better the inhomogeneity of the group action over space and time represented by the torsion tensor and the curvature tensor. The identication of these objects with such physical actions, namely, indicate a possibility of rstly setting up a weighted ane connection after analysing the images in a (non)linear way, secondly scaling the original images in turn on the basis of a nonlinear scaling set by this connection, and so forth. Another possibility would be, instead of nonlinear scaling the input image, to formulate blurring schemes for the integral invariants by means of themselves [35]. Last but not least attributing some weight to the landscape of ridges and ruts one can formulate topological scale-space theories [41, 35]. One can so to speak redistribute the connectivity or topological dimension, possibly given additional weight by energy measures, over a weighted landscape of ridges and ruts. The resulting space of images allows in turn a topological description [8, 13, 30, 31, 35] or a noncommutative geometric description [1, 2]. In the latter description method one again will encounter notions such as that of connection and that of curvature, but now based on and complying to the proper topology inicted on the reference frame, i.e. the energy states of the horizontal and vertical layers of detectors of a vision system, by the similarity jet that in turn is brought about by a suitable (non)linear scale-space paradigm.

Acknowledgements This work was 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.

Notes 1. It should be emphasised that spatio-temporal dieomorphisms of the image domain form a particular class of image transformations, and do not necessarily commute with linear scaling nor satisfy the conservation law for total energy of the input image [35].

References 1. A. P. Balachandran, G. Bimonte, G. Landi, F. Lizzi and P. Teotonio-Sobrinho, Lattice gauge elds and noncommutative geometry, Universidade de Sao Paulo, Instituto de Fisica-DFMA, Sao Paulo, SP, Brazil, 1996. 2. G. Bimonte, E. Ercolessi, F. Lizzi, G. Landi, G. Sparano and P. Teotonio-Sobrinho, Noncommutative lattices and their continuum limits, Departamento de Fysica Teorica, Facultade de Ciencias, Universitad de Zaragoza, Zaragoza, Spain, 1995. 3. J. Blom, Topological and Geometrical Aspects of Image Structure, PhD thesis, Utrecht University, 1992. 4. E. Cartan, Sur les variétés à connexion ane et la théorie de la relativité généralisée, Gauthiers-Villars, 1955. 5. R. Cipolla and A. Blake, Surface orientation and time to contact from image divergence and deformation, In Proc. ECCV'92, May 1992, Santa Margherita Ligure, Italy, pp. 187202. 6. J. Damon, Local Morse theory for solutions to the heat equation and Gaussian blurring, Jour. Di. Eqtns., 1993. 7. D. H. Eberly, Geometric Methods for Analysis of Ridges in N-Dimensional Images, PhD thesis, The University of North Carolina, Chapel Hill, North Carolina, 1994. Department of Computer Vision. 8. A. V. Evako, R. Kopperman and Y. V. Mukhin, Dimensional properties of graphs and digital spaces, Journal of Mathematical Imaging and Vision, Vol. 6, pp. 109119, 1996. 9. O. Faugeras, On the motion of 3D curves and its relationship to optical ow, In Proc. ECCV'90, April 1990, Antibes, France, pp. 107117. 10. L. M. J. Florack and M. Nielsen, The intrinsic structure of the optic ow eld, Technical Report ERCIM07/94-R033 or INRIA-RR-2350, ERCIM, July 1994. 11. L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink and M. A. Viergever, Cartesian dierential invariants in scale-space, Journal of Mathematical Imaging and Vision, Vol. 3, pp. 327348, 1993. 12. F. W. Hehl, J. Dermott McCrea and E. W. Mielke, Weyl Space-times, The Dilation Current, And Cre-


13. 14. 15. 16. 17. 18.

19. 20. 21. 22. 23. 24.

25. 26. 27. 28. 29. 30. 31.

ation Of Gravitating Mass By Symmetry Breaking, Verlag Peter Lang, Frankfurt, 1988, pp. 241311. G. T. Herman and Z. Enping, Jordan surfaces in simply connected digital spaces, Journal of Mathematical Imaging and Vision, Vol. 6, pp. 121138, 1996. B. K. P. Horn, Robot Vision, MIT Press, Cambridge MA, 1986. P. Johansen, S. Skelboe, K. Grue and J. D. Andersen, Representing signals by their top points in scalespace, In Proceedings of the 8-th International Conference on Pattern Recognition, pp. 215217, 1986. P. Johansen, On the classication of top-points in scale-space, Journal of Mathematical Imaging and Vision, Vol. 4, pp. 5768, 1994. A. Kadi and D. G. B. Edelen, A gauge theory of dislocations and disclinations, Lecture Notes in Physics, volume 174, Springer-Verlag, Berlin, 1983. S. N. Kalitzin, B. M. ter Haar Romeny, A. H. Salden, P. F. M. Nacken and M. A. Viergever, Topological numbers and singularities in scalar images; scalespace evolution properties, submitted to Journal of Mathematical Imaging and Vision, May 1996. H. Kleinert, Gauge Fields in Condensed Matter, Vol. 12. World Scientic Publishing C0. Pte. Ltd., Singapore, 1989. J. J. Koenderink, The structure of images, Biol. Cybern., Vol. 50, pp. 363370, 1984. J. J. Koenderink, Scale-time, Biol. Cybern., Vol. 58, pp. 159162, 1988. J. J. Koenderink and W. Richards, Two-dimensional curvature operators, Journal of the Optical Society of America-A, Vol. 5, pp. 11361141, 1988. J. J. Koenderink and A. J. van Doorn, Invariant properties of the motion parallax eld due to the movement of rigid bodies relative to an observer, Optica Acta, Vol. 22, pp. 773791, 1975. J. J. Koenderink and A. J. van Doorn, A description of the structure of visual images in terms of an ordered hierarchy of light and dark blobs, In Second Int. Visual Psychophysics and Medical Imaging Conf., IEEE Cat. No. 81 CH 1676-6, 1981. J. J. Koenderink and A. J. van Doorn, Local features of smooth shapes: Ridges and courses, In Proc. SPIE Geometric Methods in Computer Vision II, Vol. 2031, pp. 213, 1993. Jan J. Koenderink, A hitherto unnoticed singularity of scale-space, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 11, pp. 12221224, 1989. T. Lindeberg, Scale-space for discrete signals, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 12, pp. 234245, 1990. T. Lindeberg, Scale-space behaviour of local extrema and blobs, Journal of Mathematical Imaging and Vision, Vol. 1, pp. 6599, 1992. T. Lindeberg, Scale-Space Theory in Computer Vision, The Kluwer International Series in Engineering and Computer Science., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994. McAndrew and C. A., Osborne, A survey of algebraic methods in digital topology, Journal of Mathematical Imaging and Vision, Vol. 6, pp. 139159, 1996. M. Newman, A fundamental group for grey-scale digital images, Journal of Mathematical Imaging and Vision, Vol. 6, pp. 139159, 1996.

27

32. T. Okubo, Dierential Geometry, Marcel Dekker, Inc., New York, 1987. 33. J. F. Pommaret, Systems of Partial Dierential Equations and Lie Pseudogroups, Gordon and Breach, Paris, 1978. 34. J. H. Rieger, Generic evolutions of edges on families of diused grey-value surfaces, Journal of Mathematical Imaging and Vision, Vol. 5, pp. 207217, 1995. 35. A. H. Salden, Dynamic Scale-Space Paradigms, PhD thesis, Utrecht University, The Netherlands, 1996. 36. A. H. Salden, Invariant theory, In Gaussian ScaleSpace Theory, Dordrecht, The Netherlands, Kluwer Academic Publishers, 1996. 37. A. H. Salden, L. M. J. Florack and B. M. ter Haar Romeny, Dierential geometric description of 3D scalar images, 3D Computer Vision, Utrecht, The Netherlands, Technical Report 91-23, 1991. 38. A. H. Salden, L. M. J. Florack and B. M. ter Haar Romeny, J. J. Koenderink and M. A. Viergever, Multi-scale analysis and description of image structure, In Nieuw Archief voor Wiskunde, Vol. 10, pp. 309326, 1992. 39. A. H. Salden, B. M. ter Haar Romeny, L. M. J. Florack, J. J. Koenderink and M. A. Viergever, A complete and irreducible set of local orthogonally invariant features of 2-dimensional images, In Proceedings 11th IAPR Internat. Conf. on Pattern Recognition, The Hague, The Netherlands, 1992, pp. 180184. 40. A. H. Salden, B. M. ter Haar Romeny and M. A. Viergever, Local and multilocal scale-space description, In Proc. of the NATO Advanced Research Workshop Shape in Picture - Mathematical description of shape in grey-level images, Vol. 126 of NATO ASI Series F, Berlin, 1994, pp. 661670. 41. A. H. Salden, B. M. ter Haar Romeny and M. A. Viergever, Dynamic scale-space theories, In Proc. Conf. on Dierential Geometry and Computer Vision: From Pure over Applicable to Applied Dierential Geometry, Nordfjordeid, Norway, August 17 1995. 42. A. H. Salden, B. M. ter Haar Romeny and M. A. Viergever, Algebraic invariants of linear scalespaces, submitted to Journal of Mathematical Imaging and Vision, March 1996. 43. A. H. Salden, B. M. ter Haar Romeny and M. A. Viergever, Classical scale-space theory from physical principles, submitted to Journal of Mathematical Imaging and Vision, October 1995. 44. Santalo, L.A., Integral geometry in general spaces, In Proceedings International Congress of Mathematics, Vol. 1, Cambridge, 1950, pp. 483489. 45. Santalo, L.A., Integral Geometry and Geometric Probability, Vol. 1, Addison-Wesley Publishing Company, London Amsterdam Don Mills, Ontario Sydney Tokyo, 1976. 46. Spivak, M., Dierential Geometry, Vol. 15, Publish or Perish, Inc., Berkeley, California, USA, 1975. 47. B. M. ter Haar Romeny, Geometry-Driven Diusion in Computer Vision, Kluwer Academic Publishers, Dordrecht, 1994. 48. A. L. Yuille and T. A. Poggio, Scaling theorems for zero-crossings, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 8, pp. 1525, 1986.

28

Salden, ter Haar Romeny and Viergever and non-linear scale-space theory, medical computer vision applications, picture archiving and communication systems, dierential geometry and perception. He is author of numerous papers and book chapters on these issues, has edited recently a book on non-linear diusion theory in Center Vision and is involved in and initiated a number of international collaborations on those subjects.

Alfons H. Salden received a M.Sc. in Experimental Physics in 1992 and a Ph.D. both from Utrecht University in 1996. His main research interests are scale-space theories, invariant theory, dierential and integral geometry, theory of partial dierential and integral equations, topology and category theory.

Max A. Viergever received the M.Sc. degree in ap-

Bart M. ter Haar Romeny received a M.Sc. in Ap-

plied Physics from Delft University of Technology in 1978, and a Ph.D. from Utrecht University in 1983. After being the principal physicist of the Utrecht University Hospital Radiology Department he joined in 1989 the department of Medical Imaging at Utrecht University as associate professor. He is a permanent member of the sta of the newly established Images Sciences Institute of Utrecht University and the University Hospital Utrecht. His interests are mathematical aspects of front-end vision, in particular linear

plied mathematics in 1972 and the D.Sc. degree with a thesis on cochlear mechanics in 1980, both from Delft University of Technology. From 1972 to 1988 he was assistant/associate professor of applied mathematics at this University. Since 1988 he is professor and head of the department of Medical Imaging at Utrecht University, and as of 1996 scientic director of the newly established Image Sciences Institute of Utrecht University and the University Hospital Utrecht. He is (co)author of over 200 refereed scientic papers on biophysics and medical image processing, and (co)author/editor of 10 books. His research interests comprise all aspects of computer vision and medical imaging. Max Viergever is a board member of IPMI and IAPR, is editor of the book series Computational Imaging and Vision of Kluwer Academic Publishers, associate editor-in-chief of IEEE Transactions on Medical Imaging, editor of the Journal of Mathematical Imaging and Vision, and participates on the editorial boards of several journals.