# Unsupervised range-constrained thresholding.

**ABSTRACT** Three range-constrained thresholding methods are proposed in the light of human visual perception. The new methods first implement gray level range-estimation, using image statistical characteristics in the light of human visual perception. An image transformation is followed by virtue of estimated ranges. Criteria of conventional thresholding approaches are then applied to the transformed image for threshold selection. The key issue in the process lies in image transformation which is based on unsupervised estimation for gray level ranges of object and background. The transformation process takes advantage of properties of human visual perception and simplifies an original image, which is helpful for image thresholding. Three new methods were compared with their counterparts on a variety of images including nondestructive testing ones, and the experimental results show its effectiveness.

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**ABSTRACT:**In this paper, an algorithm for image thresholding based on semivariance analysis is presented. The rationale of the approach is to binarize an image such that it best reproduces the original image variation across several spatial scales. The method can be alternatively viewed as one identifying the binary image that best approximate the overall level of edgeness measured across multiple scales in the original image. A comparison with seven other thresholding methods is presented for 2 synthetic images and 22 Non-Destructive Testing (NDT) grey level images. The results indicate that the proposed method is highly competitive. Performance of the proposed method in relation to the image content is also discussed.Pattern Recognition Letters 04/2013; 34(5):456–462. · 1.06 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Introducing more information to improve the segmentation quality was regarded as an effective way, such as three-dimensional Otsu thresholding. However, it should be led to be very time consuming for real-time applications, and the Otsu criterion is questionable in some cases, for example, nondestructive testing. In the paper, a novel mechanism based on data field, originated from physical fields, is proposed for three-dimensional thresholding. Without any explicit criterions, an optimal threshold vector is produced using the self-adaptive evolution of data particles in the data field. And the proposed method has low time complexity. Experimental results, compared with the state-of-art algorithms and the related methods, suggest that the new proposal is efficient and effective.Neurocomputing 11/2012; 97:278–296. · 2.01 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Thresholding methods based on entropy have been proposed and developed over the years. In this paper, an improved Tsallis entropy based thresholding method is proposed for segmenting the images which presenting local long-range correlation rather than global long-range correlation. The advantage of the proposed method is to distinguish the pixels' local long-range correlation by the nonextensive parameter q. And the experimental results of various infrared images as well as nondestructive test ones show the effectiveness of the proposed method.Signal Processing 12/2012; 92(12):2931–2939. · 2.24 Impact Factor

Page 1

Unsupervised range-constrained thresholding

Zuoyong Lia,⇑, Jian Yangb, Guanghai Liuc, Yong Chengd, Chuancai Liub

aDepartment of Computer Science, Minjiang University, Fuzhou 350108, China

bSchool of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, China

cSchool of Computer Science and Information Technology, Guangxi Normal University, Guilin 541004, China

dSchool of Communication Engineering, Nanjing Institute of Technology, Nanjing 211167, China

a r t i c l ei n f o

Article history:

Received 5 November 2009

Available online 26 September 2010

Communicated by Y.J. Zhang

Keywords:

Thresholding

Image segmentation

Human visual perception

Standard deviation

Unsupervised estimation

a b s t r a c t

Three range-constrained thresholding methods are proposed in the light of human visual perception. The

new methods first implement gray level range-estimation, using image statistical characteristics in the

light of human visual perception. An image transformation is followed by virtue of estimated ranges. Cri-

teria of conventional thresholding approaches are then applied to the transformed image for threshold

selection. The key issue in the process lies in image transformation which is based on unsupervised esti-

mation for gray level ranges of object and background. The transformation process takes advantage of

properties of human visual perception and simplifies an original image, which is helpful for image thres-

holding. Three new methods were compared with their counterparts on a variety of images including

nondestructive testing ones, and the experimental results show its effectiveness.

? 2010 Elsevier B.V. All rights reserved.

1. Introduction

Image segmentation intends to extract an object from a back-

ground based on some pertinent characteristics such as gray level,

color, texture and location (Tao et al., 2008). It is a critical prepro-

cessing step in image analysis and computer vision (Huang and

Wang, 2009; Sen and Pal, 2010). Among the existing segmentation

techniques, thresholding is one of the most popular approaches in

terms of simplicity, robustness and accuracy. Implicit assumption

in image thresholding is that object (foreground) and background

have distinctive gray levels. Thresholding serves a variety of appli-

cations, such as biomedical image analysis (Hu et al., 2006; Min and

Park, 2009), character identification (Huang et al., 2008; Nomura

et al., 2009; Pai et al., 2010) and industrial inspection (Ng, 2006).

Thresholding techniques fall into bilevel and multilevel cate-

gory (Coudray et al., 2010; Horng, 2010; Malyszko and Stepaniuk,

2010; Wang et al., 2010) according to the number of segments. The

former assumes an image to be composed of two components (i.e.,

object and background), with an aim of finding an appropriate

threshold for distinguishing both parts. Thresholding result is a

binary image where all pixels with gray levels higher than

determined threshold are classified into foreground and the rest

of pixels assigned to background, or vice versa. The latter category

supposes that an image consists of multiple parts, each having

homogeneous gray level. Obviously, multiple thresholds should

be chosen to group pixels with gray level within a specified range

into one class. It can be regarded as an extension of the bilevel one.

Thresholding can also be classified into parametric and non-

parametric approaches from another perspective (Bazi et al.,

2007; Sahoo and Arora, 2004; Tizhoosh, 2005). In the parametric

approach, gray level distribution of an image is assumed to obey

a given statistical model, and optimal parameter estimation for

the model is sought by using image histogram. Fitted model is used

to approximate practical distribution. Bottom of valley in the mod-

el is regarded as the appropriate location of the optimal threshold.

This usually involves a nonlinear estimation of intensive computa-

tion. The nonparametric method determines the optimal threshold

by optimizing certain criterion, such as between-class variance

(Otsu, 1979), variance (Hou et al., 2006) and entropy (Kapur

et al., 1985; Pun, 1980). The nonparametric approach is proved

to be more robust and accurate.

Many thresholding approaches have been developed over the

last few years (Albuquerque et al., 2004; Kwon, 2004; Ramesh

et al., 1995; Sahoo et al., 1988; Wang et al., 2008). For example,

Bazi et al. (2007) proposed a parametric method, which finds the

optimal threshold through parameter estimation based on the

assumption that object and background follow a generalized

Gaussian distribution. Otsu’s method (1979) chooses the threshold

by maximizing the between-class variance of both object and

background. Sahoo et al. (1988) revealed that Otsu’s method is

one of the better threshold selection approaches for general real

world images with regard to uniformity and shape measures.

However, Otsu’s method exhibits a weakness of tending to classify

0167-8655/$ - see front matter ? 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.patrec.2010.09.020

⇑Corresponding author. Tel.: +86 13906926400; fax: +86 059183761607.

E-mail addresses: fzulzytdq@126.com, fzulzytdq@yahoo.com.cn (Z. Li).

Pattern Recognition Letters 32 (2011) 392–402

Contents lists available at ScienceDirect

Pattern Recognition Letters

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

Page 2

an image into two parts of similar size regardless of the practical

situation. After exploring the underlying reason for Otsu’s weak-

ness, Hou et al. (2006) presented an approach based on minimum

class variance, which can be regarded as a generalization of Otsu’s.

In image thresholding, one of the most efficient techniques is en-

tropy-based approach, which regards a gray level image histogram

as a probability distribution. In Pun’s method (1980), the threshold

is determined by maximizing the posteriori entropy of object and

background. Kapur et al. (1985) found some flaws in Pun’s deriva-

tions and proposed a corrected version. In 2004, Albuquerque et al.

presented an approach based on Tsallis entropy. Tsallis entropy is

applied as a general entropy form for information theory. In addi-

tion, Ramesh’s method (1995) finds the threshold by minimizing

function approximation error for image histogram with a bilevel

function. Wang et al. (2008) determined the threshold by optimiz-

ing a criterion function deduced by image histogram and Parzen

window technique. A thorough survey over thresholding is pro-

vided in the literature (Sezgin and Sankur, 2004).

Conventional nonparametric approaches utilize some criteria to

search the optimal threshold from all the gray levels, and neglect

properties of human visual perception. This makes them suffer a

limitation of being unable to obtain satisfactory results when seg-

menting some images. To eliminate the above limitation, three

unsupervised range-constrained thresholding methods are pre-

sented in this paper. Based on properties of human visual percep-

tion, they first find gray level ranges of object and background in an

unsupervised way by utilizing statistical characteristics of an im-

age. Then, an image transformation is implemented via the above

ranges. The transformation should simplify an original image and

improve segmentation performance. Finally, three existing criteria

are applied to a transformed image for threshold determination.

The performance of range-constrained methods is compared with

their conventional counterparts by testing a variety of images.

Experimental results show their superiority over the counterparts.

The remainder of this paper is organized as follows: Section 2

introduces properties of human visual perception. Three unsuper-

vised range-constrained methods are proposed in Section 3. Their

performance is detailed on a variety of images and compared with

their counterparts in Section 4. Conclusions appear in Section 5.

2. Human visual perception

Human visual perception has the following properties as de-

scribed in the literature (Arora et al., 2008).

(1) Human eye is insensitive to features present at the both

extremes of pixel intensity, whereas sensitive to distinguish-

ing features at the mid-range intensities. This characteristic

suggests a focus upon mid-region of a gray scale image, i.e.,

around the mean, when segmenting images.

(2) A lot of images may have either histograms with high inten-

sity values or more structures near a certain value (usually

the mean) than that farther from the mean. A rough estima-

tion of such a histogram exhibits a Gaussian distribution.

3. Unsupervised range-constrained thresholding methods

Conventional nonparametric thresholding approaches find the

optimal threshold via optimizing some criteria. The process ne-

glects the properties of human visual perception, resulting in

unsatisfactory segmentation for some real world images. In an ef-

fort to eliminate the limitation, the authors have tried a new

scheme in the light of human visual perception. The scheme first

finds gray level ranges of object and background via human visual

perception and image statistical characteristics in an unsupervised

way, then simplifies an original image by image transformation

based on the ranges, and finally applies the criteria of three exist-

ing methods to the transformed image, thus forming range-

constrained approaches with better segmentation performance.

3.1. Gray level ranges of object and background

For the sake of description convenience, we assume that object

(foreground) pixels have higher gray levels than background ones.

In the light of human visual perception, two gray levels should be

chosen as the lower bound for object and the upper bound for

background by using statistical characteristics of an image. After

finding the lower and upper bounds, gray level ranges of object

and background are accordingly determined, respectively. The

detailed process is as follows:

(1) Without losing generality, Let I be a gray scale image with L

levels [0, 1, ... , L ? 1]. The number of pixels with gray level i

is denoted by ni and the total number of pixels by

N = n0+ n1+ ??? + nL?1. The mean and standard deviation of

the image are defined as

l ¼1

N

X

L?1

i¼0

ini;

ð1Þ

r ¼

1

N ? 1

X

L?1

i¼0

ði ?lÞ2ni

!1

2

:

ð2Þ

(2) Compute two gray levels

Tu¼ l ? b ?r;

Tl¼ l þ b ?r;

wherebisaparameteranditsvaluecanbeautomaticallydetermined

by optimizing the proposed criterion in Eq. (13). Tuand Tlare the

upper and lower bounds for background and object, respectively.

(3) Determine gray level ranges of object and background via

the following way:

ð3Þ

ð4Þ

RF¼ ðTlL ? 1?;

RB¼ ½0 TuÞ:

Finding the upper and lower bounds needs choosing a reasonable

value for parameter b, which in turn involves a statistical criterion

to be defined.

ð5Þ

ð6Þ

Assuming that the image I is divided into three portions via two

gray levels t1and t2, where t1< t2, with three parts denoted by Cb, Cf

and Cm, where Cb is the background class with gray levels

[0, ... , t1? 1],

[t2+ 1, ... , L ? 1], and Cmthe middle class with levels [t1, ... , t2].

Cmis the middle transition region between Cband Cfas described

in (Hu et al., 2008). The mean of each class is defined as

Cf

theforeground classwithlevels

lb¼1

Nb

X

t1?1

i¼0

ini;

ð7Þ

lf¼1

Nf

X

L?1

i¼t2þ1

ini;

ð8Þ

lm¼

1

Nm

X

t2

i¼t1

ini;

ð9Þ

where Nb, Nfand Nmare the numbers of pixels in Cb, Cfand Cm,

respectively.And their respectivestandard deviation can be given by

Z. Li et al./Pattern Recognition Letters 32 (2011) 392–402

393

Page 3

rb¼

1

Nb? 1

X

t1?1

i¼0

ði ?lbÞ2ni

!1

2

;

ð10Þ

rf¼

1

Nf? 1

X

L?1

i¼t2þ1

ði ?lfÞ2ni

!1

2

;

ð11Þ

rm¼

1

Nm? 1

X

t2

i¼t1

ði ?lmÞ2ni

!1

2

:

ð12Þ

Based on the above standard deviations, the statistical criterion

could be defined as

rS¼ a ? ðrbþrfÞ þ ð1 ?aÞ ?rm;

where a is a parameter between 0 and 1. The criterion consists of

two terms, the first term standing for the sum of standard devia-

tions corresponding to Cband Cf. Standard deviation is a common

statistical measure reflecting degree of deviations between mean

and individuals. Hence, the term could represent intra-class similar-

ities of the background and foreground to some extent. The smaller

the term is, the higher the similarities. But it is worth mentioning

that both background and foreground themselves of a practical im-

age usually have some deviations on pixels’ gray levels, especially

for background. So it is unreasonable to determine the value of b

by minimizing only the first term. To solve the problem, rm, stan-

dard deviation of the transitional class, Cm, is introduced into the

criterion as a penalty term. The parameter a in Eq. (13) is a weight

balancing the contributions of the two terms.

For the image I, following steps describe the detail about auto-

matic selection of b:

ð13Þ

(1) Initialize MVS to be infinite, and i = 1, where MVS is the min-

imum value of rS, i is the temporal number of iterations.

(2) Compute two gray levels t1 and t2 via the following

equations:

t1¼ l ? 0:1 ? i ?r;

ð14Þ

t2¼ l þ 0:1 ? i ?r;

wherel andr are defined in Eqs. (1) and (2), respectively. If t1< 0 or

t2> L ? 1, then break the process; else, continue step 3. Here, the

reason using a constant 0.1 in Eqs. (14) and (15) is as follows: image

thresholding includes an implicit assumption that object and back-

ground have distinctive gray levels, that is, their difference should

be at least several gray levels. Standard deviation of an image is usu-

ally between 20 and 100, with practical values for our sample

images in experiments being as: Cell (24.532), Tile (20.829), PCB

(54.341), Gearwheel (100.05), Potatoes (83.056), Block (73.122),

Lena (47.51), Peppers (53.179), Flower (41.912) and Corn (62.083).

Therefore, 0.1 is basically the minimum coefficient for standard

deviation r in Eqs. (14) and (15).

ð15Þ

(3) Calculate the value of our statistical criterion rSby Eq. (13).

If rS< MVS, then MVS = rS, b = 0.1 ? i and i = i + 1; else,

i = i + 1. Subsequently, return to step 2.

Once b is determined, the upper and lower bounds can be calcu-

lated by Eqs. (3) and (4). Take cell image in Fig. 1(a) as an example.

Its foreground, background and middle classes are shown in

Fig. 1(b)–(d), where bright pixels are our focuses. The upper and

lower bounds are 222 and 227, and b is automatically set to 0.1.

Accordingly, gray level ranges of the background and object are

[0 222) and (227 L ? 1], respectively.

3.2. Image transformation

In order to implement image transformation, gray level ranges

of object and background must first be obtained as described in

Section 3.1. The ranges can be treated as gray level constrains on

object and background. For the image I, the process of image trans-

formation is as follows:

(1) Calculate the upper and lower bounds by Eqs. (3) and (4).

(2) Obtain gray level ranges of object and background, i.e., RF

and RB, by Eqs. (5) and (6).

(3) Implement image transformation via the following way:

8

>

where f(i, j) and ftr(i, j) are gray levels at pixel (i, j) of the original im-

age and the transformed form, respectively.

ftrði;jÞ ¼

Tu

Tl

fði;jÞ

if fði;jÞ 2 RB;

if fði;jÞ 2 RF;

otherwise;

>

:

<

ð16Þ

The transformation weakens gray level changes in both object

and background simultaneously, thus simplifying the original im-

age. The weakening effect is favorable to subsequent image seg-

mentation. Take transformed form Fig. 2(b) for the cell image as

an example. Their respective histogram is displayed in Fig. 2(c)

and (d). From Fig. 2, one can conclude that gray level changes of

object and background have been weakened dramatically, and

the transformed image becomes much simpler than the original.

Furthermore, one can observe that the transformation turns a his-

togram of unimodal distribution into an apparent bimodal one, the

latter being preferable when detecting the valley in a histogram,

for valley is usually regarded as the appropriate location of the

optimal threshold.

3.3. Method 1: Range-Constrained Ramesh’s method (RCramesh)

RCramesh recursively approximates the histogram of a given

transformed image with a bilevel function, and finds the optimal

threshold by minimizing approximation error. Here, the error is

represented by the variance of the approximated histogram. It is

worth mentioning that gray level range of the transformed image

Fig. 1. Cell image and its classes: (a) original, (b) foreground class, (c) background class, (d) middle class.

394

Z. Li et al./Pattern Recognition Letters 32 (2011) 392–402

Page 4

should be [TuTl], where Tuand Tlare the upper and lower bounds.

For a given gray level Tu6 t 6 Tl, the approximation error in RCra-

mesh can be formulated as

EðtÞ ¼

1

N1? 1

X

t

i¼Tu

ði ?l1Þ2þ

1

N2? 1

X

Tl

i¼tþ1

ði ?l2Þ2;

ð17Þ

where

l1¼1

N1

X

t

i¼Tu

ini;

ð18Þ

l2¼1

N2

X

Tl

i¼tþ1

ini;

ð19Þ

where N1is the number of pixels with levels [Tu, ... , t], and N2the

number of pixels with levels [t + 1, ... , Tl]. The optimal threshold t*

can be determined by RCramesh,

T?¼ Arg min

Tu6t6TlfEðtÞg:

ð20Þ

3.4. Method 2: Range-Constrained Tsallis method (RCtsallis)

For a transformed image, let pibe the probability of gray level i

appeared in the image, where Tu6 i 6 Tl. Assuming that the pixels

in the image are classified into two classes, A and B, by a gray level

t, where Tu6 t 6 Tl. Class A corresponds to the foreground and class

B to the background, or vice versa. Cumulative probability of each

class can be defined as

xA¼

X

t

i¼Tu

pi;

ð21Þ

xB¼

X

Tl

i¼tþ1

pi:

ð22Þ

A priori Tsallis entropy for each class is defined as

qðtÞ ¼1 ?Pt

SA

i¼Tupi=xA

q ? 1

ðÞq

;

ð23Þ

SB

qðtÞ ¼1 ?PTl

And the optimal threshold t*can be determined by RCtsallis,

i¼tþ1pi=xB

q ? 1

ðÞq

:

ð24Þ

T?¼ Arg max

Tu6t6TlfSA

qðtÞ þ SB

qðtÞ þ ð1 ? qÞSA

qðtÞSB

qðtÞg:

ð25Þ

3.5. Method 3: Range-Constrained Wang’s method (RCwang)

For a transformed image with gray levels [Tu, Tu+ 1, ... , Tl], the

number of pixels with level i (Tu6 i 6 Tl) is denoted by niand the

total number of pixels by N ¼ NTuþ NTuþ1þ ??? þ NTl. Suppose that

the pixels in the transformed image are divided into two classes, A

and B, by a level t (Tu6 t 6 Tl). A is the set of pixels with levels

[Tu, ... , t], and B the set of pixels with levels [t + 1, ... , Tl]. The

two sets constitute the corresponding probability spaces, and their

probability density function, pA(x, y) and pB(x, y), can be formulated

as follows by using Parzen window estimation:

pAðx;yÞ ¼1

N

X

t

i¼Tu

X

Ni

j¼1

1

h2

Ni

uðx;y;xj;yj;h2

NiÞ;

ð26Þ

pBðx;yÞ ¼1

N

X

Tl

i¼tþ1

X

Ni

j¼1

1

h2

Ni

uðx;y;xj;yj;h2

NiÞ;

ð27Þ

where

uðx;y;xj;yj;h2

NiÞ ¼

1

2pexp

?ðx ? xjÞ2þ ðy ? yjÞ2

2h2

Ni

!

:

ð28Þ

In Eqs. (26)–(28), hNidenotes the window width of the set com-

posed of those pixels with gray level i, (xj, yj) is the coordinate of

the jth pixel in the set. And the optimal threshold t*can be deter-

mined by RCwang,

ZZ

?2

t?¼ Arg min

Tu6t6Ti

ZZ

Xp2

Aðx;yÞdxdy þ

ZZ

Xp2

Bðx;yÞdxdy

?

XpAðx;yÞpBðx;yÞdxdy

?

;

ð29Þ

where X = [1, m] ? [1, n], m and n are the height and width of the

image, respectively. And the following equations hold:

ZZ

X

p2

Aðx;yÞdxdy ¼1

N2

X

t

i¼Tu

X

Ni

j¼1

X

t

k¼Tu

X

Nk

r¼1

Gðxj;yj;xr;yr;h2

Niþ h2

NkÞ;

ð30Þ

ZZ

Xp2

Bðx;yÞdxdy ¼1

N2

X

Tl

i¼tþ1

X

Ni

j¼1

X

Tl

k¼tþ1

X

Nk

r¼1

Gðxj;yj;xr;yr;h2

Niþ h2

NkÞ;

ð31Þ

ZZ

XpAðx;yÞpBðx;yÞdxdy¼1

N2

X

t

i¼Tu

X

Ni

j¼1

X

Tl

k¼tþ1

X

Nk

r¼1

Gðxj;yj;xr;yr;h2

Niþh2

NkÞ;

ð32Þ

where

Gðxj;yj;xr;yr;h2

Niþ h2

NkÞ ¼

1

h2

Niþ h2

Nk

uðxj;yj;xr;yr;h2

Niþ h2

NkÞ;

ð33Þ

and the definition of function u can refer to Eq. (28).

The advantages of RCramesh, RCtsallis and RCwang over their

counterparts are as follows: (1) a transformed image is used in-

stead of the original one during thresholding, which transfers

Fig. 2. Images and histograms: (a) original cell image, (b) the transformed image, (c) histogram of (a), (d) histogram of (b).

Z. Li et al./Pattern Recognition Letters 32 (2011) 392–402

395

Page 5

Table 1

Thresholds, numbers of misclassified pixels, ME values and running times obtained by applying various methods to the NDT images.

NDT imagesThresholding methods

Ramesh’s methodRCrameshTsallisRCtsallisWang’s methodRCwang

Cell

Threshold

Misclassified pixels

ME

Running times (s)

241

53,861

0.82185

1.641

222

2634

0.040192

1.844

171

7375

0.11253

3.469

226

4172

0.06366

3.672

171

7375

0.11253

281.27

225

3780

0.057678

281.47

Tile

Threshold

Misclassified pixels

ME

Running times (s)

213

46,719

0.71288

1.375

197

1711

0.026108

1.563

149

10229

0.15608

3.406

200

2703

0.041245

3.594

149

10229

0.15608

265.69

199

2257

0.034439

265.88

PCB

Threshold

Misclassified pixels

ME

Running times (s)

173

15,707

0.23967

0.359

79

2655

0.040512

0.625

161

14,426

0.22012

0.375

80

2501

0.038162

0.641

161

14,426

0.22012

203.59

80

2501

0.038162

203.86

Fig. 3. Thresholding results of cell image: (a) original, (b) ground truth image, (c) Ramesh’s method (t = 241), (d) RCramesh (t = 222), (e) Tsallis (t = 171), (f) RCtsallis (t = 226),

(g) Wang’s method (t = 171), (h) RCwang (t = 225).

Fig. 4. Thresholding results of tile image: (a) original, (b) ground truth image, (c) Ramesh’s method (t = 213), (d) RCramesh (t = 197), (e) Tsallis (t = 149), (f) RCtsallis (t = 200),

(g) Wang’s method (t = 149), (h) RCwang (t = 199).

396

Z. Li et al./Pattern Recognition Letters 32 (2011) 392–402

Page 6

Fig. 5. Thresholding results of PCB image: (a) original, (b) ground truth image, (c) Ramesh’s method (t = 173), (d) RCramesh (t = 79), (e) Tsallis (t = 161), (f) RCtsallis (t = 80),

(g) Wang’s method (t = 161), (h) RCwang (t = 80).

Table 2

Thresholds, numbers of misclassified pixels, ME values and running times obtained by applying various methods to the simple real world images.

Simple images Thresholding methods

Ramesh’s method RCrameshTsallisRCtsallisWang’s methodRCwang

Gearwheel

Threshold

Misclassified pixels

ME

Running times (s)

228

26,829

0.40938

0.5

84

106

0.0016174

0.672

13

2247

0.034286

1.094

85

131

0.0019989

1.266

202

12,664

0.19324

262.34

85

131

0.0019989

262.52

Potatoes

Threshold

Misclassified pixels

ME

Running times (s)

143

1805

0.027542

0.609

97

0

0

0.812

64

1823

0.027817

0.734

98

38

0.00057983

0.937

65

1593

0.024307

189.13

98

38

0.00057983

189.33

Block

Threshold

Misclassified pixels

ME

Running times (s)

126

12,554

0.19156

0.437

52

428

0.0065308

0.593

17

2589

0.039505

0.781

53

378

0.0057678

0.937

128

13,910

0.21225

232.56

53

378

0.0057678

232.72

Fig. 6. Thresholding results of gearwheel image: (a) original, (b) ground truth image, (c) Ramesh’s method (t = 228), (d) RCramesh (t = 84), (e) Tsallis (t = 13), (f) RCtsallis

(t = 85), (g) Wang’s method (t = 202), (h) RCwang (t = 85).

Z. Li et al./Pattern Recognition Letters 32 (2011) 392–402

397

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range of threshold selection from [0 L ? 1] to [TuTl]. This coincides

with the properties of human visual perception and improves seg-

mentation performance, (2) image transformation in the proposed

approaches weakens gray level changes in object and background,

thus simplifying the original image, which is helpful for subse-

quent image thresholding.

4. Experimental results

To evaluate the performance of our range-constrained methods,

a variety of images have been chosen as testing samples. The re-

sults yielded by our methods were compared with those obtained

by their counterparts, i.e., Ramesh’s method (1995), Tsallis

(Albuquerque et al., 2004) and Wang’s method (2008). The quality

of segmentation result is quantitatively evaluated via misclassifi-

cation error (ME) measure (Yasnoff et al., 1977), which regards im-

age segmentation as a pixel classification process. The measure

reflects the percentage of background pixels erroneously classified

into foreground, and conversely, foreground pixels erroneously as-

signed to background. For a two-class segmentation, ME can be

simply formulated as

ME ¼ 1 ?jBO\ BTj þ jFO\ FTj

jBOj þ jFOj

;

ð34Þ

where BOand FOare the background and foreground of the ground

truth image, BTand FTthe background and foreground pixels in the

thresholded image, and |?| the cardinality of a set. The value of ME

varies between 0 for a perfectly classified image and 1 for a totally

erroneously classified one. A lower value of ME means better qual-

ity. In Tsallis and RCtsallis, q = 3. In our methods, a = 0.4. All exper-

iments are performed on a notebook PC with 2.13G Intel Core 2 Duo

CPU and 3G RAM. All the images used in the experiments are of

256 ? 256 pixels and 8-bit (i.e., 256 gray levels).

4.1. Experiments on NDT images

The range-constrained methods were first applied to three NDT

images and compared with their counterparts. NDT means to de-

tect an object and quantify its possible defects without harmful

Fig. 7. Thresholding results of potatoes image: (a) original, (b) ground truth image, (c) Ramesh’s method (t = 143), (d) RCramesh (t = 97), (e) Tsallis (t = 64), (f) RCtsallis

(t = 98), (g) Wang’s method (t = 65), (h) RCwang (t = 98).

Fig. 8. Thresholding results of block image: (a) original, (b) ground truth image, (c) Ramesh’s method (t = 126), (d) RCramesh (t = 52), (e) Tsallis (t = 17), (f) RCtsallis (t = 53),

(g) Wang’s method (t = 128), (h) RCwang (t = 53).

398

Z. Li et al./Pattern Recognition Letters 32 (2011) 392–402

Page 8

effects on itself by special equipments and methods. It is used in a

broad variety of applications, such as aeronautics and astronautics,

nuclear industry, chemistry and civil constructions.

The results in terms of thresholds, numbers of misclassified pix-

els, ME values and running times for various approaches are listed

in Table 1. The table shows that segmentation results obtained by

authors’ methods have less misclassified pixels and lower ME val-

ues, implying better performance. This could be attributed to the

utilization of human visual perception in finding ranges of object

and background for subsequent image transformation. The trans-

formation dramatically simplifies the original image by weakening

gray level changes of object and background. The coincidence with

human visual perception and the image simplification are helpful

for improving segmentation performance. In addition, one can ob-

serve that conventional methods are only slightly faster than our

approaches with regard to the speed of segmentation. The range-

constrained methods reduce search space during thresholding

from the whole gray levels of an original image to a much smaller

range [TuTl] and save running time, as compared with conventional

approaches. Nevertheless, the new methods need extra time to

estimate ranges of object and background, with implementing im-

age transformation. Performance judgment over various methods

can also be evidenced by visual segmentation results shown in

Figs. 3–5. From the figures, one can conclude that the proposed

methods enjoy better visual effects, as they segment the objects

more accurately.

4.2. Experiments on other images

In this section, seven general real world images were chosen as

test specimen. These images fall into two groups. The first is for

simple images, and the second for complex ones. Table 2 lists

quantitative comparison of segmentation results for the first

group. The data show that the proposed methods achieve better

segmentations with less misclassified pixels and lower ME values.

Visual thresholding results are displayed in Figs. 6–8. The figures

indicate, in addition to a completely segmenting for objects, the

new approaches exhibit less background noise.

Segmentation results for the complex images are in Figs. 9–12.

The first two are classic gray level types, and the rest come from

Fig. 9. Thresholding results of Lena image: (a) original, (b) histogram, (c) Ramesh’s method (t = 107), (d) RCramesh (t = 119), (e) Tsallis (t = 161), (f) RCtsallis (t = 127),

(g) Wang’s method (t = 122), (h) RCwang (t = 126).

Fig. 10. Thresholding results of peppers image: (a) original, (b) histogram, (c) Ramesh’s method (t = 131), (d) RCramesh (t = 124), (e) Tsallis (t = 70), (f) RCtsallis (t = 128),

(g) Wang’s method (t = 95), (h) RCwang (t = 128).

Z. Li et al./Pattern Recognition Letters 32 (2011) 392–402

399

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famous Berkeley image database. Due to their complex structures,

quantitative measurement of segmentation quality experiences

serious difficulty at present. Only visual perception of subjective

nature is applicable to the judgment of segmentation quality.

Figs. 9–12 show range-constrained methods preserving more de-

tails of the objects, implying better results. More details relating

ranges of object and background for all the images in our experi-

ments are provided in Table 3.

4.3. Parameter selection

The range-constrained methods involve two parameters a and

b, and b is determined by our proposed statistical criterion with

a. Hence, only one parameter a is left uncertain. The parameter

is used to balance the contributions of the two terms in our crite-

rion, and smaller value implies larger contribution of the transi-

tional class. To find the reasonable value for a, a series of

Fig. 11. Thresholding results of flower image: (a) original, (b) histogram, (c) Ramesh’s method (t = 225), (d) RCramesh (t = 54), (e) Tsallis (t = 123), (f) RCtsallis (t = 55),

(g) Wang’s method (t = 122), (h) RCwang (t = 55).

Fig. 12. Thresholding results of corn image: (a) original, (b) histogram, (c) Ramesh’s method (t = 155), (d) RCramesh (t = 100), (e) Tsallis (t = 138), (f) RCtsallis (t = 101),

(g) Wang’s method (t = 131), (h) RCwang (t = 101).

Table 3

Ranges of object and background obtained by the proposed methods for image transformation under the assumption that object pixels have higher gray levels than background

ones.

ImagesRange of objectRange of background ImagesRange of object Range of background

Cell

Tile

PCB

Gearwheel

Potatoes

(227 255]

(201 255]

(90 255]

(104 255]

(113 255]

[0 222)

[0 197)

[0 79)

[0 84)

[0 97)

Block

Lena

Peppers

Flower

Corn

(67 255]

(128 255]

(135 255]

(63 255]

(112 255]

[0 52)

[0 119)

[0 124)

[0 54)

[0 100)

400

Z. Li et al./Pattern Recognition Letters 32 (2011) 392–402

Page 10

experiments on six images with different a have been carried out.

Experimental results are listed in Table 4. The table indicates that:

(1) ME values of images usually vary from small to big. The reason

is that a big a determines an unreasonable value for b, which in

turn leads to inaccurate gray level ranges of object and back-

ground. Such inaccuracy in ranges will significantly affect image

transformation and subsequent image thresholding, (2) with refer-

ence to MME values of six images, three range-constrained meth-

ods have different optimal values for a. For example, in RCtsallis

and RCwang, the optimal value for positions somewhere between

0.1 and 0.4, whereas in RCramesh, the optimal value is 0.5. How-

ever, it is worth mentioning that the difference of MME values ob-

tained by RCramesh is small when a varies between 0.1 and 0.5.

Therefore, as a general guide, the parameter ranges between 0.1

and 0.4, a 0.4 being selected for our proposed methods. Experimen-

tal data reveal a = 0.4 usually yield lowest average MME value for

the proposed methods, while a varying between 0.1 and 0.4 leads

to the same b as well as transformed image, which brings about

segmentation result with the same average MME value. When a

higher than 0.4, however, average MME value becomes larger, thus

indicating a = 0.4 acting as a turning point.

5. Conclusions

Three novel range-constrained thresholding methods are pro-

posed in this paper. In the light of human visual perception, the

methods first use statistical characteristics of an image to estimate

gray level ranges of object and background. At this stage, a new

statistical criterion for automatic selection of the parameter b is

developed. Subsequently, the gray level ranges are accordingly

determined. Automatic choice of b enhances the universality of

our methods. Then, range-constrained methods utilize the above

ranges to implement image transformation. Finally, criteria of

three existing thresholding methods are applied to the trans-

formed image for threshold selection. Image transformation by

confining gray level ranges of object and background coincides

with human visual perception and simplifies an original image,

which benefits image segmentation. In addition, experimental re-

sults show that running times of the proposed methods are com-

parative with those of their counterparts, and the reason has

been interpreted in Section 4.1. Experimental results on a variety

of real world images including NDT images show the effectiveness

of the proposed range-constrained methods.

Acknowledgements

This work is supported by National Natural Science Founda-

tion of China (Grant Nos. 60472061, 60632050, 90820004,

60875010), National 863 Project (Grant Nos. 2006AA04Z238,

2006AA01Z119), Technology Project of Provincial University of

Fujian Province (JK2010046) and Technology Project of Education

Department of Fujian Province (JA10226).

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0.04979

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0.33373

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0.66911

0.04979

0.62694

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RCtsallis

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Block

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0.06366

0.04125

0.03816

0.00200

0.00058

0.00577

0.02524

0.06366

0.04125

0.03816

0.00200

0.00058

0.00577

0.02524

0.06366

0.04125

0.03816

0.00200

0.00058

0.00577

0.02524

0.06366

0.04125

0.03816

0.00200

0.00058

0.00577

0.02524

0.81273

0.04125

0.03816

0.00200

0.00058

0.00577

0.15008

0.81273

0.35802

0.66072

0.00200

0.16295

0.00577

0.33370

0.81273

0.35802

0.66072

0.03253

0.35788

0.02237

0.37404

0.81273

0.35802

0.66072

0.03536

0.38225

0.43695

0.44767

0.81273

0.35802

0.66072

0.03536

0.38225

0.43695

0.44767

0.81273

0.35802

0.66072

0.03536

0.38225

0.43695

0.44767

RCwang

Cell

Tile

PCB

Gearwheel

Potatoes

Block

MME

0.05768

0.03444

0.03816

0.00200

0.00058

0.00577

0.02310

0.05768

0.03444

0.03816

0.00200

0.00058

0.00577

0.02310

0.05768

0.03444

0.03816

0.00200

0.00058

0.00577

0.02310

0.05768

0.03444

0.03816

0.00200

0.00058

0.00577

0.02310

0.12642

0.03444

0.03816

0.00200

0.00058

0.00577

0.03456

0.12642

0.12059

0.11636

0.00200

0.33977

0.00577

0.11848

0.12642

0.12059

0.11636

0.03253

0.38225

0.02237

0.13342

0.12642

0.12059

0.11636

0.03844

0.38225

0.09178

0.14597

0.12642

0.12059

0.11636

0.03844

0.38225

0.09178

0.14597

0.12642

0.12059

0.11636

0.03844

0.38225

0.09178

0.14597

Average MME for three methods

0.022500.022500.022500.02250 0.067350.22325 0.247780.30912 0.309120.30912

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