Robust and efficient automated detection of tooling defects

in polished stone

J.R.J. Lee, M.L. Smith *, L.N. Smith, P.S. Midha

Machine Vision Laboratory, Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England,

Frenchay Campus, Bristol BS16 1QY, UK

Received 1 December 2004; received in revised form 31 March 2005; accepted 31 May 2005

Available online 7 October 2005

Abstract

The automated detection of process-induced defects such as tooling marks is a common and important problem in machine

vision. Such defects are often distinguishable from natural flaws and other features by their geometric form, for example their

circularity or linearity. This paper discusses the automated inspection of polished stone, where process-induced defects present

as circular arcs. This is a particularly demanding circle detection problem due to the large radii and disrupted form of the arcs, the

complex nature of the stone surface, the presence of other natural flaws and the fact that each circle is represented by a relatively

small proportion of its total boundary. Once detected and characterized, data relating to the defects may be used to adaptively

control the polishing process. We discuss the hardware requirements of imaging such a surface and present a novel

implementation of a randomised circle detection algorithm that is able to reliably detect these defects. The algorithm minimizes

the number of iterations required, based on a failure probability specified by the user, thus providing optimum efficiency for a

specified confidence whilst requiring no prior knowledge of the image. The probabilities of spurious results are also analyzed,

and an optimization routine introduced to address the inaccuracies often associated with randomized techniques. Experimental

results demonstrate the validity of this approach.

# 2005 Published by Elsevier B.V.

Keywords: Circle detection; Circle Hough transform; Randomised algorithm; Surface inspection; Polished stone

www.elsevier.com/locate/compind

Computers in Industry 56 (2005) 787–801

1. Introduction

The automated visual detection and classification

of surface defects, such as scratches and tooling

marks, is a common and important problem in the

manufacturing industry. A rudimentary literature

* Corresponding author.

0166-3615/$ – see front matter # 2005 Published by Elsevier B.V.

doi:10.1016/j.compind.2005.05.006

search will reveal examples too numerous to mention.

Many industrial inspection applications demand the

accurate and timely location of circular or linear objects

or defects in the presence of noise, discontinuities and

varying levels of occlusion. The research described

in this paper attempts to respond to a particularly

demanding problem relating to the automated inspec-

tion of polished stone. The ornamental stone industry

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801788

Fig. 1. Arrangement of camera and light source.

is an important and rapidly developing sector. Recent

analyses report a world trade growth rate of around 7%

per annum, with trade in ornamental stone accounting

for more than 2% of gross national product in China,

Italy, Spain, India and South Korea in 2001 [1]. When

processing the raw material, dislodged grit from

rotating tool heads often leads to large, disjointed

circular arcs being imprinted on the surface of the slab,

particularly in softer stones such as marble. A typical

polishing line will incorporate up to 18 stages, with

each stage bearing a rotating grinding/polishing

head with progressively finer grit. As such, the severity

of these circular arcs varies considerably. Finer

scratches will only normally be visible at close range

and when they disrupt a specular reflection highlight

from a light source. However, the fact that the human

eye is naturally drawn to such geometrically definable

features makes their detection imperative. As such,

the operator or in many cases the proprietor of the

processing plant will spend several minutes tilting

the slab under a skylight or a striplight in order to

manually identify any defects. This is an important

process, as the aesthetic quality of the stone will have

a substantial impact on its value and hence its

application.

Automating the detection of such defects using

machine vision is challenging for a number of reasons.

The first task is to distinguish between variations in

albedo, such as the veins and colorations associated

with natural stone products, and variations in surface

topography. This has previously been accomplished

using photometric stereo [2,3]. However, in this case

the highly specular, crystalline nature of the surface

tends to prohibit the immediate application of a

photometric technique. Instead, we replicate the

manual technique used in the industry, by looking

for disruptions to the specular reflection from an

extended light source. A linescan camera is positioned

such that it captures an image of the light source

reflected from the polished stone slab, as shown in

Fig. 1.

The acquired image, generated from concatenated

lines with the aid of an encoder signal as the slab

moves along a conveyor, is then a function of the

glossiness or reflectance value of the slab at a given

incident angle of illumination. This effectively

removes the effects of colour variations on the

intensity of the image whilst enhancing the visibility

of natural fissures, process-induced defects and

variations in finish quality. Fig. 2 shows an image

acquired from a 300 cm2 marble slab, to the right of a

conventional image of the same slab acquired under

diffuse illumination for comparison. It should be noted

that the image contains approximately 3000 � 3000

pixels at full resolution.

It can be seen that the image on the right contains a

great deal of information on the condition of the

surface of the slab. Process-induced defects, i.e.

circular arcs with large radii, are apparent at varying

levels of severity. A poorly finished region, typically

caused when a misaligned grinding tool at an early

stage of processing removes too much material, can be

seen in the top left corner. Various natural fissures,

often associated with the presence of a vein in the

stone, are also apparent. Finally, natural variations in

reflectance due to the metamorphic nature of the stone

are represented by the intensity variation seen

throughout the image.

The focus of this paper is the detection and

quantification of process-induced scratches. As such,

the influence of the latter three artefacts can be

reduced to varying extents using standard edge

detection and filtering techniques, resulting in images

of a similar nature to that shown in Fig. 3. Note that

this relates to a small section of the total slab shown at

full resolution.

It can be seen that the process-induced scratches are

now among the dominant features of the image, and as

such the problem becomes one of circle detection.

This is particularly challenging in this case due to the

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801 789

Fig. 2. Acquired image from marble slab.

extremely large radius (in terms of pixels) of the

circles, the relatively small proportion of the circle

boundary present in each case and the disjointed

nature of the arcs.

The classical algorithm for problems of this kind is

the circle Hough transform (CHT) [4–6]. In Cartesian

coordinates, a circle with centre (x0,y0) and radius r is

Fig. 3. Image of marble slab after pre-processing.

described by

ðx� x0Þ2 þ ðy� y0Þ2 ¼ r2 (1)

As such, any edge pixel (x,y) in the image can be

parameterised as a conic surface in (x0,y0,r) space.

Peak values in the accumulator array occur where

many of these manifolds intersect. The corresponding

indices then provide the parameters of dominant

circular forms in the image. This determination-by-

voting approach has proved extremely popular, largely

due to its strong immunity to noise and occlusion.

However, the laborious process of calculating radii for

every possible combination of centre coordinates, and

for every edge pixel present in the image, makes the

CHT extremely computationally expensive. This,

together with the massive memory requirements asso-

ciated with the three-dimensional accumulator array,

makes the algorithm unacceptable for many time-

critical applications. The situation is further com-

pounded in the case of images of circular features

with very large radii, for which memory and proces-

sing times correspondingly increase. More recently

proposed alternative methods also prove inadequate in

this case. The UpWrite and similar approaches, where

small neighbourhoods of edge pixels are approxi-

mated by straight lines, grouped and analysed for

the best-fit curve, fail in the presence of complex

backgrounds or disjointed arcs [7,8]. The concept of

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801790

polling an image for circles by examining their inter-

section with random lines, proposed by Cheng and Liu

[9], is also unsuitable in applications such as those

described in this paper. In this case there is simply not

a large enough proportion of the total circle present to

ensure a reliable detection.

In this paper we show how a randomised

‘hypothesise and test’ circle detection technique can

be adapted to provide robust and efficient classifica-

tion under difficult conditions. The algorithm adap-

tively minimises the number of iterations required,

based on a failure probability specified by the user,

thus providing optimum efficiency for a specified

confidence whilst requiring no a priori knowledge of

the image. A simple optimisation routine is introduced

to address the inaccuracies often associated with

randomised techniques, and the probability of

spurious results is analysed. Whilst it would not be

possible to cover, within the bounds of this article, the

hundreds of papers published on circle detection and

modifications to the CHT, the major developments are

briefly reported in the following section. The

interested reader is also directed to Leavers [4] and

Kalviainen et al. [10], who provide excellent reviews

of significant developments in the field. The remainder

of this paper will detail our own algorithm and present

a range of experimental results.

2. Background

Many proposed improvements to the CHT have

attempted to utilise geometrical properties of the circle

in order to reduce the computational complexity and

memory requirements of the algorithm whilst retain-

ing the principal desirable property of strong noise

immunity. Perhaps the most intuitive such property

concerns the observation that a normal to the

circumference at any point will intercept the centre

of the circle. Thus, several researchers have attempted

to use the gradient orientation generated by edge

detection algorithms, such as the Sobel operator, to

reduce the space transform to a two-dimensional

(x0,y0) parameterisation [4,6]. Whilst this implemen-

tation works well under ideal conditions, its sensitivity

to noise and discretisation errors makes it unworkable

in practice [11]. More recently, Ioannou et al. [12]

used the fact that a line perpendicularly bisecting any

chord of a circle will pass through its centre, whilst

Kim and Kim [11] calculate centre positions from

pairs of intersecting chords. Both approaches use an

initial (x0,y0) parameterisation followed by filtered

histogramming to validate the circles and to locate the

corresponding radii, thus reducing the memory

requirements and computational complexity from

O(n3) to O(n2 + n), where n is the number of edge

pixels in the image. Foresti et al. [13] use an

alternative parameterisation to improve the efficiency

of the HT, based on two dependent equations of the

first order rather than the classical second order

equation. Dynamic memory allocation is used to

alleviate the problems associated with the use of a

three-dimensional accumulator array. The approach

requires edge gradient direction information, however

the authors assert that an improved peak validation

algorithm overcomes the inherent uncertainty in edge

orientation. Guil and Zapata [14] combine the use of

the gradient vectors associated with points on the

circumference of the circle with a coarse-to-fine

focussing algorithm to reduce computation time.

Huddleston and Jezekiel [15] also use edge orienta-

tion, together with analysis of proximity and circular

symmetry, in their distributed Hough transform

(DHT). The six-stage algorithm is centered around

the property that a signed curvature alone is enough to

uniquely assign an edge pixel to a particular feature,

thus reducing the parameter space from three to one

dimension. Probabilistic analysis is used to progres-

sively refine the curvature calculations as the edge

pixels are grouped into entities. However, as with all

such techniques the algorithm degrades in the

presence of edge orientation noise. Documented

variants of these approaches are too numerous for

comprehensive review in this paper, the interested

reader is again referred to Leavers [4].

The drive for increased efficiency has steered many

researchers away from the one-to-many mapping,

used in the traditional Hough transform techniques,

towards a convergent, or many-to-one, mapping.

Individual edge pixels transform into n-dimensional

manifolds in parameter space. The mapping of

multiple pixels simultaneously can be used to either

reduce uncertainty in the parameter space or to reduce

the dimensionality of the parameter space. Many

diverse implementations of this hypothesis have been

seen over the previous decade. Chutatape and Linfeng

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801 791

[16] combine this approach with the dynamic

allocation of memory and the elimination of feature

points already attributed to a line in order to reduce the

computational burden and storage requirements of the

HT. Olson [17,18] also uses a many-to-one mapping

and proposes an algorithm whereby only those pixel

groupings that share some distinguished set of j edge

pixels are mapped into the parameter space. However,

the major trend in circle detection over recent years

has been towards randomised or probabilistic algo-

rithms. Xu et al. [19] introduced the randomised

Hough transform (RHT) in 1990. This was one of the

first approaches to suggest that mapping n edge pixels,

for an n parameter curve, to a single point in the

parameter space could alleviate many of the problems

associated with traditional Hough transform techni-

ques, such as low speed and the generation of false

positives. Xu’s approach combines this many-to-one

mapping with a sampling algorithm, whereby sets of n

pixels are randomly selected to generate a system of n

equations in n unknowns. Much statistical analysis has

been done towards deriving robust stopping criteria for

the RHT, see for example Xu and Oja [20], Kalviainen

et al. [10] and Shaked et al. [21]. Kiryati et al. [22] also

proposed randomly polling a small subset of the edge

data in their probabilistic Hough transform (PHT),

choosing however to use the conventional one-to-

many mapping into parameter space. Their work

demonstrated that the random polling approach can

dramatically reduce computation time with only a

slight degradation in performance. Kiryati et al. [23]

compared the performance of the RHT and the PHT,

concluding that in their basic forms the RHT is more

efficient for analysis of high quality, low noise edge

images whilst the PHT becomes preferable in the

presence of increasing noise degradation.

Matas et al. [24] present the progressive probabil-

istic Hough transform (PPHT). The algorithm retains

the traditional one-to-many voting process but

implemented in a much more efficient algorithm.

An edge pixel is selected at random from the image,

the accumulator array is updated and the pixel is

removed from the image. If the highest cell value

modified by the pixel exceeds a noise threshold

derived from statistical analysis of random voting, the

feature is registered, all the pixels in the corresponding

feature are removed and all their votes cast are

retracted from the accumulator array. The randomised

circle detector (RCD) algorithm proposed by Chen

and Chung [25] works on a similarly progressive basis

but uses a many-to-one mapping that allows the

requirement for an accumulator array to be removed

altogether. The algorithm first picks four points at

random from the set of edge pixels V in the image and,

subject to a minimum separating distance criterion,

tests whether they are concyclic within an error margin

designed to account for discretisation. Having located

a possible circle, the algorithm determines how many

of the remaining edge pixels lie on the same circle.

This is conceptually similar to ‘hypothesise and test’

algorithms such as RANSAC [26,27]. If the number of

edge pixels found to satisfy the equation of the circle

exceeds a predetermined threshold Tr, scaled to reflect

the circumference of the possible circle, the circle is

accepted and the contributory edge pixels are deleted

from V. The algorithm repeats this procedure until the

number of edge pixels remaining (jVj) falls below a

predetermined threshold Tmin.

The simplicity, computational efficiency and

minimal storage requirements of this approach make

it more attractive than those previously discussed,

particularly as the identification of circular features

with extremely large radii prohibits the use of any

accumulator-based technique. Practical experimenta-

tion, however, identifies a number of issues that

degrade the performance of the algorithm. In a real

image, it is highly unlikely that the level of noise or

non-circular features contributing to Vwill be known a

priori. Thus, given that W edge pixels (where W � V)

do not contribute to any circular features, a manually

and arbitrarily selected Tmin could result in false

negatives (a genuine circular object remaining unde-

tected) if Tmin >W or an infinite loop if Tmin < W.

For example, suppose that we set Tmin to 500 pixels.

The algorithm stops before it detects the light,

diagonal scratch in Fig. 6a. However, when applied

to the noisier edge image shown in Fig. 8a the

algorithm detects the circular feature present but

continues searching indefinitely, or until it is drawn to

a close by an arbitrary failure count threshold. The

desirable case of Tmin = W is extremely unlikely

without prior knowledge of the image.

The use of arbitrary stopping criteria such as these

results in our having little idea of the probability of

circular features being missed, the probability of

spurious circles being detected or the propriety of the

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801792

number of iterations effected. The algorithm also

suffers from poor accuracy under certain conditions,

as the equation determined by four points selected at

random from a particular circular feature may not

represent the best fit through the feature as a whole.

This can lead to both underestimation and over-

estimation of the radii, with corresponding errors in

the location of the centre of the circle.

In this paper we address the limitations of the RCD

to provide an adaptive algorithm that allows the user to

specify the probability of a circular feature being

missed. The number of iterations required to reduce

the probability of failure to the specified level is

continually reevaluated as the set of edge pixels is

reduced. This thereby provides an adaptive, robust

stopping criterion for the algorithm and removes the

need for a maximum remaining pixel threshold Tmin.

When a genuine circle is detected, a refinement

subroutine is used to eliminate the inaccuracies often

present in randomised detection routines. The prob-

ability of spurious results being returned by the

algorithm is also analysed.

3. Algorithm and mathematical background

3.1. Mathematical preliminaries

The equation of the circle passing through the three

points (x1,y1), (x2,y2) and (x3,y3) can be expressed in

matrix form as

x2 þ y2 x y 1

x21 þ y22 x1 y1 1

x22 þ y22 x2 y2 1

x23 þ y23 x3 y3 1

���������

���������¼ 0 (2)

To find the centre coordinates and the radius of the

circle we assign coefficients of a general quadratic:

ax2 þ by2 þ cxyþ dxþ eyþ f ¼ 0 (3)

In the case of a circle, a = b and c = 0. Thus, complet-

ing the square gives:

a

�xþ d

2a

�2

þ a

�yþ e

2a

�2

þ f � d2 þ e2

4a¼ 0

(4)

where

a ¼x1 y1 1

x2 y2 1

x3 y3 1

������������ (5)

d ¼ �x21 þ y21 y1 1

x22 þ y22 y2 1

x23 þ y23 y3 1

�������

�������(6)

e ¼x21 þ y21 x1 1

x22 þ y22 x2 1

x23 þ y23 x3 1

�������

�������(7)

f ¼ �x21 þ y21 x1 y1

x22 þ y22 x2 y2

x23 þ y23 x3 y3

�������

�������(8)

The centre and radius can then be determined as

x0 ¼ � d

2a(9)

y0 ¼ � e

2a(10)

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 þ e2

4a2� f

a

r(11)

Thus, given three points selected at random we can

determine the parameters of the corresponding circle.

Chen and Chung’s algorithm works on the basis that,

whilst three non-collinear points are trivially concyc-

lic, four concyclic points provide a strong indication of

the presence of a genuine circle. Having calculated the

parameters of the circle corresponding to the first three

points, the distance between the fourth point and the

boundary of the circle can be easily determined as

follows:

d4! 123 ¼ jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx4 � x0Þ2 þ ðy4 � y0Þ2

q� rj (12)

3.2. Accuracy issues

There are various cases where the robustness of the

RCD algorithm suffers. Firstly, four contiguous edge

pixels, attributable to either noise or a genuine circular

feature, could be construed as forming a small circle.

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801 793

This case can be easily avoided by removing the scaling

factor associated with the minimum weighting thresh-

old Tr. We contend that it is not necessary to scale the

weighting threshold according to the radius of the

possible circle, particularly as this will discriminate

against extremely large circles where only a small

proportion of the circumference intersects the image.Trshould be set to reflect the smallest absolute circle

weighting we would expect to encounter; the presence

of more heavily weighted circles will not affect the

performance of the algorithm. Chen and Chung [25]

identify the case where two of the three selected edge

pixels are unacceptably close, leading to a spurious

circle intersecting the genuine feature. This is avoided

by imposing a minimum separation criterion, whereby

the randomly selected edge pixels are only accepted if

any two of the three pixels are separated by more than

this minimum distance. In practice, however, this is

simply not an issue. The spurious circle identified is

extremely unlikely to contain enough edge pixels to be

counted as a genuine feature. Furthermore, the

minimum separation criterion increases the computa-

tion required on every random selection of edge pixels

and makes reliable probability estimation impossible.

A far more common and more serious weakness of

randomised techniques occurs when a slight under-

estimation or overestimation of the circle radius leads

to multiple partial circles being detected from a single

genuine feature, as illustrated in Fig. 4.

Thus, we propose an alternative to the minimum

point separation criterion that ensures that any circles

detected are the best possible fit through the edge pixels

that form the corresponding feature. Supposewe detect

Fig. 4. (a) Underestimation of circle and

a circle that satisfies the minimum weighting criterion.

We find that U edge pixels lie within the annulus

defined by the circle parameters and the maximum

discretisation error. From the original set of three pixels

used to derive the circle parameters, we replace the first

pixel with alternatives taken exclusively from the

subset U, reevaluating the vote after each replacement

until the optimal value is located. We then repeat the

procedure for the second and third seed pixels. The

circle parameters associated with the maximum vote

returned by this subroutine will thus correspond to the

’best fit’ through the detected feature. This is a far more

constrained and efficient optimisation than would be

achieved by randomly selecting a series of three-pixel

sets from the subsetU. Note this is only effective due to

the finite width of the scratch segments detected. If the

width of the annulus were to exceed 2–3 pixels this

point-by-point optimisation could fail to resolve the

optimal circle fit. In this case a better alternative would

be to continually select one of the three seed pixels at

random, testing alternatives on a keep or replace basis,

until the algorithm settles on the optimal fit. However

we anticipate that in most applications, as in our own,

the point-by-point optimisation approach will provide

consistently good results at a low computational cost.

Although this results in a slight increase in

computation time there is no significant impact on

performance as the refinement subroutine need only

be executed when a genuine circle, with weighting

exceeding the threshold Tr, is detected. Experimental

results demonstrate that this approach effectively

eliminates the inaccuracies often associated with

randomised algorithms such as the RCD.

(b) overestimation of circle radius.

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801794

3.3. Probability of false negatives

In the following derivation we will generalise the

algorithm such that instead of picking four points at

random, we pick m points at random. Each iteration of

the circle detection algorithm can be regarded as a

Bernoulli trial, with an independent probability of

detecting a circular feature of given prominence.

Suppose that there are jVjedge pixels remaining in the

image and we are searching for circular features of

prominence U, where U � V. It is important here to

distinguish between the probability of any circular

feature being detected and the probability of a

particular circular feature being detected. In this case

we are interested in the latter. The probability that a

particular circular feature will be detected is the

probability that m points, randomly selected from V,

will lie on the feature:

p ¼ U

V� U � 1

V � 1� U � 2

V � 2� � � � � U � m

V � m

¼ U!ðV � mÞ!V !ðU � mÞ! (13)

As such, we can determine the probability of a circular

feature remaining undetected after n iterations of the

algorithm:

pfailure ¼�1�

�U!ðV � mÞ!V !ðU � mÞ!

��n

(14)

As U � Tr, we can continually tailor the number of

iterations required to achieve the desired failure prob-

ability on a ’worst case scenario’ basis (the prob-

ability of a feature being detected reduces to a mini-

mum as U! Tr). Thus we define a failure probability,

pfn, as

pfn ��1�

�Tr!ðV � mÞ!V!ðTr � mÞ!

��n

(15)

In practice the factorial terms make this incalculable

for large values of Tr and V. Thus given that Tr and V

are much larger than m and that for large values there

is little difference between sampling with or without

replacement, the probability of false negatives can be

reliably approximated as follows:

pfn ��1�

�TrV

�m�n

(16)

3.4. Probability of false positives

We can also analyse the probability of false

positives, a major concern in standard CHT techniques

due to random accumulation of votes as a result of the

one-to-many mapping. We cannot determine the

general probability of a false positive being returned,

as the form of the probability distribution will depend

on the circumferences of the individual possible

circles. We can however determine the likelihood that

a circle detected by the algorithm is a false positive. In

this case, the probability of a non-existent circular

feature being detected is a combination of:

(a) T

he probability that Tr or more randomly

distributed edge pixels are mutually concylic.

(b) T

he probability that m pixels from the pseudo-

feature will be simultaneously selected by the

algorithm within n iterations.

The analysis described in Section 3.2 is used by the

algorithm to ensure that the probability of (b) occurring

will be close to one. As such, we need only concern

ourselves with the probability of (a) occurring, and the

derivation proceeds as follows. Let I denote the com-

plete set of image pixels, V the subset of edge pixels

where V � I and Tr the minimum vote threshold as

before. Any three pixels picked at random are said to be

trivially concyclic (except when they are collinear) and

define a circle c123. The probability that a fourth edge

pixel v4 picked at random will lie on the same circle is

Pðv4 2 c123Þ ¼V � 3

I � 3(17)

However, as the number of possible pixels on the

circle boundary increases, the number of possible

outcomes becomes combinatorially explosive as the

probabilities of subsequent pixels contributing to the

circle are dependent on all previous results. As we are

in effect sampling from a finite population, this can be

formulated as a hypergeo metric distribution

H(I,V,2pr). In a population of I pixels, V are edge

pixels and I–Vare background pixels. If a group of 2pr

pixels (where 2pr is the maximum number of pixels

that could lie on the circular feature identified by the

first four points) is selected at random, we wish to

determine the probability that the chosen group will

contain Tr or more edge pixels. In order to do this we

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801 795

initially define a probability pk that the chosen group

will contain exactly k edge pixels and 2pr–k back-

ground pixels. A feature pixel can be chosen in

Vk

� �

different ways, whilst a background pixel can be

chosen in

I�V2pr�k

� �

differentways.Given that any choice of k feature pixels

can be combinedwith any choice of background pixels:

Pk ¼

Vk

� �I�V2pr�k

� �

I2pr

� � (18)

However, the factorial terms in thebinomial coefficients

againmake this incalculable in practice. As the popula-

tion I is very large and the sample size 2pr is large, we

can use a Poisson approximation to the hypergeometric

distribution with minimal error, such that

pk �e�llk

k!(19)

where l = 2Vpr/I. Thus, the probability pfp that a

circular feature has been inferred from randomly

distributed pixels is the sum of all probabilities pk,

where Tr < k � 2pr:

pfp ¼X2prk¼Tr

pk (20)

This information can then be used to provide a con-

fidence rating that a circle detected is a genuine

circular feature. Whilst this is unlikely to be necessary

in normal industrial operation it can be useful when

setting up the system, for example in choosing thresh-

old values and setting the minimum circle vote Tr.

One might question the validity of the assumption

that edge pixels pertaining to non-circular features are

randomly distributed in the image. In our case these fall

into two categories: those relating to ’surface noise’, i.e.

local variations in the reflectivity of the stone slab, and

those relating to natural defects such as pits andfissures.

The former are evidently randomly distributed. The

latter are not: they are often grouped into similarly

orientated clusters, which may contribute to false

detections. In this case, analysis of the typicalwidth of a

process-induced circular scratch is usually sufficient to

exclude such clusters from the voting process.

3.5. The algorithm

In this section, we present an outline of our

algorithm. First, we define a value for the minimum

circle vote Tr. In practice this should be fixed as the

smallest weighting wewould expect to encounter in an

image. Whilst Eqs. (19) and (20) could be used to

determine a value of Tr that will admit all but spurious

features, the substitution of typical values into these

equations indicate that there is likely to be a

considerable gap between the weightings indicative

of a genuine circle and the weightings likely to

correspond to a false positive. For the sake of

efficiency Tr needs to be fixed as high as possible,

as the number of iterations required will increase

exponentially as the ratio Tr/V decreases. We also need

to define a step size s, a minimum point separation dsand an acceptable probability of failure, pf. The

algorithm then proceeds as follows:

(1) D

efine a set V of all the edge pixels in the image

(following edge detection and thresholding

processes).

(2) E

valuate the probability of false negatives

(Eq. (16)). If pfn < pf then stop.

(3) S

elect four pixels vi (where i = 1, . . ., 4) at randomfrom V. Check that all pixel pairs satisfy the

minimum distance criteria ds. Otherwise, discard

the points and go to step (2).

(4) D

efine a circle c123 from v1, v2 and v2 (Eqs. (2)–(11)). Evaluate the distance d4!123 (Eq. (12)). If

d4!123 is greater than an acceptable discretisation

error then go to step (2).

(5) E

valuate the number of pixels U (where U � V)

that lie within an acceptable discretisation error of

the circle boundary (Eq. (12)). If U > Tr then

V = V � U. Otherwise, go to step (2).

(6) O

ptimise the detected circle using the refinement

subroutine described in Section 3.2. Store the

circle parameters x0, y0 and r and the weightingU.

Calculate the probability pfp that the circle is

spurious (Eq. (20), if required. Return to step (2).

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801796

Fig. 5. (a) Bright field image, (b) extracted edge data, and (c) detected circles.

Fig. 6. (a) Bright field image, (b) extracted edge data, and (c) detected circles.

Fig. 7. (a) Bright field image, (b) extracted edge data, and (c) detected circles.

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801 797

Fig. 8. (a) Bright field image, (b) extracted edge data, and (c) detected circles.

4. Experimental results

The algorithm was tested using a range of images

showing defects encountered in the inspection of

polished stone surfaces. Fig. 5 shows a light scratch in

the presence of noisy surface data. Note that all these

scratches will have radii of at least 6000 pixels.

Fig. 6 shows light and heavy scratches in the

presence of noisy surface data and natural flaws in the

surface of the stone.

Fig. 7 shows multiple light and heavy scratches in

the presence of noisy surface data and natural defects.

Note that the edge extraction routine is tailored to

exclude very light scratches, which are not visible to

the naked eye.

Fig. 8 shows a light scratch in the presence of

substantial natural defects, of the kind often found to

coincide with the location of veins in polished marble.

Fig. 9. (a) Large disrupted circular feature, (b) RCD circle detec-

tion, and (c) current algorithm circle detection.

5. Discussion

5.1. Technical issues

The algorithmwe have developed is an extension of

the four-point method proposed by Chen and Chung

[25]. Kim et al. [8] identify three main problems with

randomised approaches such as the four-point algo-

rithm; namely probability estimation, accuracy and

the relationship between speed and the proportion of

edge pixels belonging to circular features to the total

number present. We have comprehensively addressed

the first point with a robust stopping criterion based on

probabilistic analysis. Furthermore, we would assert

that the optimisation routine introduced in Section 3.2

overcomes the inaccuracies associated with techni-

ques of this kind. The improvements attained using

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801798

Fig. 10. (a) Detection errors in original RCD where U Tr and (b) current algorithm circle detection.

Fig. 11. Relationship between n and Tr/V.

this approach can be illustrated as follows. Fig. 9a

shows a synthesised image of a large, disrupted

circular feature. The original RCD algorithm repeat-

edly excludes smaller sections of the boundary from

the fitted circle, as illustrated in Fig. 9b (close up of the

circle fitting through the lower right segment of the

circle). The current algorithm, on the other hand,

consistently finds the best fit through all the boundary

points, thus providing a more accurate determination

as shown in Fig. 9c.

If the circle is large with respect to the minimum

weighting Tr, the errors can become considerably

more pronounced, as illustrated in Fig. 10a.

The current algorithm again returns consistently

accurate results as shown in Fig. 10b, revealing

another advantage to this approach. Not only does the

optimisation routine lead to more accurate circle

detection, but the algorithm becomes less sensitive to

the selected threshold value Tr.

However, the dependence of the speed of the

algorithm on the signal-to-noise ratio of the image

remains a concern. Fig. 11 shows the relationship

between the number of iterations required to achieve a

given probability of failure and the ratio Tr/V.

This emphasises the need for careful selection of

the minimum vote threshold Tr. However, in real

images the shape characteristics of pixel groups

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801 799

Fig. 12. Relationship between pfn and n.

relating to machining defects (i.e. circles) are often

very different from those relating to natural defects or

other noise in the image. Thus, a level of preliminary

filtering can often significantly reduce the number of

pixels not contributing to any circular feature.

The advantages of statistically determining a stopp-

ing point for the algorithm can be reiterated by consi-

dering the inverse exponential relationship between the

probability of failure, pfn, and the number of iterations

performed. This is shown graphically in Fig. 12.

It can be seen that for any value of Tr/V there will be

an optimum point at which to stop the algorithm (in

Fig. 12 this is more apparent for the higher ratios). If

the number of iterations is cut short, the probability

of failure rises rapidly. If, on the other hand, the

algorithm is allowed to continue unnecessarily,

execution speed will suffer with little effect on the

probability of failure. Our algorithm ensures that only

the minimum number of iterations necessary to ensure

a reliable detection are performed.

5.2. Implementation issues

In its current form this system would be used as a

labour saving tool by providing a fast, objective

alternative to manual inspection of the finished stone

product. In practice, the perceived severity of process-

induced scratches is related to the distance at which

they become visible to a human observer. This is not

necessarily a function of thewidth, depth or length of a

particular scratch, instead it depends directly on the

extent towhich the scratch disrupts the image of a light

source reflected from the surface, a phenomenon that

will depend on the intrinsic properties of the material

and the glossiness of the overall finish. Classification

along these lines is commensurate with the require-

ments of the end user: for example, scratches only

visible from within 1 m will not preclude the stone

from use as external cladding on the upper storeys of a

building. This method of grading the severity of

process-induced defects can be replicated by the

machine vision system, by considering the resolution

at which the defect becomes apparent. For example,

the circle detection algorithm and the precursory

processing steps can be applied to representations of

the acquired image at 10�, 5�, 2� and l� reduced

resolutions. The resolution at which the defect became

apparent, and as such the severity of the defect, can be

indicated to the operator by colour-coding the defect

on the monitor. This approach has little impact on

processing time, as a featureless image will be passed

very quickly by the adaptive algorithm. On the other

hand, if a process-induced defect is detected at l0�reduced resolution there is little to be gained from

closer inspection of the surface.

It is also hoped that the classification of defect

severity will allow the extension of the technology to

provide some level of process control on the polishing

line. It should be possible to relate the nature of the

defect to the processing stage at which the damage

occurred. Early indication of a problem at a particular

stage would allow remedial action to be taken before

further damage occurs, thus reducing wastage and

improving overall output quality. The development of

imaging technology capable of quantifying defects in

polished stone also paves the way for the introduction

of new standards, which should improve quality

assurance and aid specifiers when selecting stone for a

particular application.

6. Conclusions

In this paper we have discussed the automated

inspection of polished stone and described how a

randomized circle detection algorithm can be adapted

to provide robust defect detection in challenging

industrial environments. An overall view of the

acquisition process was given and the requirements

of the stone polishing industry were discussed. A

circle detection algorithm was developed which

allows the user to control the trade-off between speed

J.R.J. Lee et al. / Computers in Industry 56 (2005) 787–801800

and reliability by means of a failure probability. This is

an extension of the randomised four-point selection

approach proposed by Chen and Chung [25].

Statistical modeling was used to relate the size of

the circles of interest, and the total number of edge

pixels in the image, to the number of iterations

required and the likelihood of spurious results being

returned. The algorithm thus provides optimum

efficiency for a given level of confidence. A

refinement procedure was also introduced to counter-

act the inaccuracies often associated with randomised

techniques. Extensive validation was undertaken using

images of real defects encountered in the inspection of

polished stone. We conclude that this approach offers

a robust, efficient solution to the circle detection

problem in conditions where conventional techniques

fail. The development of technologies such as these

has the potential to make a substantial impact on the

ornamental and dimensional stone sectors, paving the

way for improved quality assurance and adaptive

process control.

Acknowledgement

The authors wish to express their gratitude to Paul

White at UWE for his contribution to the statistical

modeling and analysis.

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Jason Lee is a research student in the

Faculty of Computing, Engineering and

Mathematical Sciences at the University

of the West of England. He studied Elec-

tronic Engineering at the University of

Bristol, and after a period in industry

returned to academia to continue his edu-

cation. His research interests centre

around the automated inspection of nat-

ural materials.

Melvyn Smith is Reader in Machine

Vision within the Faculty of Computing,

Engineering and Mathematical Sciences

at the University of the West of England

(UWE), Bristol, UK. He received his

BEng (Hons) degree in Mechanical Engi-

neering from the University of Bath in

1987, MSc in Robotics and Advanced

Manufacturing Systems from the Cran-

field Institute of Technology in 1988 and

his PhD from the University of the West of England in 1998. He is

Co-director of the Centre for IntelligentManufacturing andMachine

Vision Systems (CIMMS) at UWE. His research interests include

machine vision, industrial manufacturing and quality control sys-

tems. He has published a book as well as numerous journal and

conference papers in connection with his work.

Lyndon Smith is an active researcher andlecturer at the University of the West of

England, Bristol, UK. His research has

been reported in refereed journal papers

and technical presentations at various

dustry 56 (2005) 787–801 801

international locations, with a current

career total of over 60 publications. He

has experience in various areas of man-

ufacturing, including advanced machine

vision techniques for inspection and

metrology. He recently completed a 1-year secondment at The

Pennsylvania State University, and he acted as the EU Organising

Chairman for the 2002 International Conference on Process Model-

ling in PowderMetallurgy and Particulate Materials, held in Newport

Beach, CA, USA. He is also on the Editorial Board of the leading

international journal ’Powder Metallurgy’ (published by the Institute

ofMaterials,Minerals andMining in theUK).Since obtaininghisPhD

in 1997 he has continued to develop his research interests, through

successful application for research funding from various sources.

Sagar Midha is Reader and Research

Leader in the Mechanical, Manufacturing

and Aerospace Engineering School

within the Faculty of Computing, Engi-

neering and Mathematical Sciences at the

University of the West of England

(UWE), Bristol, UK. He received his

BSc Eng (Mech) from Punjab University

in India and his PhD from the Loughbor-

ough University in 1975. He is the Direc-

tor of the Centre for Intelligent Manufacturing and Machine Vision

Systems (CIMMS) at UWE. His research interests include machine

vision, Modelling and Knowledge Based Systems in Manufacturing

processes including stone polishing and Concurrent Engineering. He

has edited three books and has published numerous journal and

conference papers in connection with his work.