Using Catastrophe Theory to Derive Trees from Imageswittg/preprints/papers/89.pdf · In section 2 theory on Gaussian scale space, catastrophe theory and a brief outline of the hierarchy

Using Catastrophe Theory to Derive Trees from Images

Arjan Kuijper ([email protected])IT University of Copenhagen, Department of Innovation, Rued Langgaardsvej 7, DK-2300Copenhagen, Denmark

Luc M.J. Florack ([email protected])Technical University Eindhoven, Department of Biomedical Engineering, Den Dolech 2,NL-5600 MB Eindhoven, The Netherlands

Abstract. In order to investigate the deep structure of Gaussian scale space images, one needsto understand the behaviour of critical points under the influence of blurring. We show howthe mathematical framework of catastrophe theory can be used to describe the different typesof annihilations and the creation of pairs of critical points and how this knowledge can beexploited in a scale space hierarchy tree for the purpose of a topology based segmentation. Akey role is played by scale space saddles and iso-intensity manifolds through them. We discussthe role of non-generic catastrophes and their influence on the tree and the segmentation.Furthermore it is discussed, based on the structure of iso-intensity manifolds, why creationsof pairs of critical points don’t influence the tree. We clarify the theory with an artificial imageand a simulated MR image.

Keywords: Gaussian scale space, scale space saddles, critical points, topology, deep structure,multi-scale segmentation

1. Introduction

The presence of structures of various sizes in an image demands almost au-tomatically a collection of image analysis tools that is capable of dealingwith multiple scales simultaneously. Various types of multi-scale paradigmshave been developed [65]. They can be divided into two groups: linear andnon-linear scale spaces.

1.1. SCALE SPACE

The concept of (linear) scale space has been introduced in the Western worldby Witkin [74] and Koenderink [41]. They showed that the natural way torepresent an image at finite resolution is by convolving it with a Gaussian ofvarious bandwidths, thus obtaining a smoothened image at a scale determinedby the bandwidth. This approach has lead to the formulation of various invari-ant expressions – expressions that are independent of the coordinates – thatcapture certain features in an image at distinct levels of scale [13, 14, 19–23].

Under convolution with a Gaussian features are blurred and their locationschange as a function of scale, as long as they remain well-defined. To avoidthis a much as possible, non-linear scale spaces have been introduced, in

c�

2004 Kluwer Academic Publishers. Printed in the Netherlands.

jmiv04.tex; 22/07/2004; 12:48; p.1

2 Kuijper and Florack

which e.g. the blurring on parts with a high gradient (i.e. edges) is muchsmaller than in the rest of the image [18, 63, 73].

Multi-scale approaches are nowadays becoming more and more commonand are being integrated with methods using PDEs, variational approachesand mathematical morphology [1, 8, 15, 17, 30, 31, 33, 58, 59].

1.2. DEEP STRUCTURE

In this paper we focus on linear, or Gaussian, scale space. This has the ad-vantage that each scale level only requires the choice of an appropriate scale;and that the image intensity at that level follows linearly from any previouslevel. It is therefore possible to trace the evolution of certain image entitiesover scale. The exploitation of various scales simultaneously has been re-ferred to as deep structure by Koenderink [41]. It pertains to information ofthe change of the image from highly detailed – including noise – to highlysmoothened. Furthermore, it may be expected that large structures “live”longer than small structures (a reason that Gaussian blur is used to suppressnoise). The image together with its blurred version was called “primal sketch”by Lindeberg [53–55]. Since multi-scale information can be ordered, one ob-tains a hierarchy representing the subsequent simplification of the image withincreasing scale. In one dimensional images this has been done by severalauthors [34, 36, 37, 72], but higher dimensional images are more complicatedas we will discuss below.

1.3. RELATED WORK

An essentially unsolved problem in the investigation of deep structure is howto establish meaningful links across scales. This linking can be region-wise,that is: all points that belong to a certain region are identified with that re-gion and are connected to a similar region at a larger scale, cf. multi-scalewatershed segmentation [25, 33, 60–62] or deep-structure segmentation [38].A disadvantage is that one firstly needs to define these regions.

Another way is to link points if they satisfy some constraint. Vincken etal. [43, 70, 71] built the so-called hyperstack, based on a linear scale space,but with linking essentially based on the “affection” between two potentiallycorresponding points. It appeared that this line of approach also worked wellif non-linear scale spaces were used. A drawback of the hyperstack is thecounter-intuitively linking in a fine-to-coarse direction.

A well-defined and user-independent strategy is obtained by linking pointsthat satisfy a topological constraint. This approach has been used in � -D im-ages by various authors [28, 52, 66]. They linked extrema, but noticed thatsometimes new extrema occurred, disrupting a good linking.

This creation of new extrema in scale space has been studied in detailby Damon [10–12], proving that these creations are generic in images of

jmiv04.tex; 22/07/2004; 12:48; p.2

Trees from Images 3

dimension larger than one. That means that they are not some kind of ar-tifact, introduced by noise or numerical errors, but that they are to be ex-pected in any typical case. This was somewhat counterintuitive, since blurringseemed to imply that structure could only disappear, thus suggesting that onlyannihilations could occur. Damon, however, showed that both annihilationsand creations are generic catastrophes. Whereas Damon’s results were statedtheoretically, application of these results were reported in e.g. [27, 46, 52, 53].

The main consequence is that in order to be able to use the topologicalapproach one necessarily needs to take into account these creation events.This has been done in previous work by Kuijper et al. [47, 49, 51, 50].

Apart from the aforementioned catastrophe points (annihilations and cre-ations) there is a second type of topologically interesting points in scale space,viz. scale space critical points. These are spatial critical points with vanishingscale derivative. This implies a zero Laplacean in linear scale space. AlthoughLaplacean zero-crossings are widely investigated (the “Laplacean of Gaus-sian” as edge-detector), the combination with zero gradient has only beenmentioned occasionally, e.g. by [27, 42, 45, 52].

Several authors investigated the shape of iso-intensity manifolds [27, 32,41] in scale space. Obviously, at annihilations some structure disappears.However, these points are not the only special points in relation to the iso-intensity manifolds as we showed in [47, 50]. In contrast, in [47, 50] weproved that the critical points in scale space also form special points as thesedefine so-called critical iso-intensity manifolds, i.e. iso-intensity manifoldswith self-intersection encapsulating an extremum, see Section 2.4.

Scale space critical points, together with annihilations and creations allowus to build a hierarchical structure that can be used to obtain a so-called pre-segmentation: a partitioning of the image in which the nesting of iso-intensitymanifolds becomes visible.

1.4. THIS PAPER

In the aforementioned articles [47–50] we also showed that it is sometimesdesirable to use descriptions of higher order (and thus non-generic) catastro-phes to describe the change of structure. It sometimes has a direct relationto the hierarchy tree and the pre-segmentation, in the sense that two or moreregions can be endowed with the same critical iso-intensity manifold. In thispaper we describe these catastrophes in scale space and show the implicationsfor the iso-intensity manifolds through the scale space saddles, and therebyfor both the hierarchy tree and the pre-segmentation.

Furthermore, perturbations of some higher order non-generic catastrophesare shown to describe creations of pairs of critical points. Although the latterare generic, they are commonly ignored in applications (e.g. [66, 71]). Thisis either due to the false believe that they are not supposed to exist at all, or

jmiv04.tex; 22/07/2004; 12:48; p.3


to the heuristic observation that they are rarely found. We will also discussthis issue and give an explanation why creations do not influence or harm thehierarchical structure.

In section 2 theory on Gaussian scale space, catastrophe theory and a briefoutline of the hierarchy tree is given. Catastrophes in scale space in genericcoordinates and their effects on the hierarchy are discussed in section 3. Wegive some applications in section 4 and end with a summary and discussionin section 5.

2. Background

In [50, 47] we presented a uniquely defined hierarchical structure describinga scale space image. In section 2.4 we shortly outline the basic steps. In orderto understand the essential elements, we define a Gaussian scale space insection 2.1. The structure depends on the evolution of spatial critical pointsas the scale changes. The locations of these points in scale space form one-dimensional manifolds, the so-called critical curves, containing two types ofspecial points. The first type is formed by the scale space saddles, discussedin section 2.2. The second type are the catastrophe points, presented in section2.3.

2.1. GAUSSIAN SCALE SPACE

DEFINITION 1. �� denotes an arbitrary � -dimensional image. We willrefer to this image as the initial image.

DEFINITION 2. �� denotes the �� -dimensional Gaussian scalespace image of �� .

The Gaussian scale space image is obtained by convolution of an initialimage with a normalized Gaussian kernel of zero mean and standard deviation�

�� : �� ! �#"%$'&)( *,+�-.( /021 ��34�657398Consequently, �� satisfies the diffusion equation::!; �� "< =?>�@ :BA:DC A= �� FEHG2I�KJL�� 8 (1)

Here JL�� denotes the Laplacean. Differentiation is now well-defined,since derivatives of the image up to arbitrary order at any scale are given by::MC = �� ::MC = �N�O��P��RQ ::DC = �O��TSP��U8

jmiv04.tex; 22/07/2004; 12:48; p.4

Trees from Images 5

That is, an arbitrary derivative of the image is obtained by the convolution ofthe initial image with the corresponding derivative of a Gaussian.

DEFINITION 3. Spatial critical points, i.e. saddles and extrema (maxima orminima), at a certain scale �� are defined as the points at fixed scale �� wherethe spatial gradient vanishes:

�� L�� , that is, L�� .We will refer to these points as spatial critical points to distinguish them fromscale space critical points, see Definition 6.

The type of a spatial critical point is given by the eigenvalues of the Hes-sian � , the matrix with the second order spatial derivatives, evaluated at itslocation.

DEFINITION 4. The Hessian matrix at a certain scale � � is defined by � E GNI� � �� , where each element of � is given by

�=� � � :MA:DC = :MC � �� U8

The trace of the Hessian equals the Laplacean. For maxima (minima) alleigenvalues of the Hessian are negative (positive). At a spatial saddle point� has both negative and positive eigenvalues.

Since �� is a smooth function in �� -space, spatial critical pointsare part of a one dimensional manifold in scale space by virtue of the implicitfunction theorem.

DEFINITION 5. A critical curve is a one-dimensional manifold in scalespace on which L�� .

Consequently, the intersection of all critical curves in scale space with aplane of certain fixed scale � � yields the spatial critical points of the image atthat scale.

2.2. SCALE SPACE SADDLES

DEFINITION 6. The scale space saddles of �� are defined as the pointswhere both the spatial gradient and the scale derivative vanish: �� and JL�� .

In Definition 6 we used Eq. (1). Note that it describes the critical pointsof �� in scale space. In [50, 47] it is proven that these points are indeedalways saddle points, a result of the well-known maximum principle.

jmiv04.tex; 22/07/2004; 12:48; p.5


DEFINITION 7. The extended Hessian � of �� is matrix of secondorder derivatives in scale space defined by

� �RQ � �6� J L��2J� L� � � JOJL� S�8Here � � � is the Hessian.

Note that the elements of � are purely spatial derivatives. Again, this ispossible by virtue of the diffusion equation, Eq. (1).

The fact that scale space critical points are always saddles implies that theextended Hessian has both positive and negative eigenvalues at scale spacecritical points. Furthermore, in [50, 47] we have proven that if the intensityof the spatial saddle points on a critical curve is parameterized by scale, scalespace saddles are in fact the extrema of the parameterization.

2.3. CATASTROPHE THEORY

The spatial critical points of a function with non-zero eigenvalues of theHessian are called Morse critical points. The Morse Lemma states that atthese points the qualitative properties of the function are determined by thequadratic part of the Taylor expansion of this function. This part can bereduced to the Morse canonical form by a slick choice of coordinates.

If at a spatial critical point the Hessian degenerates, so that at least oneof the eigenvalues is zero, the type of the spatial critical point cannot bedetermined.

DEFINITION 8. The catastrophe points of �� are defined as the pointswhere both the spatial gradient and the determinant of the Hessian vanish: L�� and �� .

The term catastrophe was introduced by Thom [68, 69]. It denotes a (sud-den) qualitative change in an object as the parameters on which this objectdepends change smoothly. This behaviour was already known by the termsperestroika, bifurcation and metamorphosis. The name catastrophe theorywas suggested by Zeeman [75] to unify singularity theory, bifurcation theoryand their applications and gained wide popularity. A thorough mathematicaltreatment on singularity theory can be found in the work of Arnol’d [3, 2, 4–7]. More pragmatic introductions and applications are widely published, e.g.[9, 24, 26, 57, 64, 75].

jmiv04.tex; 22/07/2004; 12:48; p.6

Trees from Images 7

2.3.1. The Catastrophe GermsThe catastrophe points are also called non-Morse critical points, since a higherorder Taylor expansion is essentially needed to describe the qualitative prop-erties. Although the dimension of the variables is arbitrary, the Thom SplittingLemma states that one can split up the function in a Morse and a non-Morsepart. The latter consists of variables representing the � “bad” eigenvalues ofthe Hessian that become zero. The Morse part contains the �� remainingvariables. Consequently, the Hessian contains a ��D�� D� sub-matrixrepresenting a Morse function. It therefore suffices to study the part of �variables. The canonical form of the function at the non-Morse critical pointthus contains two parts: a Morse canonical form of �� variables, in termsof the quadratic part of the Taylor series, and a non-Morse part. The latter canby put into canonical form called the catastrophe germ, which is obviously apolynomial of degree 3 or higher.

Since the Morse part does not change qualitatively under small pertur-bations, it is not necessary to further investigate this part. The non-Morsepart, however, does change. Generally the non-Morse critical point will splitinto a non-Morse critical point, described by a polynomial of lower degree,and Morse critical points, or even exclusively into Morse critical points. Thisevent is called a morsification. So the non-Morse part contains the catastrophegerm and a perturbation that controls the morsifications.

Then the general form of a Taylor expansion �� at a non-Morse criticalpoint of an � dimensional function can be written as (Thom’s Theorem):

�� C @ � 8 8 8 � C�� C @ � 8 8 8 � C�� @ � 8 8 8 � �� "<=?> �� @�� = C A= � (2)

where � �� C @ � 8 8 8 � C�� denotes the catastrophe germ, �� C @ � 8 8 8 � C�� @ � 8 8 8 � �� the perturbation germ with an � -dimensional space of parameters, and in theMorse part

�= �! � . In Table I the germs with �#" �

are listed. In 2D theseform, together with $&%

@ EHG2I�' CBA )( A and taking *�+ and * &+ together as *�%+ ,

the so-called Thom’s seven.These germs are the starting point of the infinite set of so-called simple

real singularities, whose catastrophe germs are given by the infinite series$�%� EHG2I� C �,� @ � �.- � and */%� EHG2I� C A ( 0( � &

@� �.- �

, and the threeexceptional singularities 1�2 EHG2I� C�3 /( + , 154 E GNI� C�3 � C ( 3 , and 176 EHG2I� C�3 �/(98 .The germs $

�� and $ &� are equivalent for �� and k even.

2.3.2. Catastrophes and Scale SpaceIn Definition 8, the number of equations defining the catastrophe point equals�F�� and therefore it is over-determined with respect to the � spatial variables.Consequently, catastrophe points are generically not found in typical images.

jmiv04.tex; 22/07/2004; 12:48; p.7


Table I. Description of non-Morse critical points for maximal 4 differentperturbation parameters. Each contain a catastrophe germ (CG) and corre-sponding perturbation term (PT).

name nickname CG PT� / Fold �� Cusp �� 0 �� / � /� 0 Swallowtail �� / � / �� Butterfly �� / � / �� 0 � 0��0 Hyperbolic Umbilic � / � � � � � � � �� / � �� /� +0 Elliptic Umbilic � / � �� / � �� /� �� Parabolic Umbilic � / � � 0 �� / � �� / �� 0 � /

In scale space, however, the number of variables equals �� and catastrophesoccur as isolated points.

Although the list of catastrophes starts very simple, it is not trivial to applyit directly to scale space by assuming that scale is just one of the perturbationparameters.

For example, in one-dimensional images the Fold catastrophe reduces toC�3 � C . It describes the change from a situation with two critical points (amaximum and a minimum) for �� to a situation without critical pointsfor �� . See e.g. Figure 1 in Section 3.1.1 for an example of such anannihilation sequence. This event can occur in two ways. The extrema areannihilated for increasing , but the opposite – creation of two extrema fordecreasing – is also possible.

In scale space, however, there is an extra constraint: the germ has to sat-isfy the diffusion equation. Thus the catastrophe germ

C 3implies an extra

term � C � . On the other hand, the perturbation term is given by @ C

, so bytaking �� scale plays the role of the perturbing parameter. This givesdirectionality to the perturbation parameter, in the sense that the only re-maining possibility for this $ A -catastrophe in one-dimensional images is anannihilation.

In higher dimensional images also the opposite – i.e. a Fold catastrophedescribing the creation of a pair of critical points – is possible. Then theperturbation �� with increasing � requires an additional term of theform � � C ( A in order to satisfy the diffusion equation, see Definition 9.

2.3.3. The Scale Space Catastrophe GermsThe transfer of the catastrophe germs to scale space has been made by manyauthors, [10–12, 16, 35–39, 46–51, 53, 55], among whom Damon’s account isprobably the most rigorous. He showed that the only generic morsifications inscale space are the aforementioned Fold catastrophes describing annihilationsand creations of pairs of critical points. These two points have opposite sign

jmiv04.tex; 22/07/2004; 12:48; p.8

Trees from Images 9

of the determinant of the Hessian before annihilation and after creation. Allother events are compounds of such events. It is however possible that onemay not be able to distinguish these generic events, e.g. due to numericallimitations, coarse sampling, or (almost) symmetries in the image.

DEFINITION 9. The scale space catastrophe germs [10–12] are defined by

� A �� EHG2I� C�3 @ � � C @ �� C �� EHG2I� C�3 @ � � C @ � � � C @ C AA � � �� B8

The quadratic term� �� is defined� �� E GNI� "< =?> A � = � C A= � ��

where � "= > A � = �� and�= �� .

Note that the scale space catastrophe germs � A and � C, and the quadraticterm

�satisfy the diffusion equation. The germs � A and � C correspond to

the two qualitatively different Fold catastrophes at the origin, an annihilationand a creation respectively. From Definition 9 it is obvious that annihilationsoccur in any dimension, but creations require at least 2 dimensions. Conse-quently, in 1D signals only annihilations occur. Furthermore, for images ofarbitrary dimension it suffices to investigate the 2D case due to the SplittingLemma.

2.3.4. The Annihilation GermSpatial critical points at any scale � for � A follow directly from � A �� : � � C A @ � � ��

��= C = �� - �

Then the critical curve is parameterized by � � � �� 8 8 8 � � �� " � . Atthe origin a catastrophe takes place. The determinant of the Hessian is givenby � �� C @ , with the constant �L� ��

� "��"=?> A � = . So two critical pointswith opposite sign approach the origin as � increases to zero. Note that tr � �� C @ �� "=?> A �

�=, which is generically non-zero at catastrophe points. This

explains the constraints on the�=

in Definition 9.

2.3.5. The Creation GermThe creation germ is a bit more complicated. Spatial critical points at anyscale � for � C follow from � C �� : �� C A @ � � CBAA ��

�C A � � A � � C @ � ��

��= C = �� - �

jmiv04.tex; 22/07/2004; 12:48; p.9


Since we look in the neighbourhood of the origin, we takeC A � � . Then

the critical curve is parameterized by � � �� 8 8 8 � � �� - � . At the ori-gin a catastrophe takes place. The determinant of the Hessian is given by�� C @ � � A � � C @ � � � � � CBAA , with the constant � � � �

� " "=?> 3 � = , sotwo critical points with opposite sign leave the origin as � increases fromzero. Note that this catastrophe is a Fold catastrophe since it describes thecreation of two critical points, although there is a striking resemblance to thedescription of the Elliptic Umbilic catastrophe. Furthermore, the descriptionof the catastrophe is essentially local: If � is taken too large, the (non-generic)degeneration of the Hessian at

C @ � @2� A has to be taken into account. We will

elaborate on these items in Section 3.

2.4. SCALE SPACE HIERARCHY

From the previous section it follows that each critical curve in �� -spaceconsists of separate branches, each of which is defined from a creation eventto an annihilation event. We set

��the number of creation events on a critical

path and��

the number of annihilation events. Since there exists a scale atwhich only one spatial critical point (an extremum) remains (see Loog etal. [56]), there is exactly one critical path with

�� , whereas all othercritical paths have

�� . That is, all but one critical paths are definedfor a finite scale range.

One of the properties of scale space is non-enhancement of local extrema.Therefore, iso-intensity manifolds (isophotes in 2D) in the neighbourhood ofa spatial extremum at a certain scale � � move towards the spatial extremumat coarser scale until at some scale � @ the intensity of the extremum equalsthe intensity of the manifold. The iso-intensity surface in scale space formsa dome, with its top at the extremum at scale � @ . Since the intensity of theextremum is monotonically in- or decreasing (depending on whether it is aminimum or a maximum, respectively), all such domes are nested. Retro-spectively, each extremum branch carries a series of nested domes, definingincreasing regions around the extremum in the input image.

In [50, 47] we have proven that these regions are uniquely related to oneextremum as long as the intensity of the domes does not reach that of the so-called critical dome. The latter is formed by the iso-intensity manifold withits top at the extremum and containing a scale space saddle (see section 2.2)that is part of the same critical curve. An example of a critical dome and itsrelated critical curve is shown in Figure 6 in Section 3.2.1.

In this way a hierarchy of regions of the input image is obtained, whichcan be regarded as a kind of pre-segmentation. It also results in a partition ofthe scale space itself. Details can be found in [44, 47, 49, 50].

The crucial role is played by the scale space saddles and the catastrophepoints. As long as only annihilation and creation events occur, the hierarchy

jmiv04.tex; 22/07/2004; 12:48; p.10

Trees from Images 11

is obtained straightforwardly. However, sometimes higher order catastrophesare needed to describe the local structure, viz. when two or more catastro-phes happen to be almost incident and cannot be segregated due to coarsesampling, numerical imprecision, or (almost) symmetries in the image. In thenext section we describe these higher order events.

3. Scale space catastrophes, scale space saddles, and iso-intensitymanifolds

In this section we discuss the appearance of catastrophe events in scale spaceand the effect on scale space saddles. Firstly, results on one-dimensional im-ages are given, because in this particular case scale space saddles coincidewith catastrophe points. Secondly, multi-dimensional images are discussed.In order to do so, we need to define the types of critical curves that are allowedwhen only generic situations are considered.

DEFINITION 10. A scale space germ is a Taylor expansion satisfying thescale space constraint (the diffusion equation), yielding critical curves withonly generic catastrophes on them.

All the catastrophes of Table I can be transformed into scale space germs byletting them satisfy the diffusion equation: Take the Laplacean of the germand add the suitable polynomial in � . Next, one of the perturbation parameterscan be set equal to � . The scale space germs will thus be a combination ofcritical curves with generic annihilations and (sometimes) creations. If theperturbation parameters are set zero, the scale space germ will contain at leastone non-generic point. These are points of interest in the remainders of thispaper.

3.1. 1D IMAGES

In 1D images the critical iso-intensity manifolds (or separatrices) are given bythe isophotes through the catastrophes points, since these points are identicalto the scale space saddles: � �� and � ; � J�� . At such pointsthe extended Hessian, Definition 7, reads

� � Q � � �� S 8It is generically non-zero at scale space saddles and �� 0� � A�� . Inone dimensional images only cuspoid catastrophes (the $ � -type of Table I)occur, of which we will discuss the Fold $ A and the Cusp $ 3 .

jmiv04.tex; 22/07/2004; 12:48; p.11


-1 -0.5 0.5 1x

-0.4

-0.2

0.2

0.4

Intensity

-1 -0.5 0.5 1x

-0.4

-0.2

0.2

0.4

Intensity

-1 -0.5 0.5 1x

-0.4

-0.2

0.2

0.4

Intensity

Figure 1. Fold catastrophe for increasing scales a) t=-1: Two extrema. b) t=0: Catastrophe atthe origin. c) t=1: No extrema.

-1 -0.8 -0.6 -0.4 -0.2s

-4

-2

2

4

Intensity

-3 -2 -1 1 2 3x

-2

-1

1

2t

Figure 2. a) Parameterized intensity of the Fold catastrophe. b) 1+1D intensity scale spacesurface of the Fold catastrophe in

� �� space. c) Segments of b), defined by the scalespace saddle intensity.

3.1.1. Fold catastropheThe generic annihilation is called a Fold and is defined by (see Definition 9and further) �� C �� C 3 � � C �B8The only perturbation parameter is given by � after the identification

@ �� .The intensity for increasing scales is shown in Figure 1. It has a scale spacesaddle if both derivatives are zero, that is,� � � � � C A � �� ; � � C ��So it is located at the origin with intensity equal to zero. The determinant ofthe extended Hessian equals � � � , indicating a saddle. A possible parame-terization of the critical curve is � C �� U�� '�� 7� �� " � and thecorresponding parameterized intensity reads �O�� '�� /" � , seeFigure 2a.

The critical dome is given by the isophotes �� C �� through the ori-gin, so � C �� O� � � �� and � C �� C ,� @2 C A � . Figure 2b shows isophotes� � ��.�� 7� � in the � C �� C �� -space, where the self-intersection of� � � gives the annihilation point. This isophote gives the separatrices ofthe different parts of the image. The separation curves in the � C �� -plane areshown in Figure 2c. For � � � , four regions exist, for � � � two remain(compare to Figure 1a-c).

At the catastrophe point the isophotes of the scale space saddle form apitchfork. Due to the causality principle it has 3 branches downwards and

jmiv04.tex; 22/07/2004; 12:48; p.12


only one upward, i.e. at the scale space saddle four separate regions changeto two separate regions. Locally the isophotes are described by �� C �� ; � @2 C�3 � C � ��E GNI�� , so the horizontally traversing branches of the scale spacesaddle isophote necessarily have branches given by �� @2 C A , describing thedisappearance of two regions.

3.1.2. Cusp catastropheAlthough all catastrophes are generically described by fold catastrophes, onemay encounter higher order catastrophes, e.g. due to numerical imprecisionor symmetries in the signal, for instance when a set of two minima and onemaximum change into one minimum, but one is not able to detect whichminimum is annihilated.

The first higher order catastrophe describing such a situation is the Cuspcatastrophe. The scale space representation of the catastrophe germ reads L� C + ��

C A �6�� A � , the perturbation term was given by @ C � A CBA , see

Table I. Obviously, scale fulfils the role of the perturbation by A . Thereforethe scale space form is given by�� C ��

C + � C A �� A � � C �where the two perturbation parameters are given by � for the second orderterm and

�for the first order term. Scale space critical points are given by� �� @

3 C 3 � �C � � � �� ; � C A � ��

If� � � the situation as sketched above occurs. The catastrophe takes place

at the origin, where two minima and a maximum change into one minimumfor increasing � . At the origin also � �� , resulting in a zero eigenvalue ofthe extended Hessian. Note that this degeneration is automatically induced bythe Cusp catastrophe. The parameterized intensity curves ( � @ � � � '�)�

A � and � A � � � �� 7� �� 0� ��

A � "�� ) are shown in Figure 3a. Note that at thebottom left the two branches of the two minima with equal intensity, given by� A , coincide. The case � � � � �� , where a morsification has taken place, isvisualized in Figure 3b. This Figure shows the remaining Fold catastrophe ofa minimum and a maximum (compare to Figure 2a), and the unaffected otherminimum.

It is this splitting that may not be discernible in practice, although it is thegeneric situation. Depending on the value and sign of

�one can find the three

different types of catastrophe shown in Figure 4a-c. With an uncertainty inthe measurement they may coincide, as shown in Figure 4d, where the ovalrepresents the possible measure uncertainty.

With the degeneration of the extended Hessian at the origin if� � � ,

also the shape of the isophotes changes as shown in Figure 5. Since one

jmiv04.tex; 22/07/2004; 12:48; p.13


-1 -0.5 0.5 1s

-1-0.75-0.5-0.25

0.250.50.75

1Intensity

-1 -0.5 0.5 1s

-1-0.75-0.5-0.25

0.250.50.75

1Intensity

Figure 3. Parameterized intensity of the Cusp catastrophe a) �� b) ��

-4 -2 2 4x

-3-2-1

123t

-4 -2 2 4x

-3-2-1

123t

-4 -2 2 4x

-3-2-1

123t

Figure 4. Critical paths in the� �� -plane. a) � �� b) �� c) �� d) detection of the

critical paths around the origin with uncertainty represented by the oval.

eigenvalue is zero, the only remaining eigenvector is parallel to theC

-axis.So there is no critical isophote in the � -direction, but both parts pass theorigin horizontally. Consequently, three regions disappear. Furthermore theannihilating minimum cannot be distinguished from the remaining minimum.

3.1.3. Higher order CuspoidsOne can easily verify that higher order Cuspoids, $ � , � � � , correspond tothe annihilation of � regions simultaneously. Morsification per perturbationparameter leads to $&� , �� catastrophes, and a complete morsification yieldsonly Fold catastrophes.

3.2. � -D IMAGES, � � �In higher dimensions the structure is more complicated, since genericallyscale space saddles do not coincide with catastrophe points. For � -D images,� � � , it suffices to investigate scale space critical points in 2+1D, since thefirst seven elementary catastrophes can locally be written in 2 dimensions.Apart from the in one dimension determined cuspoid catastrophes $ � caus-ing annihilations, also umbilic catastrophes * � occur, requiring 2 variables(see Table I). The first two types are the hyperbolic *

�+ and the elliptic * &+umbilic catastrophes.

-3 -2 -1 1 2 3x

-1

-0.8

-0.6

-0.4

-0.2

0.2

t

-3 -2 -1 1 2 3x

-1

-0.8

-0.6

-0.4

-0.2

0.2

t

-3 -2 -1 1 2 3x

-1

-0.8

-0.6

-0.4

-0.2

0.2

t

Figure 5. Critical isophotes in the� �� -plane. a) � �� b) �� c) ��

jmiv04.tex; 22/07/2004; 12:48; p.14


If we assume �� so that JL� � � , the extended Hessian,Definition 7, becomes

� � �� ;� � � � � �� ;� � ; �� ; � ; ;�� 8

The determinant is � � ; ;� � A�� P� A� �� A� ; �� A� ; � �)� � ; � � �� ;and the trace simplifies to � ; ; . Both are generically non-zero.

In this section we subsequently describe the scale space representations in2+1D of the cuspoid catastrophes $ A and $ 3 , and the umbilic catastrophes*�+ and * &+ , together with their morsifications, the appearances of scale

space saddles and the possibilities with respect to the degeneration of � .

3.2.1. Fold catastropheThe first type of catastrophes is given by the Fold catastrophe, which followsdirectly from Table I and Eq. (2) and was given in � -D in Section 2.3.4:�� C � (B�� C 3 � � C �� ( A � �� (3)

where � � . Positive sign describes a saddle – minimum annihilation,negative sign a saddle – maximum annihilation. Without loss of generalitywe take 9� � . Then ��

� � � � CBA � �� (� ; � � C � �� .� � �9� � �

C�� 9� ��

so the catastrophe takes place at the origin with intensity equal to zero andthe scale space saddle is located at � C � (B�� O� � � @3 � � ,� @@

6 � with intensity�@A 4 . The surface �� C � (B�� @A 4 through the scale space saddle is shown

in Figure 6. It has a local maximum at � C � (B�� @2 � � ,� @4 A � : the top of the

extremum dome. Recall that the coordinates have no quantitative significance.The iso-intensity surface through the scale space saddle can be visualised

by two surfaces touching each other at the scale space saddle. One part ofthe surface is related to the extremum corresponding to the scale space sad-dle. The other part encircles some other segment of the image. The surfacebelonging to the extremum forms a dome. The critical curve intersects thissurface twice. The saddle branch has an intersection at the scale space saddle,the extremum branch at the top of the dome, as shown in Figure 6.

A parameterization of the two branches of the critical curve is given by� C �� ( �� U�� '�� '� � �" � .

jmiv04.tex; 22/07/2004; 12:48; p.15


Figure 6. The critical curve contains a catastrophe point (bright dot) and a scale space saddle(dark dot). The iso-intensity surface through the scale space saddle contains two parts touchingeach other at the scale space saddle. One part is dome-shaped and intersects the critical curveat the top of the dome.

-0.2 -0.15 -0.1 -0.05s

-0.2-0.15-0.1-0.05

0.050.10.150.2Intensity

Catastrophe

Saddle-1 -0.5 0.5 1

s

-0.4

-0.2

0.2

0.4

Intensity

Catastrophe

Saddle

Figure 7. Intensity of the critical curve, parameterized by a) the x-coordinate and b) thet-coordinate. Both showing at the origin an annihilation, at the minimum the scale spacesaddle.

The intensity of the critical curve reads �� '� � �� " �

(with:�� and

:�� J�� :�� ). The scale space saddle islocated at � � @@

6 , the catastrophe at the local maximum, the connection ofthe two intensity-curves, O� � . These points are visible in Figure 7a as thelocal minimum of the parameterization curve and the connection point of thetwo curves, the upper branch representing the spatial saddle, the lower onethe minimum.

Note that an alternative parameterization of both branches of the criticalcurve simultaneously is given by � C �� '� � ( �� '�U��H�� '�� ,� @A A � . Then theintensity of the critical curve is given by �� '��

3 � A . Now: � � �0�

and: � �� '�� A � �� !�H� � �� and the latter is still equivalent toJL� � � � . The catastrophe takes place at �� , the saddle at ��

@3 . These

points are visible in Figure 7b as the extrema of the parameterization curve.The branch � � represents the saddle point, the branch � � the minimum.

jmiv04.tex; 22/07/2004; 12:48; p.16


-4 -2 0 2 4x-4

-3

-2

-1

0

1

2t

-3 -2 -1 1 2 3x

-2

-1

1

2

t

Catastrophe

Saddle

-4 -3 -2 -1 1 2s

-8

-6

-4

-2

2

4

Intensity

CatastropheSaddle

Figure 8. a) Critical paths. b) Critical paths with zero-Laplacean, catastrophe point and scalespace saddle if �

� . c) Intensity of the critical paths. The part bottom-left represents twobranches ending at the catastrophe point.

3.2.2. Cusp catastropheWith the similar argumentation as in the one-dimensional case it is also in-teresting to investigate the behaviour around the next catastrophe event. Thehigher-dimensional cusp catastrophe in scale space follows directly from Ta-ble I and Eq. (2). It is the 2-D scale space extension of the catastrophe dis-cussed in section 3.1.2 and is defined by�� C � ( ��

C + � C A �� A �� 6��( A � � � Cwhere, again, 9� � . If

� �� a fold catastrophe results. Then �� @

3 C�3 � �C � � �� (� ; � CBA � � %� ��

�� 9� � � C A � �� 9� � � �� C A �U8

The critical curves in the � C �� -plane at� �� ( �� are shown in Figure

8a. They form a so-called pitchfork bifurcation at the origin, the catastrophepoint.

Critical points are on the curves given by � C �� '� � ( �� U�� '� and� C �� ( �� U�� '�� '� � �" � .The intensities are given by � @ �� 9� �� '�� A � � with its

extremum at � � and � A �� '�)� � �� A � � � " � .The latter has an extremum at � @A . Since " � , these scale space saddlesonly occur if � � . It is therefore essential to distinguish between the twosigns of .

3.2.2.1. Case � � For positive , the curve � C � (B�� '� containssaddles if �� and minima if �� . The other curve contains minima onboth branches. At the origin a catastrophe occurs, at � C � ( ��O� � � � � � � 4�a scale space saddle, see Figure 8b. The intensities of the critical curves areshown in Figure 8c; The two branches of the minima for � � � have equalintensity. The iso-intensity manifold in scale space forms a double dome sincethe two minima are indistinguishable, see Figure 9.

jmiv04.tex; 22/07/2004; 12:48; p.17


Figure 9. 2D Surface trough the scale space saddle at a Cusp catastrophe, � � .

-3 -2 -1 1 2 3x

-2

-1

1

2

t

Catastrophe

Saddle

Saddle Saddle

-2 -1 1 2 3s

-4-3-2-1

1234

Intensity

CatastropheSaddle

Saddle

-4 -2 2 4x

-3

-2

-1

1

2

3t

Catastrophe

Saddle

SaddleSaddle

Figure 10. a) Critical paths with zero-Laplacean, catastrophe point and scale space saddleif � � � . b) Intensity of the critical paths. The part bottom-left represents two branchesending at the catastrophe point. c) Critical paths with �

�� / � � �� , zero-Laplacean,catastrophe point and scale space saddle.

A small perturbation ( � � � � �� ) leads to a generic image containing aFold catastrophe and thus a single cone. However, as argued in section 3.1.2this perturbation may be too small to identify the annihilating minimum. Wewill use this degeneration in Section 4 to identify multiple regions with onescale space saddle.

3.2.2.2. Case � � If is negative, the curve � C � (B�� '� containsa maximum if � � � and a saddle if � � � , while the curve � C � (B�� 9�� '� � �� contains saddles. Now 3 scale space saddles occur: at� C � (B�� )� � � � � ,� � and � C � ( ��)� � � � � � � @A 4� , see Figure 10a. Thecorresponding intensities are shown in Figure 10b, where again the intensitiesof the two saddle branches (and thus the scale space saddles) for � � �coincide.

The iso-intensity surfaces through the scale space saddles are shown inFigure 11. The scale space saddles at �� @A both encapsulate the maximumat the t-axis. The scale space saddle at � � � is void, i.e. it is not related

jmiv04.tex; 22/07/2004; 12:48; p.18


Figure 11. 2D Iso-intensity manifold trough the scale space saddles a) at � � � �/ and b) at� �

to an extremum. This is clear from the fact that there is only one extremumpresent.

If a small perturbation ( � � � � �� ) is added the three scale spacesaddles remain present in the generic image. Their trajectories in the � C �� -plane are shown in Figure 10c. Now a Fold catastrophe is apparent, but alsoa saddle branch containing two (void) scale space saddles, caused by theneighbourhood of the annihilating saddle-extremum pair.

3.2.2.3. Degeneration of �� The extended Hessian degenerates if itsdeterminant vanishes, i.e. if

� � �� C A � � � . This implies �� C A. Then� � � � reduces to

+ 3 C 3 � � � � . For� � � the degeneration takes place at

the origin, that is, at the cusp catastrophe. But thenC � � � � � � and � ; � �

implies � � , which is non-generic. For other arbitrary values of�, � ; � �

impliesC A � � , so it is located at � C � ( �� sgn � � � � � � � � � @A 4� ,

where � � and �� A � � � � 3 .

This special value is located at the non-annihilating saddle branch wherethe two scale space saddle points coincide, i.e. where the saddle branch touchesthe zero-Laplacean surface. This case is non-generic, since the intersection ofthe critical curve and the hyper-plane JL� � � at this value is not transverse.This value describes the transition of the case with two void scale space sad-dles to the case without scale space saddles: For

� � � �+ 3 � � 3 two void scale

space saddles occur on the non-annihilating saddle branch as shown in Figure10c. For

� � � �+ 3 � � 3 none occur since the saddle branch does not intersect

the zero-Laplacean. In other words: a Fold catastrophe in scale space occurs,regarding two scale space critical points (i.e. saddles) with different signs of�� and controlled by the perturbation parameter

�.

jmiv04.tex; 22/07/2004; 12:48; p.19


3.2.3. Hyperbolic umbilic catastropheThe hyperbolic umbilic catastrophe germ is given by

C 3 � C ( A . Its scale spaceaddition is � C � . The perturbation term contains three terms:

@ C � A ( � 3 ( A . Obviously scale takes the role of

@. The scale space hyperbolic umbilic

catastrophe germ with perturbation is thus defined by�� C � (B�� C 3 � C ( A �� C �6�� ( A � ��(where the first three terms describe the scale space catastrophe germ. The set� � �F� form the extra perturbation parameters. Then ��

� � � � C A ��6��( A�� C ( � � �( �� ; �� C � �

�� 9� � � �C � C � 4� � � ( A

�� 9� � � �� C �� U8One can verify that at the combination � � �F� � � � � �7� four critical pointsexist for each � � � . At � �� the four critical curves given by � C � ( �� 63 � � � �� and � C � (B�� annihilate simultaneously atthe origin (see e.g. Kalitzin [38]). This is non-generic, since this point is ascale space saddle and also �� .

Morsification takes place in two steps. In the first step one perturbationparameter is non-zero. If �� and � � � , the annihilations are sepa-rated. At the origin a Fold catastrophe occurs with critical curves � C � ( �� 63�� . On one of these curves both a scale space saddle at � C � (B�� + � � ,� 3 /

@ A 6 � , and the other catastrophe at � C � (B�� O� � � � � ,� 36 A � arelocated. At the latter the critical curves � � � � � � A ��U�� '� 36 Aannihilate in a (non-generic!) Cusp catastrophe.

If � � and � �� , the double annihilation breaks up into two Foldannihilations with symmetric non-intersecting critical curves. A scale spacesaddle is not present.

Finally, if both and � are non-zero, this complete morsification resultsin the generic case with two critical curves, each of them containing a Foldannihilation. One the two critical curves contains a scale space saddle, locatedat � C � ( �� @+ � � A��3 ,� 3 /

@ A 6 � � /@6 / �The extended Hessian degenerates for

C � � . Then follows from � ; ��that

C � �� and from �� also �9�� , which is a non-generic situation.

jmiv04.tex; 22/07/2004; 12:48; p.20


3.2.4. Elliptic umbilic catastrophesThe elliptic umbilic catastrophe germ is given by

C 3 � � C ( A . Its scale spaceaddition is � � C � . The perturbation term contains three terms:

@ C � A ( � 3 ( A . Obviously scale takes the role of

@. The scale space elliptic umbilic

catastrophe germ with perturbation is thus defined by�� C � ( �� C 3 � � C ( A � � C � � � ( A � �� ( (4)

where the first three terms describe the scale space catastrophe germ. The set� � �F� form the extra perturbation parameters. Now �� CBA � �� ( A� � � � � �

C ( � � ( �� ; � � � C � � ��

C � � � C � � � � � ( A��.� � �9� �� C �U8

The combination � � �F� � � � � �7� gives two critical points for all � �� on the critical curves � C � ( �� )� �� and � C � ( ��P�� . At the origin a so-called scatter event occurs: the criticalcurve changes from y-axis to x-axis with increasing � . Just as in the hyperboliccase, in fact two Fold catastrophes take place; in this case both an annihilationand a creation.

The morsification for �� , � �� leads to the breaking into two criticalcurves without any catastrophe: �� implies

C � ( � � , but then�� .The morsification for �� , � � � leads to only one catastrophe event

at the origin: the Fold creation. The sign of determines whether the criticalcurve contains a maximum – saddle pair or a minimum – saddle pair. Withoutloss of generality we may choose � � . For the moment we assume � �� tocompare this case with the Fold annihilation. Then the generic creation germis defined as �� C � (B�� C 3 � � C � � � C ( A ��( A � �� (5)

The scale space saddle is located at � C � ( �� @3 � � @@ 6 � and its inten-sity is �� @3 � � @@ 6 �L� @A 4 . The surface �� C � (B�� @A 4 has a local saddle at� C � (B�� )� � � @2 � � @4 A � , see Figure 12. At creations newly created extremumdomes cannot be present, which is obvious from the non-creation of newlevel-lines. Whereas annihilations of critical points lead to the annihilationsof level-lines, creations of critical points are caused by the rearrangement ofalready existing level-lines.

This fact becomes clearer if we take a closer look at the structure ofthe critical curves. The critical curve containing the creation is given by

jmiv04.tex; 22/07/2004; 12:48; p.21


Figure 12. Iso-intensity surface of the scale space saddle of the creation germ. a) Unperturbed.b) Perturbed.

Figure 13. Critical curves of Eq. (4) with � � in

� �� -space. a)�� : Degeneration

at the connection of the two critical paths b) � ��

� / � � : Morsification with twocatastrophes on one of the critical curves. c) �

� � �� / � � : Morsification without catastrophes

on the critical curves.

� C � (B�� . The other critical curve given by � C � ( �� @2 � � @4 A � �U��

represents two branches connected at the second catastrophe, see Figure 13a.This point is located at � C � ( �� @2 � � @4 A � , is an element of both curves andobviously degenerates the extended Hessian. At this point two saddle pointsand the created extremum go through a Cusp catastrophe resulting in onesaddle. Note that ignoring this catastrophe one would find a sudden changeof the extremum into a saddle point while tracing the created critical points.Obviously this catastrophe is located between the creation catastrophe andthe scale space saddle. The latter therefore does not invoke a critical domearound the created extremum.

The intensity of the creation pair is given by �� '�� # � � �� - � ,the intensity of the other pair by �� '� � @A @ 2 � � "

@4 A . The intensities of

both paths are shown in Figure 14a. A close-up around the catastrophe pointsis given in Figure 14b.

Note that the intensity curve at the bottom-left of Figure 14a-b containstwo saddle branches with equal intensity. Figure 14b shows that at the catas-trophe in the origin two curves are created. The saddle curve (the left curve)remains, the extremum one (the lower curve) is annihilated at the second

jmiv04.tex; 22/07/2004; 12:48; p.22


-0.05 0.05 0.1 0.15s

-0.04

-0.02

0.02

0.04

Intensity

Catastrophes

Saddle

-0.02 0.02 0.04 0.06s

-0.01-0.005

0.0050.010.0150.020.0250.03Intensity

Catastrophes

Saddle

s

Intensity

Catastrophes

Saddle

s

Intensity

Catastrophes

Saddle

Figure 14. a) Intensities of critical paths,�� . b) Close-up at both catastrophes,

�� . c)

Intensities of critical paths,�� / 0 � � . d) Close-up at both catastrophes,

�� / 0 � � .

catastrophe with one of the two saddle branches with equal intensity. Theother saddle branch continues and contains a scale space saddle.

A complete morsification by taking � � � � � � � resolves the scatter. Itcan be shown that the Hessian has two real roots if and only if

� � � �

@3 A � � .

At these root points subsequently a creation and an annihilation event takeplace on a critical curve as shown in Figure 13b. If

� � � �

@3 A � � the critical

curve doesn’t contain catastrophe points, see Figure 13c.If we take � � @A + � � the creation is approximately at � � 8 � � � � � � 8 � � � ,� � 8 � � � �

� �and the annihilation is at � C � ( �� #��

�D�� 7� . The intensity curves

at this situation are visible in Figure 14c-d. Figure 14c shows that the twosaddle curve have different intensities and do not coincide. One curve doesn’tcontain catastrophes, but only one scale space saddle. The other curve con-tains two catastrophes. A close-up around the catastrophes is shown in Figure14d.

Due to this morsification the two critical curves do not intersect eachother. Also in this perturbed system the minimum annihilates with one ofthe two saddles, while the other saddle remains unaffected. The scale spacesaddle remains on the non-catastrophe-involving curve. That is, the creation– annihilation couple and the corresponding saddle branch are not relevantfor the scale space saddle and thus the scale space segmentation.

The iso-intensity surface of the scale space saddle due to the creation germdoes not contain a dome-shaped surface connected to some other surface, butshows only two parts of the surface touching each other at a void scale spacesaddle, recall e.g. Figure 12.

3.2.5. Higher order UmbilicsOne can verify that of the higher order Umbilic catastrophes, * %� , � � �

,the *

�� describe the various annihilations in two dimensions, the compoundof (several) Fold catastrophes. The * &� introduce complicated scatter-likebehaviour which also morsify into Fold catastrophes, but now a combinationof both annihilations and creations.

3.3. MORSIFICATION SUMMARY

All non-Fold catastrophes morsify to Fold catastrophes. The morsificationgives insight in the structure around the catastrophe point regarding the crit-

jmiv04.tex; 22/07/2004; 12:48; p.23


Figure 15. 2D test images a: Artificial image built by combining two identical blobs andadditive noise. b: 181 x 217 artificial MR image.

ical curves and the scale space saddles. In one-dimensional images, catas-trophes and scale space saddles coincide. Therefore, at higher catastrophesthe extended Hessian necessarily degenerates. These catastrophes, however,give insight in the case where more than two critical points are involved in acomplicated annihilation or at several annihilations at almost the same scalespace position, without having the ability of distinguish between the Foldpairs.

In higher dimensional images, the cuspoid catastrophes (the $ � ) give thesame insight, but also allow the assignment of a scale space saddle, and conse-quently a scale space segment, to more than one extremum. Furthermore themorsification of the Cusp catastrophe showed that it is generic to encounterscale space saddles that are not connected to some dome shaped iso-intensitymanifold: the so-called void scale space saddles.

The morsified * &+ catastrophe describes the creation of two critical pointsand the annihilation of one of them with another critical point. So whiletracing the critical branches of a critical curve both an annihilation and acreation event are traversed.

4. Applications

In this section we give some examples to illustrate the theory presented in theprevious sections. To show the effect of a cusp catastrophe in 2D, we firstlytake a symmetric artificial image containing two Gaussian blobs and addnoise to it. This will make the non-generic symmetric image generic, but in asense “almost non-generic”. This image is shown in Figure 15a. Secondly, theeffect is shown on the simulated MR image of Figure 15b. Note that also inthis case an almost symmetric, thus non-generic, situation occurs. This imageis taken from the web site http://www.bic.mni.mcgill.ca/brainweb.

jmiv04.tex; 22/07/2004; 12:48; p.24


Figure 16. Example of a Cusp catastrophe: a: Critical paths in scale space. b: Segmentaccording to a Fold catastrophe. c: Segment according to a Cusp catastrophe.

4.1. ARTIFICIAL IMAGE

Of the noisy image of Figure 15a, a scale space image was built containing41 scales ranging exponentially from $ �� to $ / �� . The calculated critical pathsare presented in Figure 16a. Ignoring the paths on the border, caused by theextrema in the noise, the paths in the middle of the image clearly show thepitchfork-like behaviour, typical of a non-generic Cusp catastrophe, recallFigure 4. Note that since the symmetric image is perturbed, instead of a cuspcatastrophe a fold catastrophe occurs. The scale space saddle on the saddlebranch and its intensity define a closed region around the lower maximum,see Figure 16b. For details on how the hierarchy and the segmentation isobtained, cf. the algorithm presented in [50, 47]. However, if the noise wereslightly different, one could evidently have found the region around the uppermaximum instead. Knowing that the image should be symmetric and observ-ing that the critical paths indeed are pitchfork-like, it is thus desirable to labelthe catastrophe as a Cusp catastrophe. Then the scale space saddle (and itsintensity) defines the two regions around both involved extrema, see Figure16c. This image one would rather expect given Figure 15a. In this sense onecould consider the labeling of the tree as a stabilizing factor.

4.2. SIMULATED MR IMAGE

Subsequently, we took the 2D slice from an artificial MR image shown inFigure 15b. The scale space image at scale � 8 � � with the large structuresremaining is shown in Figure 17a. The hierarchical structure belonging tothis image is shown in Figure 17b.

Now 7 extrema are found, defining a hierarchy of the regions around theseextrema as shown in Figure 18a. In this case is it visually desirable to identifya region to segment

�

@with more or less similar size as region

� 3 . This isdone by assigning a Cusp catastrophe to the annihilation of the extremum ofsegment

� 3 , in which the extremum of segment�

@is also involved. Then

jmiv04.tex; 22/07/2004; 12:48; p.25


Cat.6

Cat.4

Cat.5

Cat.3

Cat.2

Cat.1

S0S1

S2 S1

S3 S1

S4 S3

S6 S2

S5 S1

S7 S2

Figure 17. a) Image on scale 8.4 b) Hierarchy derived from a).

S1S2

S3

S4 S5S6

S7

S0

S1S2

S3

S4 S5S6

S7

S0

Figure 18. a) Segments of the 7 extrema of a, assuming that only generic catastrophes occur,which is the actual case in fact. b) Idem, with the iso-intensity manifold of � � chosen equallyto � � , i.e. after changing the label of a generic event into a non-generic one.

the value of the scale space saddle defining segment� 3 also defines an extra

region around the extremum in segment�

@. This is shown in Figure 18b,

reflecting the symmetry present in Figure 17a.Taking a closer look at the intensities along the critical paths, Figure 19a,

it can be seen that the part belonging to�

@almost gets a point of inflection,

i.e. a degeneracy signaling a cuspoid situation due to the close presence ofthe annihilation on another nearby path.

Next, we see that some other symmetry is induced in the tree: identifyingthe intensities of

�

@and

� 3 , the parts� + and

�

8 also become related. This isvisually clear from Figure 18b, but also from Figure 19b. There we see twocritical paths that are close in intensities and scale of annihilation.

We note that in this example several creation – annihilation events oc-curred, as described by the morsification of the * &+ catastrophe. These didnot influence the result, as explained earlier.

jmiv04.tex; 22/07/2004; 12:48; p.26


10 20 30 40scale

125130135140145150155Intensity

5 10 15 20 25scale

130

135

140

145

Intensity

Figure 19. Intensities of the critical paths belong to segments a) � � and � � . b) � 0 and � � .

5. Summary and Discussion

In this paper we investigated the (deep) structure on various catastrophe eventsin Gaussian scale space and their impact on iso-intensity manifolds. Althoughit is known that pairs of critical points are annihilated or created (the latter ifthe dimension of the image is 2 or higher), it is important to describe thelocal structure of the image around these events. The importance of this localdescription follows from its significance in building a scale space hierarchy.This algorithm depends on the critical curves, their catastrophe points and thescale space saddle points. We therefore embedded the mathematically knowncatastrophes as presented in section 2 in the framework of linear scale spaceimages and obtained scale space germs.

Firstly, annihilations of extrema can occur in the presence of other ex-trema. In some cases it is not possible to identify the annihilating extremumdue to numerical limitations, coarse sampling, or symmetries in the image.Then the event is described by a Cusp catastrophe instead of a Fold catas-trophe. This description is sometimes desirable, e.g. if prior knowledge ispresent and one wishes to maintain the symmetry in the image. The scalespace hierarchy can easily be adjusted to this extra information. We gaveexamples in section 4 on an artificial image and a simulated MR image. Wediscussed the $ 3 and the *

�+ for this purpose, but the higher order catastro-

phes in the sequences $ � � � � � and *��

can be dealt with in a similarfashion.

Secondly, the morsification of the * &+ catastrophe was discussed, showinghe successive appearance of a creation – annihilation event on a critical curve.This doesn’t influence the hierarchical structure nor the pre-segmentation, butis only important with respect to the movement of the critical curve in scalespace. We showed that this appearance heavily depends on the morsificationparameters. They explain that in most cases creations can be ignored, sincethey can be regarded as nothing but a local protuberance of the critical curve.

The theory described in this paper extends the knowledge of the deepstructure of Gaussian scale space. It embeds higher order catastrophes withinthe framework of a scale space hierarchy. It explains how these events can in

jmiv04.tex; 22/07/2004; 12:48; p.27


principle be used for segmentation, interpreted and implemented, e.g. if priorknowledge is available.

References

1. L. Alvarez and J. Morel. Morphological approach to multiscale analysis: From principlesto equations. In ter Haar Romeny, [30], pages 101–112, 1994.

2. V. I. Arnold, editor. Geometrical Methods in the Theory of Ordinary Differential Equa-tions, volume 250 of Grundlehren der mathematischen Wissenschaften: A Series ofComprehensive Studies in Mathematics. Springer-Verlag, Berlin, 1983.

3. V. I. Arnold. Catastrophe Theory. Springer-Verlag, Berlin, 1984.4. V. I. Arnold, editor. Ordinary Differential Equations. Springer-Verlag, Berlin, 1992.5. V. I. Arnold, editor. Dynamical Systems VI: Singularity Theory I, volume 6 of

Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1993.6. V. I. Arnold, editor. Dynamical Systems VIII: Singularity Theory II & Applications,

volume 39 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1993.7. V. I. Arnold, editor. Dynamical Systems V: Bifurcation Theory and Catastrophe Theory,

volume 5 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 1994.8. R. van den Boomgaard and A. Smeulders. The morphological structure of images:

The differential equations of morphological scale-space. IEEE Transactions on PatternAnalysis and Machine Intelligence, 16(11):1101–1113, 1994.

9. J. W. Bruce and P. J. Giblin. Curves and Singularities. Cambridge University Press,1984.

10. J. Damon. Local Morse theory for solutions to the heat equation and Gaussian blurring.Journal of Differential Equations, 115(2):386–401, 1995.

11. J. Damon. Generic properties of solutions to partial differential equations. Archive forRational Mechanics and Analysis, pages 353–403, 1997.

12. J. Damon. Local Morse theory for Gaussian blurred functions. In Sporring et al. [67],pages 147–162, 1997.

13. J. Duncan and N. Ayache. Medical image analysis: Progress over two decades and thechallenges ahead. IEEE Transactions on Pattern Analysis and Machine Intelligence,22(1):85–105, 2000.

14. L. M. J. Florack. Image Structure, volume 10 of Computational Imaging and VisionSeries. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.

15. L. M. J. Florack. Non-linear scale-spaces isomorphic to the linear case with applicationsto scalar, vector and multispectral images. International Journal of Computer Vision,42(1/2):39–53, 2001.

16. L. M. J. Florack and A. Kuijper. The topological structure of scale-space images. Journalof Mathematical Imaging and Vision, 12(1):65–80, February 2000.

17. L. M. J. Florack, R. Maas, and W. J. Niessen. Pseudo-linear scale space theory.International Journal of Computer Vision, 31(2/3):247–259, 1999.

18. L. M. J. Florack, A. H. Salden, B. M. ter Haar Romeny, J. J. Koenderink, and M. A.Viergever. Nonlinear scale-space. Image and Vision Computing, 13(4):279–294, May1995.

19. L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever. Scaleand the differential structure of images. Image and Vision Computing, 10(6):376–388,July/August 1992.

jmiv04.tex; 22/07/2004; 12:48; p.28


20. L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever. Carte-sian differential invariants in scale-space. Journal of Mathematical Imaging and Vision,3(4):327–348, 1993.

21. L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever. Generalintensity transformations and differential invariants. Journal of Mathematical Imagingand Vision, 4(2):171–187, May 1994.

22. L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever. Linearscale-space. Journal of Mathematical Imaging and Vision, 4(4):325–351, 1994.

23. L. M. J. Florack, B. M. ter Haar Romeny, J. J. Koenderink, and M. A. Viergever. TheGaussian scale-space paradigm and the multiscale local jet. International Journal ofComputer Vision, 18(1):61–75, April 1996.

24. A. T. Fomenko and T. L. Kunii. Topological Modeling for Visualization. Springer-Verlag, Tokyo, 1997.

25. J. M. Gauch. Image segmentation and analysis via multiscale gradient watershedhierachies. IEEE Transactions on Image Processing, 8(1):69 –79, January 1999.

26. R. Gilmore. Catastrophe Theory for Scientists and Engineers. Dover, 1993. Originallypublished by John Wiley & Sons, New York, 1981.

27. L. D. Griffin and A. Colchester. Superficial and deep structure in linear diffusionscale space: Isophotes, critical points and separatrices. Image and Vision Computing,13(7):543–557, September 1995.

28. L. D. Griffin, A. Colchester, and G. Robinson. Scale and segmentation of grey-levelimages using maximum gradient paths. Image and Vision Computing, 10(5):389–402,1992.

29. L.D. Griffin and M. Lillholm, editors. Scale Space Methods in Computer Vision, volume2695 of Lecture Notes in Computer Science. Springer -Verlag, Berlin Heidelberg, 2003.

30. B. M. ter Haar Romeny, editor. Geometry-Driven Diffusion in Computer Vision, vol-ume 1 of Computational Imaging and Vision Series. Kluwer Academic Publishers,Dordrecht, 1994.

31. B. M. ter Haar Romeny, L. M. J. Florack, J. J. Koenderink, and M. A. Viergever, ed-itors. Scale-Space Theory in Computer Vision: Proceedings of the First InternationalConference, Scale-Space’97, Utrecht, The Netherlands, volume 1252 of Lecture Notesin Computer Science. Springer-Verlag, Berlin, July 1997.

32. R. D. Henkel. Segmentation in scale space. In Computer Analysis of Images andPatterns. Lecture Notes in Computer Science, vol. 970. Springer-Verlag, pages 41-48,1995.

33. P. T. Jackway. Gradient watersheds in morphological scale-space. IEEE Transactionson Image Processing, 5(6):913–921, 1996.

34. P. Johansen. On the classification of toppoints in scale space. Mathematical Imagingand Vision, 4(1):57–67, 1994.

35. P. Johansen. Local analysis of image scale space. In Sporring et al. [67], pages 139–146,1997.

36. P. Johansen, M. Nielsen, and O.F. Olsen. Branch points in one-dimensional Gaussianscale space. 13:193–203, 2000.

37. P. Johansen, S. Skelboe, K. Grue, and J. D. Andersen. Representing signals by theirtoppoints in scale space. In Proceedings of the International Conference on ImageAnalysis and Pattern Recognition (Paris, France, October 1986), pages 215–217. IEEEComputer Society Press, 1986.

38. S. Kalitzin. Topological numbers and singularities. In Sporring et al. [67], pages 181–190, 1997.

39. S. N. Kalitzin, B. M. ter Haar Romeny, A. H. Salden, P. F. M. Nacken, and M. A.Viergever. Topological numbers and singularities in scalar images. Scale-space evolu-

jmiv04.tex; 22/07/2004; 12:48; p.29


tion properties. Journal of Mathematical Imaging and Vision, 9(3):253–296, November1998.

40. M. Kerckhove, editor. Scale-Space and Morphology in Computer Vision, volume 2106of Lecture Notes in Computer Science. Springer -Verlag, Berlin Heidelberg, 2001.

41. J. J. Koenderink. The structure of images. Biological Cybernetics, 50:363–370, 1984.42. J. J. Koenderink. A hitherto unnoticed singularity of scale-space. IEEE Transactions on

Pattern Analysis and Machine Intelligence, 11(11):1222–1224, 1989.43. A. Koster. Linking Models for Multiscale Image Segmentation. PhD thesis, Utrecht

University, 1995.44. A. Kuijper. The Deep Structure of Gaussian Scale Space Images. PhD thesis, Utrecht

University, 2002.45. A. Kuijper. On manifolds in Gaussian scale space. In Griffin and Lillholm [29], pages

1–16, 2003.46. A. Kuijper and L. M. J. Florack. Calculations on critical points under Gaussian blurring.

In Nielsen et al. [59], pages 318–329, 1999.47. A. Kuijper and L. M. J. Florack. Hierarchical pre-segmentation without prior knowledge.

In Proceedings of the 8th International Conference on Computer Vision (Vancouver,Canada, July 9–12, 2001), pages 487–493, 2001.

48. A. Kuijper and L. M. J. Florack. The relevance of non-generic events in scale space.In Proceedings of the 7th European Conference on Computer Vision (Copenhagen,Denmark, May 28–31, 2002), pages 190–204, 2002.

49. A. Kuijper and L. M. J. Florack. Understanding and modeling the evolution of criticalpoints under Gaussian blurring. In Proceedings of the 7th European Conference onComputer Vision (Copenhagen, Denmark, May 28–31, 2002), pages 143–157, 2002.

50. A. Kuijper, L. M. J. Florack, and M. A. Viergever. Scale space hierarchy. Journal ofMathematical Imaging and Vision, 18(2):169–189, April 2003.

51. A. Kuijper and L.M.J. Florack. The hierarchical structure of images. IEEE Transactionson Image Processing, 12(9):1067–1079, 2003.

52. L. M. Lifshitz and S. M. Pizer. A multiresolution hierarchical approach to image segmen-tation based on intensity extrema. IEEE Transactions on Pattern Analysis and MachineIntelligence, 12(6):529–540, 1990.

53. T. Lindeberg. Scale–space behaviour of local extrema and blobs. Journal of Mathemat-ical Imaging and Vision, 1(1):65–99, 1992.

54. T. Lindeberg. Detecting salient blob-like image structures and their scales with a scale-space primal sketch: a method for focus-of-attention. International Journal of ComputerVision, 11(3):283–318, 1993.

55. T. Lindeberg. Scale-Space Theory in Computer Vision. The Kluwer International Seriesin Engineering and Computer Science. Kluwer Academic Publishers, 1994.

56. M. Loog, J. J. Duistermaat, and L. M. J. Florack. On the behavior of spatial critical pointsunder Gaussian blurring, a folklore theorem and scale-space constraints. In Kerckhove[40], pages 183–192, 2001.

57. Y.-C. Lu. Singularity Theory and an Introduction to Catastrophe Theory. Springer-Verlag, Berlin, second corrected printing edition, 1976.

58. E. Nakamura and N. Kehtarnavaz. Determining number of clusters and prototypelocations via multi-scale clustering. Pattern Recognition Letters, 19(14):1265–1283,1998.

59. M. Nielsen, P. Johansen, O. Fogh Olsen, and J. Weickert, editors. Scale-Space Theoriesin Computer Vision, volume 1682 of Lecture Notes in Computer Science. Springer -Verlag, Berlin Heidelberg, 1999.

60. O. Fogh Olsen. Multi-scale watershed segmentation. In Sporring et al. [67], pages191–200, 1997.

jmiv04.tex; 22/07/2004; 12:48; p.30


61. O. Fogh Olsen. Generic Image Structure. PhD thesis, University of Copenhagen,Denmark, 2000.

62. O. Fogh Olsen and M. Nielsen. Multi-scale gradient magnitude watershed segmentation.In ICIAP’97 - 9th International Conference on Image Analysis and Processing, volume1310 of Lecture Notes in Computer Science, pages 6–13, September 1997.

63. P. Perona and J. Malik. Scale space and edge detection using anisotropic diffusion. IEEETransactions on Pattern Analysis and Machine Intelligence, 12(7):629–639, 1990.

64. T. Poston and I. N. Stewart. Catastrophe Theory and its Applications. Pitman, London,1978.

65. A. H. Salden. Dynamic Scale-Space Paradigms. PhD thesis, Utrecht University, 1996.66. A. Simmons, S. R. Arridge, P. S. Tofts, and G. J. Barker. Application of the extremum

stack to neurological MRI. IEEE Transactions on Medical Imaging, 17(3):371–382,June 1998.

67. J. Sporring, M. Nielsen, L. M. J. Florack, and P. Johansen, editors. Gaussian Scale-Space Theory, volume 8 of Computational Imaging and Vision Series. Kluwer AcademicPublishers, Dordrecht, second edition, 1997.

68. R. Thom. Stabilite Structurelle et Morphogenese. Benjamin, New york, 1972.69. R. Thom. Structural Stability and Morphogenesis. Benjamin-Addison Wesley, 1975.

translated by D. H. Fowler.70. K. L. Vincken. Probabilistic Multiscale Image Segmentation by the Hyperstack. PhD

thesis, Utrecht University, 1995.71. K. L. Vincken, A. S. E. Koster, and M. A. Viergever. Probabilistic multiscale im-

age segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence,19(2):109–120, 1997.

72. T. Wada and M. Sato. Scale-space tree and its hierarchy. In ICPR90, volume II, pages103–108, 1990.

73. J. A. Weickert. Anisotropic Diffusion in Image Processing. Teubner, Stuttgart, 1998.74. A. P. Witkin. Scale-space filtering. In Proceedings of the Eighth International Joint

Conference on Artificial Intelligence, pages 1019–1022, 1983.75. E. C. Zeeman. Catastrophe Theory: Selected Papers, 1972-1977. Addison-Wesley

Publishing Company, 1977.

Authors’ Vitae

Arjan Kuijperreceived his M.Sc. degree in applied mathematics in 1995 with a thesis onthe comparision of two image restoration techniques, from the University ofTwente, The Netherlands. During the period 1996–1997 he worked at ELTRAParkeergroep, Ede, The Netherlands. In the period 1997-2002 he has beena Ph.D. student and associate researcher at the Institute of Information and

jmiv04.tex; 22/07/2004; 12:48; p.31


Computing Sciences of Utrecht University. In 2002 he received his Ph.D.degree with a thesis on “Deep Structure of Gaussian Scale Space Images”and worked as postdoc at Utrecht University on the project “Co-registrationof 3D Images” on a grant of the Netherlands Ministry of Economic Affairswithin the framework of the Innovation Oriented Research Programme. SinceJanuari 1st 2003 he has been working as an assistant research professor atthe IT University of Copenhagen in Denmark funded by the IST Programme“Deep Structure, Singularities, and Computer Vision (DSSCV)” of the Euro-pean Union. His interest subtends all mathematical aspects of image analysis,notably multiscale representations (scale spaces), catastrophe and singularitytheory, medial axes and symmetry sets, and applications to medical imaging.

Luc M.J. Florackreceived his M.Sc. degree in theoretical physics in 1989, and his Ph.D. de-gree in 1993 with a thesis on image structure, both from Utrecht University,The Netherlands. During the period 1994–1995 he was an ERCIM/HCMresearch fellow at INRIA Sophia-Antipolis, France, and INESC Aveiro, Por-tugal. In 1996 he was an assistant research professor at DIKU, Copenhagen,Denmark, on a grant from the Danish Research Council. From 1997 untilJune 2001 he was an assistant research professor at Utrecht University atthe Department of Mathematics and Computer Science. Since June 1st 2001he is with Eindhoven University of Technology, Department of BiomedicalEngineering, currenlty employed as an associate professor. His interest sub-tends all structural aspects of signals, images and movies, notably multiscalerepresentations, and their applications to imaging and vision.

jmiv04.tex; 22/07/2004; 12:48; p.32

Documents

Using Catastrophe Theory to Derive Trees from Imageswittg/preprints/papers/89.pdf · In section 2 theory on Gaussian scale space, catastrophe theory and a brief outline of the hierarchy