THE PERFORMANCE OF THE GENERALISED HOUGH TRANSFORM:

CONCAVITIES, AMBIGUITIES AND POSITIONAL ACCURACY

E R DaviesMachine Vision Group, Department of Physics

Royal Hollonay & Bedford New CollegeEgham Hill, Egham, Surrey TH20.0EX, UK

ABSTRACT

This paper studies the use of the generalisedHough transform to locate objects possessingconcavities. Hhen locating objects with internalsymmetries, ambiguities can arise which nay not beresolvable in the presence of occlusions. On theother hand concavities can help to lake objectlocation more accurate. The boundary orientationdistribution can be used to analyse the situation:it shows that enhanced accuracy of location alongone axis may result in reduced accuracy in aperpendicular direction. Finally, accuracy oflocation is independent of the position of theobject localisation point.

This paper starts with the observation that aproblem arises when the GHT is used to detectobjects possessing concavities. In principle thiscan lead to ambiguities in the location of suchobjects, and it was felt necessary to investigatethe problem closely in order to find how its effectscould be minimised.

In section II we briefly describe the operationof the GHT. Then in section III we consider theproblem posed by concavities, following this insection IV by a discussion of the effects ofsymmetries. In sections V and VI we investigate howthe accuracy with which an object can be located isaffected by its shape and by errors in theestimation of edge orientation.

I INTRODUCTION

Although the Hough transform was originallydevised for the detection of straight lines as longago as 1962 (Hough 1962), it only came into wide usein the image processing community after it was're-discovered' by Rosenfeld in 1969 and furtherdeveloped by Duda and Hart in 1972. Subsequent workapplied the technique to the detection of circles,at the same time making it more efficient by showinghow locally available edge orientation informationcould be used to cut down the number of votesaccumulated in parameter space (ICimme et al 1975).Later, the technique was applied to other specifictypes of curve such as ellipses and parabolas beforeBallard finally generalised it so that it could beapplied to the detection of arbitrary shapes(Ballard 1981). The resulting 'generalised Houghtransform' (GHT) retained the facility for makinguse of local edge orientation information, and isthus a highly efficient procedure.

More recently, Davies has found that oneadvantage of using the GHT rather than the basicHough transform to detect straight edges is thatthis enables objects such as squares and rectanglesto be detected directly - i.e. without furtherhigh-level processing to deduce the presence of theshape (Davies 1986). He has also shown how toensure that optimal sensitivity is attained whileusing the technique to detect objects possessingstraight edges (Davies 1987b). Finally, he hasdemonstrated that the GHT is exceptionally robustand is therefore particularly suitable forindustrial (e.g. automated inspection) applications(Davies 1984).

The author is grateful to the UK Science andEngineering Research Council for financial supportduring the course of this research.

II THE GENERALISED HOUGH TRANSFORM

The action of the GHT may be summarised asfollows. First, a Sobel or similar edge enhancementoperator is applied to the image, and the resultingintensity gradient image is thresholded to find thelocations of the most significant edge pixels. Thelocation and orientation of each edge pixel are thenused to estimate the position of a localisationpoint L within every object of a specific typesupposed to be present in the original image. Allsuch estimated locations of L are accumulated in aparameter space which is, for the GHT, congruent toimage space. Finally, peaks are sought in parameterspace which indicate the presence and position ofobjects of the chosen shape. For objects of othershapes, the points accumulated in parameter spaceare essentially incoherent and do not focus onpeaks: such objects effectively contribute noise inparameter space, and are unlikely to interfere withthe process of locating the chosen type of object(Ballard 1981).

The precise way in which the position of L isestimated after locating an edge pixel can in simplecases be analytic. For example, in the case of acircle, L is found merely by moving a distance equalto R along the edge normal. For complex curves, ifthe edge normal orientation is 6, we move a distanceR(6) in a direction <?(9) from the edge location,the values of R and <fi being obtained from a lookuptable.

As outlined so far, the orientation of the curvehas to be known in advance. However, if objectorientation is able to vary, we merely adopt thestrategy of augmenting parameter space in anotherdimension, each plane in parameter space then beingused to detect the object in one of its possibleorientations. In what follows we simplify thediscussion by assuming the orientation of the objectis known. Finally, because of possible size andshape variations, it will often be preferable tosave computation by using the GHT to look for

327AVC 1987 doi:10.5244/C.1.44

features rather than for whole objects: suitablefeatures include holes, corners, lines, circulararcs and so on. Again, we ignore such complicationsin Hhat follows.

Ill THE EFFECTS OF CONCAVITIES ON OBJECTDETECTION BY THE GHT

Before proceeding to study the effects ofconcavities on object detection, it is relevant tooutline the problems that arise when objects possessstraight edges. Basically, for a simple convexobject, each edge pixel gives rise to a single votein parameter space. As described in section II,this vote will be placed at the estimated positionof the object localisation point L. However, whenan object possesses a straight edge, this strategyseverely limits the number of votes accumulated at Lby the pixels on a particular straight edge -typically only one or two votes being accumulated:hence sensitivity is markedly reduced. Daviesshowed that this problem could only be eliminated byaccumulating a line of points in parameter space foreach straight edge pixel, the appropriate length ofthe line being equal to the length of the straightportion of the boundary in the object beingdetected, and the direction being that of thestraight portion of the object boundary (Davies1987b). At a particular edge pixel, this directionis estimated by taking the local edge orientation.

The rationale for this parametrisation of thestraight line is as follows. For a given edge pixelwhose direction indicates it is on the straightportion of the object boundary, we accumulate a voteat all possible positions of the localisation pointwhich would confirm this as being a valid straightedge pixel. Though this rationale might seem cleverbut somewhat ad hoc, Davies was able to confirm itsvalue on the basis that the GHT has to approximate aspatial matched filter if sensitivity is to beoptimised (Davies 1967b).

The situation when concavities arise is closelyrelated to that for straight edges. In particular,when a single concavity is present, each of the edgepixels within the concavity has the same orientation

as one other edge pixel on the object boundary.(But note that certain concavities can act as doubleconcavities in this respect - see Figure 1.) Thismeans that, for a particular range of edge orient-ations, it is not known locally at the edge pixelwhether it is on that portion of the boundary withinthe concavity or elsewhere. Thus there is anambiguity in our local knowledge of the position ofthe edge pixel relative to the localisation point -just as there is a multiple ambiguity in the case ofpixels on a straight edge - and there is no choicebut to accumulate two votes in parameter space forsuch edge pixels. One could argue that the case ofa concavity is more extreme than that of a straightedge, but in fact the straight edge gives rise to amultiple ambiguity which involves accumulating agreat many points in parameter space for each edgepixel, and this is computationally more serious.

For more complex types of concavity - multipleholes, sawtooth edges and so on - the localambiguity increases, and with it the number of votesthat have to be accumulated per edge pixel, for atleast a proportion of local edge orientations.Indeed, for spirals of many sorts and all cases ofholes, at least two votes have to be accumulated peredge pixel for all possible edge orientations.

Though a potential problem has arisen withconcavities, we see that the rationale that enabledus to cope with straight edges also permits us tocope with concavities, and at the same time toattain the maximum sensitivity. It is clear thatsensitivity is maximised since every edge pixelcontributes an equal amount to the peak, at L, inparameter space: thus all the available 'signal' isutilised.

IV SYmETRIES THAT ARISE FROM CONCAVITIES

Though we apparently solved the concavityproblem successfully in the previous section, thiswas achieved by accumulating additional points inparameter space. Such points lead to additional'clutter' in parameter space which could conceivablyfocus into unwanted peaks. This means that it isnecessary to examine whether the additional pointsthat are accumulated could make phantom objectsappear. Unfortunately, there are situations whenthis can happen. In particular, if any objectpossesses symmetries between separate parts of itsshape, then spurious peaks can appear in parameterspace. Consider the v-shaped object shown inFigure 2. There is a 2-fold translational symmetry

Figure 1 A shape with a structured concavity

This diagram shows a concavity which must be classedas a 'double' concavity: thus there are two pointsin the concavity with the same orientation as pointP on the convex part of the boundary.

Figure 2 A v-shaped object and its transform

(a) shows a v-shaped object, and (b) shows itstransform. The main peak appears at the localis-ation point L, and the subsidiary peaks appear atequal distances D either side of it: D is equal tothe distance between the two prongs of the v.

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between the two prongs of the v, and hence there isa local ambiguity for a particular set of pixelsabout their location within the object. In thiscase it is not the pixels within the concavity thatpose the problem, but those on the two convexregions related by symmetry. This nay at firstappear paradoxical, but clearly such a situation canonly arise when a concavity is present - it is anindirect effect of a concavity that convex regionsrelated by symmetry can arise. On applying ournormal rule for accumulating points in parameterspace, we find that there are (even in this simplecase of 2-fold symmetry) three main peaks inparameter space. The largest is that correspondingto L, at which all points on the boundary of theobject have contributed: the other two have equalsize corresponding to the much smaller numbers ofedge pixels that have contributed to them(Figure 2). A variation on this situation is shownin Figure 3 where the symmetry may be described interms of two translation vectors in two dimensions.

In such cases, the interpretation strategy thatmust be applied is to look first for the largestpeaks in parameter space, and then to check whetherother peaks appear that could possibly have arisenfrom such symmetries. Since these subsidiary peakswill be at a predictable distance from the mainpeak, they in fact provide useful additionalevidence that the given type of object is present.

It is interesting to consider how detection ofsuch an object will change if it becomes partiallyoccluded. The main possibilities are: (1) the mainpeak will diminish in size; (2) either or both ofthe subsidiary peaks will dissappear. If only oneof the prongs of the v-shape is visible in theoriginal image, then only one subsidiary peak willbe found in parameter space. Now the GHT is knownto be highly robust and to have excellent capabilityfor detecting objects that are even quite grosslyoccluded. In this case, to interpret an image inwhich only one prong of the v-shape is visible, wemust know that it is possible for two small roughlyequal peaks to appear in parameter space, either ofwhich may correspond to L. If one of these islarger than the other, it is most likely that itwill correspond to L, though some ambiguity willremain. In such cases symmetry has permitted thesuppression of information that could have been usedto locate the object unambiguously. This phenomenonwill clearly generalise to more complex objectshapes, where higher order symmetries can bepresent.

Although symmetry would appear generally to be adisadvantage in locating an object (especially insituations where occlusions can occur), this is notactually so. Consider for example the followingcase where a periodic structure is being sought(Figure 4). In that case available edge locationsprovide a barrage of peaks in parameter space, andhence a relatively large amount of ambiguity in theevent of occlusions. However, ignoring possibleocclusions, there is also an enormous amount ofinformation on the longitudinal location of theobject. In fact there is much more information onlongitudinal location than there would be for anobject of equal size containing no concavities.Furthermore, there is much greater accuracy oflongitudinal location than for lateral location,again because of the presence of concavities. Henote also that the additional accuracy is not justdue to the increase the number of edge pixels forthat size of object, but arises partly from thechanged distribution of edge orientations within theshape.

HOW ACCURACY OF OBJECT LOCATION IS AFFECTEDBY SHAPE

In this section we model the situation describedabove by examining the distribution of objectboundary orientations. For a circle this will beuniform. It will also be uniform for an object suchas a washer which contains a hole and for which thisconcavity has not changed the capability formeasurement of object location - supposing that wenormalise to the situation for objects of constanttotal boundary length. (The reason it is necessary

Figure 3 An object containing holes and itstransform

In this case the original object (a) is a bracketcontaining holes, and the symmetry is describable interms of two translation vectors in two dimensions,(b) shows the transform.

L

Figure 4 A periodic structure and its transform

(a) shows a periodic comb-like structure, whichgives rise to a barrage of peaks in parameterspace (b). The periodicity provides an enormousamount of information on the longitudinal locationof the object, though there is a risk that thetransform will be misinterpreted if gross occlusionoccurs.For a linear n-fold symmetry, 2n-l peaks appear inparameter space; their magnitudes are proportionalto the relevant binomial coefficients, f**"1)(i=0,l,...,2n-2), but (as here) the central peak atL normally has increased size because of thenon-periodic parts of the shape.

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to normalise our results in this particular way isthat on multiplying object linear dimensions by M,we are effectively multiplying the number ofindependent Measurements of x and y object centrelocation by M, so accuracy is increased, or errorsare reduced relatively, by a factor VM.) For amore instructive example, take the E-shape ofFigure 5. In this case the distribution of objectboundary orientations 6 is f(6), which is shown inFigure 6. Taking N K as the number of verticalboundary points giving information on the xcoordinate, and N y as the number of horizontalboundary points giving information on the ycoordinate, we find that N ~ 2NK, which means thatthe y coordinate can be found ~1.4 times moreaccurately than the x coordinate. Finally, takingthe comb shape shown in Figure 4 we obtain thedistribution of Figure 7: this has N, I O N , and sothe x coordinate can be measured ~3 times moreaccurately than the y coordinate.

To summarise, we note that object locationaccuracy depends strongly on object boundary length,and that a preponderance of one or another orient-ation in the distribution of boundary orientationsmeans that object location will be defined moreaccurately along one axis than another; in addition,increased accuracy of location parallel to thispreferred axis means reduced accuracy of locationalong a perpendicular axis, for objects of givenboundary length.

It will be clear from the above discussion thatwe can obtain further information on the accuracy ofobject location by detailed analysis of the boundaryorientation distribution f(6). Note first that therequired information has periodicity n, since anedge at a particular angle of gives the samepositional information as one which is inverted - orwhich has orientation <x+ ». If we expand f(6) as afourier series,

f(8) = ae/2 + a(cos 6 + a^cos 29 + ...

+ b,sin 9 + b^sin 29 + ... (1)

where

Figure 5 An E-shaped object and its transform

Extended contributions to the transform (b) fromeach of the straight sides of the original shape (a)are suppressed in this diagram in order to clarifythe effects of symmetry.

(1/ir)

(1/*)

cf(8> cos n6 de

f(6) sin n6 d9

(2)

(3)

then the above discussion, and the requirement thatwe should match f(9) to a bivariate normal errordistribution at L, shows that we are only interestedin the variations produced by the terms in cos 29,sin 29. In particular, the term a./2 gives novariation; the terms in cos 9, sin 6 give avariation with the wrong periodicity; and the higherorder terms give a variation which is faster thanrequired for the present purpose. Thus a good modelof the positional accuracy problem can be obtainedby examining how the terms in cos 29, sin 28modulate the constant term. Particularly importantare the amplitude of the variation and the axisalong which the variation is largest. By writingthe relevant terms in the form

g(8) = ao/2 + cacos(28 - *)

we find the amplitude as

the high accuracy axis being given by

V = arctan(b4/aa)

Carrying out this computation for a shape inwhich a proportion p of the boundary has (edgenormal) orientation close to ei or <* + •» and aproportion q = 1 - p has orientation «•+ */2 or* + 3ir/2, we find:

(4)

(5)

(6)

a. = (1/ir)I f(9) d8

= k(p + q) = k

= (1/ir) I f(9) cos 29 d9-it

= k(p - q)cos 2of

= (1/*) f f(8) sin 28 d8= k(p - q)sin 2 of

Hence

ca = k(p - q)

and * = 2<X

(7)

(8)

(9)

(10)

(11)

If the object has perimeter N pixels, the effectivenumber of pixels providing information about itslocation parallel to the u-axis (at an angle <x tothe x-axis) is N u - pN and the number providing

f(6)

n 2-rc

e

Figure 6 Distribution of boundary orientations forthe E-shaped object

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information about its location along a perpendicularaxis is N v = qN. From equations 7-9 He now deduce

P = <ao + ca)/(2ao)

q = <ac - ca)/(2a.)

(12)

(13)

He then get the following values for the effectivenumbers of pixels:

N(ao + ca)/(2ao)

N(a. - cx)/(2ao)

(14)

(15)

These formulae clearly give the expected resultswhen ca= 0 (q = p) and when cx= ao (q = 0), and arevalid for any object (such as a line, square,rectangle, box or comb) which has all its edgesorientated along or perpendicular to a given axis.However, the model presented here indicates thatthese formulae are more generally valid for objectsof all shapes and orientations. Applying theseideas to some of the features that are commonly usedfor complex object location and recognition tasks,we find that small holes and corners both have avery small coefficient ca and hence are guaranteedto provide reasonable accuracy in all directions.

The fact that we can draw ©-distributions inthis way, and deduce measurement accuraciesindependently of the method of detection, indicatesthat we have found a fundamental and useful shapedescriptor. It also means that our results willhave more value than they would have if they wereonly relevant to some particular specialisedtechnique. On the other hand, the philosophy wasobtained from consideration of the GHT, implyingperhaps that it is capable of achieving results thatare limited more by fundamental properties of shapethan by its own characteristics. In this respect wenote that the GHT is not an arbitrary technique butis derived from, and closely related to spatialmatched filtering, which in turn is known to giveoptimal signal-to-noise ratio in locating objects(Davies 1987a); in addition, accuracy will bestrongly dependent on the possibility of discerningobject positions when the reasons for error arefundamentally linked to the presence of noise.

VI THE EFFECTS OF ORIENTATION ERRORS

In spite of what was said in section V about theGHT - namely that its likeness to a spatial matchedfilter makes it highly efficient at suppressingimage noise - there is some doubt about whether itcould in practice realise the object locationaccuracies indicated above. The main reason for

f(e)

this doubt is the fact that every edge pixel is usedto provide an independent estimate of the locationof L, this estimate being limited by the accuracywith which edge orientation can be measured locally.It has frequently been pointed out that an error iQin the estimation of edge orientation will at adistance R lead to a positional error R46 (Ballard1981, Davies 1987a). This clearly results in anoverall lowering of the accuracy of location, thoughintuition suggests that if L is located in aparticular position - e.g. near the centroid of theobject boundary (Davies 1987a) - then the overallerror will be minimised. It should also be pointedout that edges are frequently 2-3 pixels thick, andsome radial error will consequently arise from thissource: however, for objects whose linear dimensionsare more than "40 pixels in size, orientation errorsin the location of L will predominate - if onlybecause common edge detection operators such as theSobel permit orientation to be estimated only within~1°. He shall therefore ignore radial errors andconcentrate on azimuthal errors in L in whatfollows.

To obtain a clearer view of these azimuthalerror3, consider Figure 8. In this case we have acircle, and the localisation point is at anarbitrary position near one end of it. Rather thanestimating the effects of azimuthal errors directlyon L, we here examine the problem from a differentpoint of view. If the estimated orientation at aparticular edge pixel is subject to error SQ, thenthere is a local uncertainty about which part of theobject boundary the edge pixel is on. In fact wecan mark out a section B of the boundary at whichthe edge pixel could be. (For simplicity we hereassume that there is equal probability of the edgepixel being at any of these locations and zero of itbeing elsewhere on the boundary. This simplyifyingassumption helps the calculation, but will not

A2TC

e

Figure 7 Distribution of boundary orientations forthe comb shape

Figure 8 Effect of edge orientation errors on thetransform of a circle

Here the transform of a circle is being calculatedfor localisation point L. Errors in edge orient-ation at edge point E mean that is not certain whichof the points within error arc B it corresponds to.As a result the estimated position of the localis-ation point L may be any of the points on theinverted error arc I. The final error distributionat L is the result of accumulating all possibleinverted arcs of type I.

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affect the generality of our conclusions.) Next, wemake successive assumptions that the edge pixel isat each of these locations and Bark the correspond-ingly deduced positions of L. For a circle, this•eans that L Bust lie on an arc which is an invertedfora of the error arc B. Repeating this for alledge pixels gives an isotropic set of error arcssimilar to that due to B. It is noteworthy that thefact that some edge pixels are closer to L thanothers has had no particular effect.

This is a somewhat remarkable result anddeserves cement. First, if L is taken as thecentre of the circle, the situation revertsnaturally to the usual situation, wherein theaccuracy can be calculated in tens of the azimuthalerror subtended at the centre as a result of theedge orientation errors. Second, we can alwaysassume a best case position for L (here we wouldtake the centre) and then deduce another locationfor L by adding a constant vector to L or by addingit directly to each of the vectors from the edgepixels to the original L. Thus the reason for theerrors being independent of L is that no additionalerror can result frc* adding a constant vector to agiven vector.

However, it is also necessary to determine whythe previous view of the situation was erroneous.Imagine we select a position of L on the circun-ference of the circle. Then it appears thatorientation errors have no affect for nearby edgepixels. But in fact they have a marked effect,since as soon as it is admitted that we do not knowexactly what the local edge orientation is we do notknow which of a number of edge pixels we areconsidering. Thus we ought to accumulate a range ofpossible locations for L in parameter space. Thismeans at least that there is an upper limit on theaccuracy with which an edge pixel can predict theposition of L. On the other hand there could be alower limit to the accuracy, which results from edgeorientation errors adding to the basic inaccuracy atlarge distances. However, we have seen, bycalculating the azimuthal error as resulting fromthe ambiguity in the positions on the circumferenceat which the located edge pixel can be taken to lie,that we obtain a lesser error, which should beregarded as a lower limit on the error of location.

As an example, we consider the situation for anellipse with semi-axes a and b and high eccentric-ity. In this case the various arcs of constantangular error on the boundary vary widely in length.If p is the local radius of curvature, then thelength of the corresponding arc, and of theresulting error arc at L, is pie. This will varyfrom (a*/b)ie to (ba/a)/6 for various positions ofthe edge pixel on the ellipse. The result is thatthe distribution of points in parameter space near Lis a very much flattened roughly elliptic shapewhose ratio of major to minor axes is ~(a/b)*. Asthe original ellipse becomes more elongated theapproximation we have taken eventually breaks down,since (a"/b)rf8 becomes greater than 2a. However, itis clear that in that case the longitudinal errorbecomes equal to 2a - the length of the correspond-ing straight line - while the lateral error becomesso small that it will be dominated by radial errorsdue to the width of the object boundary. Thus themodel we have adopted gives exactly the rightextrapolation to the case of a straight line, asspecified by Davies (1987b). On the other hand acalculation based on working out the arc lengthsubtended at L by azimuthal errors resulting fromthe relevant distance from the edge pixel wouldpredict totally the wrong distribution of errors inL (namely, L would be known more accurately alongthe length of the line than laterally).

A. appraisal of GHT positional accuracy

The ideas of this section have so far shown thatthe position of the localisation point is not arelevant factor in finding the accuracy with whichthe GHT can estimate object location. They havealso shown that edge orientation accuracy isimportant in helping to determine the position of anobject: i.e. the GHT is capable of extracting someedge orientation information to refine the measuredposition of an object. He now need to work out howthese results relate to those of section V.

In fact, section V predicted the relativeaccuracies with which object location could be foundin two perpendicular directions. It did this on thebasis that the edge pixels have a given orientationdistribution f(8). However, we now note that thiscalculation was based on a model such that a givenedge pixel gives a measure of object location normalto the edge, with radial accuracy given by vr, butgives no information on object location in theazimuthal direction. Thus the GHT is well able tomeet this requirement and indeed should enablesomewhat higher accuracies to be achieved. To thisextent the model of section V is inadequate.However, it does permit us to obtain simply alower limit on the available accuracy of objectlocation. For this we assume a basic accuracy of<rr, and modify this in two directions taking accountrespectively of the effective numbers of pixels, Hwand Nv (see section V). Thus the error o> will inthe two directions be divided by factors VN^ and«/N .

VII CONCLUDING BEHARKS

This paper has examined the effects ofconcavities on the performance of the GHT. Perhapssurprisingly, this seemed to lead to problems onlyin cases when specific symmetries arose. Thenadditional peaks in parameter space appeared whichcould lead to potential ambiguities which might notbe resolvable in cases of gross object occlusion.On the other hand the additional peaks could inprinciple be used to help with object location andidentification, and need not be regarded asdisadvantageous. Next, it was found that thedistribution of edge orientation values permittedthe accuracy of object location in differentdirections to be estimated in a straightforwardmanner: this approach appeared fundamental andindependent of the particular method used to locateobjects - thus representing an upper limit on theavailable accuracy of object location assuming thatlocal edge orientation information is Ignored. Itwas shown that the GHT is well able to meet thisaccuracy, and should in fact be able to exceed it iflocal edge orientation information is taken fullyinto account.

The theory developed here shows how the accuracywith which an object may be located depends on(a) the total number of boundary pixels and(b) their relative orientations. It also makesclear that, for a given boundary length, enhancedaccuracy of location parallel to one axis is matchedby reduced accuracy of location along a perpen-dicular axis. In addition, small objects need notbe less accurately locatable than large objects if,because of their more complex shapes, they havesimilar boundary lengths. For example, a smallobject with many holes or with a sawtooth edge willbe very accurately locatable.

In addition, it has been found that the accuracyof location obtainable using the GHT is independentof the position of the object localisation point.

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Finding the limits of object location accuracyis vital not just directly - e.g. in getting robotsto assemble precision parts - but also forinspection purposes, e.g. when the dimensions ofcomplex machinery must be measured as accurately aspossible (within the available resolution) byprecise location of individual components andfeatures. In this respect the paper has been ableto ascertain a number of the inherent measurementlimitations of the GHT and related techniques.

REFERENCES

1 Ballard, D. H. (1981) "Generalising the Houghtransform to detect arbitrary shapes", PatternRecoqn.. 13, no. 2, 111-122

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3 Davies, E. R. (1986) "Reduced parameter spacesfor polygon detection using the generalised Houghtransform", Proc. 8th Int. Conf. on PatternRecoqn.. Paris (28-31 Oct 1986), 495-497

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5 Davies, E. R. (1987b) "A new parametrisation ofthe straight line and its application for theoptimal detection of objects with straightedges". Pattern Recoqn. Lett.. 6, no. 1, 9-14

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8 Kinae, C , Ballard, D. and Sklansky, J. (1975)"Finding circles by an array of accumulators",Comm. Assoc. Comput. Mach.. 18, no. 2, 120-122

9 Rosenfeld, A. (1969) Picture Processing byComputer. Academic Press, New York

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