differential and integral geometry of linear scale-spaces · 2007-08-25 · differential and...

Journal of Mathematical Imaging and Vision 9, 5–27 (1998)c© 1998 Kluwer Academic Publishers. Manufactured in The Netherlands.

Differential and Integral Geometry of Linear Scale-Spaces

ALFONS H. SALDEN, BART M. TER HAAR ROMENY AND MAX A. VIERGEVERImage Sciences Institute, Utrecht University Hospital, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands

[email protected]

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

Keywords: linear scale-space theory, similarity jet, differential geometry, integral geometry, affine connection,metric, structure equations, Bianchi identities, torsion, curvature, translation vector field, affine rotation vectorfields, superposition principles

1. Introduction

Our aim is to quantify the differential and integral ge-ometry of spatio-temporal input images. In order toachieve our goal this geometric quantification is carriedout for similarity jets of linear scale-spaces of thoseimages [36, 43, 47]. A reason for considering thesesimilarity jets, instead of merely the initial input im-ages, is the desire for reproducibility of differential andtopological measurements despite perturbations of theinput images due to small scale uncorrelated noise con-tributions, Euclidean movements and similarity trans-formations.

In the past decades several attempts have been madeto describe spatio-temporal images based on variousmathematical and physical disciplines, namely:

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

The first three approaches concern local and multi-local description methods, whereas the remainingmethods are heading for a global description. All theseapproaches for describing spatio-temporal images havetheir advantages and disadvantages. We make these as-pects explicit for the local and multi-local descriptionmethods.

6 Salden, ter Haar Romeny and Viergever

In the classical continuum mechanical approach[9, 10, 14, 23] the assumption of the existence of aninvertible (time-dependent) transformation of a spatialconfiguration mapped bijectively onto another (diffeo-morphisms of spatial configurations over time) allowsa quantification of optic flow including rigid motions.But this assumption and that of constancy of total den-sity are not always met in real images due to the finite-ness of the resolution properties of the vision systemor the dynamics of the scene that evidently involvesmorphisms of spatial configurations over time. For ex-ample, a change in topology over time represented bya jump in the Euler-number, indicating the number ofholes of a surface, excludes the presence of spatial(point) correspondence. Another example is an imagegradient field that is initially constant over space, butover time becomes inhomogeneous and singular due tonon-integrable deformations of the input image. Moregenerally, the spatio-temporal dynamics over an en-semble of input images consists not only of a totallyintegrable deformational spatio-temporal part, but alsoof non-exact (non-integrable) parts due to the breakingover space and time of the (in)homogeneity of e.g., theGalilean transformation group (scenes in which thenumber of objects are not the same can be describedby such a symmetry breaking).

In the differential and integral geometric approach[5, 35, 41] the breaking of spatio-temporal symmetriesover input images is associated to so-called Volterraprocesses, i.e., deformation, insertion or removal ofspatio-temporal configurations. These processes areintensively studied in defect theory [17] and gauge fieldtheories [19], and can be nicely quantified in terms ofa Burgers vector field and Frank vector fields related tothe torsion and the curvature of the processes. Thesevector fields can display so to speak the image forma-tion over space-time and scale [35, 41]. In gray-valuedinput images normally there does not exist any orderof contact, i.e., the left and right derivatives of the in-put image along any direction in the image domainare not the same. This lack of order of contact boilsdown to the fact that the image gradient field is multi-valued and not—as naively assumed in most applica-tions of linear scale-space theories—infinitely manytimes continuously differentiable. The latter structureis boldly inflicted on the image data. In defect fieldtheories [19] such defects in the order of contact areidentified as manifestations of dislocations and discli-nations of a lattice. In order to be able to quantifythese kinds of defects in an input image an externalobserver using still a linear scale-space paradigm can

take advantage of reflecting boundary conditions alonghyperplanes going through a point of interest [35, 36,41, 43]. If one now computes connection one-formsand frame vector fields after linear scaling under re-flective boundary conditions each pair of subimagesbordered by the initial image boundary and the hyper-plane, then one can quantify defects through integralinvariants indicating how much the image is curved andtwisted. Although the computational complexity re-duces considerably in the fully discretized linear case,computational capacity can still become, in practice, alimiting factor. The question arises whether it is possi-ble to derive and measure such image formation aspectsby means of torsion and curvature machines withoutintroducing blurring of the image defined on the inter-section of a half space and the spatio-temporal imagedomain. Adopting the viewpoint of an internal observer[35, 41], who introduces an affine connection, onecan circumvent capacity problems and derive torsionand curvature aspects for the similarity jet of a linearscale-space despite single-valuedness of the differen-tial structure of the input image [35, 41]. Moreover, onecan exploit this image-induced affine connection to de-fine new scaling paradigms, which only locally assumee.g., a Minkovski space-time to exist with a Lorentzmetric structure, but with a global structure determinedby the (metric)-affine connection [35, 41]. Degeneracyof the connection can cause undesirable singularitiesin related geometric objects, such as the torsion tensorand curvature tensor. But by choosing a proper affineconnection, essential singularities due to the vanishingof structures of the similarity jet of the spatio-temporalinput images can be avoided. It should be reckonedthat the machines related to a particular connection forquantifying the Volterra process are intrinsically multi-local operators.

In the invariant-theoretic approach [36, 39, 40, 42]essential singularities do not occur, unless rationalproperties of the similarity jet are used to read out themorphology over space-time and under linear scaling.The morphology of the image under linear scaling andsmall perturbations is described in terms of a completeand irreducible set of (multi)-local invariants. The im-portance of multi-local invariants cannot be underes-timated, for they naturally pop up as components ofvector fields in the slot-machines or in these machinesthemselves measuring some twist or curvature of theinput image.

Having indicated some of the advantages anddrawbacks of the differential and integral geometricmethod for describing spatio-temporal input images

Differential and Integral Geometry of Linear Scale-Spaces 7

the similarity jet of a spatio-temporal input image isdefined in Section 2. In Section 3 a summary of differ-ential and integral geometry is presented. Finally, inSection 4 the geometric theory presented in Section 3is applied to geometries living on the similarity jet ofspatio-temporal input images.

2. The Similarity Jet

The linear scale-space theory is treated in order to ob-tain a similarity jet of a spatio-temporal gray-valued in-put image. For extensive expositions on the continuous,semi-discrete and discrete linear scale-space theory thereader is referred to [21, 29, 36, 43, 47].

Assume that space-time(M, g) can be modeled bythe product of Riemannian manifolds(Mx, gx) and(Mt , gt ), i.e.,M = Mx×Mt andg = gx⊗gt . HereMx

is aD-dimensional manifold representing the space ofpositions, andMt a one-dimensional manifold repre-senting the space of times. Furthermore, the spatialand temporal metric tensor fieldsgx and gt are bi-linear forms on the product of corresponding tangentbundles and coincide with the standard inner productonRD andR, respectively. So the space-time mani-fold (M, g)can be represented by a(D, 1)-dimensionalEuclidean product space. Now letL0 be a (not neces-sarily smooth) spatio-temporal input image of a densityfield defined on a(D, 1)-dimensional (bounded) space-time domainD⊂M onto a one-dimensional manifoldN representing the density values:

L0 :D ⊂ M → N.

In case of a bounded domainD this domain canbe a product of aD-dimensional spatial domainDx

and a one-dimensional temporal domainDt , i.e.,D=Dx × Dt . The reason for explicitly stating that space-time has a product topology of Euclidean spaces is thatspace and time are physically not on an equal foot-ing in the Newtonian context. However, in a visionsystem one may assume them to be commensurable,because the realization of the observed field over hori-zontal and vertical layers allows such an interpretation(if so, it would, of course, be more than natural to as-sume space-time to be a Minkovski space-time with aLorentzian metric structure. However, this would onlyaffect the physical interpretation, but not the line ofreasoning in the sequel).

If the vision system performs a smoothing of a den-sity field observed over space-time, then the system

may generate a two-parameter scale-space of imagesL of the input imageL0 [21, 36, 43]:

L :D ×Rx ×Rt → N,

with initial condition

lim(sx,st )↓ (0,0)

L(x, t, sx, st ) = L0(x, t),

where the positionsx ∈ Dx, timest ∈ Dt , spatial scalesx ∈ Rx and temporal scalest ∈ Rt are all independentphysical observables having only meaning within thevision system.

If the smoothing of the field satisfies a law of conser-vation of total density, then, according to the divergencetheorem, the smoothing by the vision system shouldbe governed by a system of spatio-temporal diffusionequations:

∂L

∂sx− ∂2L

∂xi ∂xi= 0,

∂L

∂st− ∂2L

∂t∂t= 0.

Note that, whenever the space-time domainD isbounded, the initial conditions have to be supple-mented by boundary conditions (for examples, seeagain [36, 43]). The solution of the above Cauchy prob-lem, i.e., the diffusion equation and the initial-boundaryvalue condition, can be found by deriving their Green’sfunctions. These functions can be obtained by reducingthe diffusion equations to so-called Weber differentialequations in terms of coordinates and energies [36, 43]

ξ = x√sx,

τ = t√st,

3 = s−ax s−b

t L , a, b ∈ R,

that are invariant under the two-parameter(λ, µ)-scaling group of point transformations, the so-calledgroup of similarity transformations:

(x, t, sx, st , L) = (λx, µt, λ2sx, µ2st , λ

2aµ2bL),

with λ,µ ∈ R+. After this reduction procedure thefundamental geometric and physical object constructed


on the basis of the spatio-temporal input imageL0 isthe continuous similarity jetj∞30:

Definition 1. The similarity jet j∞30 is defined by:

j∞L0 = {ξ, τ,3En, Em},with

3En, Em = sn+D

2x s

m+12

t L En, Em,

where

L En, Em =(L0 ∗ Gx

En) ∗ Gt

Em,

andn = |En| ∈ Z+0 andm = | Em| ∈ Z+0 are the spa-tial order and temporal order of differentiation, respec-tively, and Gx and Gt are the spatial and temporalGreen’s functions consistent with the above Cauchyproblem.

These Green’s functions can be derived by us-ing the normalized Gaussian as Green’s function onMx ≡RD andMt ≡R, and by applying, subsequently,the method of images or Fourier transform techniques[36, 43]. The found similarity jet is a unique and com-plete physical object for quantifying the dynamics of ascene (see Section 4), as it represents, given an inputimage, the states of a vision system under linear scaling(latent in those of the vertical and horizontal layers) interms of measurable physical entities, namely distancesin lengths and energies, with which units of measureare associated, and a certain topology [35].

We conclude our discussion of linear scale-spacetheory by pointing out some of its drawbacks andvirtues in describing the geometry of the similarity jetsof spatio-temporal input images.

In practice fully discretizing the continuous theorybecomes indispensable, for in the continuous and semi-discrete case the kernels are defined at infinitely manypoints in the image domain, and consists of an infinitesum of translated Gaussians in the case of boundaryconditions [36, 43]. In the fully discretized case thesupport of the kernel is, for any integer scale, definitelybounded, such that, even in the case of diffusion withboundary conditions, the number of “Gaussians” con-tributing to the kernel, is also finite. Furthermore, inte-gration over the image domain can be simply replacedby summations over a finite number of measurementsof the initial input image.

Linear scale-space theory enables the application ofmodern differential and integral geometry [35, 41]. An

external observer can quantify the lack of order of con-tact, the multi-valuedness (non-differentiability) of theinput image or better the torsion and curvature in theinitial image formation process over scale and on ahyper-tube in space-time. An internal observer onlyaware of the outcomes of linear scaling can also detectthe torsion and curvature of a geometry determined bythe similarity jet by reading out the similarity jet ac-cording to a rule determined by the input image underthat scaling. Note that an external observer can detectmulti-valuedness of the similarity jet, whereas the in-ternal observer does not. Instead the internal observerconcludes that the induced affine connection, that isthe rule for relating structures over the image domain,implies a non-trivial geometry of the image formationprocess under linear scaling.

If the image domain is curved, twisted and has a dif-ferent lattice symmetry in the fully discretized case oflinear scale-space theory [36, 43], and if the scalingparadigm has to be also invariant under other symme-tries than similarity and Euclidean invariance as postu-lated in linear scale-space theory [35, 41, 47], then thescaling has to be adjusted to these geometric aspects inthe image formation. The most obvious difference be-tween the linear and nonlinear scaling theories is latentin the supposed connection living on the image domain,which is needed to relate image properties with eachother. For example, in the linear case the connection isassumed to be globally Euclidean, whereas in the non-linear case it is assumed to be locally Euclidean, but(hardly reckoned or made explicit) non-flat and twistedaccording to properties of the input image. The geom-etry of the similarity jets obtained through the use ofthe nonlinear scaling mechanisms, however, does notyield any new insights in the differential and integralgeometric description of spatio-temporal input images,except for the imposed additional symmetries.

3. Differential and Integral Geometry

The differential and integral geometry of manifolds isformally quantified by means of (metric or non-metric)connection one-forms, torsion tensors, curvature ten-sors, affine translation vectors and affine rotationvectors. For extensive expositions on all the abovesubjects the reader is referred to [4, 32, 44–46]. Therelations between these geometric entities and physi-cal objects encountered in defect theory, general rel-ativity and gauge field theories [17, 19] are pointedout. This relationship will encourage us to combine


the above mathematical and physical disciplines withlinear scale-space theory (see Section 4).

3.1. Differential Geometry

First, the system of Cartan structure equations for amanifold with a general affine connection is stated.The differential geometry is caught by the framefield, the connection one-forms, the torsion two-form,the curvature two-form and the Bianchi identitiesover the manifold. Relations between these geomet-ric (mathematical) objects and those physical objectsencountered in defect theory and gauge field theoriesare briefly indicated.

Second, manifolds with a metric connection are de-fined as those, for which the metric tensor is com-patible with the connection. After imposing a dual-ity constraint between the connection one-forms, thatmay define the metric tensor, and a frame vector field,the implications of (in) compatibility for the remainingconnection-one forms are made partly explicit togetherwith those for the related torsion tensor and curvaturetensor.

Let M be ann-dimensional base manifold and con-sider the affine frame bundleB ≡ P(M, π, A(n,R))whereP is the total space consisting of all affine framesVp at each pointp ∈ M , π : P → M is the projec-tion andA(n,R) = GL(n,R)FT(n,R) the full affinegroup, whereGL(n,R) is the general linear group andT(n,R) the translational group, acting on the rightsemi-directly on each local affine frameVp.

Definition 2. A local affine frameVp at a pointp ofan-dimensional manifoldM is defined as

Vp = (x, e1, . . . ,en); x, ei ∈ TpM,

whereTpM is the tangent space to the manifoldM atpoint p.

In order to compare and relate the local affine framesover the manifoldM an affine connection0 is specifiedwhich defines in turn the covariant derivative operator.

Definition 3. An affine connection0 on ann-dimen-sional manifoldM is defined by one-formsωi andωi

jon the tangent bundleT M:

∇x = ωi ⊗ ei ,

∇ei = ω ji ⊗ ej ,

where⊗ denotes the tensor product, and∇ the covari-ant derivative operator.

Note that, the connection one-forms and the framevector field need not be related through some set ofconstraints. This freedom will be crucial in describingthe similarity jet (Definition 1) in Section 4.

Despite the seemingly immense freedom of choiceof an affine connection0 the connection one-formsωi

andω ji do satisfy so-called structure equations:

Theorem 1. The structure equations for an affineconnection0 are given by:

Dωi = dωi +ωik ∧ ωk = Äi ,

Dω ji = dω j

i +ω jk ∧ ωk

i = Ä ji ,

where d denotes the ordinary exterior derivative, Dthe covariant exterior derivative operator, α ∧ β theexterior product of tensorsα and β, Äi the torsiontwo-forms andÄ j

i the curvature two-forms.

Proof: See [46]. 2

Note that, the torsion two-formsÄi and the curvaturetwo-formsÄ j

i are related to the componentsTijk and

Rjikl of the torsion tensorT and curvature tensorR as

follows:

Äi = 1

2Ti

jkωj ∧ ωk,

Äji =

1

2Rj

iklωk ∧ ωl ,

with

T = Tijkω

j ⊗ ωk ⊗ ei ,

R = Rjiklω

i ⊗ ωk ⊗ ωl ⊗ ej .

Applying again the covariant exterior derivativeoperatorD to these structure equations yields the inte-grability conditions for the affine connection0, i.e., theBianchi identities:

Theorem 2. The Bianchi identities for the affine con-nection0 of the manifold M are given by:

DÄi = Äij ∧ ω j , DÄ j

i = 0.

Proof: See [32, 46]. 2


The above structure equations and the Bianchi iden-tities now constitute a system of so-called Cartanstructure equations. This system is closed, which im-plies that the torsion and curvature tensor together withthe connection one-forms form the fundamental differ-ential geometric objects for assessing equivalence oftwo manifolds with an affine connection and for de-riving higher order differential geometric objects. Thelatter geometric objects are obtained by taking covari-ant derivatives of the torsion tensor and the curvaturetensor. For irreducible components of these derivedgeometric objects see [4]. In defect theory and gaugefield theories [12, 17, 19] the above geometric objectsand identities have particular physical counterparts. Forexample, the Bianchi identities can be identified withconservation laws for dislocation densities and discli-nation densities, or conservation laws for the angularmomentum density and the energy-momentum density,respectively.

Now one can subdivide the above manifolds with anaffine connection into those with an affine connectioncompatible with a metric tensor (metric-affine connec-tion) and those with an affine connection incompati-ble with a metric tensor [12, 19]. In order to define ametric-affine connection one assumes that the connec-tion one-formsωi determine a metric tensorγ , that theconnection one-formsωi and the frame vector fieldsei

are each duals, and that the covariant derivative of themetric tensorγ vanishes identically.

Definition 4. A metric tensorγ on ann-dimensionalmanifold M with affine connection (Definition 3) isdefined by:

γ = ωi ⊗ ωi .

Definition 5. For an n-dimensional manifold withaffine connection (Definition 3) the connection one-formsωi and the frame vector fieldsej are dual, if andonly if

ωi (ej ) = δij ,

whereδij is the Kronecker delta-function.

Definition 6. An n-dimensional manifold with ametric-affine connection is a manifold with an affineconnection (Definition 3) and metric tensor (Defini-tion 4), for which the following compatibility conditionholds:

∇γ = 0.

This means that the angles between and lengths ofvectors measured by the metric tensorγ under paralleltransport associated with the affine connection0 arepreserved.

The question arises how to express the connectionone-formsωi

j in terms of the one-formsωi and theframe vector fieldsej given the compatibility constraint(Definition 6) and the duality constraint (Definition 5).In order to establish this relationship, quantify the com-ponent functionsep

i of the frame vector fieldsei withrespect to a local reference frameV0

p = (x0, e0i ) as:

ei = e0pep

i ,

and the componentsejq of the one-formsω j through

the duality constraints:

eipep

j = gij ,

epi ei

q = gpq ,

wheregpq are the components of the metric tensor liv-ing on the reference tangent bundle spanned locallyby the reference frame vector fieldse0

p. On the basisof the definition of an affine connection (Definition 3)and the above representations of the frame vector fieldsei it is easily verified that the connection one-formsωi

jare directly related to so-called connection coefficients0i

jk :

ωij = 0i

jkωk, 0i

jk = ej (log E), E = (eip

).

Note that, the connection coefficients evidently directlyrelate to the well-known Weberfractions often encoun-tered in any field of exact science. In this case thecomponents of the torsion tensorT and the curvaturetensorR can be expressed by means of the frame vectorfieldsei and the connection coefficients0i

jk as follows:

Tkjk =

1

2

(0i

jk − 0ik j

),

Rijkl = ek0

ij l − ej0

ikl + 0m

jl0imk− 0n

kl0in j .

Note that, the torsion need not be identically zero fora manifold with a metric-affine connection. If the tor-sion tensor vanishes one speaks of Riemannian mani-folds, else of Cartan-Einstein manifolds. Furthermore,the reference frame may be twisted and curved dueto the connection and possibly the metric living onthe reference frame bundle. An external observer isaware of these reference properties, but will consider


them as unimportant for the essential dynamics of ascene. Similarly, the latter reference frame intricaciesare eliminated by an internal observer only being sen-sible to twist and curvature through an image-inducedconnection.

Summarizing, the affine connections on a manifoldsatisfy systems of Cartan structure equations. Theseequations determine, through the torsion two-formsand the curvature two-forms and the connection one-forms, the equivalence relations for a manifold. Choos-ing a metric on the basis of the connection one-formsωi and imposing the duality constraint between theseforms and the frame vector fieldsej , one can constructa metric connection, such that the components of theconnection one-formsωi

j are fully determined by theformer geometric objects. Note that, the differentialgeometry treated here mainly concerns that from theviewpoint of an internal observer [17]. However, inSection 4 examples are also presented of differentialgeometry (Riemannian geometry) from the viewpointof an external observer.

3.2. Integral Geometry

In this subsection integral invariants are defined forn-dimensional manifoldsM with an affine connection0, and those with an affine connection0 and a metrictensorγ .

First, consider ann-dimensional manifoldM with anaffine connection0 (metric or non-metric) as defined inthe previous Section 3.1. Two fundamental integral in-variants, based on the connection one-forms(ωi , ωi

j )

and the frame vector fieldsek, are the affine transla-tion vector and the affine rotation vectors belonging toan affine displacement around an infinitesimally smallcontour [4]:

Definition 7. The affine translation vectorb and theaffine rotation vectors are defined by:

b =∮

C∇x,

fi =∮

C∇ei ,

whereC is a infinitesimally small closed loop on atwo-dimensional submanifoldS of M with the sameinduced affine connection0.

One may obtain the submanifoldS by settingD − 2 of the connection one-formsωi equal to zero.

Using Stokes’ theorem and the structure equations(Theorem 1) the affine translation and rotation vectorscan be written as [4]:

b =∫

Sc

Äi ⊗ ei ,

fi =∫

Sc

Äji ⊗ ej ,

where Sc⊂ S is an infinitesimally small patch withboundaryC. The nonvanishing of the translation vectorb indicates the presence of torsion on the given mani-fold, whereas that of the affine rotation vectorsfi indi-cates the presence of curvature. The above vectors con-stitute the integral versions of the structure equations(Theorem 1) and form measures for the inhomogene-ity of the affine group action. The Bianchi identities(Theorem 2) imply that the vectors satisfy the follow-ing superposition principles [19]:

B =∑

b,

Fi =∑

fi .

For integral invariants of higher degree and interest-ing dimensional restrictions, the reader is again referredto Cartan’s memoir [4]. These superposition princi-ples play a crucial role in quantifying the geometry ordynamics at critical points and along ridges and rutsas will be shown in Section 4. Furthermore, it is alsoworthwhile to mention that these integral invariants arevery stable within the context of the linear scale-spaceframework given a non-degenerate (metric) affine con-nection.

Second, consider ann-dimensional manifoldM withan affine connection0 and a metric tensorγ as definedin the previous subsection. For such a manifold thedefinitions of the above integral invariants remain thesame. But because of the existence of the metric tensorit is possible to decompose, e.g., the affine rotationvectorsfi into “rotational” partsf rot

i , “dilational” partsf dilα and “shearing” partsf shear

α :

fi = f roti + f dil

i + f sheari ,

with

f roti =

∫Sc

Äj ||i ej ,

f dili =

∫Sc

Äk)(kδ

ji ej ,


where|| and( ) denote the full anti-symmetrization andsymmetrization operator, respectively. Besides theselocal integral invariants there exist other integral in-variants, e.g., the volume measure and the Euler char-acteristic [32, 46], that can be constructed on the basisof a metric connection, the covariant derivatives ofthe torsion tensor and those of the curvature tensor[4].

In defect theory [17] and gauge field theory [19] onerefers to the translation vectorband the rotation vectorsfi as the Burgers vector field and the Frank vector fields,respectively. They are conceived as manifestation ofdislocations and disclinations along defect lines createdby Volterra processes (see Figs. 1 and 2). Furthermore,the superposition principles for those vectors coincidewith Kirchhoff’s law for currents.

Summarizing, the differential and integral geometryof manifolds with affine connection and possibly ametric structure is captured by means of the Cartan

Figure 1. Burgers vector fieldb due to a dislocation caused bya displacement fieldu. The underlying Volterra process breaks thesquare lattice symmetry (a) into (b) by removing lattice points andmoving the horizontal lattice lines closer to each other. Traversingthe circuitC in (a) one returns to the initial starting point, whereas in(b) one ends up being Burgers vector fieldb away from the startingpoint.

Figure 2. Frank vectorf due to a wedge disclination. The under-lying Volterra process breaks the square lattice symmetry (a) into(b) by cutting the initial square lattice (a) open along the negativex-axis and inserting new lattice points (b) such that the lattice sur-faces are smoothly continued along the cutting line except at theorigin. Traversing the circuitC in (a) and measuring the change in avector carried along one observes no change in that vector, whereasin (b) one experiences the Frank vector fieldf .

structure equations. The chosen frame fields, connec-tion one-forms and metric structure fully determinethrough these equations the curvature tensor, the tor-sion tensor and their covariant derivatives, as wellas integral invariants, such as the affine translationvector and the affine rotation vectors. Some of thephysical counterparts for the frame fields, the connec-tion one-forms and the Cartan structure equations arebriefly indicated. The latter observation encourages usto proceed in applying the just presented formalismto geometries determined by similarity jets in the nextsection. As pointed out in the introduction, in generalthere does not exist any order of contact in structures re-lated to an input image implying that the above analysisholds only locally. However, although the internal ob-server in a linear scale-space cannot detect any lack oforder of contact for the similarity jet, because of the reg-ularization property of the linear scaling paradigm, onecan read out the similarity jet according to a rule speci-fied by a (metric)-affine connection such that the imageformation under linear scaling can still be expressed interms of non-trivial curvature and torsion aspects.

4. Applications

Manifolds with a metric connection, and those withaffine connection and metric tensor together with theirgeometries are constructed on the basis of the similarityjet (Definition 1) (see Sections 4.1 and 4.2). First, how-ever, some preliminary remarks should be made aboutrestrictions imposed by the linear scaling paradigm,and some auxiliary geometric objects and machinesshould be defined.

The first remark concerns the value of imposing addi-tional invariance conditions other than those underlyingthe linear scaling paradigm. Although the fundamen-tal symmetries underlying the linear scaling paradigmconcern Euclidean invariance and self-similarity, thereis no reason to adopt an external observer point of viewand to require additional invariance of the similarityjet such as that under the group of (monotonic) gray-value transformationsL0 = f (L0) 6= constant, unlessone is really interested in the Euclidean geometry ofisophotes and flowlines [11, 38]. One should realizethat the linear scaling paradigm is intrinsically not in-variant under such a group of transformations nor is in-tended to be so. For scaling paradigms adjusted to theabove invariance condition and heading for an analysisof other types of image structures the reader is referredto [35, 41, 47].


The second remark concerns the degeneracy of ametric connection and consequently the introductionof essential singularities of derived geometric objectssuch as torsion and curvature tensors [35, 41, 47].These singularities occur especially in geometric scale-space theories in which the invariance under the groupof gray-value transformations is taken care of. Thenthe pure metric relations, e.g., the affine curvatures ofcurves or nets, and thus the connection can becomesingular. However, restricting to pure classical affineinvariance one can derive affine image flows that do notyield essentially singular scale-spaces [35, 41].

In order to describe objects such as branching pointsand defect lines in the similarity jet a local referenceframe is needed, that is a frame which at least is invari-ant under the group of similarity transformations (seeSection 2). The simplest local dual reference framesatisfying this invariance condition can already beconstructed on the basis of only the spatio-temporalpositions and scales (see Theorem 3).

Theorem 3. A local dual reference frame V∗0 =(ξ, τ,dχα) constructed on the basis of the similarityjet and invariant under the similarity group is given bythe following one-forms dχα:

dχ i0 =

dxi

√sx,

dχ D+10 = dsx

sx,

dχ D+20 = dt√

st,

dχ D+30 = dst

st.

Proof: Show thatdχα0 = dχα0 under the similaritygroup (Definition 1). 2

The local reference frameV0, consisting of framevectorse0

β , can be derived from the following dualityconstraint:

dχα0(e0β

)= δαβ .Note that the Greek indices denote spatio-temporalvariables and scales, whereas the Latin indices denoteonly spatial variables. Furthermore, that the metrictensorg0= dχα0 ⊗ dχα0 constructed on the basis of theabove one-formsdχα has componentsgαβ = δαβ . The

latter metric property is a consequence of the requiredinvariance of physical measurements under a similaritygroup action and duality constraint. The latter metricityalso considered by Eberly [7] plays, however, no rolein describing the relevant geometry of the similarity jetas to be demonstrated shortly. However, if one acceptsit as an additional structure on the similarity jet it canbe used to read out particular Riemannian geometriesas done by Eberly in [7].

Having constructed a local self-similar referenceframe one can continue defining manifolds on the im-age domain with a geometry brought about by endow-ing it with an image dependent connection (seeSections 3, 4.1 and 4.2).

In the next subsection most of the manifolds con-cern isophotes and flowlines of gray-valued images3

represented by the similarity jet at a fixed spatial andtemporal resolution and a fixed time.

Definition 8. An isophoteCi of a gray-valued image3 at a fixed spatial and temporal resolution,s0

x ands0t ,

respectively, and a fixed timet0 is defined by:

Ci = {ξ0 |3(ξ0) = constant}.

Definition 9. A flowline Ci of a gray-valued image3 at a fixed spatial and temporal resolution,s0

x ands0t ,

respectively, and a fixed timet0 is defined by:

C f ={ξ0

∣∣∣∣ dξ0

dp= 3(ξ0(p))

},

wherep is an arbitrary parameter.

In order to quantify their classical Euclidean differ-ential geometries in terms of the similarity jet extensiveuse has been made of the method of implicit differen-tiation [11, 22].

Another object studied in both the following sub-sections and determined by the similarity jet is ann-dimensional netN on aD-dimensional manifoldM .

Definition 10. An n-dimensional netN on a D-dimensional manifoldM is defined by the set of in-tegral curves forn≤ D vector fieldsvi ∈ T M definedon 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 repre-sented by the similarity jet and inherits all the image


formation (morphological) aspects brought about bylinear scaling.

In order to connect nets (Definition 10) over scalesand space-time, so-called projection one-forms areneeded [20]. As the geometry of the similarity jet is con-tained in3, let us first proposeδ log

(33u

) = 3α3δχα,

with 3u≤3 being the dynamic unit (dynamic innerscale) of the vision system, as a natural order of varia-tion of the system activity as function of the dimension-less canonical coordinatesχ ; it can also be conceivedas a connection one-form on the input image. The latterconcept yields a solution to the commensurability (as-pect ratio) problem between space-time and the imagegray-values. The point of view of an internal observeris adopted, who is only aware of or adjusts to the firstorder similarity jet (the internal observer has no otherreference frame available than that induced by the firstorder image structure). This observation fixes the con-nection on the similarity jet without bothering aboutthe reference frame on the system statesξα as neededin Eberly’s approach [7]. With our connections identi-cally vanishing there is no topological or better phys-ical information latent in the similarity jet other thanthe trivial one of a constant input image. Furthermore,labeling the space-time events attains only meaning inthe context of the input image. The desired connectionone-forms for isophotes on the image domain comeabout by considering the following variational princi-ple:

δ3

3= 0,

and decomposing it into three parts, the first part due tothe spatial scaling of the input image,δss3

3, the second

part due to the temporal scaling of the input image,δts3

3

and the third part due to the space-time behavior of theinput image,δst3

3:

δ3

3= δss3

3+ δts3

3+ δst3

3= 0;

δss3

3= 3i

3δχ i + 3D+1

3δχ D+1 = 0,

δts3

3= 3D+2

3δχ D+2+ 3D+3

3δχ D+3 = 0,

δst3

3= 3i

3δχ i + 3D+2

3δχ D+2 = 0.

Solving the above variational problems under the as-sumption of proximity and equal luminance [20] thedesired projection one-forms are derived easily.

Definition 11. The projection one-forms for the sim-ilarity jet are defined by:

ωD+1 = −3D+13i

32dχ i + 3i3i

32dχ D+1,

ωD+2 = −3D+33D+2

32dχ D+2+ 3D+23D+2

32dχ D+3,

ωD+3 = −3D+23i

32dχ i + 3i3i

32dχ D+2.

One could, of course, divide each one-form above bythe component function of its last term yielding otherprojection one-forms:

ωD+1 = −3D+13i

3i3idχ i + dχ D+1,

ωD+2 = −3D+33D+2

3D+23D+2dχ D+2+ dχ D+3,

ωD+3 = −3D+23i

3i3idχ i + dχ D+2.

The second term of these one-forms are readily iden-tified with translations along the spatial scale axis, thetemporal scale axis and the time axis. The first termin ωD+3 corresponds to the translation in the spatialdomain due to a velocity. Unfortunately, these alter-native one-forms can become singular and physicallynot measurable as in the case of so-called drift veloc-ities introduced by Lindeberg [28] as soon as one ap-proaches extrema. These one-forms or velocities askfor adaptive sampling and further analysis in the neigh-borhood of such points. The similarity jet, however,is not hampered by such a singular behavior, neitherare the projection one-forms (Definition 11) or relatedvector fields with the same component functions (thatare not satisfying some duality constraint). The latterproperty of the mentioned geometric objects will play acrucial role in reading out the geometry of the similar-ity jet or better the morphology of the image formationover time and under linear scaling at critical points(see Section 4.2.2). The non-degeneracy also occursby definition for multi-local invariants of the similar-ity jet computed around a critical point between twopatches bordered by ridges and ruts [35]; for exam-ple, measuring the jump in the average image gradientfield between the two patches will certainly not yieldessential singularities.

However, one might object that the linear scalingparadigm is not concerned with connecting points, butrather in connecting images and subimages, whereas


the nonlinear scaling paradigms, such as the shorteningflows, are concerned with connecting points. A sim-ilar reasoning can be applied for the space-time dy-namics of the image. Realizing this difference in theobjectives of the scaling paradigms and the dynam-ics of a space-time field one is immediately inclinedto replace the above projection one-forms by the fol-lowing very simple one-forms, namelyωD+i = dχ D+i

wherei = 1, 2, 3. It should be clear that the analysisof the above singularities on the basis of these one-forms becomes trivial and not hampered by all kindsof degeneracies in the similarity jet.

As soon as space, time and the scale variables areviewed as incommensurable physical entities, thenone is urged to come up with a machine for readingout the geometry other than a metric connection. Thishappens as soon as, e.g., space-time is viewed as a prod-uct of two manifolds with their own metric structuresas is the case in linear scale-space (see Section 2). Ageometric machine that is not troubled by such a com-patibility constraint, but still capable of comparing ge-ometric objects living on a product manifold is the Liederivative with respect to some vector field [33, 46]:

Definition 12. The Lie derivativeL XY of a geometricobjectY with respect to vector fieldX is defined at eachpoint p on a manifoldM by:

(L XY)p = limt→0

1

t

(θ−t∗

(Yθ(t,p)

)− Yp),

whereθt∗ : TpM → Tθ(t,p) an isomorphism.

Note that the geometric object compared along acertain trajectory by the Lie derivative (Definition 12)can be, e.g., a connection, a torsion tensor or a cur-vature tensor defined in the previous section living oneither submanifold. For expressions of Lie derivativesin terms of component functions of the vector field andthe geometric object, and their well-known propertiesthe reader is referred to [33].

4.1. Manifolds with Metric Connection

Spatio-temporal geometric objects are constructed onthe basis of the similarity jet in order to derive for arelated manifold with a metric connection the funda-mental differential and integral geometric properties.First, at one level of spatial and temporal scale the Eu-clidean geometry of space curves, hypersurfaces andnets are quantified in terms of the similarity jet (see

Section 4.1.1). Second, a metric-affine geometry ofnets is formulated and quantified in terms of the simi-larity jet again at fixed scales (see Section 4.1.2).

4.1.1. Euclidean Geometry of Space Curves, Surfacesand Nets. Consider a space curveC in D-dimen-sional Euclidean spaceED at fixed levels of spatialand temporal resolutions and at a fixed time.

Definition 13. A space curveC in D-dimensionalEuclidean spaceED is defined as a collection of vec-tors x ∈ ED that are twice differentiable with respectto an arbitrary parameterp:

C ≡ {x ∈ C2(ED) | x = x(p) ∧ p ∈ [0, 1]}.

If curve C is a non-degenerate curve parametrizedby Euclidean arclength parameters, then in the neigh-borhood of regular points the frameV , i.e., a positivelyoriented orthonormal basis, is given by:

V = (x; e1, . . . ,en);

e1 = dx

ds, ds=

(dxi

dp

dxi

dp

) 12

dp,

where x is determined in the reference system andthe remaining unit frame vector fieldse2, . . . ,en areobtained by means of the Gramm-Schmidt orthonor-malization process applied to the vectorsd1x

ds , . . . ,dnxdsn .

Now the Euclidean metric connection one-formsωi andωi

j for its metric connection (see Section 3) are givenby:

ω1 = ds, ω2 = · · · = ωn = 0,

ωji = −ωi

j , ωkk+1 = kk ds, ωk

l = 0, ∀l > k+ 1,

where the latter one-formsω ji can be simply expressed

in the following matrix form:

(ω

ji

) = dsd log E

ds,

where the matrixE hasn columns defined by the com-ponent functions of the frame vector fieldsei :

E = (e1, . . . ,en).

Higher order Euclidean differential geometric invari-ants are subsequently straightforwardly derived by tak-ing derivatives of the found curvatureskk with respectto Euclidean arclength parameters. Furthermore, space


curves can be described in terms of a Euclidean canon-ical expansion:

x(s) =∞∑

k=0

1

k!

(dkx

dsk

)sk.

Do realize that the above curvatureskk are invariantzero-forms being connection coefficients for the rota-tion group and have nothing to do with the curvaturetwo-form encountered in Section 3.

Although the frame field construction is impossi-ble, e.g., at inflection points, the Euclidean curvaturesmight still be measurable. Whether these curvaturescan be computed is set by the local Euclidean structureleft and right from those points. In order to derive in-tegral invariants for (segments of) a curve the framesand the curvatures form appropriate observables. Forexample, the Euclidean curvatures integrated over acurve segment are global invariants unaffected by theassociated group action.

We give two examples of the above geometry forspace curves defined on the basis of the spatial layersof the similarity jet, i.e., isophotes and flowlines of atwo-dimensional gray-valued image, and flowlines ofa three-dimensional gray-valued image.

Example 1. For a planar curve the Euclidean framevector fields are given by:

ei1 =

dxi

ds,

ei2 =

(de1

ds· de1

ds

)− 12 dei

1

ds,

and the connection one-forms are given by:

ω1 = ds, ω2 = 0,(ω

ji

) = ( 0 kds

−kds 0

),

with the Euclidean curvaturek given by:

k = de1

ds· e2,

where · denotes the standard inner product on aEuclidean space.

Choose in case of a two-dimensional gray-valuedimage3 at a pointξ0 of an isophote (Definition 8)image dependent local Cartesian coordinatesv andw,

in which the curve can be described byξ = (v,w(v))such thatwv = dw

dv = 0. Applying the method of im-plicit differentiation introduced and exploited notablyby [11, 22] yields the following variations up to secondorder:

3(v,w(v)) = 30 ∈ R,δv3 = 3vδv +3wwvδv

= 0,

δvv3 = 3vvδ2v + 23wvwvδ

2v

+3www2vδ

2v +3wwvvδ2v

= 0,

with e.g.,

3v = ∂3

∂v,

and using the requirement,wv = 0, one obtains up tosecond order the following description of the isophotew(v) = −3vvv

2

23w. Thus thew coordinate is a conse-

quence of a projection of a vector onto the local unitnormal image gradient defining the second frame vec-tor field e2, and the coordinatev that onto the firstframe vector fielde1. In terms of the similarity jet theEuclidean frame vector fieldsei

j and the curvature fieldki of the isophotes (Definition 8) are given by:

ei1 =

(εki3k

(3 j3 j )12

),

ei2 =

(3i

(3 j3 j )12

),

ki = 3i εi j3 j pεpq3q

(3 j3 j )32

.

Note that the derivatives with respect to the coordinatesv andw are directional derivatives:

∂

∂v= ei

1|ξ = ξ0 · ∇ξ , ∂

∂w= ei

2|ξ = ξ0 · ∇ξ ,

with

∇ξ =(∂

∂ξ i

)∣∣∣∣sx,t=s0

x ,t

.

Here the viewpoint of an external observer is adoptedby assuming the curve to be embedded in a higherdimensional Euclidean space.


For the flowlines of a two-dimensional gray-valuedimage3 the Euclidean frame vector fieldsef

i andthe flowline curvature fieldk f are given by:

ef1 =

(3i√3i3i

),

ef2 =

def1

ds,

k f = ef2 ·

def1

ds,

with

d

ds= 3i√

3 j3 j

∂

∂ξ i.

Note that the latter derivative operator is not equivalentto the directional derivative∂

∂w, and that immediately

the viewpoint of an internal observer is adopted.Integral invariants, such as the total rotation index

12π

∫C kds for the image of isophotes or flowlines can

be computed by considering fully discretized similarityjets [36, 43]. One may even consider a total rotationindex on a submanifold consisting of a set of segmentsof, e.g., isophotes that lie within a region bounded byridges and ruts [7, 25, 35] (see also Example 4).

Example 2. For a three-dimensional space curveCthe frame vector fieldsei are given by:

e1 = dx

ds,

e2 =(

de1

ds· de1

ds

)− 12 de1

ds,

e3 = e1× e2,

and the connection one-forms by:

ω1 = ds, ω2 = ω3 = 0,

(ωi

j

) = 0 kds 0

−kds 0 tds

0 −τds 0

,with the Euclidean curvaturek and torsiont :

k =((

e1× de1

ds

)·(

e1× de1

ds

)) 12

,

t = −e2 · de3

ds.

The frame vector fieldsei , the curvature fieldk and thetorsion fieldt for three-dimensional flowlines can bestraightforwardly expressed in terms of a correspond-ing similarity jet as:

e1 =(

3i

(3n3n)12

),

e2 =(

de1

ds· de1

ds

)− 12 de1

ds,

e3 = 1

t

(ke1+ de2

ds

),

d

ds= 3i√

3 j3 j

∂

∂xi,

k = ((x(1) × x(2)) · (x(1) × x(2)

)) 12 ,

t =(x(1) × x(2)

) · x(3)k2

,

where thexi = di xdsi are given in terms of image proper-

ties as:

x(1) = e1,

x(2) = (3s3s)3p3pj − (3p3pq3q)3 j

(3n3n)2,

x(3) = 3p3pq3qk

(3n3n)32

+ 3p3q3pqk

(3n3n)32

−3(3p3pq3q)3r3rk

(3n3n)52

− 2(3p3pq3qr )3r3k

(3n3n)52

− (3p3q3r3pqr)3k

(3n3n)52

− 4(3p3pq3q)

23k

(3n3n)72

.

Again realize that the flowline coincides with the in-tegral curve corresponding to the normalized gradientvector field of the differential invariant3.

Consider now hypersurfacesS in Euclidean spaceED. The Euclidean frame fieldV for such a surfaceSand its Euclidean connection one-forms are given by:

V = (x; e1, . . . ,eD),

ωD = 0, ω1, . . . , ωD−1 6= 0,

ωji = 0, j = 1, . . . D − 1,

ωDi = hi j ω

j , hi j = h ji .

Heree1, . . . ,eD−1 form an orthonormal basis spanningthe tangent bundle to the surfaceSandeD its unit nor-mal bundle. The infinitely many times differentiable


functionshi j on S define the elementary symmetricpolynomialspk

pk = (n− k)!

n!δ

i1...i kj1... jk

h j1i1. . . h jk

ik.

with

δi1...i kj1... jk=

0 if iα = iβ, jα = jβ; α 6= β,{ j1, . . . , jk} 6= {i1, . . . , i k}

1 if j1, . . . , jk even permutation

of i1, . . . , i k

−1 if j1, . . . , jk odd permutation

of i1, . . . , i k

that in turn determine the principal curvatures [46].

Example 3. For an isophote of a three-dimensionalgray-valued image3 an appropriate frame fieldV toposition the surface is readily obtained [37]:

V0 = (x; u, v, w),

with

ui = εi jk3 j3km3m

(εi jk3 j3km3mεi pq3p3qr3r )12

vi = εi jk w j uk

wi = 3i

(3 j3 j )12

It is shown how the desired so-called Frenet frame fieldV [4] can be realized by applying a suitable rotationaround thew-axis to the unit tangent vectorsu andv. This rotation must bring the directions of the newx1- andx2-axis in the tangent plane to the surface inline with the principal directions corresponding to theprincipal curvaturesκ1 andκ2, respectively. Perform-ing such an operation yields a canonical Euclidean de-scription of the surface with respect to the coordinatesystem(x1, x2, x3) in terms of derivatives of the prin-cipal curvatures with respect to both thex1- and thex2-axis:

x3(x1, x2)

=2∑

i=1

∞∑p=0

1

(p+ 2)!

∂ pκi

∂xi1 . . . ∂xi pxi xi xi1 . . . xi p .

The explicit quantification of the principal curva-tures and directions in terms of the similarity jet of a

three-dimensional gray-valued image follows straight-forwardly through the application of the method ofimplicit differentiation [11, 22]. Using the gaugingconditions,3u = 0 and3v = 0, the solution of thevariational principle up to second order gives the fol-lowing second order description of the surface in termsof u- andv-coordinates:

w(u, v) = 1

2

(∂2w

∂u2u2+ 2

∂2w

∂u∂vuv + ∂

2w

∂v2v2

),

with

∂2w

∂u∂u= −3uu

3w

,

∂2w

∂u∂v= −3uv

3w

,

∂2w

∂v∂v= −3vv

3w

.

Note that, again, the derivatives of the similarity solu-tion concern directional derivatives with respect to theframe vector fields constituting frameV0. The meancurvature H and the Gaussian curvatureK of thesurface are then given by:

H = −(3uu+3vv

23w

),

K =(3uu3vv −32

uv

32w

).

These curvatures are in turn directly related to the prin-cipal curvaturesκi through the eigenvalue problem forthe components of the second fundamental form:

det

(∂2w∂u∂u − κ ∂2w

∂u∂v

∂2w∂u∂v

∂2w∂v∂v− κ

)= κ2− 2Hκ + K = 0.

Thus the principal curvatures can be expressed as:

κ1,2 = H ±√

H2− K .

In order to find the rotation angleθ for which theframe lines up with the principle curvature directionsconsider an arbitrary unit tangent vectorφ:

φ ≡ u cosθ + v sinθ,


and normal curvatureκφ along that direction:

κφ ≡ −φi3i j φ j

(3p3p)12

.

The principal directions are now obtained by substi-tutingφ into the expression forκφ and solving for angleθ the following equation:

κi = κφ.

This is consistent with the fact that the extrema ofthe normal curvature coincide with the principal cur-vatures. Using elementary properties of trigonometricfunctions and the expression for the mean curvatureHthe equation reduces to:

H − κi =(3uu−3vv

23w

)cos 2θ +

(3uv

3w

)sin 2θ,

or

sin(2θ + δ) = λi cosδ

with

δ = arctan

(3uu−3vv

3uv

),

λi = (H − κi )

(3uv

3w

)−1

.

Solving this trigonometric equation for one of the prin-cipal curvatures, the rotation angleθi for letting theu-andv-axis be parallel to thex1- andx2-axis is givenby:

θi = −1

2δ + arcsin(λi cosδ).

Using the method of implicit differentiation to find adescription of the surface in terms ofw = w(u, v)and applying the derived rotation over angleθi to gofrom the coordinates(u, v, w) to the new coordinates(x1, x2, x3)a canonical Euclidean invariant descriptionis established:

x3(x1, x2)

=w(x1 cosθi − x2 sinθi , x1 sinθi + x2 cosθi ).

Finally, consider an arbitraryn-dimensional net em-bedded in aD-dimensional Euclidean spaceED. TheEuclidean frame fieldV for such a netN and itsEuclidean connection one-forms are given by:

V = (x; e1, . . . ,en), ei = vi√vi · vi

,

ωi = dsi , ωij = 0i

k jωk,

wheree1, . . . ,en form an orthonormal frame field span-ning the tangent bundle to the local Euclidean spaceED and toN simultaneously,dsi infinitesimally smallEuclidean displacements along the integral curves onN generated by vector fieldsei (satisfying the dualityconstraint (Definition 5)), and the connection coeffi-cients0i

jk are given by:

0ik j =

(∂ log E

∂sk

)i

j

, E = (e1, . . . ,en) .

The torsion tensorT and the curvature tensorRof the net, endowed with the above Euclidean connec-tion, can easily be expressed in terms of the connectioncoefficients and derivatives with respect toei = d

dsi .Similarly, the integral invariants for these nets can beobtained (see Section 3).

Example 4. Consider a two-dimensional net definedby isophotes and flowlines of a two-dimensional gray-valued image3. Choose on this net as Euclidean framevector fieldsei :

e1 = ei1,

e2 = ef1 ,

where the frame vector fieldse1 ande2 are unit tangentvector fields to the isophotes and flowlines, respectively(see Fig. 3 and also Example 1). Next choose on the

Figure 3. Input image3 (left) and its Euclidean frame fieldV(right) with the first frame vector field tangent to the isophotes.


net the following Euclidean connection one-forms:

ω1 = dsi , ω2 = dsf ,

ωij = 0i

k jωk,

with connection coefficients:

0ik j =

(∂ log E

∂sk

)i

j

; E = (e1, e2) , sk = si , sf ,

wheresi is the Euclidean arclength along the isophoteand sf the Euclidean arclength along the flowline.These coefficients are readily expressed in terms of theEuclidean isophote curvatureki and flowline curvaturek f (see Example 4):

(0i

1 j

) = ( 0 ki

−ki 0

),

(0i

2 j

) = ( 0 −k f

k f 0

).

One observes that the connection is not symmetric, andthat the torsion tensor and the curvature tensor are notvanishing. It is easily shown that the component func-tions of these tensors can be expressed in terms of linearcombinations of derivatives of the flowline curvatureand the isophote curvature as already noticed in [38].In Fig. 4 the locations where the isophote and flowlinecurvature field are vanishing are computed on the ba-sis of the Euclidean connection induced by the inputimage in Fig. 3.

The question arises which are the essential physi-cal objects of the above net invariant under diffeomor-phisms of the image domainE2. It is clear that the setof (non)-isolated singularities ofL0 is one of them, for

Figure 4. From left to right the curves on which the isophote curva-ture fieldki and the flowline curvature fieldk f vanish together withthe isophotes of the input image depicted in Fig. 3.

the vanishing of the image gradient is not influencedby such a diffeomorphism. Not so obvious is that forthe landscape of ridges and ruts of the input imageL0 [7, 25, 35] (i.e., the singular curves of the steepestlines of descent and ascent in the input image).1 Thetopological equivalence of these singularity sets canbe explained by the fact that across them the integralcurves of the image gradient have opposite convexity;the curvature vector to the flowlines flips across ridgesand ruts along the isophotes. Consequently, the con-nection on the flowlines at the ruts and ridges alongthe isophotes is completely degenerate implying thatany order of derivative with respect to the Euclideanarclength parametersi of the flowline curvaturek f isvanishing. Because taking all orders into account andthe fact that to a finite order there will always be non-ridge or non-rut points for which they are zero, it isimpossible to distinguish on the basis of a pure localanalysis between ridges, ruts and the borders of theirinfluencing zones consisting of, e.g., inflection points[35]. Thus it is impossible to tell on the basis of apure local analysis whether all, a particular subset ornone of the disjoint components of the zero-crossingsof the flowline curvature are parts of the landscape ofridges and ruts.

But computing instead the torsion or equivalently thetranslation vector of the normalized flowline curvaturevector field highlights the characteristics of ridge or rutpoints, i.e., the multi-valuedness of image structures, inthis case the normalized flowline curvature vector field,defined by the single-valued similarity jet. In Fig. 5 thelength of the translation vector fieldb corresponding tothe frame vector fieldsei in Example 4 for a discretizedgray-valued input imageL0 is computed by means of

Figure 5. Left frame: a 256× 256 pixel-resolution discrete inputimage L0(x, y) = 1 − y2, x < 0 and L0(x, y) = 1 − 1

2((y −x)2 + (y+ x)2)), x ≥ 0 with its center as origin. Right frame: theEuclidean length of the translation vector|b| for a linearly scaledversion of that image. Note that the non-isolated set of singularitiesx ≤ 0 andy = 0 will instantaneously disappear upon continuouslinear scaling.


linear scale-space theory. It seems that the ridge (end)points on the non-isolated singularity set are detected(realize that ridges and ruts can be discerned on thebasis of the isophote curvatureκ1 [7, 25, 35]. Ifκ1 > 0andκ2 = 0, then the points possibly belong to ridges;if κ1 < 0 andκ2 = 0, then to ruts). However, onthe basis of Figs. 3 and 4 one might object that thelength of the translation vector field corresponding tothe above frame fields on the outer components of thezero-crossings of the flowline curvature field is non-vanishing implies that one did not yet use the propermachine to point out ruts and ridges. In order to findridges and ruts one actually has to investigate, as men-tioned in the beginning, the normalized flowline cur-vature vector field (preserving its orientation) and readout the change of this orientation field along the integralcurve belonging to the normalized flowline curvaturevector field, i.e., an isophote. Applying the half-spacemethod used by an external observer [35] one detectsa reversal of the orientation field across ridges and rutsalong isophotes, but across inflection points, for exam-ple, one observes no change in the orientation field.Alternatively, a topological method introduced in [35]can be used. It wraps a strip around a (non) isolatedcritical point and locates the extrema of the length ofthe image gradient field while covering by means of theencircling tip-point of the strip the whole, e.g., spatialimage domain. Applying one of both methods aboveone obtains a branching pattern (and thus connectiv-ity relations) of the landscape of ridges and ruts, andother (non) isolated singularity sets for free and withina linear scale-space setting.

Although in the above example the affine transla-tion vector and the affine rotation vectors can be com-puted for also continuous scale-space theories otherintegral invariants such as the Euler characteristic canonly be retrieved in the case of (semi)-discretized linearscale-spaces. In the continuous case the computationof the latter image properties can become a mathemati-cal nuisance and an undesirable action from a physicalpoint of view given the finite resolution properties of avision system and uncertainty relations underlying anymeasurement.

4.1.2. Metric-affine Geometry of Nets.Assumingspace and time commensurable physical entities a(D+1)-dimensional spatio-temporal net (Definition 10)on a (D + 1)-dimensional manifoldM representingspace-time is endowed with a metric-affine connection(Definition 6). Note that the assumption of a separate

Euclidean connection on the spatial and the temporalpart of the image domain has been dropped. The ge-ometry comes about through the connection one-formsthat are not constructed on the basis of a Galilean framework [4] (see Example 5). In this framework the cou-pling of space-time with the external source fields areonly reminiscent in the connection one-formsωi

j .

Example 5. Choose a(2+ 1)-dimensional net on a(2+1)-dimensional gray-valued image at fixed spatialscale and temporal scale with a metric-affine connec-tion brought about by the following connection one-forms ωi defined on the basis of the similarity jet(Definition 1):

ω1 = ∂ log3

∂χ idχ i ,

ω2 = εi j∂ log3

∂χ idχ j ,

ω3 = −(∂ log3

∂χ3

)∂ log3

∂χ idχ i

+(∂ log3

∂χ i

∂ log3

∂χ i

)dχ3,

whereωi refers to the spatial part of the image, andω3

to the spatio-temporal part coinciding with space-timeprojection one-forms. Computing on the basis of theduality constraint the frame vector fieldsei and sub-sequently the connection coefficients as in Example 4,one immediately observes that the dynamics capturedby the connection one-forms, the torsion tensor andthe curvature tensor inherit typical image-dependent“accelerations”, and other spatio-temporal image for-mation aspects from the imposed metricity.

As stressed in the introduction of this section ametric connection can be inconsistent with the un-derlying physical postulates about the commensura-bility or topology of space and time. Another ratherdisturbing fact is that in practice the metric connec-tions or derived geometric objects all tend to becomeessentially singular at certain locations. In particularpoints, e.g., critical points and top-points, a geomet-ric description can become unfeasible. A last problemconcerns the fact that in practice the dynamics of ascene represented by an input image needs to be spatio-temporally updated. Especially, if the capacity of thevision system is limited and the analysis should be per-formed real-time, then the consecutive images shouldbe spatio-temporally bounded. The question arises how


to connect the similarity jets of these bounded spatio-temporal input images. In the next subsection the abovementioned problems are tackled by constructing suit-able affine connections (Definition 3).

4.2. Manifolds with Affine Connection

In Section 4.2.1 two examples of affine geometry ofnets are treated. In Section 4.2.2 the similarity jet isprovided with an affine connection in order to quantify ageometry at critical points. Finally, in Section 4.2.3 thegeometry of a set of similarity jets of spatio-temporalinput images is captured by supplementing this set withadditional connection one-forms.

4.2.1. Affine Geometry of Nets.Two examples ofaffine geometries on nets are considered, becausespace-time is modeled by the product of Euclideanspaces (see Section 2). In Example 6 a metric-affineconnection on space is permitted. The additionalconnection-one forms come about by applying Liederivatives (Definition 12) with respect to a velocityvector field of these spatial one-forms, and of a one-form with the same component functions as that vectorfield. In Example 7 the duality constraint and metricityis completely dropped in order to avoid the generationof singular geometric objects determined by the con-nection.

Example 6. Choose a(2, 1)-dimensional net on(2, 1)-dimensional Euclidean space-timeM and endowthe spatial part of the net with a metric-affine connec-tion by taking the connection one-formsωi equal tothose in Example 5. The frame vector fieldsei livingon the spatial subtangent space of space-time are deter-mined by the duality constraint, but are, of course, notequal to those found in Example 5. The other connec-tion one-formω3 is used to construct a correspondingframe vector fielde3 which can be imposed a separateduality constraint. Now in order to obtain the connec-tion one-formsωαβ other than those related to the met-ric connections living on the separate submanifolds Liederivatives are taken with respect to the frame vectorfieldseα of the componentsωα( ∂

∂χγ) of the connection

one-formsωα:

ωαβ =(

Leβωα

(∂

∂χγ

))dχγ .

Evidently, the “connection coefficients”0αβγ =(Leβω

α( ∂∂χγ)) in this case reflect an absolute change in

the order of magnitude. Subsequently, the torsion andcurvature tensor given this connection can be straight-forwardly computed and can be identified with spatio-temporal creation and annihilation of image details dueto external fields. The morphology of an input imagejumping from isophote to isophote within the similar-ity jet is so to speak operationalized by an affine trans-lation vector field and an affine rotation vector field,such as the Burgers and Frank vector field found indefect theory [17]. These morphological entities sat-isfy the superposition principle at branching points asmentioned in Section 3.

Note that taking the alternative projection one-formω3 (frame vector fielde3) on space-time given in theintroduction of this section one could construct a mo-mentum like vector field, for examplep = log3e3

where log3 could be interpreted as some (topologi-cal) mass for the external fields yielding the similaritysolution3. The commutator of the Lie derivatives ofthis vector fieldp and the spatial frame vector fieldei can subsequently related to “uncertainty relations”within the similarity jet. These uncertainty relationsin turn define the structure functions of the nonlinearLie-algebra underlying the image formation under thelinear scale-space paradigm.

In the next example a completely affine connectionis set up for a similar net as in the previous exam-ple. This affine connection will not yield undesirableessential singularities in derived geometric objects asone would prefer to on the basis of the measurabilityof the similarity jet.

Example 7. Choose the same connection one-formsωα as in Example 6, but without imposing duality normetricity on either submanifold of space-time. Next setthe components of the frame vector fieldseα equal tothose ofωα. Finally, define the connection one-formsωαβ again as the Lie derivatives of the components ofconnection one-formsωα with respect to the vectorfields eβ . Now the connection even if it is vanishingdoes ensure that the torsion and the curvature and theirderivatives remain measurable. One actually quantifiesthe torsion and curvature of the similarity jet [33, 35].

4.2.2. Geometry at Singularities.Normally singu-larity or catastrophe theory is used to describe the sim-ilarity jet at critical points and top-points [16, 20, 28,34]. But here the geometry is induced by a connec-tion constructed on the basis of the similarity jet thathas on critical sheets appropriate projection one-forms


related to drift velocity vector fields as introduced byLindeberg [28]. The reader not familiar with the def-initions of critical points, top-points and drift velocityvector fields should consult the references listed above.

We restrict ourselves to a(D, 1)-dimensional net onthe similarity jet of aD-dimensional spatial input im-age. For other type of images the procedure to followis the same. We study the geometry of the similar-ity jet along critical curves and in the neighborhood oftop-points. Now constrain the above net to one withan affine connection by specifying first the followingconnection one-forms spanning the dual tangent spaceto the spatial translation group:

ωk = (Ak)pqBpqidχi , k = 1, . . . , D,

with the components of matrixA and matrixB givenby:

A =(∂2 log3

∂χ i ∂χ j

),

B =(

∂3 log3

∂χ i ∂χ j ∂χk

).

In general these connection one-forms are not vanish-ing on critical curves. A supplementary connectionone-formωD+1, that is not hampered by singularitieson critical curves but still relates to the drift velocityintroduced by Lindeberg [28], is given by:

ωD+1 = (detAA−1G)i dχi + detAdχ D+1,

with

G =(∂2 log3

∂χ D+1∂χ j

).

Use now the components of these connection one-forms to construct the frame vector fieldseα, and de-fine subsequently the additional connection one-formsωαβ as Lie derivatives of the components of the one-formsω with respect to the vector fieldseβ as done inSection 4.2.1. Now the geometry and thus the morphol-ogy of the similarity jet along critical curves can nicelybe expressed in terms of measurable torsion and curva-ture aspects (if the reader prefers to impose a Euclideanmetric connection on the spatial domain, then he hasonly to proceed as in the previous subsection). In asimilarity jet the critical curves merge upon increasing

scale, for example two Gaussian blobs with three ex-trema merge into one blob with one extremum. In prac-tice these critical curves can be viewed as defect linesin defect theory [19] and the bifurcations as branchingpoints that can be characterized by their integral invari-ant properties, namely the affine translation vector andthe affine rotation vectors (see Section 3). It is inter-esting to attach to these critical curves the landscape ofridges and ruts over scale in order to obtain a particulardecomposition of the first order similarity jet into cells.The poset structure (the structure of partially orderedsets), connectivity relations and (local) topological di-mensions of this jet can subsequently be nicely read outby logical or topological filtering methods [24, 35].

Note that it can happen that the second and higher or-der parts of the similarity jet are vanishing too at criticalpoints and that these critical points form non-isolatedsingularity sets. Fortunately, one almost never encoun-ters them in similarity jets, because continuous linearscaling blurs them away instantaneously and becausethey disappear abruptly under perturbations of the in-put image. In the discrete linear scale-spaces, however,one may find them over a whole range of scales. Butbeyond half the outer scale all non-isolated singulari-ties are blurred away.

Furthermore, as mentioned in the introduction of thissection other projection one-formωD+1 can be consid-ered for reading out the similarity jet along the scaleaxis. Taking the connection one-form dictated by thelinear scaling paradigm, i.e.,ωD+1 = dχ D+1, the mor-phology of the input image under the linear scalingparadigm can be followed by tracing the transitions inthe differential and integral geometry across scale inthe neighborhood of (not at) the critical points. In thiscase one can still rely on only the first and second orderinformation content of the similarity jet instead of in-vestigating 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 ordomain is bounded. The latter boundedness is due tothe finite resolving capacity of a vision system. Nowthe question arises how to update such an input image.First of all one should note that it is possible to varythe spatial domainDx as well as the temporal domainDt with the present momentT0 running as a functionof the external field30. Assume the spatio-temporaldomains, for convenience, to be independent of thepresent momentT0 and of the external field, and to beof fixed extents. Thus the scene dynamics is studied at


every present momentT0 on spatio-temporal domainsDx,t (T0)of fixed lengths. Note that the present momentis the upper bound of the temporal domain. Next let thetemporal domainDt (T0) at present momentT0 overlapwith the temporal domainDt (T ′0)at the present momentT ′0 ≥ T0. Now squeeze the time-difference between thetwo present moments, but keepT ′0 fixed. At the presentmomentT ′0 the transition between both scene dynam-ics can be quantified in terms of their similarity jetsas follows. Conceive the present moment as an addi-tional degree of freedom as Koenderink did [21] andassume the classical scaling paradigm also applicableto this set of similarity jets. Extending our ideas ofSection 4 leads to an updated (metric)-affine scene dy-namics represented by the updated similarity jets. Thedual frame of the reference system is supplemented bythe one-form:

dχ D+4 = dT√ST,

dχ D+5 = dST

SuT

,

where ST is the scale parameter with respect to thepresent momentT . In order to measure the updatedscene dynamics two additional connection one-formsare needed. On the basis of a fourth variational prin-ciple allowing to connect similarity jets under presentmoment scaling:

δT S3

3= 3D+4

3δχ D+4+ 3D+5

3δχ D+5 = 0,

and the proximity and equal total luminance condition[20] and the assumption that the present moment shouldbe connected through the vector fieldχ D+4, one has thefollowing two additional connection one-forms:

ωD+4 = −3D+53D+4

32dχ D+4

+3D+43D+4

32dχ D+5,

ωD+5 = dχ D+5.

Having constructed the necessary connection one-formsωα the related frame vector fields and the otherconnection one-formsωαβ can be found analogously asin the previous Sections 4.2.1 and 4.2.2. Of course, theconnection between the updated similarity jets can alsobe supplemented by the natural onesωD+4 = dχ D+4

andωD+5 = dχ D+5.

5. Conclusion and Discussion

The (updated) dynamics of a scene observed by a visionsystem at various resolutions of time and space is quan-tified by imposing (metric)-affine connections on a setof similarity jets of the spatio-temporal input imagesof the scene. The similarity jets are obtained by ap-plying linear scale-space theory to the spatio-temporalinput images. The observed dynamics is expressed interms of differential and integral invariants of the im-ages at different spatial and temporal scales. Alter-natively, the whole similarity jet can be described bysimilar invariants. Especially, the torsion two-forms,the curvature two-forms, the affine translation vectorand the affine rotation vector appear to be appropriatemachines to read out the dynamics or the similarity jetstructure. The importance of these machines is demon-strated at critical point and top-points, and for ridgesand ruts. The close relation between our differentialand integral geometric theory of scene dynamics, de-fect theory and gauge theory is pointed out by showingthe close relation between branching points and defectlines, and critical points and non-isolated singularitysets, respectively.

Up to now space-time is modeled globally as a(D, 1)-dimensional Euclideanly flat and non-twistedspace, whereas an internal observer can verify thisspace-time property only locally. It makes sense, there-fore, to generalize linear scale-space theory in this con-text [35, 41, 47]. It is evident that our formalism isreadily extended to nonlinear scale-space theories, andalso permits the construction of other theories. If theinternal observer, for example, experiences locally aMinkovski space-time structure it is reasonable to ad-just the scale-space theories for such a physical fact.Finally, it may be crucial to exploit the loss of orderof contact or the inhomogeneity of the group actionover space and time represented by the torsion tensorand the curvature tensor. The identification of theseobjects with such physical actions, namely, indicate apossibility of firstly setting up a weighted affine con-nection after analyzing the images in a (non)linear way,secondly scaling the original images in turn on the ba-sis of a nonlinear of scaling set by this connection, andso forth. Another possibility would be, instead of non-linear scaling the input image, to formulate blurringschemes for the integral invariants by means of them-selves [35]. Last but not least, attributing some weightto the landscape of ridges and ruts one can formulatetopological scale-space theories [35, 41]. One can, so


to speak, redistribute the connectivity or topologicaldimension, possibly given additional weight by energymeasures, over a weighted landscape of ridges and ruts.The resulting space of images allows in turn a topologi-cal description [8, 13, 30, 31, 35] or a noncommutativegeometric description [1, 2]. In the latter descriptionmethod one again will encounter notions such as thatof connection and that of curvature, but now based onand complying to the proper topology inflicted on thereference frame, i.e., the energy states of the horizon-tal and vertical layers of detectors of a vision system,by the similarity jet that in turn is brought about by asuitable (non)linear scale-space paradigm.

Acknowledgments

This work was supported by the Netherlands Organi-sation of Scientific Research, grant number 910-408-09-1, and by the European Communities, H.C.M. grantnumber ERBCHBGCT940511.

Note

1. It should be emphasized that spatio-temporal diffeomorphismsof the image domain form a particular class of image transfor-mations, and do not necessarily commute with linear scaling norsatisfy 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 fields and noncommutativegeometry,” 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 theircontinuum limits,” Departamento de Fysica Teorica, Facul-tade de Ciencias, Universitad de Zaragoza, Zaragoza, Spain,1995.

3. J. Blom, “Topological and geometrical aspects of image struc-ture,” Ph.D. thesis, Utrecht University, 1992.

4. E. Cartan,Sur les Varietesa Connexion Affine et la Theorie dela Relativite Generalisee, Gauthiers-Villars, 1955.

5. R. Cipolla and A. Blake, “Surface orientation and time to contactfrom image divergence and deformation,” inProc. ECCV’92,Santa Margherita Ligure, Italy, May 1992, pp. 187–202.

6. J. Damon, “Local Morse theory for solutions to the heat equationand Gaussian blurring,”Jour. Diff. Eqns., 1993.

7. D.H. Eberly, “Geometric methods for analysis of ridges in n-dimensional images,” Ph.D. thesis, Department of ComputerVision, The University of North Carolina, Chapel Hill, NorthCarolina, 1994.

8. A.V. Evako, R. Kopperman, and Y.V. Mukhin, “Dimensionalproperties of graphs and digital spaces,”Journal of Mathe-matical Imaging and Vision, Vol. 6, No. 2/3, pp. 109–119,1996.

9. O. Faugeras, “On the motion of 3D curves and its relationship tooptical flow,” in Proc. ECCV’90, Antibes, France, April 1990,pp. 107–117.

10. L.M.J. Florack and M. Nielsen, “The intrinsic structure of the op-tic flow field,” Technical Report ERCIM-07/94-R033 or INRIA-RR-2350, ERCIM, July 1994.

11. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, andM.A. Viergever, “Cartesian differential invariants in scale-space,”Journal of Mathematical Imaging and Vision, Vol. 3,pp. 327–348, 1993.

12. F.W. Hehl, J. Dermott McCrea, and E.W. Mielke,Weyl Space-times, The Dilation Current, and Creation of Gravitating Massby Symmetry Breaking, Verlag Peter Lang: Frankfurt, 1988,pp. 241–311.

13. G.T. Herman and Z. Enping, “Jordan surfaces in simply con-nected digital spaces,”Journal of Mathematical Imaging andVision, Vol. 6, No. 2/3, pp. 121–138, 1996.

14. B.K.P. Horn,Robot Vision, MIT Press: Cambridge MA, 1986.15. P. Johansen, S. Skelboe, K. Grue, and J.D. Andersen, “Repre-

senting signals by their top points in scale-space,” inProceed-ings of the 8th International Conference on Pattern Recognition,1986, pp. 215–217.

16. P. Johansen, “On the classification of top-points in scale-space,”Journal of Mathematical Imaging and Vision, Vol. 4, pp. 57–68,1994.

17. A. Kadi and D.G.B. Edelen,A gauge theory of dislocationsand disclinations, Lecture Notes in Physics, Vol. 174, Springer-Verlag: Berlin, 1983.

18. S.N. Kalitzin, B.M. ter Haar Romeny, A.H. Salden, P.F.M.Nacken, and M.A. Viergever, “Topological numbers and sin-gularities in scalar images; scale-space evolution properties,”Journal of Mathematical Imaging and Vision, accepted.

19. H. Kleinert,Gauge Fields in Condensed Matter, World Scien-tific Publishing Co.: Singapore, Vol. 1–2, 1989.

20. J.J. Koenderink, “The structure of images,”Biol. Cybern., Vol.50, pp. 363–370, 1984.

21. J.J. Koenderink, “Scale-time,”Biol. Cybern., Vol. 58, pp. 159–162, 1988.

22. J.J. Koenderink and W. Richards, “Two-dimensional curvatureoperators,”Journal of the Optical Society of America-A, Vol. 5,pp. 1136–1141, 1988.

23. J.J. Koenderink and A.J. van Doorn, “Invariant properties ofthe motion parallax field due to the movement of rigid bod-ies relative to an observer,”Optica Acta, Vol. 22, pp. 773–791,1975.

24. J.J. Koenderink and A.J. van Doorn, “A description of thestructure of visual images in terms of an ordered hierarchyof light and dark blobs,” inSecond Int. Visual Psychophysicsand Medical Imaging Conf., IEEE Cat. No. 81 CH 1676-6,1981.

25. J.J. Koenderink and A.J. van Doorn, “Local features of smoothshapes: Ridges and courses,” inProc. SPIE Geometric Methodsin Computer Vision II, 1993, Vol. 2031, pp. 2–13.

26. J.J. Koenderink, “A hitherto unnoticed singularity of scale-space,”IEEE Trans. Pattern Analysis and Machine Intelligence,Vol. 11, pp. 1222–1224, 1989.


27. T. Lindeberg, “Scale-space for discrete signals,”IEEE Trans.Pattern Analysis and Machine Intelligence, Vol. 12, pp. 234–245, 1990.

28. T. Lindeberg, “Scale-space behaviour of local extrema andblobs,” Journal of Mathematical Imaging and Vision, Vol. 1,pp. 65–99, 1992.

29. T. Lindeberg,Scale-Space Theory in Computer Vision, TheKluwer International Series in Engineering and ComputerScience., Kluwer Academic Publishers: Dordrecht, TheNetherlands, 1994.

30. McAndrew and C.A. Osborne, “A survey of algebraic methods indigital topology,”Journal of Mathematical Imaging and Vision,Vol. 6, No. 2/3, pp. 139–159, 1996.

31. M. Newman, “A fundamental group for gray-scale digital im-ages,” Journal of Mathematical Imaging and Vision, Vol. 6,No. 2/3, pp. 139–159, 1996.

32. T. Okubo,Differential Geometry, Marcel Dekker Inc.: NewYork, 1987.

33. J.F. Pommaret,Systems of Partial Differential Equations andLie Pseudogroups, Gordon and Breach: Paris, 1978.

34. J.H. Rieger, “Generic evolutions of edges on families of dif-fused gray-value surfaces,”Journal of Mathematical Imagingand Vision, Vol. 5, No. 3, pp. 207–217, 1995.

35. A.H. Salden, “Dynamic scale-space paradigms,” Ph.D. thesis,Utrecht University, The Netherlands, 1996.

36. A.H. Salden, “Invariant theory,” inGaussian Scale-Space The-ory, Kluwer Academic Publishers: Dordrecht, The Netherlands,1996.

37. A.H. Salden, L.M.J. Florack, and B.M. ter Haar Romeny, “Dif-ferential geometric description of 3D scalar images,” 3D Com-puter Vision, Utrecht, The Netherlands, Technical Report 91-23,1991.

38. A.H. Salden, L.M.J. Florack, B.M. ter Haar Romeny, J.J.Koenderink, and M.A. Viergever, “Multi-scale analysis and de-scription of image structure,” inNieuw Archief voor Wiskunde,1992, Vol. 10, pp. 309–326.

39. A.H. Salden, B.M. ter Haar Romeny, L.M.J. Florack, J.J.Koenderink, and M.A. Viergever, “A complete and irreducibleset of local orthogonally invariant features of 2-dimensional im-ages,” inProceedings 11th IAPR Internat. Conf. on PatternRecognition, The Hague, The Netherlands, 1992, pp. 180–184.

40. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Lo-cal and multilocal scale-space description,” inProc. of the NATOAdvanced Research Workshop Shape in Picture—MathematicalDescription of Shape in Gray-Level Images, Vol. 126 of NATOASI Series F, Berlin, 1994, pp. 661–670.

41. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Mod-ern Geometry and Dynamic scale-space theory,” inProc. Conf.on Differential Geometry and Computer Vision: From Pure overApplicable to Applied Differential Geometry, Nordfjordeid, Nor-way, August 1–7, 1995.

42. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Al-gebraic invariants of linear scale-spaces,” submitted toJournalof Mathematical Imaging and Vision, March 1996.

43. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Lin-ear scale-space theory from physical principles,”Journal ofMathematical Imaging and Vision, accepted.

44. L.A. Santalo, “Integral geometry in general spaces,” inProceed-

ings International Congress of Mathematics, Cambridge, Vol. 1,1950, pp. 483–489.

45. L.A. Santalo, “Integral Geometry and Geometric Probabil-ity,” Addison-Wesley Publishing Company: London, 1976,Vol. 1.

46. M. Spivak, Differential Geometry, Publish or Perish, Inc.:Berkeley, California, USA, Vol. 1–5, 1975.

47. B.M. ter Haar Romeny,Geometry-Driven Diffusion in ComputerVision, Kluwer Academic Publishers: Dordrecht, 1994.

48. A.L. Yuille and T.A. Poggio, “Scaling theorems for zero-crossings,”IEEE Trans. Pattern Analysis and Machine Intel-ligence, Vol. 8, pp. 15–25, 1986.

Alfons H. Saldenreceived his M.Sc. in Experimental Physics and aPh.D. from Utrecht University in 1992 and 1996, respectively. Hismain research interests are scale-space theories, invariant theory,differential and integral geometry, theory of partial differential andintegral equations, topology and category theory.

Bart M. ter Haar Romeny received his M.Sc. in Applied Physicsfrom Delft University of Technology in 1978, and a Ph.D. fromUtrecht University in 1983. After being the principal physicist ofthe Utrecht University Hospital Radiology Department he joined, in1989, the department of Medical Imaging at Utrecht University asan associate professor. He is a permanent member of the staff of thenewly established Images Sciences Institute of Utrecht Universityand the University Hospital Utrecht. His interests are mathematicalaspects of front-end vision, in particular linear and non-linear scale-space theory, medical computer vision applications, picture archivingand communication systems, differential geometry and perception.He has authored numerous papers and book chapters on these issues,has recently edited a book on non-linear diffusion theory in ComputerVision and is involved in and initiated a number of internationalcollaborations on those subjects.


Max A. Viergever received his M.Sc. in Applied Mathematics in1972 and a D.Sc. degree with a thesis on cochlear mechanics in

1980, both from Delft University of Technology. From 1972 to 1988he was assistant/associate professor of applied mathematics in Delft.Since 1988 he is professor and head of the department of MedicalImaging at Utrecht University, and as of 1996 scientific director of thenewly established Image Sciences Institute of Utrecht University andthe University Hospital, Utrecht. He is (co)author of over 200 ref-ereed scientific papers on biophysics and medical image processing,and (co)author/editor of 10 books. His research interests compriseall aspects of computer vision and medical imaging. He is a boardmember of IPMI and IAPR, is editor of the book series Computa-tional Imaging and Vision of Kluwer Academic Publishers, associateeditor-in-chief of IEEE Transactions on Medical Imaging, editor ofthe Journal of Mathematical Imaging and Vision, and participates onthe editorial boards of several journals.

differential and integral geometry of linear scale-spaces · 2007-08-25 · differential and...

Documents