corner detection via topographic analysis of vector-potential · corner detection via topographic...

Corner detection via topographic analysis of vector-potential

Bin Luo a,b, A.D.J. Cross a, E.R. Hancock a,*

a Department of Computer Science, University of York, York YO1 5DD, UKb Anhui University, Anhui, People's Republic of China

Received 22 September 1998; received in revised form 23 December 1998

Abstract

This paper describes how corner detection can be realised using a new feature representation based on a magneto-

static analogy. The idea is to compute a vector-potential by appealing to an analogy in which the Canny edge-map is

regarded as an elementary current density residing on the image plane. In this paper, we demonstrate that corners are

located at the saddle-points of the magnitude of the vector-potential. These points correspond to the intersections of

saddle-ridge and saddle-valley structures, i.e. to junctions of the edge and symmetry lines. We describe a template-based

method for locating the saddle-points. This involves performing a non-minimum suppression test in the direction of the

vector-potential and a non-maximum suppression test in the orthogonal direction. Experimental results using both

synthetic and real images are given. We investigate the angle and scale sensitivity of the new corner detector and

compare it with a number of alternative corner detectors. Ó 1999 Elsevier Science B.V. All rights reserved.

Keywords: Corner detection; Topographic analysis; Vector-potential; Saddle-point detection

1. Introduction

Corners are important dominant points in dig-ital images. In many computer vision tasks, suchas image registration, image matching (Costabileet al., 1985), object recognition (Liu and Srinath,1990b; Han and Jang, 1990) and motion analysis(Dreschler and Nigel, 1992), accurate corner de-tection is essential. Broadly speaking, there aretwo corner detection strategies adopted in the lit-erature. The ®rst of these is based on the analysisof pre-segmented contours, while the second isbased on the di�erential analysis of the raw grey-scale image. However, in both cases it is the rate of

change of contour angle that is used to charac-terise corner features.

1.1. Related literature

In the case of boundary-based corner detectionfrom pre-segmented contours there are three pro-cessing steps. Firstly, the image is pre-segmented.Secondly, boundaries of the object in the image areextracted and chain-coded. Finally, algorithms aredeveloped for identifying corners in the chain-codes. A common technique is to search for cor-ners at the intersection points or junction pointsbetween straight line segments (Xie et al., 1993). Infact, the use of chain-codes to provide a digitalcharacterisation of corners abound in the litera-ture (Freeman and Davis, 1977; Beus and Tiu,1987; Koplowitz and Plante, 1995; Rosenfeld and

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Pattern Recognition Letters 20 (1999) 635±650

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Johmston, 1973; Rosenfeld and Weszka, 1975). Agood review is provided by Liu and Srinath(1990a). However, it must be stressed that theweak link in the contour-based method of cornerdetection is the prior availability of a reliable im-age segmentation.

Grey-scale corner detection algorithms can bedivided into two groups. Template-based cornerdetectors (Rangarajan et al., 1989; Mehrotra et al.,1990) exploit the similarity between a given tem-plate of speci®c orientation for each image sub-window. Because multiple orientation templatesare used, the technique is computationally expen-sive. Gradient-based corner detectors (Kitchenand Rosenfeld, 1982; Singh, 1990; Deriche andGiraudon, 1993; Noble, 1988; Wang and Brady,1995; Harris and Stephens, 1988), on the otherhand, rely on measuring the curvature of an edgethat passes through a given image neighbourhood.The strength of the corner response depends onboth the edge-strength and the rate of change ofedge-direction. Gradient-based corner detectiontechniques are more likely to respond to noise thantheir contour-based counterparts, and often per-form quite poorly.

Focusing in more detail on gradient-basedcorner detection, Wang and Brady (1995) describea curvature-based corner detector which measuresthe strength of corners in terms of the so-called``total curvature''. The total curvature is propor-tional to the second derivative of the image in-tensity along the edge-tangent direction, and isinversely proportional to the edge-strength. Themethod o�ers the attractive feature of exploitingan explicit measure of curvature as well as pro-viding a degree of false corner suppression. How-ever, one of its weaknesses is that it cannot controlfalse responses when signi®cant image noise ispresent. The popular Plessey operator (Harris andStephens, 1988) is another curvature-based cornerdetector which is based only on ®rst derivatives ofthe image intensity. Although it is quite robustwith respect to noise, it can only perform well on`L'-junctions or right-angle corners. The localisa-tion of other junction types is poor. The SUSANcorner detector (Smith and Brady, 1997), on theother hand, performs a local template comparison.A degree of robustness is achieved by computing

template correlation statistics for a circular maskcentred on the corner candidates. In particular, itexamines the number of pixels within the maskwhich have a similar brightness to the template.The number of agreement counts is taken as a votefor the corner hypothesis. High voting pixels aretaken to have a signi®cant corner strength. Themethod also performs a non-maximum suppres-sion test to reject false-positives.

1.2. Paper outline

Our aim in this paper is to present a new cornerdetection method which exploits both template-based and gradient-based concepts. We appeal to agradient-based image representation which hasalready been shown to provide a convenient to-pographic representation for edge and symmetryfeatures (Cross and Hancock, 1997, 1998). Therepresentation commences from the Canny edge-map, i.e. the directional gradient of the image in-tensity. By rotating the edge-gradient vectors wecompute a ®eld of edge-tangents. The tangentvectors are smoothed by performing an averagingprocess in which the vectors are weighted accord-ing to the inverse of their distance from the imagepoint in question. This weighting function is sug-gested by magneto-statics. As a result, we refer tothe resulting gradient ®eld as the vector-potential.According to this representation, local maxima ofthe magnitude vector-®eld which exhibit direc-tional consistency (i.e. ridges) correspond to edges.Directionally consistent local minima (i.e. ravines)are symmetry lines. In this paper, we extend thistopographic picture to include corners. These aresaddle-structures where directionally consistentridges and ravines intersect.

It is important to contrast our method ofanalysis with Haralick's topographic primal sketch(Haralick et al., 1983). Whereas Haralick focuseson the topographic structure of grey-scale features(i.e. a scalar image representation), we analyse thetopography of a vector-®eld representation. Themain advantage is that we are able to exploit di-rectional consistency in the localisation of topo-graphic structure and, hence, improve therobustness of feature detection. With the topo-graphic representation in hand, the main practical

636 B. Luo et al. / Pattern Recognition Letters 20 (1999) 635±650

problem that confronts us is the localisation of thesaddle-points. This is more di�cult than localisingridges and ravines since we are concerned withidentifying point features rather than contourfeatures. In the case of ridges and ravines, we canexploit constraints on compatible continuity ordirectionality. In the case of saddle-points theconstraints are more subtle, since we are seekinglocations which are consistent with being thejunctions between saddle-ridges and saddle-valleys.

Based on this observation, we develop topo-graphic tests for the consistent saddle-structurein the vector-potential. This is e�ectively a tem-plate-based method. We search for consistentvalley structure in the direction of the vector-potential and consistent ridge structure in theorthogonal direction. In other words, the direc-tional template characterises local saddle-struc-ture as the intersection of ridge (i.e. edge) andravine (i.e. symmetry) structures. Moreover,computation is simpli®ed since the template is®xed to be in the direction of the vector-poten-tial. In this way we avoid explicit computationof directional second derivatives of the imageintensity. As we will demonstrate experimentally,this o�ers advantages in terms of improved noisesensitivity.

The outline of this paper is as follows. In Sec-tion 2, we review the vector-potential representa-tion. Section 3 introduces the topographicrepresentation of features in the vector-potential.In Section 4, we confront some of the practicaldi�culties associated with saddle-localisation.Real-world experimental examples are presentedin Section 5. Questions of algorithm sensitivityand comparison are addressed in Section 6. Fi-nally, Section 7 provides some conclusions andidenti®es avenues for future investigation.

2. Image representation using vector-potential

In this section, we review the feature-represen-tation recently reported by Cross and Hancock(1997). The starting point is to compute the Cannyedge-map (Canny, 1986). Accordingly, we com-mence by convolving the raw image I with a

Gaussian kernel of width r. The kernel takes thefollowing form:

Gr�x; y� � 1

2pr2exp

�ÿ x2 � y2

2r2

�: �1�

With the ®ltered image in hand, the Canny edge-map is recovered by computing the gradient

E � rGr � I : �2�In order to compute a vector-®eld representationof the edge-map, we will need to introduce anauxiliary z dimension to the original x±y co-ordi-nate system of the plane image. In this augmentedco-ordinate system, the components of the edge-map are con®ned to the image plane. In otherwords, the edge-vector at the point �x; y; 0� on theinput image plane is given by

E�x; y; 0� �oGr�I�x;y�

oxoGr�I�x;y�

oy

0

0B@1CA: �3�

For an ideal step-edge, the resulting image gradi-ent will be directed along the boundary normal. Amore convenient representation is the edge-tangentvectors which ¯ow along the object boundary.Accordingly, we re-direct the edge-vectors so thatthey are tangential to the original planar shape bycomputing the cross-product with the normal tothe image plane z � �0; 0; 1�T. The tangent vectorat the point �x; y; 0� on the input image plane isde®ned to be

j�x; y; 0� � z ^ rGr � I�x; y�: �4�To be more explicit, in terms of its components thetangent vector is given by

j�x; y; 0� �ÿ oGr�I�x;y�

oy

oGr�I�x;y�ox

0

0B@1CA: �5�

In our previously reported work, the key ideaunderlying the image representation was to char-acterise edges and symmetry lines using topo-graphic structures in the edge-tangent ®eld. Edgescorresponded to locations where the tangent vec-tors re-enforce one-another. In other words, theboundaries are identi®ed as local maxima of thetangent ®eld. Symmetry points are those at which

B. Luo et al. / Pattern Recognition Letters 20 (1999) 635±650 637

there is cancellation between diametrically op-posed tangent vectors. Axes of symmetry are linesof local minimum in the tangent ®eld. At the levelof ®ne detail, intensity ridges or ravines give rise tolocal symmetry axes.

Unfortunately, since the raw gradient vectorsare likely to be noisy we must develop a means ofsmoothing the tangent ®eld so that we can performthe required topographic analysis. To realise thisgoal, we appeal to magneto-statics to develop ameans of smoothing the tangent ®eld. According-ly, we compute an analogue of the vector-potentialby regarding the edge-tangents as a ®eld of ele-mentary currents. The vector potential is found byintegrating over volume and weighting the con-tributing currents according to inverse distance. Inother words, the vector-potential at the pointr � �x; y; z�T in the augmented space in which theoriginal image plane is embedded is

A�x; y; z� � lZ

V 0

j�x0; y0; z0�jrÿ r0j dV 0; �6�

where r0 � �x0; y 0; z0�T and l is the permeabilityconstant which we set equal to unity. Since thecontributing edge-tangent vectors (or currents) aredistributed only on the image plane, the volumeintegral reduces to an area integral over the imageplane. As a result, the components of the vector-potential are as follows:

A�x; y; z�

�ÿ R R oGr�I�x0 ;y0�

oy01��

�xÿx0�2��yÿy0�2�z2p dx0 dy0R R oGr�I�x0 ;y0�

ox01��

�xÿx0�2��yÿy0�2�z2p dx0 dy0

0

0BBBB@1CCCCA:�7�

The structure of the vector-potential deservesfurther comment. In the ®rst instance, the com-ponents are con®ned to the x±y plane for all valuesof the auxiliary co-ordinate z. However, as wemove away from the image plane the role of thisauxiliary dimension is to average the generatingcurrents over an increasingly large area of theoriginal image plane. In other words, the role ofthe auxiliary z-dimension is to allow us to performvolume integration of the edge-tangent vectors. By

sampling the vector-potential for various x±yplanes at increasing sampling height z above theimage plane, we induce a scale-space representa-tion. We exploit this property to produce a ®ne-to-coarse image representation as we sample thevector-potential at increasing sampling heightsabove the physical image plane.

In order to develop the appropriate di�erentialoperators for feature characterisation from thevector-potential we have taken the magneto-staticanalogy one step further and have appealed to thegeometry of the associated magnetic ®eld. Ac-cording to magneto-statics, the magnetic ®eld isthe curl of the vector-potential. It is important tostress that because it is less computationallytractable than the vector-potential, the magnetic®eld is never used directly in our image represen-tation. The role of the magnetic ®eld is to providean auxiliary representation. The geometry of themagnetic ®eld allows us to understand the di�er-ential structure of the vector-potential. Accordingto our representation of the image structure,symmetry lines follow the local minima of thevector-potential. In other words, they connectimage points where there is strong cancellationedge tangent vectors associated with symmetricallyplaced object boundaries. By contrast, edge-contours follow the local maxima of the vector-potential. According to our representation, theedge-lines connect points where there is strongdirectional re-enforcement between edge-tangentvectors. Symmetry lines can be interpreted aslocations where the magnetic ®eld is perpendicularto the sampling image plane. Edges are locationswhere ®eld lines are tangential to the relevantsampling plane. When viewed from the perspectiveof the di�erential structure of the vector-potential,symmetry lines are locations where the componentof the curl in the image plane vanishes, i.e.z ^ r ^A�x; y; z� � 0, edges are locations wherethe transverse component of the divergence van-ishes, i.e. r � �z ^A�x; y; z�� 0.

Corners, or points of locally maximum boun-dary curvature, can be viewed as edge-locationswhere there is a local symmetry axis associatedwith a rapid change in boundary direction. Whenviewed from the perspective of our image repre-sentation, corners therefore correspond to


locations where both the edge and symmetryconditions are simultaneously satis®ed. From atopographic viewpoint, corners are located whereboundary lines and symmetry lines meet. In otherwords, we are interested in locating points wherethere is a local maximum of the magnitude of thevector-potential in one direction and a local min-imum in the orthogonal direction. As a result,corner detection can be treated as saddle-pointdetection.

3. Topographic representation

In Section 2, we established that corners aresaddle-points in the magnitude of the vector-po-tential. We therefore focus on the analysis of thescalar quantity

g�x; y�z � jA�x; y; z�j: �8�The topographic structure of the vector-potentialcan be characterised using the Hessian matrix

Hg � gxx gxy

gxy gyy

� �; �9�

where the second derivatives are given by

gxx � o2g�x; y�zo2x

;

gxy � o2g�x; y�zoxo2y

;

gyy � o2g�x; y�zo2y

:

The eigen-structure of the Hessian matrix canbe used to gauge the curvature of the surface. Thetwo eigen-values of H are the maximum andminimum curvatures. The orthogonal eigen-vec-tors of H are known as the principal curvaturedirections. The mean-curvature (K) of the surfaceis found by averaging the maximum and minimumcurvatures. Finally, the Gaussian curvature (H) isequal to the product of the two eigen-values. As aresult,

H � gxxgyy ÿ g2xy �10�

and

K � gxx � gyy

2: �11�

The signs and zeros of the mean and Gaussiancurvatures can be used to categorise the localsurface geometry into a number of distinct topo-graphic classes. These classes are summarised inTable 1. In this paper, we are interested in thesaddle-structures which are labelled as hyperbolicfeatures in the table. These features are charac-terised by the condition H < 0. In particular, weare interested in points that are consistent withbeing the intersections of edge and symmetry lines,i.e. in the intersections of saddle-ridges and sad-dle-valleys. The joint condition for the intersec-tions is

K 6� 0 ^ H < 0: �12�

By searching for the intersection of consistentsaddle-ridges, we overcome some of the problemsof localising saddle-points, for which K � 0 andH < 0. This can prove di�cult since there are noconstraints from the directionality of the desiredfeature. In particular, we mitigate this di�cultyand realise the corner localisation process usingtemplates to search for the junctions betweensymmetry lines and edge-lines.

4. Implementation

In this section, we describe two aspects of theimplementation of our corner detector. The ®rst ofthese is the means by which we compute the vec-tor-potential and the resulting computationalcomplexity of the method. The computation isrealised using fast Fourier transforms. The second

Table 1

Curvature classes

Class Symbol K H Region-type

Dome D ÿ + Elliptic

Ridge R ÿ 0 Parabolic

Saddle-ridge SR ÿ ÿ Hyperbolic

Plane P 0 0 Hyperbolic

Saddle-point S 0 ÿ Hyperbolic

Cup C + + Elliptic

Valley V + 0 Parabolic

Saddle-valley SV + ÿ Hyperbolic


implementational detail concerns the practicalmeans by which we locate saddle-structures.

4.1. Computing the vector-potential

Key to our implementation is the fact that thevolume integrals appearing in the de®nition of the

vector-potential (Eq. (7)) can be replaced by spa-tial-convolutions with a sampling height-depen-dent ®lter. Speci®cally, we invoke the Fourierduality between convolution in the spatial domainand multiplication in the frequency domain. In thisway, the discretised version of the vector-potentialcan be computed using just three 2D Fourier

Fig. 1. Topographic representation and corner detection.


transforms and a pair of frequency-domain con-volutions.

Our basic goal is to compute the vector-poten-tial at a given sampling height above the imageplane. The 2D integrals appearing in the de®nitionof the vector-potential can be discretised to givethe following x and y components:

Ax�x; y; z� � ÿX

x0

Xy0

oGr � I�x0; y 0�oy0

� 1��xÿ x0�2 � �y ÿ y0�2 � z2

q ; �13�

Fig. 2. Topographic representation at di�erent sampling heights.


Ay�x; y; z� �X

x0

Xy0

oGr � I�x0; y 0�ox0

� 1��xÿ x0�2 � �y ÿ y0�2 � z2

q : �14�

The double summation can be replaced by aconvolution with a composite ®lter. For instance,the x-component of the vector-potential is asfollows:

Ax�x; y; z� � ÿ�Vx�r; z� � I��x; y�: �15�

Fig. 3. Topographic representation at di�erent sampling heights.


We exploit the commutative properties of convo-lution to compute the composite ®lter Vx�r; z�. The®lter is found by convolving the appropriate di-rectional derivative of the Gaussian with the in-verse Euclidean distance operator, i.e.,

Vx�r; z��x; y�

�X

x0

Xy0

oGr�x0; y0��oy0

� 1��xÿ x0�2 � �y ÿ y0�2 � z2

q : �16�

For practical reasons, we would like to realisethe computation of the components of the vector-potential using fast Fourier transforms. Our basicstrategy is to exploit the Fourier duality betweenconvolution in the spatial-domain and multiplica-tion in the frequency domain. Schematically, weutilise the identity

Ax �Fÿ1 F�Vx�r; z�� F�I �� 17�to compute each of the components of the vector-potential in turn. In this way the vector-potentialcan be obtained using three separate Fouriertransform operations. The ®rst of these involvescomputing the Fourier transform of the raw imageF�I �. Two separate weighted spatial frequencydistributions are then constructed by multiplyingthe components of the image Fourier transformwith the Fourier representation for each of the

composite ®lters in turn, i.e. we computeF�Ax�r; z�� and F�Ay�r; z��. Finally, the two spa-tial components of the vector-potential are ob-tained by inverse Fourier transformation of thetwo weighted frequency distributions.

Since the computation is implemented usingfast Fourier transforms, the time complexity is ofthe order �N ln�N��2 where N is the linear imagedimension. In fact, we have implemented the al-gorithm on an SGI Indy workstation where it iscapable of processing 256 ´ 256 pixel images at therate of 25 frames per second.

4.2. Localising saddles

Based on the results presented in Section 3, wemake the following observations concerning thetopographic structure of the vector-potential in theproximity of corners:· There is a local minima of the magnitude of the

vector-potential in the direction of the vector-potential.

· There is a local maxima of the magnitude of thevector-potential in the orthogonal direction.

· At the locations of corners, the magnitude ofvector-potential along both the contour andits orthogonal direction changes rapidly.

· At the locations of corners, the magnitude ofthe Gaussian curvature is signi®cant.

Based on the ®rst two observations, we search forsaddle-points that are consistent when viewed

Fig. 4. Corner detection results.


from a ®nite support neighbourhood. In practicewe localise consistent topographic structure usinga simpli®ed form of template convolution. Ourtemplate tests for orthogonal maxima and minima

using directional second derivatives and subse-quent non-maximum suppression and non-mini-mum suppression tests. The saddle-points arecorner candidates. Because of image noise and

Fig. 5. Corner detection at di�erent sampling heights.


other imperfections, the points detected by oursaddle-template are not always the locations oftrue corners in the image. To overcome thisproblem we can appeal to the directional consis-tency of the derivatives of the vector-potential tore®ne the corner estimates. The aim is to search fororthogonal ridge and ravine structures. To meetthis goal, we used directional second derivativeoperators to compute a corner ``strength'' mea-sure. This measure captures the directional varia-tions of the vector-potential along the contourdirection and in the orthogonal direction.

To be more formal, suppose that V �x; y; z� �r2jjjA~j is the second derivative of the magnitude of

the vector-potential in the direction of the vector-potential. The magnitude of this quantity will takeon a maximum value when there is a local mini-mum or valley structure in the magnitude of thevector-potential. Further suppose that R�x; y; z� �r2?jA~j is the second derivative of the magnitude of

the vector-potential in the direction perpendicularto the local orientation of the vector-®eld. Themagnitude of this quantity exhibits a local maximawhen there is a ridge structure or local maxima inthe vector-potential. Using these two operators,we search for corners by computing the followingcorner strength measure which gauges consistentsaddle-structure:

C � jV �x; y; z�j � jR�x; y; z�j: �18�The measure is an approximation to the Gaussiancurvature. It is large in value when there are or-thogonal ridges and valleys in the magnitude ofthe vector-potential. Corners are selected by

thresholding this aggregate measure of cornerstrength.

5. Experiments

In this section, we provide some experimentalevaluation of the corner detection algorithm. Theexperimental work is divided into two parts. Wecommence with some examples on binary imageryto illustrate some of the properties of the repre-sentation. Next we furnish real-world examples.To illustrate the properties of our vector-potentialrepresentation and corner detection algorithm, weuse a simple binary image of ``E''. Fig. 1(a) is themagnitude of the vector-potential for the binaryimage, Fig. 1(b) is the direction of the vector-potential. Fig. 1(c) shows the detected corners.For this simple image, the results are all correct.The magnitude of the vector-potential is displayedas a function of the sampling height z in Fig. 1(a)to emphasise the topographic structure. Here thesaddle-structure associated with the corners isclear. The ridge and ravine structure of the edgeand symmetry lines is also evident. In Fig. 1(b) wedisplay the vectorial representation of A�x; y; 0�.The main feature to note from this ®gure is thatthe direction of the vector-potential changesrapidly at the corner locations.

As explained earlier, we can endow our imagerepresentation with a scale-space dimension bysampling the vector-potential at increasing sam-pling heights above the image plane. In Figs. 2 and3, we provide some qualitative examples of this

Fig. 6. Cog-wheel test images.


scale-space sampling. In each case the left-hand®gure is an elevation map showing the magnitudeof the vector-potential while the right-hand ®gureis the vector-®eld. The main feature to note fromthese examples is that as the scale or samplingheight is increased, so the saddle-structures be-come shallower.

We now turn our attention to real-world scenes.To provide some comparison, we have providedsome experimentation with the SUSAN cornerdetector (Smith and Brady, 1997). Fig. 4(a) is theoriginal INRIA o�ce image. In Fig. 4(b) we showthe result of applying the algorithm reported inthis paper, while Fig. 4(c) shows the result of ap-

Fig. 8. Saddle-structures for di�erent opening angles.

Fig. 7. Comparison for di�erent sampling heights.


plying the SUSAN corner detector. The resultsobtained with our algorithm are generally cleaner,and there are fewer false positives. There are alsosome interesting qualitative di�erences in the de-tected corners. For instance in our algorithm, themeeting of the line-like horizontal bars and thickervertical bars of the window are detected as singlejunctions. In the case of SUSAN, double cornersare returned. The result of our algorithm is moreperceptually intuitive and may prove more usefulfor higher level matching problems.

Finally, we provide some examples of the scale-space detection of corners in the INRIA o�cescene. Fig. 5 shows the results of corner detectionat a number of di�erent sampling heights. The leftcolumn of the ®gure shows the magnitude of thevector-potential, while the right-column shows thedetected corners superimposed on the originalimage. As we move from the top row of the ®gureto the bottom row, the sampling height z of thevector-potential increases. As the sampling heightincreases, then so only the dominant corners re-main. However, the majority of the signi®cantcorners persist over the full set of samplingheights.

6. Performance analysis

Our ®nal piece of experimental work is aimed atmeasuring the noise sensitivity of our corner de-tector and comparing it with some of the alterna-tive corner detectors reviewed in the introductorysection of this paper. The speci®c algorithms usedin this comparison are the SUSAN corner detector(Smith and Brady, 1997), Wang and Brady's cor-ner detector (Wang and Brady, 1995) and thePlessey corner detector (Harris and Stephens,1988). To realise this comparison, we have gener-ated synthetic images of cog-wheels (see Fig. 6)and have added salt-and-pepper noise with knownproportion. By increasing the number of spikes onthe circumference of the cog, we can systematicallyreduce the opening angle of the corners. The syn-thetic ®gure provides ground-truth data in whichthe number of target corners is known. We focuson two aspects of the noise sensitivity of our cor-ner detector. The ®rst of these is the scale-depen-

dance of the corner detection process. The secondis the error rate for false positives and false nega-tives.

We commence our evaluation by measuring theaccuracy of the corner detection process as afunction of sampling height (i.e. spatial scale) andcorner opening angle. Fig. 7 shows the fraction ofcorrectly detected corners as a function of sam-pling height z. The di�erent curves are for di�erentopening angles. The main conclusion that can bedrawn from this plot is that our corner detectordegrades with increasing sampling height. There isalso an indication that we encounter di�cultieswith small opening angle corners.

Fig. 9. Comparison of noise sensitivity.


In order to understand in a qualitative way theangle systematics involved in corner detection, wehave generated a series of synthetic examples. Theresulting plots of the vector-potential magnitude

are shown in Fig. 8. As the angle increases, so thedepth of the saddle decreases. At small openingangles, the width of the saddle becomes very nar-row and hence di�cult to localise.

Fig. 10. Noise sensitivity for various opening angles.


The second aspect of our sensitivity study is toprovide some comparison with the alternativecorner detectors under conditions of controllednoise. The plots in Fig. 9 show the probability ofboth true positives, i.e. the fraction of genuinecorners that are correctly detected, and the prob-ability of false positives, i.e. the fraction of de-tected corners which are false alarms. Theprobabilities of true positives and false positivesare plotted as a function of the probability orfraction of added noise. In each case, the solidcurve is the result of the proposed algorithm, thelong dashed curve is for the SUSAN corner de-tector, the dashed curve is for Wang and Brady'scorner detector and the dotted line is for thePlessey corner detector. From the plot of Fig. 9(a),we can draw the conclusion that the proposed al-gorithm consistently outperforms the alternativesin the sense that it has a higher probability of truepositives for small portions of added noise. Whenmore noise is added to the image, the performanceof our method is almost identical with that of theWang and Brady, and the Plessey corner detectors.Our method always outperforms SUSAN in termsof its probability of true positives.

The results shown in Fig. 9(b) provide moreconclusive support for our proposed corner de-tector. Here we see that the probability of falsepositives for our proposed method is consistentlysigni®cantly lower than that for the alternativecorner detectors. The only exception occurs at lownoise levels where the Plessey corner detector o�ersslightly better performance.

Next, we aim to compare the performance ofthe corner detectors for di�erent opening anglesand di�erent sampling heights. We choose a smallopening angle of 30°, a medium opening angle of90° and a large opening angle of 120°. FromFig. 10, we draw the conclusion that, whenmeasured in terms of the fraction of true posi-tives, our method performs a little worse thanthat of the other corner detectors for smallopening angles, a little better than the others atmedium opening angles, and signi®cantly betterthan the alternatives at large opening angles.When gauged in terms of the fraction of falsepositives, our method is better than the alterna-tives at both small opening angles and large

opening angles. However, it performs a littleworse than the Plessey corner detector at mediumopening angle.

7. Conclusions

In this paper, we have presented a corner de-tection method that is based on the topographicanalysis of a vector-potential image representa-tion. According to our representation, corners aresaddle-points where saddle-ridges and saddle-val-leys intersect. Experimental results reveal themethod o�ers performance advantages over theSUSAN corner detector, Wang and Brady's cor-ner detector and the Plessey corner detector. Themost striking of these is to o�er better control overfalse positives.

There are a number of ways in which the ideaspresented in this paper could be developed. Weclearly have a means of computing a curvaturescale-space. In this respect, our work is similar tothat of Asada and Brady (1986) and Mokhtarianand Mackworth (1992). Our next step is to emu-late this work by investigating how the new cornerrepresentation can be used for shape matching andrecognition.

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