Pattern Recognition Letters 33 (2012) 793–797

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Pattern Recognition Letters

journal homepage: www.elsevier .com/locate /patrec

Ridler and Calvard’s, Kittler and Illingworth’s and Otsu’s methodsfor image thresholding

Jing-Hao Xue a,⇑, Yu-Jin Zhang b

a Department of Statistical Science, University College London, London WC1E 6BT, UKb Department of Electronic Engineering, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o

Article history:Received 27 February 2011Available online 11 January 2012Communicated by D. Coeurjolly

Keywords:Image thresholdingIterative selectionDiscriminant analysisMinimum error thresholdingMixture of Gaussian distributionsOtsu’s method

0167-8655/$ - see front matter � 2012 Elsevier B.V. Adoi:10.1016/j.patrec.2012.01.002

⇑ Corresponding author. Tel.: +44 20 7679 1863; faE-mail addresses: jinghao.xue@ucl.ac.uk (J.-H. Xu

(Y.-J. Zhang).

a b s t r a c t

There are close relationships between three popular approaches to image thresholding, namely Ridlerand Calvard’s iterative-selection (IS) method, Kittler and Illingworth’s minimum-error-thresholding(MET) method and Otsu’s method. The relationships can be briefly described as: the IS method is an iter-ative version of Otsu’s method; Otsu’s method can be regarded as a special case of the MET method. Thepurpose of this correspondence is to provide a comprehensive clarification, some practical implicationsand further discussions of these relationships.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

In this correspondence, we aim to discuss the close relation-ships between three approaches to image thresholding, namelyRidler and Calvard (1978)’s or Trussell (1979)’s iterative-selection(IS) method, Kittler and Illingworth (1986)’s minimum-error-thresholding (MET) method and Otsu (1979)’s method.

With assumptions of bimodal or multimodal probability densityfunctions of grey levels x, these three approaches are widely usedin practice and highly cited by scientific publications. They are cov-ered in some popular textbooks such as that written by Gonzalezand Woods (2002, 2008). The MET method is ranked as the bestin a comprehensive survey of image-thresholding methods bySezgin and Sankur (2004) recently. Otsu’s method is implementedas the default approach to image thresholding in some commercialand free software such as MATLAB (The MathWorks, Inc.) andGIMP (www.gimp.org).

The popularity of all three approaches is not a coincidence.Recently, Xu et al. (2011) prove that, for image binarisation (or

two-level thresholding), Otsu’s optimal threshold is the threshold tthat equals the average value of the two class means, denoted byl0(t) and l1(t), for the two classes separated by t. That is,t = {l0(t) + l1(t)}/2. This result is in fact the iterative rule underly-ing the IS method. Such a link between the IS method and Otsu’s

ll rights reserved.

x: +44 20 3108 3105.e), zhang-yj@tsinghua.edu.cn

method has also been built by other studies, such as Reddi et al.(1984) and Magid et al. (1990).

Indeed, as we shall clarify more comprehensively in this corre-spondence, these three approaches are closely related to eachother: briefly speaking, the IS method is an iterative version ofOtsu’s method; Otsu’s method can be regarded as a special caseof the MET method.

We shall show that, between the IS method, Otsu’s method andthe MET method, the links can be readily built from the perspectiveof using a Gaussian-mixture distribution to model the grey-leveldistribution of an image, as indicated by Kurita et al. (1992) andKittler and Illingworth (1986), among others. Such a perspectiveis different from, and complementary to, that of Reddi et al.(1984), Magid et al. (1990) and Xu et al. (2011).

In this context, although this correspondence may mainly revi-sit some results from various classical literature, our intentions aretwofold. First, we intend to provide the practitioners with a morecomprehensive clarification and some practical implications ofthe close relationships between these three popular approaches.Secondly, we intend to encourage further discussions about effec-tively applying, extending and evaluating the established image-thresholding approaches.

2. Relationships between the three approaches

Here we only consider image binarisation, but the discussionspresented in the following sections can be readily generalised tomulti-level thresholding.

794 J.-H. Xue, Y.-J. Zhang / Pattern Recognition Letters 33 (2012) 793–797

Hence we assume that, in an image of N pixels, there are onlytwo classes, C0 and C1.

Let {p(x), x = 0, . . . ,T}, for grey levels x, denote the normalisedgrey-level histogram constructed from the N pixels, such thatPT

x¼0pðxÞ ¼ 1 and, by abuse of notation, we shall also use p(x) todenote the probability density function of x.

In addition, let y denote the class indicator of x, for exampley = 0 for x 2 C0 and y = 1 for x 2 C1. Hence, the histogram (or moreprecisely the probability density function), p(x), can be modelledby a two-component mixture distribution: pðxÞ ¼

P1y¼0pypðxjyÞ,

where py is the prior probability for Cy (and thus p1 = 1 � p0) andp(xjy) is the class-conditional distribution of x within Cy.

Image binarisation is a technique that uses a threshold t to par-tition the image into two classes C0ðtÞ and C1ðtÞ, whereC0ðtÞ ¼ fi : 0 6 xi 6 t;1 6 i 6 Ng and C1ðtÞ ¼ fi : t < xi 6 T;1 6 i 6Ng. That is, C0ðtÞ includes all the pixels with grey levels x no biggerthan t and C1ðtÞ consists of the remaining pixels.

Now we discuss the relationships between three pairs of the ISmethod, the MET method and Otsu’s method, respectively.

2.1. The IS method and the iterative version of the MET method

In each of Sections 2.1, 2.2 and 2.3, we shall first investigate therelationship between two of the three approaches, and then dis-cuss some practical pitfalls and implications for the use of the cor-responding two approaches.

2.1.1. The relationshipBased on the Bayes discriminant rule under which the expected

misclassification error rate is minimised, an optimal threshold t isthe grey level x such that p(y = 0jx) = p(y = 1jx) or equivalently, gi-ven p(y = 0jx) > 0 and by using a discriminant function,

logpðy ¼ 1jxÞpðy ¼ 0jxÞ ¼ log

p1pðxjy ¼ 1Þp0pðxjy ¼ 0Þ ¼ 0: ð1Þ

Let us assume that, for each class y, the class-conditional distribu-tion p(xjy) is a Gaussian Nðly;r2

yÞ distribution, where ly and r2y

are the mean and variance for class Cy. It follows that Eq. (1)becomes

logp1

p0� log

r1

r0� ðx� l1Þ

2

2r21

þ ðx� l0Þ2

2r20

¼ 0: ð2Þ

As shown in Kittler and Illingworth (1986), solving this quadraticequation for x leads to an iterative version of the MET method.

Let us further assume that p1 = p0 and r21 ¼ r2

0, i.e., the two clas-ses C1 and C0 are of equal sizes and equal variances. It follows thatEq. (2) degenerates into (x � l1)2 = (x � l0)2, or simply

x ¼ l0 þ l1

2: ð3Þ

This can lead to the IS method with an iterative rulet = {l0(t) + l1(t)}/2, where ly(t) can be estimated by the samplemean of class CyðtÞ determined by threshold t.

In short, the IS method is a special case of the iterative versionof the MET method, in which case equal sizes and equal variancesare further assumed for the two classes.

2.1.2. Some practical pitfalls and implicationsIn practice, an inappropriately-selected initial value often re-

sults in the failure of an iterative algorithm. This is unfortunatelyalso the case for the iterative MET method and the IS method;see Kittler and Illingworth (1986), Ye and Danielsson (1988) andXu et al. (2011) for illustrative examples. Hence, Ye and Danielsson(1988) propose to use the IS method to set the initial threshold forthe iterative MET method, and their experiments show that this

strategy makes the latter more robust. The strategy and its positiveinfluence may be justified by the fact that the IS method can beviewed as a special case of the iterative MET method.

Furthermore, both methods can be derived from the discriminantfunction logfpðC1jxÞ=pðC0jxÞg ¼ 0. From this perspective, the itera-tive MET method is based on the Gaussian-based quadratic discrim-inant analysis (QDA), and the IS method is based on a special case ofthe Gaussian-based linear discriminant analysis (LDA). The QDA re-verts to the LDA, if equal variances are assumed; in Eq. (2), the qua-dratic term x2 disappears when r2

0 ¼ r21. Therefore, some practical

guidelines and pitfalls associated with the LDA and the QDA maybe applied to the IS method and the iterative MET method; this mer-its further empirical investigation, although beyond the scope of thiscorrespondence.

2.2. Otsu’s method and the MET method

2.2.1. The relationshipLet us assume that, for a candidate threshold t, p(xjy; t) is a

Gaussian N lyðtÞ;r2yðtÞ

� �distribution, where ly(t) and r2

yðtÞ arethe mean and variance of class CyðtÞ determined by t.

Under this assumption, Kurita et al. (1992) show that the ruleadopted by the MET method for an optimal threshold is equivalentto the search for the threshold t that provides the largest maximumlog-likelihood (or equivalently likelihood), which is based onp(x,y; t) and on the factorisation p(x,y; t) = p(y; t)p(xjy; t). That is:

t ¼ argmaxt maxXðtÞ

XN

i¼1

logfpðyi; tÞpðxijyi; tÞg" #( )

; ð4Þ

where the parameters XðtÞ ¼ p0ðtÞ;l0ðtÞ;r20ðtÞ;l1ðtÞ;r2

1ðtÞ� �

areestimated by their maximum-likelihood estimators (i.e. their sam-ple estimators). This methodology can be traced back to Kittlerand Illingworth (1986).

Under further assumptions that p1(t) = p0(t) and r21ðtÞ ¼ r2

0ðtÞ,Kurita et al. (1992) also show that Otsu’s method is equivalent tothe search for the threshold t that provides the largest maximumlog-likelihood based on p(xjy; t). That is,

t ¼ argmaxt maxXðtÞ

XN

i¼1

log pðxijyi; tÞ( )" #

; ð5Þ

where XðtÞ ¼ l0ðtÞ;l1ðtÞ;r2WðtÞ

� �, in which r2

WðtÞ, denoting r21ðtÞ

and r20ðtÞ, is estimated by the within-class variance. In fact, Eq.

(5) can be obtained through the degeneration of Eq. (4), under thosefurther assumptions that p1(t) = p0(t) and r2

1ðtÞ ¼ r20ðtÞ.

In short, Otsu’s method can be viewed as a special case of theMET method, in which case equal sizes and equal variances are fur-ther assumed for the two classes.

2.2.2. Some practical pitfalls and implicationsIn practice, when those further assumptions of equal sizes and

equal variances are approximately satisfied, Otsu’s method is pre-ferred because its model is more parsimonious (Kurita et al., 1992).However, when the assumptions are apparently violated, Otsu’sthreshold tends to split the class of a larger size (Kittler andIllingworth, 1985, 1986), as illustrated in the left-hand panel ofFig. 1 for two Gaussian classes with equal variances but distinctsizes, and to bias towards the class of a larger variance (Kittlerand Illingworth, 1986; Xu et al., 2011), as illustrated in the right-hand panel of Fig. 1 for two Gaussian classes with equal sizes butdistinct variances. In other words, a more equal-sized and equal-spreaded partition is favoured by Otsu’s method in this case. Suchpatterns may be explained by the assumptions of equal sizes andequal variances that are underlying the derivation of Otsu’smethod in subSection 2.2.1.

50 100 150 200

010

0030

0050

00

Grey level

Freq

uenc

y

tOtsu

50 100 150

020

0060

0010

000

Grey level

Freq

uenc

y

tOtsu

Fig. 1. Otsu’s binarisation of simulated data for two Gaussian classes. Left-hand panel: two classes with equal variances but distinct sizes (5%:95%); right-hand panel: twoclasses with equal sizes but distinct variances ðr2

0 ¼ 256;r21 ¼ 16Þ. Otsu’s thresholds tOtsu, indicated by solid lines, split the class with a larger size (see the left-hand panel),

and bias towards the class with a larger variance (see the right-hand panel).

J.-H. Xue, Y.-J. Zhang / Pattern Recognition Letters 33 (2012) 793–797 795

Moreover, Otsu’s method is based on Fisher’s LDA; both Otsu’smethod and Fisher’s LDA use a within-class variance r2

W and/or abetween-class variance r2

B (Otsu, 1979). As we know, Fisher’sLDA, under the assumptions of two class-conditional Gaussian dis-tributions with equal variances, is equivalent to the Gaussian-based LDA; the MET method, as shown in Section 2.1, is based onthe Gaussian-based QDA. In this context, as with the IS methodand the iterative MET method, some practical guidelines and pit-falls related to the LDA and the QDA may also be applied to Otsu’smethod and the non-iterative MET method.

2.3. The IS method and Otsu’s method

2.3.1. The relationshipFrom the relationships presented in Sections 2.1 and 2.2, we can

observe that the IS method is an iterative version of Otsu’s method.As we mentioned in Section 1, such a link has been built from dif-ferent perspectives, including that proposed in an early work byReddi et al. (1984).

Otsu’s original method based on optimising r2WðtÞ or r2

BðtÞ isslow for multi-level thresholding. Hence, Reddi et al. (1984) pro-vide an iterative version of Otsu’s method, which can be used asa fast algorithm for searching for optimal multiple thresholds.For binarisation, this iterative Otsu method applies the same iter-ative rule, t = {l0(t) + l1(t)}/2, as that of the IS method.

The iterative rule can be derived from differentiating eitherr2

WðtÞ or r2BðtÞ. As with Reddi et al. (1984), let us use r2

BðtÞ, whichis given by

r2BðtÞ ¼ p0ðtÞfl0ðtÞ � lTg

2 þ p1ðtÞfl1ðtÞ � lTg2

¼ p0ðtÞl20ðtÞ þ p1ðtÞl2

1ðtÞ � l2T ; ð6Þ

where lT = p0(t)l0(t) + p1(t)l1(t) is independent of t and denotesthe total mean of grey levels x.

Using a continuous representation of p(x), we can write

p0ðtÞ ¼Z t

0pðxÞdx; p1ðtÞ ¼

Z T

tpðxÞdx: ð7Þ

l0ðtÞ ¼1

p0ðtÞ

Z t

0xpðxÞdx; l1ðtÞ ¼

1p1ðtÞ

Z T

txpðxÞdx: ð8Þ

It follows from differentiating r2BðtÞ with respect to t that

dr2BðtÞ=dt ¼ 2l0ðtÞtpðtÞ � l2

0ðtÞpðtÞ � 2l1ðtÞtpðtÞ þ l21ðtÞpðtÞ: ð9Þ

Given p(t) > 0 and l0(t) – l1(t), with simple algebra, we can obtaint = {l0(t) + l1(t)}/2 from setting dr2

BðtÞ=dt ¼ 0, as shown in Reddiet al. (1984).

As we have mentioned, the same result can also be obtained bydifferentiating r2

W ðtÞ, because the sum of r2WðtÞ and r2

BðtÞ, denotedby r2

T , is a constant independent of t; r2T represents the total vari-

ance of grey levels x for an image. Some variants of the differenti-ation of r2

WðtÞ can be found in Magid et al. (1990) based on acontinuous representation of r2

WðtÞ, and in Dong et al. (2008) andXu et al. (2011) based on discrete representations.

In short, the IS method is an iterative version of Otsu’s method.Indeed, an exhaustive search for t = {l0(t) + l1(t)}/2 should providethe same threshold as that of Otsu’s method.

2.3.2. Some practical pitfalls and implicationsIn practice, due to the characteristics of an iterative algorithm

for optimisation, the IS method is not only subject to the initial va-lue of the threshold, as shown in Xu et al. (2011) among others, butalso subject to certain local extrema, because r2

BðtÞ, or equivalently�r2

W ðtÞ, may not be unimodal with respect to t, as illustrated inKittler and Illingworth (1985) and Lee and Park (1990). In Fig. 2we also provide an illustrative example, where: (top panel) Otsu’smethod supplies a suboptimal threshold at about 156; (middle pa-nel) the within-class variance r2

WðtÞ, which is used by Otsu’s meth-od for searching for an optimal threshold, exhibits both global andlocal minima, and hence �r2

WðtÞ is not unimodal; and (bottom pa-nel) for a range of appropriate initial values the IS method canreach a better final threshold at about 130.

In other words, the IS method may provide an optimal thresh-old quite different from, although often similar to, that providedby Otsu’s method. In addition, a global extremum located by Otsu’smethod may not be a better threshold than a local extremum lo-cated by the IS method, in particular when the sizes of two classesare highly distinct from each other. In this case a simple ‘‘valleycheck’’, which only checks whether p(t) < p(ly(t)), may help tochoose a better extremum (Kittler and Illingworth, 1985; Xueand Titterington, 2011a).

3. Further Discussions

Our interpretations of the IS method, the MET method andOtsu’s method mainly follow that by Kittler and Illingworth(1986) and Kurita et al. (1992), based on statistical mixture modelsand the maximum likelihood estimation. There exist other inter-pretations of one or two of these methods from various perspec-tives, such as those in Kittler et al. (1985) based on simplestatistics without using a histogram, in Yan (1996) based on a gen-eral weighted-cost function, in Morii (1991) and Jiulun and Winxin

100 150 200

020

0040

0060

0080

0010

000

Grey level

Freq

uenc

y

tOtsu

80 100 120 140 160 180 200

0.0

0.2

0.4

0.6

0.8

1.0

Threshold t

Res

cale

d w

ithin

−cla

ss v

aria

nce

80 100 120 140 160 180 200

8010

012

014

016

018

020

0

For the IS method

Initial threshold

Fina

l thr

esho

ld

Fig. 2. Binarisation of simulated data for two Gaussian classes. Top panel: Otsu’sthreshold, tOtsu, for two classes with equal variances but highly-distinct sizes(2%:98%). Middle panel: the within-class variance r2

W ðtÞ used by Otsu’s method forsearching for an optimal threshold tOtsu; rescaled for illustrative purposes. Bottompanel: final thresholds obtained by the IS method versus corresponding initialvalues of the thresholds.

796 J.-H. Xue, Y.-J. Zhang / Pattern Recognition Letters 33 (2012) 793–797

(1997) based on entropies and in Xue and Titterington (2011b)based on hypothesis tests, among others.

In subSections 2.1.1, 2.2.1 and 2.3.1, we have provided ourinterpretations of these established approaches and thus a morecomprehensive clarification of the relationships between them. InsubSections 2.1.2, 2.2.2 and 2.3.2, we have discussed some practi-cal pitfalls and implications for the use of these approaches. Be-sides these, we shall present some further discussions aboutextending and evaluating them effectively, as follows.

First, as shown before, Otsu’s method and the MET method canbe derived from using a mixture of Gaussian distributions to model

the grey-level distribution, based on different assumptions aboutthe Gaussian distributions. Therefore, some of their extensionscan be and have been developed by using a mixture of other distri-butions, such as Poisson distributions (Pal and Bhandari, 1993),generalised Gaussian distributions (Bazi et al., 2007; Fan et al.,2008), skew-normal and log-concave distributions (Xue and Titter-ington, 2011c), Laplace distributions (Xue and Titterington, 2011a),and certain variants of Rayleigh (Xue et al., 1999), Nakagami-gamma, log-normal and Weibull distributions (Moser and Serpico,2006), to name but a few.

Secondly, although, as with most other automatic image-thres-holding approaches, Otsu’s method and the MET method are innature a clustering (or unsupervised-learning) approach, they weremotivated by and based on discriminant analysis, a supervised-learning approach to the search for optimal separation through set-ting the discriminant function logfpðC1jxÞ=pðC0jxÞg ¼ 0. For semi-automatic image thresholding, some pixels with class labels yknown can be collected. In this case, some semi-supervised learn-ing techniques (Chapelle et al., 2006) can be adapted to imagethresholding.

Thirdly, the three approaches discussed in this correspondenceare usually based on assumptions of bimodal or multimodal prob-ability density functions of grey levels x. Hence, they often performpoorly in unimodal cases, as empirically demonstrated by Rosin(2001) and Medina-Carnicer and Madrid-Cuevas (2008) for exam-ple. Nevertheless, as shown by a piece of recent work (Medina-Carnicer et al., 2011), with a certain transformation of the imagehistogram, the performance of Otsu’s method can be improvedfor edge detection, a common unimodal-thresholding application.

Last but not the least, a variety of measures have been taken toevaluate image-thresholding methods, or more generally image-segmentation methods, in comparative studies (Sahoo et al.,1988; Rosin and Ioannidis, 2003; Sezgin and Sankur, 2004). Zhang(1996) categorises these measures into three groups: the analyti-cal, empirical goodness and empirical discrepancy groups; such acategorisation has also been discussed for edge detection (Fernán-dez-García et al., 2004; Ortiz and Oliver, 2006). The empirical-discrepancy measures require a ‘reference image’ (also called ‘goldstandard’ or ‘ground truth’), while the empirical-goodness mea-sures do not. Zhang et al. (2008) propose a hierarchy of evaluationmeasures, which classifies empirical-discrepancy measures assupervised and empirical-goodness measures as unsupervised,and provide a survey of the unsupervised measures.

In practice, each measure has its advantages and disadvantages.Examples of the disadvantages include: a supervised measure findsin general no ‘gold standard’ available for real images; an unsuper-vised measure favours certain methods that explicitly or implicityuse the measure as a criterion for searching for an optimal segmen-tation. In our context, a widely-used measure for performancecomparison between image-thresholding methods, the grey-leveluniformity measure (Levine and Nazif, 1985), is basically equiva-lent to Otsu’s method (Ng and Lee, 1996) and thus always favoursthe latter. Such an equivalence is also indicated in Zhang and Gerb-rands (1994), based on the link between the uniformity measureand the ‘goodness’ criteria proposed by Weszka and Rosenfeld(1978).

4. Summary

In this correspondence, we have provided a comprehensiveclarification of the close relationships between three popular im-age-thresholding approaches. That is, in short, Ridler and Calvard’sIS method is an iterative version of Otsu’s method; Otsu’s methodcan be regarded as a special case of Kittler and Illingworth’s METmethod. It was our expectation that such a clarification could help

J.-H. Xue, Y.-J. Zhang / Pattern Recognition Letters 33 (2012) 793–797 797

the practitioners to understand more comprehensively the charac-teristics, thresholding performances and pitfalls of these ap-proaches, and thus facilitate the application, extension andevaluation of them.

Acknowledgements

The authors are grateful for the referees’ and the Area Editor’sconstructive comments, in particular those on the unimodal thres-holding and the evaluation of image-thresholding methods.

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