Linearised Euclidean Shortening Flow of Curve Geometry Alfons H. Salden Bart M. ter Haar Romeny Max A. Viergever Imaging Sciences Center, Utrecht University Hospital, Room E.01.334, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands, email:
[email protected] November 8, 1997
Abstract
1 Introduction
The geometry of a space curve is described in terms of a Euclidean invariant connection, torsion and curvature. The torsion and curvature of the connection quantify the multi-valuedness or the loss of order of contact in the formation of the curve geometry. In order to retain a stable and reproducible geometry of the curve that is slightly aected by non-uniform protrusions of the curve a linearised Euclidean shortening ow is applied. (Semi)-discretised versions of the ow are proposed to physically realise a (semi)discrete geometry of the curve. Imposing special ordering relations the torsion and curvature in the curve formation is retrieved on a multi-scale basis. The dierential and integral geometry of the multiscale representation of the geometry of the curve under the ow is nally also quanti ed by imposing an appropriate connection, and calculating torsion and curvature aspects of this connection.
Our aim is to smooth and describe a planar or space curve embedded on a surface of an object in threedimensional Euclidean space. The curve is assumed to be extracted by a bi-perspective camera system [43] and being e.g. a shadow boundary, or a curve extracted from e.g. a three-dimensional medical image such as a vessel tree. The ordering of the points of the curve is consistent with a particular chain encoding [34] of discontinuity or singularity sets in greyvalued images. But having such a representation of a space curve one is immediately confronted with several problems which we'll address in the sequel:
How to describe the geometry of the curve? How
to describe the fact that the formation of the curve left and right from any point is dierent implying possibly no order of contact at all? Here order of contact refers to the order of differentiability of the curve with respect to an parameter. Increasing even the resolution properties of the camera system one still notices that at any level of scale the formation of the curve left and right from a given curve point remain totally dierent.
Keywords: dierential geometry, intergral geometry, curve geometry, similarity jet
1
In one view the chain encoding will be dierent age gradient eld it is clear that the ow on knots are
to that in another because of change in view, limited resolution properties of the camera system and uctuations in the luminance eld. How can one still retain a stable and reproducible description of the curve geometry despite non-uniform noise in the chain encoding? Furthermore, how does one derive a nite but complete curve geometry on a bounded scale range? One would like to reduce the computational load for performing such a smoothing and description to a minimum. In the context of such multi-scale representations of the curve geometry it's also worthwhile to know how to handle and interpret the ows on open curves, branching curves, selfintersecting curves and knots. Knowing that geometry of the curve and its formation at inner scale, i.e. the highest resolution properties available to a camera system, are quite unstable and that a multi-scale representation of that geometry yields a stable and reproducible curve geometry how can this multi-scale representation of the curve geometry be quanti ed? Smoothings or better relaxations of shapes or greyvalued images, endowed with some geometric structures such as a metric and a connection, by a ow, e.g. Euclidean shortening ows of simply closed planar curves, have thoroughly been investigated in mathematics [12], [33], [33], [26], [1], [28], [7], [8], [6], [9], [2], [74]. As such smoothings involve the solution of nonlinear Cauchy problems, i.e. initialboundary value problems, it is necessary to quasilinearise these problems in order to obtain an exact approximative Green's function [21], [64]. Convolving the initial shape with this Green's function yields an approximative solution to the considered ow problem. The limit shapes under the dierent ows are up to now not fully classi ed. For example, for a simply closed planar curve it's well known that the curve evolves to an in nitesimally small circle. For knots [62], however, almost nothing is known except that one might conjecture that the largest-scale knot types may persist under certain ows. As knots can be viewed as branching curves for a particular im-
not the most general ows one can come up with for grey-valued images such as those of vessels and nger prints. Smoothing grey-valued images by means of nonlinear ows that preserve discontinuity sets of not necessarily constant co-dimension have, however, already been proposed [79], [64]. Even a relaxation of the formation of such grey-valued images using modern geometry have been explored in [64]. In order to describe shapes or images dierential and integral geometry has been extensively studied and developed in the past [16], [71]. The evolution of shapes and their properties under ows are normally again described in terms of geometry [18] or in terms of deformation theory [81], [61]. Note that these descriptions are carried out in the context of modern geometry and topology meaning that the space of observations whatever they may be can be endowed with a non- at connection, i.e. observation space is twisted and curved not only in a geometric sense but perhaps also in a topological sense. This statement sounds quite cryptic but will become transparent shortly. Also in computer vision the shortening ows attained a lot of attention. They were applied to mainly planar curves that were either implicitly de ned by a level-set of a two-dimensional grey-valued input image [36], [38], [39], [3], [73], [63], [67], [76], [64], or that were de ned by an ordered set of extracted point-sets or polygons from a grey-valued input image [45], [56], [52], [53], [51], [54], [50], [55], [13], [14], [27], [64]. In the rst approach to the curve ow problem the problem is translated into an equivalent ow problem for the level-sets of a grey-valued image or derived image, whereas in the second one really formulates the ow problem on the extracted polygonal con gurations. But the deep structure of the multi-scale representations of shapes have scarcely been addressed in computer vision. In [44], [20], [64] the authors propose logical, topological and geometric machines to read out that deep structure. For the partial topological equivalence and modern geometry of scale-spaces the reader is advised to consult [64], [69] As this paper is mainly concerned with the geometry of the multi-scale representations of curves and their formation let's brie y elaborate on the dierent proposed shortening ows for curves pointing out
2
their objectives, assumptions and implications.
curvatures of the curve. Another assumption underlying the nonlinear morphological processing of the curve is the so-called comparison principle:
The geometric planar curve shortening ows [76],
[64] based on the nonlinear evolution of greyvalued images are developed in order to smooth images, such that certain symmetries of the landscape of isophotes and owlines are being preserved under the ow. One assumes that one of the isophotes is a representative for the curve under study. But normally one applies edge detectors to extract the curve and associates to the interior a positive grey-value and to the exterior a negative grey-value. The border between the positive and negative grey-valued regions in the two-dimensional image is then identi ed with a contour of a planar object in space. Further assumptions are that the ow on the image is invariant under Euclidean movements of the image plane or projection center, causal and invariant under spatial homogeneous grey-value transformations to restrict the class of possible ows. Of course, the admitted grey-value transformations and Euclidean movements should still allow an accurate extraction of the contour on the basis of e.g. the grey-value characteristics between Mach bands in the images [34]. A "faithful" representation of the contour under all these transformations is severely limited by the resolution properties of the camera system. The scale-space of the curve that is an embedding into a family of simpli ed curves is assumed to be recursively obtained from the input curve C by means of a sequence of lters T (semigroup property):
C1 C 2 ; T C1 T C2 ; 8 2 R+0 : The latter principle which requires e.g. a planar curve that is contained in the closure of the interior of another curve having possibly points tangent to it remains that throughout the process. This topological principle can also be stated in terms of similar relations for the corresponding evolution of a grey-valued image with two particular isophotes coinciding with the above curves. The Cauchy problems related to these curve shortening ows are subsequently solved by means of a (Osher-Sethian) nite dierence method [59], [4] or by means of mathematical morphology [72]. Unfortunately both methods do not yield an exact approximation of the Green's functions consistent with the Cauchy problems [21], [68]. Of course, the regions in the image where the approximation to the Green's function can be applied must allow the ow problem be stated in terms of a parabolic Cauchy problem in the sense of Petrovskiy. But with some tricks the methods can be extended to the other regions where the inequality does not hold (we will not address the speci c method of continuation in this paper). Another possibility is regularising the derivatives with respect to e.g. the gauge coordinate v (a coordinate axis along the tangent vector to an isophote in a two-dimensional grey-valued image) such that derivatives become convolutions with anisotropic Gaussian derivatives: @ dG(x; y; s) ? dG(~x; y; s) ; @v dy dx~ with 1 e?(x2 +y2 )=4s ; G(x; y; s) = 4s 1 e?(x2 +y2 )=4s ; G(x; y; s) = 4s
T0 C = C; T + C = T (T C ); ; 2 R+0 ; where T0 is the identity operator and and scale parameters. The scale-space is here generated by solving the Cauchy problem related to the curve shortening ow problem. The simpli cation of the curve at coarser scales can then be expressed in e.g. the decreasing number of corners that can be detected as extrema in the 3
where
x = y =
essarily) one-dimensional discontinuity or nonisolated singularity sets, in a two-dimensional grey-valued image can be formulated also for space curves in three-dimensional grey-valued images. The evolution of a space curve (with co-dimension two) is then governed by a nonlinear dierential equation with respect to the three-dimensional input grey-valued image. The local ow imposed on a level set enveloping the curve is geometrically determined by its smallest principal curvature multiplied with the unit normal vector. The co-dimensional ows higher than one also satisfy the comparison principle but disjointness is violated. For example, two circles that are linked initially will after nite time develop a nonempty intersection, and subsequently the joined circles become fat and stay joined. Of course, one could also consider space curves and discontinuity sets as intersection of the level sets of two scalar functions [22], [67]. Applying the ordinary level set approach for surfaces in three-dimensional grey-valued input images the smoothing of the shapes of interest occurs by smoothing the two level sets and determining their intersection. As this generalisation of the mean curvature ow yields a degenerate system of partial dierential equations the theory of viscosity solutions cannot be employed for analysing this system. This approach yields de nitively no fatting of linked shapes: the above circles will join and disconnect over time before vanishing into separate points. It seems as that the higher co-dimensional ows on grey-valued images are more proper in a topological sense (satisfying inclusion relations) than those proposed by [22], [67] but this is only a matter of taste. An example, however, can be given in which the latter ows (as well as those considered in this paper) may be more attractive. Having a knot in three space the higher co-dimensional ow will yield intersections and cause fattening. But perturbing the knot slightly might yield another number of intersections at each time of the evolution and certainly no sta-
p(G(G xL)(LG) L) (x; y; s0)x; p(Gi L)(Gi L) i
i
(x; y; s0 )y; p(G(G xL)(LG) L) i i x~ = (x; y; s0 )y; (Gy L) (x; y; s0 )x: y~ = p (Gy L) (Gi L)(Gi L) Note that the anisotropy is directly controlled by the acquired information about the image gradient eld at a level of scale s0 . Furthermore, that the adapted Green's functions to operationalise the blurring satisfy normalisation and that Einstein summation convention is used (summing over twice occurring subindices). Finally, that in case of vanishing image gradients the rule of l'Hôpital should be used. All the approximating ows yield an in nitesimally small circle-like curve for simply closed planar curves in the case of Euclidean shortening ows. Self-intersecting curves develop normally cusps as x2 = y3 , but need not to have in nitesimally small circles as limit shapes (see for more details [1]; the limit shapes obviously also depend on the curve shortening ow under study). Recall that a true cusp arises from the projection of a smooth space curve (illuminated iron strip) on one of its normal planes [42]. In order to break up a planar curve in its constituent parts a so-called entropy ow on planar curves [37] is proposed. The goal of the author is to generate a partial ordered set of the segments that on the basis of a certain criterion belong to dierent curve formation processes. The result of the entropy ow is that topological interesting sites where the changes in curve formation are predominant, are highlighted and classi ed. Recently one has extended the level set approach for hypersurfaces [5] to surfaces with arbitrary co-dimension. This means that the above approach for smoothing planar curves or (not nec4
bility of the limit shape, because, for example, the self-linking number of the knot may be also aected by the perturbation. The change in selflinking number due to the perturbation causes consequently the number of joints generated over scale to be instable. The other proposed method permits self-passages and removal of kinks of the knot and thus allows also to smooth the topological aspects in the knot formation. According to these authors adding obstructions or not to the ow as in the ows above should therefore be motivated. On the basis of physical grounds one might impose or reject e.g. the conservation of knot invariants such as the (self)-linking number. Alternatively, one can apply anisotropic scalespace theories [79] and dynamic scale-space theories [64]. Both theories allow to smooth space curves and discontinuity sets in higher dimensional grey-valued images preserving or enhancing them as much as possible. The dynamic scale-space theories, however, also allow to smooth the image formation, i.e. the topological transformations of a reference state of a camera system yielding a particular not necessarily smooth image gradient vector eld. On the basis of the curvatures of image formation that substantiate the existence of a space curve such as a knot or some other discontinuity set, the smoothing, recombination and splitting of such shapes is operationalised. Changes in knot type or the co-dimensionality of components of such a discontinuity set can then partially be constrained by subjecting the ow to particular user-de ned fusion and ssion rules. Koenderink [45] proposed a linear planar curve ow in order to smooth a bounded not necessarily dierentiable planar open curve of in nite length and in order to nd a hierarchical structure of a curve as that for two-dimensional greyvalued images [41]. So his objective was to nd a stable and reproducible geometric and topological structure of a curve even when this curve would be subjected to small scale protrusions. Small scale has only meaning in relation to the
ratio of the smallest length and the largest length measurable by a camera system. Thus in nities in length measurements can never occur as the resolution capacities of the system are nite. Thus applying a ow to fractal curves makes no sense in this context, but realising that such a curve can be generated by a certain rule one could instead try to smooth the instruction itself as done for curve and image formation in [64], [70] for length measurements then do not come into play. The linear curve ow is based on a Euclidean invariant (not canonical) local description of the curve with respect to the local Frenet frame eld, i.e. the curve is given by (x; y(x)) with x- and yaxis locally tangent and orthogonal to the curve, respectively, and x a parameter measuring the Euclidean distance between a point on the curve and a point further on the curve projected on the x-axis. Furthermore, the scale parameter is assumed to be completely not in uenced by the curve description, whereas this is the case for the curve shortening ows mentioned above. Note that the Euclidean arclength parameter is a function of this parameter and the x-coordinate, whereas the x-coordinate is of course not. Furthermore, that only a ow on the local description is applied and the global eects of the ow on the curve cannot be discerned on the basis of a local analysis. The linear curve ow is modelled by the isotropic ordinary diusion equation and the local curve description as initial condition [45]. The solution space to this linear Cauchy problem is generated by convolution of the initial condition with the so-called fuzzy-derivative operators Gn = ddxnGn with G the normal distribution that satisfy the following semi-group and cascade property:
Gp (; p ) Gq (; q ) = Gp+q (; p + q );
8p; q 2 Z+0; p ; q 2 R+0 , where denotes ordinary convolution. Note that the solution space is only de ned if the input image satis es the
5
following growth condition:
luminance eld over the image domain. It means that he advocates rather to follow the evolution of a discretised curve represented by a set of pixels with particular energy values as ridges or ruts than the evolution of the curve under its own properties. However, the latter does not mean that the nonlinear ows formulated above for a density eld do not make sense. The only difference between the linear scale-space ow on an image and these ows with respect to greyvalued images is that they are invariant under a larger group of transformations. The latter property of the nonlinear ows can become in particular situations advantageous, e.g. whenever camera systems possess dierent sensitivity pro les and one wants to perform with these systems recognition tasks yielding similar results despite those dierent camera properties. If such invariance conditions have to be imposed it is obvious that the nonlinear curve shortening ows are to be favoured and that it makes no dierence whether to de ne it actually on the curve or on the grey-valued as the density eld values on isophotes are irrelevant and information exchange among isophotes is undesirable. Mokhtarian [56], [52], [53], [51], [54], [50], [55] proposed analogous to Koenderink a local linear curve ow for smoothing path-based parametric representations of space curves. His objective was to obtain a higher degree of match between so-called curvature scale-space images (that are curves of vanishing Euclidean curvatures) of the input curve and the same input curve perturbed by (non)uniform noise. On the basis of these images Mokhtarian hoped to give a stable and reproducible description of contours of threedimensional objects such that object and pattern recognition tasks became more robust. The initial curve is represented at every point on the curve as a canonical form, i.e. the component functions x and y are given in terms of the Euclidean arclength parameter s at every scale t. The ow is de ned as in the nonlinear curve shortening ows above but now with respect to the initial local curve parametrisa-
y0 G (x; ) M > 0; 8 2 R+0 :
It's worthwhile to compare this semi-group and cascade property with that of the nonlinear curve shortening ow above. The dierence between them is that in the linear curve ow the scale parameter is a real free parameter, whereas in the nonlinear curve shortening ow it's intrinsically coupled to the initial condition, i.e. the intrinsic geometry of the input curve. Another aspect of the linear curve ow is that all the solutions yn = y0 Gn satisfy a so-called maximum-principle: inf (yn(x; 0)) yn (x; ) sup (yn (x; 0)); (x; ) 2 E R+0 . This maximum-principle resembles a lot the comparison principle applied in the nonlinear curve shortening ows above. The maximum-principle directly concerns the evolution of a function yn , i.e. a component of the local curve representation no two functions. The comparison principle on the contrary concerns the conservation of the inclusion relations for two curves under the nonlinear curve shortening ow. However, the comparison principle also holds for the linear curve ow considering the evolution of the ordinary ordering relations for two component functions yn and zn . Open unbounded (satisfying growth condition) and not self-intersecting curves tend for in nite scales to straight lines, i.e. the x-axes. For bounded and/or disconnected segments and points forming a "curve" the local limit shapes are again x-axes. Concerning the global eects of the linear curve ow one might for e.g. a bounded non-closed curve segments expect that they each evolve towards an x-axis obtained by just averaging the tangents over the segment. A remark made by Koenderink [42], chap. 9, concerns the relation between the physical eld, i.e. the luminance density eld, and the initial curve de ned by that eld. He proposes not to blur directly the curve but instead to relax the 6
tion. The smoothing is performed by means of a Green's function G of s such that (x; y)(s; ) = ((x0 ; y0 ) G)(s; ). In order to compare the curvature eld at dierent scales a normalisation of the ow is carried. The latter normalisation differs considerably from the global normalisations such as conservation of enclosed area or total length proposed in the nonlinear curve shortening ows. As can be concluded from the previous discussion on the comparison-principle and the maximum-principle this linearised (normalised) ow satis es the latter principle together with a related semi-group property. It should be emphasised that again the semi-group properties of this ow and the above ows are not the same. Here the semi-group property concerns the local normalised vectorial ow. Under the local linear curve shortening ow of Mokhtarian a planar curve shrinks to an in nitesimally small circle, for space curves one obtains (normalised) torsion and curvature scalespace images and an analysis of the formation of cusps for self-intersecting planar curves under the ow is feasible.
linear curve ow takes into account a sense of walking around the curve and yields instead a planar ellipse for (space) curves. The considered curve ows are shown to yield for all classical groups perfectly symmetric in nitesimally small polygons. In the case when one sticks to only Euclidean ows and rescale the symmetric polygons by imposing conservation of length they are nothing but equal-sided polygons (the discrete analogue of a circle of xed radius or curvature being a triangle). Although adding to these mathematical and computer vision approaches a new curve shortening ow paradigm seems to be super uous (for curve ow paradigms more adapted to its formation itself and in line with the thoughts of Darboux see [19], [64]), several questions arise when one has studied in depth the above curve shortening ows. Let's repeat these research questions already made manifest in our goals presented in the beginning of this introduction: As a curve consists of segments that are possibly glued together how to describe the dierential and integral geometry of the defects in the order of contact of the curve? This question is almost never addressed in the vision literature (except in [37], [64]) where one rather lingers on doing classical dierential and integral geometry of curves and surfaces [42]. But it would rather bold to say that the latter author is not aware of this image or curve formation problem. How to build a multi-scale representation of a curve that is self-intersecting, open, ending, branching and/or knotted, and has defects in the order of contact? Especially, how to combine Mokhtarian's approach and that of Bruckstein in order to obtain a non-local ow that takes into account the global initial Euclidean arclength parametrisation without normalising the ow in the Moktharian manner? In which sense does Darboux's ow [19] dier from our proposed linearised Euclidean curve shortening ow? How to build a multi-scale representation of the curve formation using our proposed linearised
Bruckstein [13], [14] developed curve shortening
ows for closed planar polygons governed by a circulant matrix (linear) transformation in order to simplify the convergence analysis (shrinking to polygonal circles, ellipses, etcetera), and to overcome problems like non-convexity and selfintersection. The number of vertices of the polygonal curve is assumed to be constant, the curve parametrisation and canonical representation are assumed to be periodic and to be invariant under classical transformation groups, namely the group of Euclidean movements, the group of (unimodular) a ne movements and the group of projectivities in the plane. With respect to the initial polygonal curve are not made further stipulations meaning that the curve can be e.g. selfintersecting. It is related to the seminal work of Darboux [19] but not completely. Darboux's 7
Euclidean shortening ow? Another method than that proposed by [37] for extracting the most relevant constituent parts of a curve then comes available. Also the description of the socalled cusps [56] that are normally encountered as the projection of the rim of a surface or space curves, and that of junctions that are the projections of outer contours of surfaces before and behind one another can be revisited. In the case of cusps it is, for example, from the point of view of curve formation (probably due to the surface formation of disconnected patches) not logical to assume that the cusp can be described left and right from it in one of the above standard canonical forms [42], e.g. y3 = ?x2 . Note that there also so-called cusps of surfaces [64]. Assuming the rim of a surface near such cusp in an orthographically projected image to be described by a very crude planar curve forgetting about correspondences how can one quantify such visual events? How to obtain a complete geometric analysis within a nite scale range? Working in a continuous setting confronts us with the problem of digging into and exploring an in nite set of data, whereas the discritisation of the curve already suggests that we are overdoing the analysis quite a bit by choosing a continuous setting for it. How to build a fully discrete scale-space on the basis of the continuous ow? Last but not least how to describe the dierential and integral geometry of a multi-scale representation of a curve or the formation of a curve? Upon the ow on the curve several important events in the scale-space occur such as "cusp" annihilation that need to be quanti ed or highlighted. Our paper is organised as follows. Euclidean differential and integral geometry of space curves and their formation is summarised in section 2. Classical and modern geometry are used to quantify the dierential and integral intrinsic properties of curves that may be defected. Examples on defected greyvalued images are added to point out the dierence
between classical and modern geometry. These examples are chosen such that they immediately carry over to the problem of quanti cation of curve formation. In section 3 our linearised Euclidean shortening ow is presented for non-closed, branching, knotted and self-intersecting curves. A comparison between Darboux's ow and ours is made in order to point out some remarkable phenomena in the multiscale representations of tame knots. The linearisation comes about by using the initial Euclidean arclength parametrisation of the curve at every level of scale as basis for smoothing the initial curve by means of ordinary Green's functions. The curves are represented by periodic point-sets or polygons following [13] that are acquired by chain encoding [34], or polygonal approximations on a square or cubic lattice before being parametrised by their Euclidean arclength. Similar ows are proposed to generate a multi-scale representation of the defects in the formation of a curve. The latter is done by putting an a-symmetry into the curve ow via a special point ordering relation on the curve. Subsequently (semi-)discretised versions of the curve (formation) ows are proposed to reduce the computational load. Finally, the dierential and integral geometry of the multi-scale representation of a curve and its formation are proposed to be quanti ed in terms of again twists and curvatures.
2 Euclidean Geometry In subsection (2.1) the dierential geometry of arbitrary space curves is treated, whereas in subsection (2.2) their integral geometry is studied. Here the geometry of space curves is studied in a classical and modern geometric sense. For the classical geometry of space curves the ordinary curvature functions and the Euclidean arclength parameter su ce to de ne all relevant dierential or integral properties of a smooth space curve. If the order of contact of the curve is nite, then the modern geometry of space curves can be captured in terms of a torsion two-form, curvature two forms, covariant derivatives of the corresponding tensors and related irreducible density elds. The latter geometry is crucial in describing the curve formation. Besides these local curve (formation) properties 8
one can also construct topological or integral invariants for the curve (formation) such as rotation indices and self-linking numbers in case of tame knots. Although such curve (formation) invariants play an important role in characterising the dynamics of curve ows in section 3.2 a thorough treatment will follow in a forthcoming paper.
semi-directly on the right to each local a ne frame
Vp .
De nition 1 A local a ne frame Vp at a point p of a n-dimensional manifold M is de ned as Vp = (x; e1 ; : : : ; en ); x; ei 2 Tp M; (1) where Tp M is the tangent space to the manifold M at point p.
2.1 Dierential Geometry
In order to compare and relate the local a ne frames over the manifold M an a ne connection ? is speci ed which de nes in turn the so-called covariant derivative operator. De nition 2 An a ne connection ? on a ndimensional manifold M is de ned by one-forms !i and !ij on the tangent bundle TM : rx = !i ei ; (2) j (3) rei = !i ej ; where denotes the tensor product, and r the covariant derivative operator. Thus for any space of observations, whether it is a grey-valued image or some spatial con guration, a particular concept of a manifold, a related frame eld and connection is chosen or in icted by the observations themselves. Recall that !i (ek ) 2 T (n; R) or one-form !i is a machine for quantifying translations (not necessarily distances) in the tangent spaces, and that !ij (ek ) 2 GL(n; R) or one-forms !ij are machines for quantifying general linear transformations in the tangent spaces. Furthermore, that the whole a ne group can be represented by a subgroup of the projective group. But giving such a projective representation would certainly not facilitate the interpretation of the dierent aspects in the torsion and curvature of the curve formation. Now the connection one-forms !i and !ij satisfy so-called structure equations: Theorem 1 The structure equations for an a ne connection ? are given by: D!i = d!i + !ki ^ !k = i ; (4) j j j j k D!i = d!i + !k ^ !i = i ; (5)
Firstly, the system of Cartan structure equations for a manifold with a (metric) 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. Secondly in the subsequent paragraphs, the theory is applied to describe the classical dierential geometry of smooth curves and to describe the modern dierential geometry of the distortion or formation of curves for which there might be no order of contact at. The reader should be cautious not to consider classical and modern dierential geometry to be on an equal footing as he would be inclined to claim upon reading e.g. [42] where much emphasis is put on making the classical one comprehensible. How to make the jump from classical to modern dierential geometry will also be one of the endeavours in this subsection by giving very simple examples for the formation of grey-valued images that immediately can be applied to that of curves with no or a nite order of contact. A complete and extensive exposition on dierential geometry, however, is out of the context of this paper. Therefore, the reader not familiar with dierential geometry in all its beauty is referred to standard textbooks like [17], [16], [18], [24], [75]. In the sequel only the elementary notions are de ned, theorems stated and examples given (for more sophisticated applications of dierential geometry see [64]). Let M be a n-dimensional base manifold and consider the a ne frame bundle L P (M; ; A(n; R)) where P is the total space consisting of all a ne 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 a ne group, where GL(n; R) is the general linear group and T (n; R) the translational group, acting 9
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.
p 3
p 2
Vp 3
Vp 2
δ
d
Proof 1 See [75]
p 0
Vp 2 p 1
Vp 0
Vp The torsion two-forms i and the curvature two1 j forms i are so to speak machines that can read out the inhomogeneity in the a ne group action over Figure 1: A closed circuit C = (p0 p1 p2 p3 p0 ) on the the manifold. The torsion two-forms i and the cur- manifold M and the frame eld V along this circuit. vature two-forms i j are related to the components Tjk i and Rikl j of the more well known torsion tensor If d = dt @t@ and = d @ @ are vector elds generT and curvature tensor R as follows: ating from p0 the circuit C , then the change of the frame eld along the upper part C + = (p0 p3 p2 ) and the lower part C ? = (p0 p1 p2 ) of the circuit C can be 1 (6) expressed as:
i = 2 Tjk i !j ^ !k ; Z Z 1 j j k l Vp ? + Vp = [; d] Vp ; (10)
i = 2 Rikl ! ^ ! ; (7) C? C which in terms of the torsion two-form and the curvature two-forms reads: with [; d] x = i (d; )ei ; (11) i [; d] ej = j (d; )ei : (12) T = Tjk i !j !k i ; (8) R = Rikl j !i !k !l j : (9) The following two simple examples show that the existence of such modern geometric machines as the torsion and curvature two-form in the human visual system are indeed very likely. Let's make the function of these geometric machines more explicit by studying within a 2- Example 1 In image perception [34] one disdimensional submanifold S , parametrised by uh = tinguishes notions like luminance and brightness uh (t; ), of manifold M the variation of the frame of objects and relates the Mach band eect to the eld in the tangent spaces around p0 for an in nites- impulse response of the human visual system. Let imally small closed circuit C = (p0 p1 p2 p3 p0 ) in S by us explain the Mach band eect as a manifestation quantifying this variation with respect to the local of the connection on the system activity due to the frame Vp0 (see gure 1). Now in general the local perceived luminance pattern. Next we compute the frame Vp2 is not equal to local frame Vp2 . curvature associated with this connection and put
10
it in a topological context of image formation. But rst of all let us recall some of the basic notions above. The luminance of an object in the context of image perception is not determined by the luminances of the other objects and that of the background, whereas the brightness is. A white object within a black or grey background has dierent levels of brightness for those backgrounds. Note that in these formulations clearly total luminances associated to particular regions in the image are meant. There exist several luminance to contrast models in the literature [34] to quantify the change in luminances of the object and its surrounding needed to be just noticeable. One of these models being Weber's law1 states that if the luminance Io of an object is just noticeably dierent from the luminance Is of its surrounding then the following ratio should be constant:
= 4 log I = jIs I? Io j ; I I e ; o
with
tor. This constant is assumed to be proportional to the change in contrast c, i.e. c = a1 + a2 , where ai ; i = 1; 2 are constants. Apparently the brightness across the interface of two levels of luminance (two strips in the image domain with two dierent almost homogeneous intensity levels) are extremal at the Mach bands. Realise that over the image domain the frame eld can be taken equal to (I n^ )(x) where n^ is the unit normal eld belonging to the detector array and ^ dual to n^, i.e. ^(^n) = 1. Doing so one recognises that the frame eld physically just embodies a ux- eld. As a de nition for the brightness of objects we now propose:
(15)
where k parametrises possibly overlapping neighbourhoods along the owlines, i.e. the integral curves of the image gradient eld. In case of a grey-level bar chart these neighbourhoods are rectangular windows containing detectors reading out the total luminances. In this de nition the contrast can apparently associated a parity. Furthermore, the window sizes need not to have the same dimension as the thickness of one of the bars and may vary. Last but not least the order (kk + 1) does matter. Reversing the order one notices that there are two brightness values attributed on either side of the interface, for example, if k = 1 then on the left and right these are:
B12 = log I1 + I1 I? I2 ; 1 I 2 ? I1 B21 = log I2 + I 2
(13)
where I e is the ground energy state of the detec-
4kk+1 log I = Ik ?I Ik+1 ; k
(16) (17)
Note that the order actually relates to the paths (strips) on the detector array generated by the vector elds d and (see also gure 2). For a human eye these vector elds can be constructed by means of spherical frame elds e and e if the distance from the center of the lens to the retina is taken unity. For the image below the paths are generated by the ordinary frame elds e^x and e^y on two-dimensional Euclidean space. The contrast can now be interpreted as a manifestation of the connection on the grey-level bar chart. This connection ! we de ne as:
!kk+1 = (4kk+1 log I )^ :
(18)
Bkk+1 (x) = [log Ik + 4k;k+1 log I ](x); (14) The impulse response of the human visual system can The choice for a logarithmic law in the luminance to contrast model resides in the fact that the total luminance fallen onto a detector aects the total system state of the detector causing some neuronal activity yielding a grey-valued image. The total possible activities of the detector can be quanti ed in terms of a so-called partition function for the system state. Thus actually the opposite of the logarithm of the perceived luminance, represented as a micro-canonical ensemble, can be considered as a free energy available to the detector. 1
subsequently be easily understood in terms of our brightness de nition where the lateral inhibition is caused by the brightness at left and right from the impulse, whereas the brightness in the center is due to just a superposition of the brightness on either side of the middle. Alternatively, the impulse response is caused by the connection ! on the array of detectors induced by the external luminance eld.
11
2 1 C
Figure 2: Circuit integral around C E 2 , where E 2 two-dimensional Euclidean space, reading out the curvature R12 = ?R21 in the grey-valued image formation. More interesting, of course, is to measure the change in connection across the interface of two levels of luminance (see gure 2) and express it in terms of a curvature output R12 , for example, if k = 1 as:
R12 = ?R21 = !12 (^e12 )^n ? !21 (^e12 )^n (19) = (I1 ? I2 ) I1 + I1 n^; 1
Another aspect is that the circuit integral in gure 2 forgets what happens at the other interface. Thus we may equally well take the dierence of the connection output multiplied by n^ . In case of occlusions yielding e.g. junctions in a grey-valued spherical image the above path dependency is crucial to read out and quantifying the cutting and pasting procedures for constructing the image. The ordering of the interfaces can be obtained by means of a topological operation in which along a circuit around the junction the magnitudes of above curvatures and others de ne the ordering of the edges connected to the junction. According to these authors the multi-valuedness locally of the curvature is a common thing to happen in ordinary images or shapes and should be appreciated and welcomed if one would like to unravel the topological aspects in the dynamics involved in image or shape formation processes. Some readers may object that junctions are destroyed by certain scale-space theories, but forget that the borders between the dierent greyvalues can still be retrieved by applying modern geometry of the smoothed images [66]. A further dwelling on the above topological aspects is out of the context as even more sophisticated machines similar to the above ones have to be applied to point them out.
2
in which e^12 is a unit normal vector eld directed from pixel 1 to 2 perpendicular to their interface in the image plane. In homogeneous regions in the greyvalued image this machine gives vanishing output and at interfaces between noticeably dierent levels of luminance a high output. One might object that one has build a sophisticated edge-detector. Indeed, but one that has incorporated the full geometry involved in the image formation. Notice that the curvature R12 can be conceived as a directed topological current in which the conductivity is determined by Ohm's law for parallel resistors. Here topological current stands for the transition going from one detector state to another experienced across their interfaces. Total (perceived) luminances can in this context be understood as topologically conserved quanta as well.
Applying the covariant exterior derivative operator
D to the structure equations yields the integrability
conditions for the a ne connection ?, i.e. the Bianchi identities. Theorem 2 The Bianchi identities for the a ne connection ? of the manifold M are given by: D i = j i ^ ! j ; (20) j D i = 0: (21) Proof 2 See [75], [57] The geometric meaning of these identities will become more transparent in the next subsection on integral geometry. In order to de ne a metric connection assume that the connection one-forms !i determine a metric ten-
12
sor , 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. De nition 3 A metric tensor on a n-dimensional manifold M with a ne connection (De nition 2) is de ned by:
= !i !i : (22) De nition 4 For a n-dimensional manifold with a ne connection (De nition 2) the connection oneforms !i and the frame vector elds ej are dual, if and only if: !i (ej ) = ji ; (23) where ji is the Kronecker delta-function.
De nition 5 A n-dimensional manifold with a
metric-a ne connection is a manifold with an a ne connection (De nition 2) and metric tensor (De nition 3) for which the following compatibility condition holds: r = 0: (24) The latter compatibility condition means that the angles between and lengths of vectors measured by the metric tensor under parallel transport associated with the a ne connection ? are being preserved. The reader is warned not to conclude from this that the torsion tensor has to vanish identically [40], [64]. 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 a n-dimensional manifold with a metric connection (De nition 5) and satisfying the duality constraint (De nition 4). In order to establish this relationship quantify with respect to a local reference frame Vp0 = (x0 ; e0i ), which are attached to the camera system, the component functions epi of the frame vector elds ei as: ei = e0p epi ; (25)
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 de nition of an a ne connection (De nition 2) and the above representations of the frame vector elds ei it is easily veri ed that the connection oneforms !j i are directly related to so-called connection coe cients ?jk i :
!j i = ?jk i !k ; ?jk i = (ej (log E ))ki ; E = (epi ): (27) Here again the interpretation conveyed in example 1 is forced upon us, namely that Weber fractions are just particular connections. Furthermore, that a connection is a generalisation of a Weber fraction. Now 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 coe cients ?jk i as follows: (28) Tjk k = 21 (?jk i ? ?kj i ); Rjkl i = ek ?jl i ? ej ?kl i + (29) m i n i ?jl ?mk ? ?kl ?nj :
Example 2 Consider the following two-dimensional grey-valued image I (see gure 3):
p
p
I (x; y) = 0, i2 = ?1 and < is the real part of a complex number. Performing the circuit integral with respect to the normalised image gradient vector eld along the upper part C + and the lower part C ? of the circuit with only p2 2 Q = [?x0 ; x0 ] fy = 0g one experiences only on Q a curvature output R12 of this eld, namely:
R12 = ?R21 = 2ey ;
(31)
where ey is the unit vector in the direction of the positive y-axis. The normalised image gradient vector eld clearly reverses direction at the the branchj j and the components e q of the one-forms ! through cut segment Q. Furthermore, one notices that at the end-points (x0 ; 0) and (?x0 ; 0), i.e. the boundary @Q the duality constraints: of the branchcut, that only once a reversion of the epi epj = i j ; epi eq i = gpq ; (26) image gradient along the circuit occurs, whereas at 13
the points in the interior Qo = Q @Q the are opposite to each other two reversions. This observation implies one can use the a ne displacement operator to asses the dimensionality of components of discontinuity and singularity sets.
topological transformations of the reference state of a camera system yielding a particular image or curve. The reader constructs easily any kind of knot or link as a ridge in a three-dimensional grey-valued image by putting Gaussian blobs along the parametrisation of those objects and ensuring that the width of the blobs is signi cantly smaller than the distance between a pair of dierent points on the objects. At some distance from the cutline (ridge) the natural borders of the in uence zone of the cutline show up (taken in nity or the boundary of the image domain also as borders). We have so to speak constructed a ribbon knot touching upon on itself and covering the whole image domain. Again our modern geometric analysis clearly detects the cutlines. Note that other nonlinear techniques [29] can also be applied to nd these discontinuity or singularity sets. However, our modern geometric approach allows a straightforward generalisation of the machines needed [64]. In the sequel a curve C is a set of vectors x of n-dimensional manifold M parametrised by an arbiFigure 3: Image I with branch-cut discontinuities on trary parameter p 2 R: Q for the normalised image gradient eld. More general branching or bifurcation sets exist, for example junctions in ngerprint or vessel images. The formation of these kinds of shapes of not necessarily constant co-dimension can also be described by means of dierential geometry. But the cutting and pasting procedure at, for example, a junction should be again unravelled by means of topological methods (see also [69], [64], [66]). Nevertheless, in order to trace endpoints of the branch-cut and alike other type of curvatures obtained by e.g. scale-space methods [69] can still be used. These methods allow us to discern between dierent dimensional singular simplexes and to construct so-called CW -complexes for grasping the topological aspects involved in the image or curve formation (see for elaborate expositions on these matters also textbooks on dierential topology [32] and algebraic topology [30]). In this context it can be interesting to study ridges and courses [46], [20], [69], [64] as natural skeletons and boundaries of objects [65]. It is advantageous to conceive the manifestation of the
C fx 2 M jx = x(p)g:
(32)
In the rst paragraph the classical dierential geometric description of a curve immersed in Euclidean space is recalled. The torsion and curvature tensor of such a curve appear to vanish, and the description of the curve is completely determined by the frame eld and the connection one-forms. For an extensive exposition on classical geometry with numerous pictures of curves and surfaces the unfamiliar reader has to turn to [42]. In the second paragraph requirements of smoothness and single-valuedness imposed on the generation of the curve are dropped in order to de ne a non- at connection, which implies a non-zero torsion and curvature tensor in the curve formation. The latter approach is quite common in defect and gauge eld theory [35], [40]. Thus in the sequel classical dierential geometry is used to describe smooth curves locally and modern dierential geometry is reserved for the quanti cation of curve formation.
14
2.1.1 Classical Dierential Geometry
In order to come to the classical dierential geometry of a curve assume the curve to be immersed in Euclidean space E n , which can be identi ed with Rn endowed with the ordinary inner product. If curve C is a map of S 1 into E n , parametrised by Euclidean arclength parameter s, then in the neighbourhood of regular points the Euclidean Frenet frame V , an positively oriented orthonormal basis, is given by:
V = (x; e1 ; : : : ; en ); with
dxi dxi ; ds = e1 = dx ds dp dp
21
(33)
Higher order Euclidean dierential geometric invariants of the curve are subsequently straightforwardly derived by taking derivatives of the found curvatures kk with respect to the arclength parameter s. These curvatures are then Euclidean invariant zeroforms or functions on Euclidean space. It should be emphasised not to confuse these curvatures with the covariant derivatives of the torsion or curvature tensor mentioned above. The curve can subsequently be given a Euclidean canonical description as follows:
x(s) =
1 1 dk x X
k k s ; k ! ds k=0
(41)
(34) which can readily be expressed in terms of the derived curvatures and the Euclidean frame through use of the above structure equations. Although where the remaining unit frame vector elds the the construction of a frame has become unfeasible e2 ; : : : ; en are obtained by applying the Grammat in ection points, the higher order Euclidean n ? 1 dx Schmidt orthonormalisation process to dx ds ; : : : ; dsn?1 . curvatures are still computable. In the two examples Now the connection is determined by the frame below the dierential geometry of planar and space eld Vp , and the connection one-forms !i and !ji are curves are summarised for later convenience. For through the structure equations given by: more illustrative matters the reader is again referred to [42]. 1 2 n ! = ds; ! = : : : = ! = 0; (35) !ij = ?!ji ; !kk+1 = kk ds; (36) Example 3 The classical dierential geometry k !l = 0; 8l > k + 1; (37) for a planar curve can be captured by the Euclidean frame eld V given by: where V = (x; e1 ; e2 ); (42) E ; E = (e : : : e ): (38) !ij = ds d log 1 n ds with In this case the connection is at, that is the tor de de ? 12 de1 dx sion two-form i and the curvature two-form ij are e1 = ds ; e2 = ds1 ds1 (43) vanishing on the curve: ds ;
dp;
d!i + !ki ^ !k = i = 0; d!ij + !kj ^ !ik = i j = 0:
(39) and connection one-forms !i and !ji given by: (40) !1 = ds; !2 = 0; (44) 0 kds j Note that this is the point of view in Koenderink's book [42], namely choosing the connection at in ac!ij = (45) ?kds 0 i ; cordance with Euclidean space. In Cartan's book [18] non- at connections are standing central by allow- where the Euclidean curvature k is given by: ing structure above the base-manifold, i.c. Euclidean space to be twisted and curved! We'll consider such 1 (46) k = de structures in the next subsection. ds e2 : 15
Example 4 The classical dierential geometry for a
y
space curve can be captured by the Euclidean frame eld V given by:
V = (x; e1 ; e2 ; e3 );
P
ε
(47)
φ
φ
L
R
with -a
e1 = dx ds ; de1 de1 ? 21 de1 e2 = ds ds ds ; e3 = e 1 e 2 ;
O
a
x
(48) z
(49)
Figure 4: Bi-perspective projection in binocular cam(50) era system.
The Cartesian coordinates (x; y; z ) xed to a biperspective camera system are related to those of the binocular system, ( L ; R ; ) where is the angle between the positive y-direction and the viewing direc!1 = ds; !2 = !3 = 0; (51) tion and the angle between the positive z -direction 0 0 kds 0 1j and the viewing direction, as: 1 0 !ij = @ ?kds 0 tds A ; (52) 0 1 B arctan x+y a C 0 ? ds 0 i @ RL A = BB arctan x?y a CC : (55) z A @ arctan y where the Euclidean curvature k and torsion t are given by: Here a is the distance between the projection centers of the two cameras. s de1 After manipulating the bi-perspective coordinates 1 presented in [43] the parametrisation of space curves e ; (53) k = e1 de 1 ds ds is given by means of the elevation angle : de 3 (54) t = ?e2 ds : x() = (xi ( R (); L (); )): (56)
and connection one-forms !i and !ji given by:
In practice the problem of coordinatisation of a curve might arise. The coordinatisation, of course, depends on the particular geometry of the camera system. In case of a binocular camera system [43] the parametrisation of a space curve can initially be available in terms of elevation angles (see gure (4)).
Using symbolic packages one readily computes the Euclidean dierential geometric invariants of the space curves in terms of entities measured by the binocular system. But in each camera the projected curve points (see gure 4) have a ne coordinates (x; z) = (x=y; z=y) in the image plane given by:
16
(x; z) = ?R(tan ; tan );
(57)
where = R ; L , R is the Euclidean distance between the projection center and the image plane, and where the Cartesian coordinate x-axis and z-axis in the image planes are in line with the x-and z -axis of the reference frame xed to the camera-system. Thus a parametrisation of space curves can still completely be given in terms of the elevation angle. The type of curves that can be described in biperspective coordinates is limited to those coinciding with discontinuities in the properties of surfaces of real objects that persist themselves in both views obtained by the camera system. One might think of discontinuities in the surface shape that cause under homogeneous lightning conditions discontinuities in the brightness levels across them in both views. Other possibilities are that the colour or texture properties on the surface and thus in both views are changing abruptly. A aw in the above parametrisation might seem the importance of the gauging of the camera system, i.e. the distance (translation vector) between the projection centers, the focal lengths and the angles that the surface normals to the image planes make with each other should be known to the system. These authors do not deny the fact that such a calibration of the camera system asks for more architectural efforts than the epi-polar geometry of a camera system advocated nowadays by so many researchers [23]. As mentioned in [64] one should be very careful with the latter choice of geometry as it becomes almost impossible to point out how the discontinuity sets of the perceived luminance eld in the images do merge and split in order to achieve a reasonable "match" between the views. The reason lies in the fact that some important camera parameters such as the focal distances are unknown. Even more cumbersome it becomes to track the morphisms of the images if the viewing angles are unknown in particular in relation to the niteness of the resolution properties of a camera system. In the latter case the similarity rules for nding matches become yet more untractable. Another problem not taken into account is the fact that the physics involved in the image formation are completely dierent in the two views. This dierence in the formation of the left and right image is caused by the re ectance and absorption properties
of their surfaces in relation to the position of the camera system and light sources and by the interposition of the objects themselves leading to occlusions. Having a completely calibrated camera system one knows which "point" in space-time is xated and to which points in the images it corresponds. Occlusions and alike can subsequently better be quanti ed in terms of so-called topological currents and curvaturers [64] coming about by Euclidean movements of the camera system and taking every pair of view as a product of visual events. But if one is only heading for the larger scale information (the most pronounced surface discontinuities projected in corresponding structures in the images) epi-polar geometry or elliptic geometry [64] can still be more advantageous than bi-perspective geometry.
2.1.2 Modern Dierential Geometry The above classical dierential geometry of space curves hinges on a local analysis, whereas Cartan [18] already noted that a multi-local analysis can yield a non- at connection. Such a non- at connection implies a non-vanishing torsion and curvature tensor of the curve formation. A non- at connection on an arbitrary curve is ensured by the multi-valuedness of the connection at a point. For example, a polygon or a curve consisting of piecewise continuous segments not necessarily glued together but still parametrised by a Euclidean parameter display a jump in their connection at its vertices or discontinuities. At those locations the frame eld and the connection one-forms are multi-valued. The twist and curvature involved in the curve formation can then be captured by the structure equations satisfying the Bianchi identities, or by computing the dierence in connection on either side of an point of the curve.
Example 5 Consider two disconnected smooth
segments of a curve "parametrised" by Euclidean arclength s with respect to a barycentric coordinate system. Analogously example 1 taking the distance to the geometric center as level of energy one observes that at the transition at the endpoints between consecutive segments a very sharp jump occurs in the curvature R's output. Choosing a Eu-
17
clidean frame eld and at connection as in example theorem 1 the a ne translation and rotation vectors 1 and example 2 for each segment separately again can be written as [18]: the twist and curvature in the curve formation can Z be quanti ed at the transition of the disconnected
i ei ; (60) b = segments. Note that the segments need not be S c Z disconnected, only the order of contact should be
i j ej ; (61) fi = nite to retrieve curvatures in the curve formation. Sc where Sc S is an in nitesimally 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 a ne 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 2.2 Integral Geometry the inhomogeneity of the a ne group action. In this subsection integral invariants are de ned for The Bianchi identities (Theorem 2) imply that the an arbitrary curve with a local Euclidean metric con- vectors satisfy the following superposition principles nection ? that is multi-valued. Firstly, the integral [40]: X X geometry of manifolds of arbitrary dimension with a (62) B out = bin ; Fiout = fiin ; metric connection is discussed. Secondly, the theory is applied to the above type of curves. Firstly, consider a n-dimensional manifold M with where the sums are taken over the currents indexed a metric connection ? as de ned in the previous sub- by "in" arriving at the point of interest, for examsection 2.1. Two fundamental integral invariants, ple a junction. Note that these currents are associbased on the connection one-forms (!i ; !ji ) and the ated an orientation or direction. This superposition frame vector elds ek , are the a ne translation vector principle just re ects the fact that the formation of and the a ne rotation vectors belonging to an a ne parts of objects is the result of an indeterminate sum displacement around an in nitesimally small contour of morphisms related to the formation of the other parts. These superposition principles have physical [18]: counterparts in the conservation laws for topological De nition 6 The a ne translation vector b and the currents, such as Kirchho's law for electric currents [40]. Note that the a ne displacement or a ne transa ne rotation vectors are de ned by: port is not the same as parallel transport as encounI tered in ordinary textbooks [18]: parallel transport rx; (58) hides so to speak the torsion of observation space! b = IC Now it's easy to de ne for an arbitrary curve the rei ; (59) Euclidean translation vector and the Euclidean rotafi = C tion vectors as: where C is a in nitesimally small closed loop on a b = rx(+) ? rx(?); (63) 2-dimensional submanifold S of M with the same infi = rej (+) ? rej (?); (64) duced a ne connection ?.
In order to quantify these torsion and curvature aspects in the following subsection the a ne transport proposed by Cartan [18] is considered to obtain integral versions of them.
One may obtain the submanifold S by setting D ? 2 where the plus and minus sign denote the segments of of the connection one-forms !i equal to zero. Using the curve on either side of the point of interest. The Stokes' theorem and the structure equations given in above vectors depend on the sense of traversing the 18
circuit. Euclidean physical observables independant of the sense of traversing the circuit come about by taking their absolute values. Other invariants, such as rotational indices R kdsintegral 1 for planar curves [75], and (self)-linking 2 C numbers, Vassiliev invariants and Möbius energies for space curves [15], [60], [78], [10], [25] can readily be de ned. Although these topological invariants and energies come into play in tracing the transitions of the knot type and the limiting shape or phases under curve ows, as will be demonstrated in the next section, a thorough treatment of such properties of curves is out of the context of this paper. An exact treatment of the problem of what a shape will be in the limit under a particular ow requires very di cult mathematical concepts such as those of spectral sequences encountered in knot theory. One may even doubt whether one actually applies the right ow for the task one likes to accomplish. As discussed above the space curves or knots can be just components of more general bifurcation sets of non-constant codimension in e.g. vessel images. The ows to ensure a certain relaxation of the shape or image formation in that case do not only aect the knot component but also the complementary components possibly of another dimensionality [64]. Both components will in uence the transitions of the topological aspects of the bifurcation set in a (non)-cooperative manner. Nevertheless, the a ne transport de ned by Cartan allows also to read out the curvature of other geometric or topological objects such as the integral invariants related to the image or shape formation mentioned above. For example, various orders of curvature in e.g. knot and link invariants for sets of knots, implicitly de ned on Euclidean space by a particular grey-valued image, can be computed by taking successive circuit integrals of these invariants. Again this shows the importance of such a ne transports in the description of the topological aspects in the image or shape formation. Needless to say that those integral invariants deployed to generate knot invariants and alike are intimately related to the a ne displacement proposed by Cartan.
3 Linearised Euclidean Curve Shortening Flow In subsection 3.1 the Euclidean shortening ow of curves is linearised by a freezing the Euclidean arclength parameter. The ow is subsequently generated by just a convolution of a normalised Gaussian with the input curve that is represented as a vector distribution on the initial arclength parametrisation. Certain aspects of this continuous ow are addressed among which the boundary conditions, initialisation, and properties of the ow. Next it is demonstrated that through this ow an exact quanti cation of the dierential and integral geometry for ordered point-sets and polygonal curves that are closed, open, (self)-intersecting and/or branching is feasible. In order to achieve some of these ows re ective boundary conditions and periodicity requirements are realised by imposing suitable ordering relations on the curves. Subsequently, it is pointed out how to use our linearised shortening paradigm to nd a smoothed version of the curve formation process. The ow yields a multi-scale representation of the torsion and curvature involved in the curve formation. In subsection 3.2 and subsection 3.3 the linearised shortening ow for curves is proposed to be (semi)discretised in order to speed up the numerical algorithms and to reduce the computational load. In particular Darboux's ow is compared with our discretised linearised Euclidean shortening ow both with respect to knots. In subsection 3.4 the geometry of the curve and its formation under linearised Euclidean shortening ow is proposed to be quanti ed in geometric terms as found in section 2.
3.1 Continuous Flow
Let's recall the nonlinear Cauchy problem [76] governing the Euclidean shortening ow of smooth curves: De nition 7 The Euclidean shortening ow of a simply closed planar curve C , that is su cient differentiable with respect to an arbitrary parameter, is
19
governed by:
@C = 4C ; 4 = @ 2 ; 2 [0; T ]; @ @s2
with initial condition
Example 6 Consider a ?circle? of radius ? s R s
parametrised by x0 (s) = R cos 2R ; sin 2R . (65) Under the linearised Euclidean shortening ow the linear scale-space of this curve can be expressed as:
x(s; ) = x0 (s) exp ? 4 2R2 :
(70)
C (; 0) = C0 ; (66) In the subsections 3.1.1 up to 3.1.3 including initial where C0 is the initial curve, is the Euclidean evo- conditions, boundary conditions, properties of the lution parameter, T 2 R is the lifetime of the curve linearised Euclidean curve shortening ow are preC0 and s is the Euclidean arclength parameter along sented and examined. In the last subsection 3.1.4
the curve. It is clear that the arclength of the curve evolves under the ow, for it is coupled to the evolution of the curve [76]. Instead of using the evolved arclength parameter to smooth the curve here the initially established Euclidean arclength parameter s = s0 is frozen in order to de ne a linearised version of the ow analogous [56], [13]. Freezing the Euclidean arclength parameter is not so strange if one realises that one also freezes e.g. the x and y coordinates in the linear scale-space theories with respect to grey-valued images on at spaces (we'll see shortly that this freezing plays a crucial role in order that the ow satis es the semi-group property). This linearisation yields a linear Cauchy problem, i.e. a linear diusion equation with initial condition. A so-called linear scale-space of the input curve C0 can then readily be derived [64]: De nition 8 A linear scale-space of an input curve C0 is a one-parameter family of curves C given by:
x(s; ) = (x0 G)(s; ); with
(67)
2 (68) G(s; ) = p 1 exp ? 4s ; 4 where the representation of the initial curve C0 is given with respect to a barycentric coordinate system, that is a Cartesian coordinate system with origin Xc given by: R x (s)ds (69) Xc = C0R 0 ds :
special ordering relations are proposed that are useful in the smoothing of curves ot its formation.
3.1.1 Initial Conditions
In order to apply the linearised Euclidean shortening ow paradigm to a set of detected points on the curves a representation x0 is needed with respect to a barycentric Cartesian coordinate system. In case of a nite number N of points its origin Xc is given by:
Xc = N1
N X q=1
Xq ;
(71)
where Xq are the coordinates of the detected points with respect to a Cartesian coordinate system xed to a camera system (see section 2). The two simplest representations of a curve underlying the detected points come about by conceiving the curve as a set of ordered points or as a polygon parametrised by Euclidean arclength. This initial parametrisation is realised by measuring the Euclidean distance between successive points, or measuring this distance along line segments between two successive vertices of the polygon. Furthermore, in order to overcome the in niteness of the support of the Green's function the representation of the input curve is assumed to be periodic. Consequently, the canonical Euclidean description of an ordered point-set with respect to a barycentric coordinate system can be given by:
C0
20
x0 (s) =
1 X
q=?1
X~ q (s ? sq );
(72)
with
X~q = Xq ? Xc ;
(73)
satisfying periodicity condition:
x0 (s) = x0 (s + nL); n 2 Z:
(74)
Similarly, the canonical Euclidean description of a polygon with respect to a barycentric coordinate system can be given by:
x0 (s) =
1 X
q=?1
X~q (s);
(75)
with, for sq s sq+1 ,
taken care of by associating to each curve point right from the start a Gaussian kernel of a width comparable to the lengths of the edges of the one- or twodimensional faces in the two- or three-dimensional images, respectively. Associating to the middle of each edge a Gaussian distribution of amplitude equal to the distance between geometric center and that middle and of width equal to half the length of the supporting edge one can come up with a curve representation that can be perturbed by varying slightly the amplitudes and widths. On the propagation of noise in this linear scheme and alike the reader is referred to [11].
3.1.2 Boundary Conditions
X~ q+1 ? X~ q X~q (s) = X~ q + s q ; (76) In the continuous ow above the curve was taken to (X~q+1 ? X~ q ) (X~q+1 ? X~ q ) be of in nite length by wrapping a rope as it were satisfying periodicity condition: X~q (s) = X~q (s + nL); n 2 Z:
(77)
Note that the support of the Green's function G is in nite implying that the representation of e.g. a simply closed curve has to be chosen periodic, s 2 R and 2 R+0 . Thus there are no boundary conditions imposed on the linearised ow. The boundary conditions can, nevertheless, be hidden in the order relations on the curve as will be shown shortly. However, the initial curve still has to satisfy a natural growth constraint for each component of the vector function as encountered in linear scale-space theory for grey-valued input images:
x0
G (s; ) M > 0; 8 2 R+0 ;
in which k k is the standard norm on Euclidean space En. Subsequently, the input curve represented on a nite lattice can be smoothed and can be associated a linear scale-space of curves. Uncertainties in localisation of the points of the curve can be captured by allowing e.g. each point on the lattice to uctuate within a cell enveloping it. Of course, this freedom in the localisation can be restricted and can be
in nitely many times around the curves, i.e. the periodicity condition in the initial condition. It can, however, happen that the curve has nite length or has an open end. In the sequel we'll cope with these two situations by supplementing the Cauchy problem with proper boundary conditions that preserve the total information under the ow. The Green's functions associated to these new Cauchy problems are simply given. Assuming that a curve of in nite length has one endpoint at s = 0 then it is necessary to supplement the Cauchy problem for the linearised Euclidean shortening ow with a boundary condition. Imposing conservation of total probability [64] leads to the addition of a re exive boundary condition at s = 0. The Green's function GHr;t with domain of convolution H = R+0 centered at a distance t from the endpoint is for this Cauchy problem given by [64]:
GHr;t (s; ) = G(s ? t; ) + G(s + t; );
(78)
where G is the normalise Gaussian kernel. For an open bounded curve with two endpoints at s = 0 and s = L the Green's function GIr;t with domain of convolution I = [0; L] centered at a distance t from s = 0 is the same for the Cauchy problem with re ective boundary conditions at both endpoints, is
21
given by [64]:
C 1 to C 2 without knowing anything about C 0 ,
and consider this as a basic concept of scale-space (79) theories. A consequence of such an axiom is that information about s0 is lost when one wants to go m=?1 from C 1 to C 2 . If one really is convinced that (G(s ? t ? 2mL; ) + G(s + t ? 2mL; )): information about s0 is lost, then one is justi ed to conclude that: 3.1.3 Linearised Flow Properties C 2 = (C 0 G)(s0 ; 2 ) In the following we'll brie y indicate the properties 6= ((C 0 G)(s0 ; t1 )) G(s1 ; t2 ? t1 ) of the linearised ow (for comparison with those for the nonlinear ows the reader is asked to consult [4]). = C ( 1 ) G(s1 ; t2 ? t1 ); Euclidean Invariance The vectorial distribution where s1 is the canonical Euclidean arclength for the initial curve with respect to a barycentric parametrisation of C ( 1 ) forgetting how it arose from coordinate system is not aected by the group of the initial parametrisation s0 and 0 , and t a consisEuclidean movements. Note that due to a Euclidean tent function of . Thus forgetting about the demovement the components of the vectorial distribu- pendence on s0 and 0 the semi-group property is tion may change, but that the vector with respect to violated. However, knowing s0 always, as it should the geometric center, of course, remains the same. since it is part of the initial condition of the ow, one can nevertheless state, that the semi-group property for the convolution operation with Green's functions Morphological Invariance Changing the contrast in the grey-valued images the linearised or nonlinear is still satis ed: Euclidean shortening ow are not in uenced. It's dn G clear that the shape in a grey-valued image that is Gp (; p ) Gq (; q ) = Gp+q (; p + q ); Gn = dsn staying the same under a change in contrast is the net of owlines and isophotes or any shape constructed 8p; q 2 Z+0; p ; q 2 R+0 , where denotes the testing from it. In this context it seems worthwile to restrict (convolution) of a vector valued distribution. ourselves to a smaller set of shapes, for example, The reader is warned not to conclude immediately ridges, ruts and other discontinuity sets [64] as they from the above reasoning that the knowledge about will remain identi able. Our selected initial curve in the dependence on s0 of geometric objects such as the grey-valued images consisting of such singular curvatures is always lost under nonlinear Euclidean simplexes as ridges is then morphologically invariant curve shortening ow. It's only hidden as the example below demonstrates. and identi able.
GIr;t (s; ) =
1 X
Causality or Semi-Group Property Freez- Example 7 Consider a circle C0 of radius R 2 R+ :
s s x(s) = R cos 2R ; sin 2R ; to satisfy this principle. As already mentioned in the introduction the semi-group property says in which s is the initial canonical Euclidean arclength that C 2 can be obtained from C 0 or from any parameter. As the radius R is constant the evolution C 1 ; 0 < 1 < 2 . Having C 0 , the initial curve, of the curvature simpli es to [76], [64]: parametrised by s0 , the smoothed versions C 1 @k = k3 : and C 2 are obtained by convolution of the initial @ curve with Green's functions G(s0 ; 1 ) and G(s0 ; 2 ), ing the Euclidean arclength parameter for all scales
and taking it equal to s = s0 is crucial in order
respectively. Now one may state that the semi-group Applying some elementary theory of dierential equaproperty implies that it should be possible to go from tions the evolution of the curve x can be expressed 22
as a function of s and as follows: r 2 1 R ? 2 2 x(s; ) = R2 x(s); 2 0; 2 R : Clearly, the whole ow of the circle between intermediate levels of scales can be expressed in terms of the initial parameters s and . Thus the statement that there is a loss of information about s, i.e. how it de nes together with and R the curvature at higher or intermediate scales, is not justi ed as the ow is completely deterministic (thus causal)! Writing down a Cauchy problem, linear or nonlinear, one is always in this position.
to the geometric center and the tangent vector averaged over all points will be unique. The position vector lying consequently in a plane for each initial Euclidean arclength can then be described in terms of polar coordinates. Preserving the total length it is clear that the limit curve is a circle. Of course, for very symmetric curves the limit behaviour will dier as will be demonstrated in the sequel.
Similarity Invariance This invariance is by many authors [76] often confused with notions like scale-invariance or normalisation of the linear scale-space of curves. In the sequel we'll try to clarify this matter for once and for all. Maximum-Principle Considering the Euclidean Let's rst de ne the similarity group related to the distance to the geometric center the following in- linearised Euclidean shortening ow: equalities hold for the points on a spatially bounded De nition 9 The similarity group corresponding to curve: the linearised Euclidean shortening ow is de ned as a one-parameter group of transformations given by: 0 kxn (s; )k sup (kxn (s; 0)k); (x; ) 2 E n R+0 . Note that the comparison principle (~s; ~; x~) = (s; 2 ; 2a x); 2 R+ ; a 2 R+ ; (80) for morphological scale-space theories [4] cannot be formulated for our linearised Euclidean curve where a is a free parameter. shortening ow. One may even doubt whether it's Now the keystone for understanding the true physreally geometrically and topologically advantageous ical and geometric meaning of the above invariance to require a comparison principle similar to that lies in the reduction of the linear diusion equation in higher co-dimensional ows [5] on grey-valued governing our linearised ow to a Weber dierential images as was discussed in the introduction and will equation [58], [64] after requiring the solution space be in section 3.3. The requirement of such a principle to our ow problem to be invariant under the simican become quite a nuisance in case of knots. larity group. Choosing the free parameter a = ? n+1 2 the solution space of the reduced ow problem conCurve Simpli cation It is obvious that all sists of a complete set of Hermite functions. Expressthe points at in nite scale tend to the geometric ing them in terms of the original variables s and center Xc : one obtains the Green's functions Gn as solutions to our linear Cauchy problem. lim kx(s; ) ? Xc k = 0; 8s: !1 It's self-evident that the observations latent in the Computing the curvatures of the curve as function of linear scale space that are invariant under the simthe initial Euclidean arclength parameter s one can ilarity group can only be constructed on the basis show that: of its own constituting elements. This (self)-similar observation space we've coined similarity jet [64] is lim !1 jk1 (s1 ; ) ? k1 (s2 ; )j = 0; 8s1 ; s2 ; de ned as follows: p
lim d ki (s; ) = 0; i = 2; : : : ; n ? 1; p 2 N : !1 dsp
De nition 10 The similarity jet j 10 of an input
The latter can be explained by the fact that the plane spanned by the position vector with respect
curve is de ned by:
23
j 1 0 = f; n g ;
(81)
with where
2 xn ; = ps ; n = n+1 2
(82)
xn = x0 Gn ; n 2 Z+0:
(83) From a physical point of view this jet represents an observation space consisting of topological quanta invariant under the similarity group. Analogous for the similarity jet of a grey-valued image [64] in which the entity is a total energy associated with a labelled spatio-temporal region in the image plane, the entity denotes a vectorial amplitude and no density eld as that is tested by the so-called Green's function G. One could say we have a vectorial integral invariant not aected by the similarity group as well as the group of Euclidean movements. Note that these topological quanta occur also in the solution of nonlinear shortening ow problems.
Euclidean Invariant Parameter Upon blur-
ring the initial Euclidean arclength parameter s becomes a Euclidean invariant parameter for the evolved curve. Thus the evolution of Euclideanly invariant geometric entities under the linearised ow can be derived by identifying s, for every 2 R+0 , with a Euclidean invariant parameter, which is not the Euclidean arclength parameter. It is clear that for computing the curvatures one has to correct for this typical parametrisation. As the relevant intrinsic properties are Euclidean dierential geometric invariants and these properties are themselves already invariant under the similarity group they need not further considered in the sequel of this section. However, in subsection 3.4 where the deep structure of the linearised Euclidean shortening ow of the geometry of the curve and its formation are studied it becomes really necessary to take care of the similarity group actions.
3.1.4 Ordering Relations
In order to perform linearised Euclidean shortening ows with re exive boundary conditions at endpoints
it is advantageous to impose periodicity requirements and ordering relations consistent with these boundary conditions and removing the problem of performing (an in nite number of) convolutions with shifted ordinary Gaussians. For example, an open curve with two re exive endpoints is parametrised as a simply closed curve with a periodicity constraint, namely x0 (s) = x0 (2nL s); n 2 Z. For a branching curve with endpoints and loops similar parametrisations can be carried out. It's obvious that the provided ordering relations are not unique and dierent orderings yield dierent evolutions under the linearised shortening ow. Instead of using only ordering relations in case of a branching curve to represent the underlying curve, the branching point can be used as geometric center for a separate linearised Euclidean shortening ow of each branch. The geometry of the curve at the bifurcation point is latent in the multi-valuedness of the transitions going from one segment to one of the others. As noticed in section 2 the formation of an arbitrary curve de nitely involves twists, curvatures and higher order covariant derivatives of these entities. These geometric objects were eectively quanti ed in terms of suitable integral invariants (see subsection 2.2). These properties can now smoothed by just assuming locally a re exive boundary condition to the left and to the right of the curve. This local re exive boundary condition rstly requires that two representations are constructed locally, namely x(+) and x(?) for the cut curve starting at the point of interest, P , wrapped around the curve in positive and negative sense, respectively. Secondly, the condition requires the local symmetry relation x(s) = x(?s) to hold. After one turn around the curve and arriving at P again one has to return, and so forth. Thus we have actually an open bounded curve of which we compare the behaviour of the endpoints. Forgetting about the base-points in the Euclidean space one can still compare analogous [31] the multi-valuedness of the tangent and cotangent geometric objects, such as the curvature elds at points. In the following paragraphs the continuous linearised Euclidean shortening ow is applied to planar curves, an unknotted space curve and planar curve
24
formation.
100
50
Applications to planar curves In the sequel the
-100
continuous linearised Euclidean shortening ow is applied to planar curves represented as vectorial pointset distributions (72) with particular ordering relations. These ordering relations are crucial, because they allow us to deal with various types of curves including simply closed, open or branching curves. The ordering relations correspond to physical structures in the underlying grey-valued. For example, the texture information latent in the grey-value data can dictate which type of ordering is applicable. In gures 5, 6 and 7 the grey-values inside the two curves having the origin in common are assumed to be the same. The only dierence between the latter two is that the other input curve is really symmetric in the origin and stays a lemniscate over all scales, whereas the second input curve develops a cusp before becoming pear shaped. If, however, the grey-values enclosed by both curves are dierent, then it's reasonable to consider the ows on the curves touching each other in the origin to be independent yielding disconnected curves for non-zero scales tending to in nitesimally small circles. In the experimental setup the number of times the curves is wrapped around itself is restricted to two in either direction as then the errors made were negligible.
-50
50
100
-50
-100
Figure 6: Ordered point-set representing a selfintersecting planar curve symmetric in the origin, and its smoothed version obtained upon applying a continuous linearised Euclidean shortening ow.
Figure 7: Ordered point-set representing a selfintersecting planar curve a-symmetric in the origin, and its smoothed version obtained upon applying a continuous linearised Euclidean shortening ow.
In gures 8, 11, 14 and 17 special periodic ordered point-sets are chosen to represent a simply closed symmetric curve, a simply closed a-symmetric curve, open curve and a branching open curve, respectively. Only the coordinates of the vertices, endpoints and branching points of the shapes are used in the initial canonical Euclidean arclength parametrisation of the curve with respect to corresponding barycentric coordinate systems (see section 3.1). After having established this initial curve parametrisation several representatives of the linear scale-spaces of those input curves are computed. The scales are chosen such Figure 5: Ordered point-set representing a closed pla- that the simpli cation of the curve is evident. The nar curve that is touching itself at the origin, and its smoothed versions of the initial curves above are dissmoothed version obtained upon applying a continu- played in gures 9, 12, 15 and 18. The smoothed ous linearised Euclidean shortening ow. curves were obtained by measuring the original length 100
50
-100
-50
50
100
-50
-100
25
of the enveloping polygons, sampling the evolved distributions of the curves for a nite number and equally distant initial Euclidean arclengths between si , the initial value of the Euclidean arclength, and sf = si + L, the nal arclength with L the total length of the initial curve. Next the found points are joined by straight line segments. Subsequently, for the same scales and at the same sampling points the Euclidean curvature elds of the initial curves were computed. In gures 10, 13, 16 and 19 these elds are depicted for the mentioned curves. Of course, the number n of windings in the periodic representation, taken into account in the computation of the ow and the intrinsic properties, depends on the scale relative to the squared total length of the curve (for relatively low scales one can restrict one-self to n = 1 to ensure negligible numerical errors in performing the ow and computing the intrinsic properties of the curve). Here we have chosen n = 2 to suppress Figure 9: The smoothed versions of the planar curve the errors as much as possible and checked whether representation in gure 8 obtained upon applying the they were really negligible to the erros made in the continuous linearised Euclidean shortening ow. localisation of the vertices. Now let us more closely analyse the obtained results for the various curves. t = 1500
t = 400 1.5
0.75
1
0.5
0.5
0.25
-1
-1.5
-0.5
1
0.5
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-0.75 -0.5 -0.25
0.25
0.5
0.75
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-1
-0.75
-1.5
t = 5000
t = 10000 0.3
0.4
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Ke 350
Ke
t = 400
t = 1500
5
300 250
4
200 3 150 2
100
1
50 50
Ke
100
150
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250
s 50
t = 5000
100
Ke
150
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s
t = 10000
3.101
1.95
50
1.9
100
150
200
250
s
3.099 1.85
3.098
1.8
3.097 3.096
1.75 50
Figure 8: Ordered point-set representing a simply closed planar curve symmetric in the origin.
100
150
200
250
s
3.095 3.094
Figure 10: The Euclidean curvature eld (46) as a function of the initial Euclidean arclength parametrisation corresponding to the smoothed curves in gure 9. For the simply closed symmetric curve in gure 26
8 the chosen scales are such that for = 400 one perceives actually eight maxima and eight minima in the Euclidean curvature eld (see gure 10) that clearly corresponds to our chosen initial curve representation. If one would start of with a real polygonal representation as de ned in the previous section one would, of course, never nd these numbers of extrema. These numbers would be the same as those found for the point distribution at scales = 1500; 5000; 10000 . Apparently as also noticed in subsection 3.1 the initial curve representation of the curve could be better given right from the start a certain inner scale related to the length of the edges connecting the vertices. Furthermore, the simpli cation of the initial curve is latent in the changing number of extrema as well as the fact that the Euclidean curvature over the curves varies less and less with increasing scale. Because of the pure symmetry of the initial curve with respect to the center of Figure 12: The smoothed versions of the planar curve mass the number of extrema above = 1500 is not representation in gure 11 obtained upon applying decreasing. But the same holds for nonlinear ows; the continuous linearised Euclidean shortening ow. only in the limit both ows will deform the initial curve to an in nitesimally small circle. Ke t = 400 t = 1500 1
t = 400 1
0.5
0.5
-1.5
-1
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0.5
1
-1
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1
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t = 5000 0.6
t = 10000 0.3
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Ke
t = 1500
350 300
5
250
4
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3
150 2
100 1
50 50
100
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s 50
100
Ke Ke
t = 5000
150
200
250
t = 10000
3.75
2.2
3.5 50
100
150
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250
s
3.25
1.8
50 1.6
Figure 11: Ordered point-set representing a simply closed planar curve a-symmetric in the origin.
s
100
150
200
250
s
2.75
1.4
2.5
1.2
2.25
Figure 13: The Euclidean curvature eld (46) as a function of the initial Euclidean arclength parametrisation corresponding to the smoothed curves in gure 12. For the simply closed a-symmetric curve in gure 27
11 similar remarks can be made as for the simply closed symmetric curve above, but now it is more evident that really the number of extrema in the Euclidean curvature eld (see gure 13) with increasing scale is decreasing. Moreover, on the basis of the evolution of that eld and that number both curves can be distinguished and recognised.
40 30 20 10 -100
-50
50
100
-10 -20 -30
Figure 16: The Euclidean curvature eld (46) as a Figure 14: Open bounded polygonal curve with two function of the initial Euclidean arclength parametriendpoints represented by an ordered periodic point- sation corresponding to the smoothed curves in gure set. 15.
For the initial curve representation of the open bounded polygonal curve in gure 14 the periodic representation is such that the kernel wraps around the endpoints, i.e. x(s + ds) = x(s ? ds) at the endpoints with initial Euclidean arclength s. Evidently, the curves evolves towards an in nitesimally small straight line segment. Furthermore, as on either side of the curve segments their convexity is opposite the curvature eld at various scales depicted in gure 16 for initial arclengths larger than the total length of the initial curve show at both endpoints a change of sign in the value of the Euclidean curvature. One could say that across the curve segment and thus also Figure 15: The smoothed versions of the planar curve at the endpoints the curvature eld is double-valued. representation in gure 14 obtained upon applying This in turn can be viewed as a manifestation of the the continuous linearised Euclidean shortening ow. twist or curvature of the curve formation. 28
50
25
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-20
20
40
-25
-50
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Figure 17: Branching polygonal curve with one bifurcation and three endpoints represented by an ordered periodic point-set equivalent to a simply closed curve wrapped around the polygonal curve such that each point is covered twice. Figure 19: The Euclidean curvature eld (46) as a function of the initial Euclidean arclength parametrisation corresponding to the smoothed curves in gure 18. For the branching curve in gure 17 one also reckons the same simpli cations as for the other curves, but one also notices topological transitions in the numer of endpoints and branching points for all scales greater than zero. Instead of using the periodic representation one might prefer to retain the branching point and consider as the geometric center for representing three separate curve segments represented as the one above and follow the evolution of this set of segments under the linearised ow. The Euclidean curvature eld (see gure 19) is then multi-valued for every scale at the branching point, i.e. the geometric center for the curve segments. Again the geometric properties in the curve formation going from one segment to another can nicely be expressed in classical and modern geometric terms (see also the paragraph on the application to curve formation). Note that the branching point splits into three endpoints, but becomes again one branching point for in nite scales with three inFigure 18: The smoothed versions of the planar curve nitesimally straight edges connected to it. Each representation in gure 17 obtained upon applying edge has a direction which is just the average of the the continuous linearised Euclidean shortening ow. unit tangent vector eld along the curve segment be29
tween the branching point and the outer endpoint.
Application to space curve The experimental setup for a linearised Euclidean shortening ow analyses of space curves is similar to that of planar curves above. The initial curve representation is that of a simply closed (unknotted) curve (see gure 20). The evolution of the curve with respect to a barycentric coordinate system, the evolution of the Euclidean curvature eld and that of the Euclidean torsion eld are computed (see gures 21, 22 and 23, respectively). Choosing proper scale ranges one observes that again the numbers of extrema in the Euclidean curvature and torsion eld are decreasing with increasing scale. This demonstrates again the curve simpli cation properties of our curve ow. Furthermore, one need not to be afraid for intersections of the curve under the ow causing self-passages. Still the curvature and torsion elds can be computed at intersections, but apparently they are multi-valued (see also paragraph in section 3.3 for a discussion related to self-passages of knots under our ow).
Figure 20: Simply closed polygonal space curve.
Figure 21: Smoothed versions of an equally distant ordered vectorial point distribution of the simply closed polygonal space curve in gure 20 obtained by performing a linearised Euclidean shortening ow.
Figure 22: The evolution of the Euclidean curvature (53) as a function of the initial Euclidean arclength parametrisation for the vectorial point distribution for the simply closed space curve in gure 21. 30
the jump in order of contact at the cusp over scale. Analogous example 2 we de ne the curvature of the connection over scale as follows:
R12 = ?R21 = k1 ? k2 ;
Figure 23: The evolution of the Euclidean torsion (54) as a function of the initial Euclidean arclength parametrisation of the vectorial point distribution for the simply closed space curve in gure 21.
Application to curve formation Splitting a
where k1 and k2 are the Euclidean curvatures at the cusp-like point for the clockwise and anti-clockwise traversed curve segments. In gure 25 we have plotted the measured curvature R12 at the cusp over scale. It's clear that the most predominant curvatures occur where the circular segments of the curve are connected. Repeating the same for the other points of the curve one would obtain a R12 -curvature eld over scale with a certain characteristic deep structure. In this context one could also consider the change of convexity or more explicitly the change of sign of the inner product between the Euclidean radius vectors again adopting the viewpoint of Gri ths [31]. The latter analysis allows us to distinguish between topologically dierent curve formation processes during the ow. At the cusp in gure 24 we observe clearly a change in convexity, whereas at the other joints between the circular curve segments we observe a conservation of the polarity at that scale. As this change in polarity cannot be retrieved on the basis of the R12 -curvature output we examine this aspect of curve formation over scale more closely in subsection 3.4.
curve into parts by applying entropy scaling mechanisms [39] can alternatively be realised by using the linearised Euclidean curve shortening ow. Asymmetries in the curve formation process are most predominant where the curve experiences the most dramatic changes in the order of contact, for example cusps [50]. Here we consider not a space curve forming cusps in the projection on the image plane. Instead we consider the curve formation under our ow on either side along a planar curve at a cusp-like point (see gure 24). The initial curve representation is such that at the cusp the curve is cut yielding locally a curve like the one in gure 14. We choose the cusp to be once the initial endpoint of the curve segment upon traversing the original curve anti-clockwise and once to be the initial endpoint of the curve segment upon traversing the original curve clockwise. Furthermore, we choose the initial curve representation for both segments analogous the one for the curve displayed in gure 14. Smoothing the curve formation Figure 24: Simply closed polygonal planar curve with along the clockwise and anti-clockwise parametrised two circular curve segments tangent to each other at curve segments at the cusp boils down to comparing the cusp having opposite convexity. 31
[64]:
Gsemi (s; ) = exp (?2 )
X
2r+s ; (84) r0;s+r0 (s + r)!r! r; s 2 Z; 2 R+0 :
The n-th order semi-discrete derivative of the semidiscrete Green's functions Gsemi n are then recursively given by[64]: Figure 25: The strength of the curvature R12 over p 1 semi scale at the cusp in gure 24. Gsemi n (s; ) = 2s Gn?1 (s ? 2 ; ) ? (85) 1 Gsemi (s + p2 ; ); In order to achieve a real-time assessment of the 2s n?1 geometry of the curve and its formation under the semi G0 = Gsemi : (86) linearised Euclidean shortening ow it becomes indispensable to discretise the continuous ow. For curves Notice that the dierence of Gaussians (DOGS) here this can be done analogously for grey-valued images is the superposition of two semi-discrete Green's func[47], [48], [64] by rstly discretising the spatial coordi- tions, one shifted over a distance of p2 in the posinates and subsequently the scale parameter 3.3. The tive direction and the other in the opposite direction. initial and boundary conditions considered and the Taking now suitable scaling limits [64] of the above properties mentioned in the previous subsection read- semi-discrete Green's functions one obtains just the ily carry over to related conditions and properties for continuous versions listed in the previous subsection. the (semi)-discretised ows. For comparison of discretisation issues involved in nonlinear (Euclidean) shortening ows with those of our ow the reader is 3.3 Discretised Flow kindly requested to consult in addition [59], [4]. Fully-discretising subsequently the semi-discretised In the sequel the curve is always represented as linearised Euclidean shortening ow by modeling a set of points or polygon along the edges and ver- the evolution parameter by integer values Z+0 the tices of a (square or cubic) lattice. The latter choice Green's function G at scale , centered at lattice site of representation will appear advantageous in realis- s0 and at lattice site s is equal to [64]: ing (semi)-discrete derivative Green's functions. Note that it is possible to put the kernel at other positions GDiscr (s0 ; s; ) = (87) 1 than the lattice points or edges to compute the lin2 ( + s ? s0 ) earised Euclidean curve shortening ow. One then p 12 ( +s?s0 ) q 12 ( ?s+s0 ) ; should not infer that one has attained some hyperacs; s0 2 Z; 2 Z+0; p + q = 1; p = 21 : curacy, since the structure formed under the ow is merely due to the available initial data of the ow. The fully-discretised n-th order derivatives are given by the same recursive relation as for the semidiscretised ow. Furthermore, suitable scaling lim3.2 Semi-Discretised Flow its relate both the discretised and continuous scaling Semi-discretising rst the linearised Euclidean short- mechanisms [64]. ening ow with respect to an input curve represented All this discretisation now allows a real-time exby an ordered set of sites on a (square or cubic) lat- act analysis of the ow, because the support of the tice yields as semi-discrete Green's function G [48], kernels in the fully-discretised case involves a nite 32
number of lattice points for every nite scale value . Moreover restricting winding number n to a nite number then is physically substantiated because the transport velocity of information in any direction is bounded.
Application to knots Darboux's ow [19] is de ned as follows:
Comparing both ows in gures 26 and 27 one immediately observes that a positively oriented Darboux's ow leads to an irreversible transition of the knot type, i.e. from a knotted state (trefoil at = 400) to an unknotted state (triangle like curve at = 1500). The triangle like shape in the unknotted state one can blame on the fact that the ow is directed. The discretised linearised Euclidean shortening ow on the contrary appears to lead to a uctuation between a knotted state (trefoil at = 2551) and an unknotted state (circle like curve at = 2570). An explanation can be sought in the periodicity that seemingly hampers the latter ow, i.e. at suitable even scales the initial tame knot type felt at every vertex is ampli ed. In Darboux's ows the latter recurrence property of discretised linearised Euclidean shortening ow is absent due to the particular choice of direction for exchange of information. So one might think of Darboux's ow as a more proper ow for also scaling the curve formation since the knot type obviously comes into play during knot formation [64].
De nition 11 Darboux's ow is a discretised ow de ned by: x+ (n; + 1) = 1=2 (x(n; ) + x(n + 1; )); (88) with x+ (N; + 1) = 1=2 (x(N; ) + x(1; )); (89) in which n indicates the n-th point in the ordered point-set of length N . and 2 N the scale parameter. Apparently Darboux's ow is characterised by a preferred direction for information exchange indicated by the +-index. Thus one can distinguish positively and negatively oriented Darboux's ows. The limit curve is an inscribed semi-regular polygon of an ellipse depending on the the odd or even number of points. It can be easily shown that the discretised linearised Euclidean shortening ow X (n; ) and the Darboux's ows x+ (n; ) and x? (n; ) are for even scales directly related as follows: X (n; + 2) = 21 (x+ (n; + 1) + x? (n; + 1)): (90) Note that the equality holds between the second iteration step in the discrete linear scheme and the rst step in the combined positive and negative oriented Darboux's ows. Thus the comparison of both ows can best be performed for only even scales! In the experiment below we considered both ows on a spatially discretised trefoil [62] slightly disturbed by uniform noise in the coordinates. The discretised linearised Euclidean shortening ow was computed by means of above Darboux's ows. Realise that after Figure 26: From top to bottom row a trefoil repretwo even iteration steps in the linearised ow one sented by a periodic point-set and two scaled versions iteration step in Darboux's ow is performed. under the Darboux ow. 33
and gure 29 the evolution for both ows are shown as function of scale . Using now a famous theorem of Milnor [49], which says that for any knot the total curvature K has at least to be larger than 4, we conclude that the positively orientated Darboux's ow causes the trefoil to collapse into an unknot, whereas the discretised linearised Euclidean shortening ow ensures the unknotting to be reversed for scale ranges that become smaller and smaller for higher scales. For in nitely high scales one may say that the probability of the trefoil being in a knotted state will be of zero measure for the discretised linearised Euclidean shortening ow.
Figure 27: From top to bottom row a trefoil represented by a periodic point-set and six scaled versions under the linearised and fully discretised Euclidean shortening ow. In order to demonstrate that the discretised linearised Euclidean shortening ow and the positively orientated Darboux's ow yield for the trefoil an unknot for in nite scales we computed the evolution of the total curvature of a tame knot version of the successive smoothed versions of the trefoil. In gure 28
Figure 28: The evolution of the total curvature of the trefoil depicted in gure 26 under Darboux's ow. Here the horizontal axis denotes the scale parameter and the vertical axis the weighted total curvature K=2.
34
Figure 29: The evolution of the total curvature of the trefoil depicted in gure 27 under the linearised and fully discretised Euclidean shortening ow. Here the horizontal axis denotes the scale parameter and the vertical axis the weighted total curvature K=2. Let us conclude this paragraph pointing out some subjects for further study and research related to ows on knots and links. In this context the reader can imagine that one can construct such sewed knots and links by connecting suitable discontinuity or singularity sets in grey-valued images [64], [66] as also indicated in subsection 2.1. On the basis of our above analysis it is important to describe the transitions of the states of knots or links under a particular ow [64]. The knot and link invariants, such as the self-linking number and the Gauss integral, seem to be the appropriate candidates for characterising the topological structure of the ow dynamics. One could, for example, compute the (self)-linking numbers (or generalised Gauss integrals) of knots and links living at two dierent scales and of so-called framed knots or links in which the framing can be chosen to be dependant on higher Euclidean (self)-similar jet structures. The latter is allowed as we only require invariance under the group of Euclidean movements and that of similarity transformations. Summarising the transition of the knot and link invariants under a particular ow can help us sieve the knot and link formation subjected to this
ow. Note that this kind of granulometry can actually be carried out for tame knots (polygonal curves or ordered point-sets), because self-linking numbers and the Gauss integral can be computed exactly [15]. In a forthcoming article we'll address this granulometric classi cation of knots and alike in more depth. Having a similarity jet of a curve or an image at our disposal one can even think about studying nonconstant higher dimensional bifurcation sets. Again generalised Vassiliev invariants [78], [80], [77] and alike can be considered for sieving structures and determining new ows. Concerning the type of ow no preference can be made in advance. Whether (self)-passages of knots and links should be prohibited depends on the geometry and topology one would like to relax. Requiring knot and link invariants to be preserved under the ow forces us to relax and study other physical entities living on the knots and links than those involved in the knot and link formation. One might, for example, derive ows that minimise the so-called Möbius energies and preserve (self)-linking numbers simultaneously [60], [78], [10], [25]. Alternatively one might, as in this paragraph, not want to constrain the ow on knots and links in order to allow e.g. the larger scale (large ball containing just) primitive knots [62] attached to each other to survive as long as possible and the smaller (singular) ones to be annihilated instantaneously. Furthermore, one might be interested in relaxing or scaling the formation of the knots and links itself [64]. In the latter case one needs a quanti cation of the topological currents similar to the a ne translation or rotation vector elds mentioned in section 2 to accomplish such a scaling. In addition it is possible to read out the curvature of topological quanta that are related to the formation of knots, links and more general singularity sets. Instead of pursuing the latter more attractive scaling mechanism one can also unravel, as done in the next subsection, the geometry of the linearised Euclidean curve shortening ow, if the grey-valued image in which the knots and links appear as special discontinuity or singularity sets, can freely be subjected to spatially homogeneous monotonic greyvalue transformations. Of course, the curvatures in
35
the curve formation can be incorporated in the ow Now a frame eld V on the multi-scale representafollowing [64]. tion of the curve invariant under the Euclidean group and the similarity group is given by:
3.4 Geometry of Linearised Flow
Conceiving the multiscale representation of a curve as result of a dynamical process [18], [69], then the connection on the similarity jet (De nition 10) of the curve can be described analogous for grey-valued input images [69], [64] by a connection that is invariant under the similarity group. The geometry consistent with the connection can subsequently be captured by twists and curvatures as in section 2. In order to quantify geometric objects determined by the similarity jet (De nition 10) a local reference frame is needed, that is itself at least invariant under the group of similarity transformations (see subsection 3.1). The simplest dual local reference frame satisfying this invariance condition can readily be constructed on the basis of only the spatial positions and scales. Theorem 3 A local dual reference frame V0 = (; d 0 ) constructed on the basis of the similarity jet (De nition 10) and invariant under the similarity group is given by the following one-forms d 0 :
V = (e1 ; : : : ; en ; en+1 ); (94) with the rst n frame vector elds e1 ; : : : ; en hav-
ing zero scale components and spatial components are coinciding with those of the unit Euclidean frame vector elds of a n-dimensional curve (see section 2), and the last frame vector eld en+1 is given by:
2 i en+1 = @@s2 ; 1 ;
(95)
where s is the initial Euclidean arclength parameter. As non-vanishing connection one-forms ! and ! , which are also invariant under the Euclidean and similarity group, one can choose: 2 i !1 = ds; !n+1 = @@s2 d i + d n+1 ; ! = ? ! ; ? = (e (log V )) :
(96) (97)
Note that here no duality of the frame vector elds ei and the connection one-forms ! is required. This connection allows a geometric quanti cation of the similarity jet of the curve or its formation i dx i in terms of torsion and curvature tensors and their p (91) d0 = ; i = 1; : : : ; n; covariant derivatives, and integral invariants as sugd0n+1 = d : (92) gested in section 2. to curve formation Let us rst deProof 3 Applying the similarity group (De nition Application ne the polarity of the curve formation under the 10) to the dual reference frame above the following linearised Euclidean curve shortening ow as the sign equality d ~ 0 = d 0 holds. of the inner product of the Euclidean curvature vecThe local reference frame V 0 , consisting of frame tors ~k1 and ~k2 for the clockwise and anti-clockwise vectors e0 , can be derived from the following duality traversed curve segments, respectively. Computing this polarity for several scales at the cusp-like point constraint: there appears a jump: at the rst scales the curvad 0 (e0 ) = : (93) ture vectors are anti-parallel, whereas beyond a certain scale they become parallel. Calculating, subseNote that the Greek indices denote spatial positions quently, the curvature of the polarity by means of and scale, whereas the Latin indices denote only spa- a circuit integral performed on the similarity jet of tial positions. Furthermore, that the metric tensor the curve we detect the scale at which this topologg0 = d 0 d 0 constructed on the basis of the above ical transition occurs (see gure 30). Of course, one one-forms d 0 has components g = . could study these type of transitions in the similarity 36
not only for the cusp for all points of the curve arriving at a true topological hierarchical ordering of the topological transforamtions involved in the forming of the initial curve.
nal of Dierential Geometry, 23:175196, 1986.
[2] J. Altschuler, S. and A. Grayson, M. Shortening space curves and ow through singularities. Journal of Dierential Geometry, 35:283 298, 1992. [3] L. Alvarez, F. Guichard, P. L. Lions, and J. M. Morel. Axiomes et équations fondamentales du traitement d'images. C. R. Acad. Sci. Paris, 315:135138, 1992.
Figure 30: To the left the polarity of the curve formation at the cusp-like point in gure 24 under the linearised Euclidean curve shortening ow. To the right the scale localisation of the change in polarity at the cusp in gure 24.
[4] L. Alvarez and J. M. Morel. Formalization and computational aspect of image analysis. Acta Numerica, pages 163, 1994. [5] L. Ambrosio and H. M. Soner. Level set approach to mean curvature ow in arbitray codimension. Instituto di Matematiche Applicate U. Dini, Via Bonanno 25 Bis, 56100 Pisa, Italy. [6] S. Angenent. On the formation of singularities in the curve shortening ow. J. Dierential Geometry, 33:601633, 1991.
4 Conclusion
A robust method for a multi-scale description of (the formation of) curves is proposed that is based on a [7] S. Angenent. Parabolic equations for curves on linearisation of the Euclidean shortening ow, special surfaces, part I. curves with p-integrable curvaordering relations on the curve and application of difture. Annals of Mathematics, 132:451483, 1991. ferential and integral geometry. Whether the curve is open, closed, intersecting or branching does not [8] S. Angenent. Parabolic equations for curves prohibit us to quantify classical geometric properties, on surfaces, part II. intersections, blowup, and such as curvatures, torsions and rotation indices, nor generalized solutions. Annals of Mathematics, modern ones such as the twist, curvatures and higher 133:171215, 1991. order properties of the curve formation process. A (semi-)discretisation of the ow is proposed on a lat- [9] S. Angenent, G. Sapiro, and A. Tannenbaum. On the a ne heat equation for nonconvex tice in order to realise a nite computational load for curves. Technical report, MIT - LIDS, 1994. establishing the geometry of the curve or the curve Submitted. formation. The similarity jet of the curve (formation) under the linearised Euclidean shortening ow is proposed to be described in similar geometric terms as [10] D. Bar-Natan. On the Vassiliev knot invariants. Topology, 34(2):423472, 1995. that for the curve (formation) at one level of scale. [11] J. Blom, B. M. ter Haar Romeny, A. Bel, and J. J. Koenderink. Spatial derivatives and the propagation of noise in Gaussian scale-space. J. [1] U. Abresch and J. Langer. The normalized curve of Vis. Comm. and Im. Repr., 4(1):113, March shortening ow and homothetic solutions. Jour1993.
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