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Unsupervised range-constrained thresholding

Article (PDF Available)inPattern Recognition Letters 32(2):392-402 · January 2011with36 Reads
DOI: 10.1016/j.patrec.2010.09.020 · Source: DBLP
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.
Unsupervised range-constrained thresholding
Zuoyong Li
a,
, Jian Yang
b
, Guanghai Liu
c
, Yong Cheng
d
, Chuancai Liu
b
a
Department of Computer Science, Minjiang University, Fuzhou 350108, China
b
School of Computer Science and Technology, Nanjing University of Science and Technology, Nanjing 210094, China
c
School of Computer Science and Information Technology, Guangxi Normal University, Guilin 541004, China
d
School of Communication Engineering, Nanjing Institute of Technology, Nanjing 211167, China
article info
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
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. 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.
Ó 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 Ó 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
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 n
i
and the total number of pixels by
N = n
0
+ n
1
+ + nL
1
. The mean and standard deviation of
the image are defined as
l
¼
1
N
X
L1
i¼0
in
i
; ð1Þ
r
¼
1
N 1
X
L1
i¼0
ði
l
Þ
2
n
i
!
1
2
: ð2Þ
(2) Compute two gray levels
T
u
¼
l
b
r
; ð3Þ
T
l
¼
l
þ b
r
; ð4Þ
where b is a parameter and its value can be automatically determined
by optimizing the proposed criterion in Eq. (13). T
u
and T
l
are the
upper and lower bounds for background and object, respectively.
(3) Determine gray level ranges of object and background via
the following way:
R
F
¼ðT
l
L 1; ð5Þ
R
B
¼½0 T
u
Þ: ð6Þ
Finding the upper and lower bounds needs choosing a reasonable
value for parameter b, which in turn involves a statistical criterion
to be defined.
Assuming that the image I is divided into three portions via two
gray levels t
1
and t
2
, where t
1
< t
2
, with three parts denoted by C
b
, C
f
and C
m
, where C
b
is the background class with gray levels
[0, ... , t
1
1], C
f
the foreground class with levels
[t
2
+1,... , L 1], and C
m
the middle class with levels [t
1
, ... , t
2
].
C
m
is the middle transition region between C
b
and C
f
as described
in (Hu et al., 2008). The mean of each class is defined as
l
b
¼
1
N
b
X
t
1
1
i¼0
in
i
; ð7Þ
l
f
¼
1
N
f
X
L1
i¼t
2
þ1
in
i
; ð8Þ
l
m
¼
1
N
m
X
t
2
i¼t
1
in
i
; ð9Þ
where N
b
, N
f
and N
m
are the numbers of pixels in C
b
, C
f
and C
m
,
respectively. And their respective standard deviation can be given by
Z. Li et al. / Pattern Recognition Letters 32 (2011) 392–402
393
r
b
¼
1
N
b
1
X
t
1
1
i¼0
ði
l
b
Þ
2
n
i
!
1
2
; ð10Þ
r
f
¼
1
N
f
1
X
L1
i¼t
2
þ1
ði
l
f
Þ
2
n
i
!
1
2
; ð11Þ
r
m
¼
1
N
m
1
X
t
2
i¼t
1
ði
l
m
Þ
2
n
i
!
1
2
: ð12Þ
Based on the above standard deviations, the statistical criterion
could be defined as
r
S
¼
a
ð
r
b
þ
r
f
Þþð1
a
Þ
r
m
; ð13Þ
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 C
b
and C
f
. 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,
r
m
, stan-
dard deviation of the transitional class, C
m
, 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:
(1) Initialize MVS to be infinite, and i = 1, where MVS is the min-
imum value of
r
S
, i is the temporal number of iterations.
(2) Compute two gray levels t
1
and t
2
via the following
equations:
t
1
¼
l
0:1 i
r
; ð14Þ
t
2
¼
l
þ 0:1 i
r
; ð15Þ
where
l
and
r
are defined in Eqs. (1) and (2), respectively. If t
1
<0or
t
2
> 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).
(3) Calculate the value of our statistical criterion
r
S
by Eq. (13).
If
r
S
< MVS, then MVS =
r
S
, 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., R
F
and R
B
, by Eqs. (5) and (6).
(3) Implement image transformation via the following way:
f
tr
ði; jÞ¼
T
u
if f ði; jÞ2R
B
;
T
l
if f ði; jÞ2R
F
;
f ði; jÞ otherwise;
8
>
<
>
:
ð16Þ
where f(i, j) and f
tr
(i, j) are gray levels at pixel (i, j) of the original im-
age and the transformed form, respectively.
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
should be [T
u
T
l
], where T
u
and T
l
are the upper and lower bounds.
For a given gray level T
u
6 t 6 T
l
, the approximation error in RCra-
mesh can be formulated as
EðtÞ¼
1
N
1
1
X
t
i¼T
u
ði
l
1
Þ
2
þ
1
N
2
1
X
T
l
i¼tþ1
ði
l
2
Þ
2
; ð17Þ
where
l
1
¼
1
N
1
X
t
i¼T
u
in
i
; ð18Þ
l
2
¼
1
N
2
X
T
l
i¼tþ1
in
i
; ð19Þ
where N
1
is the number of pixels with levels [T
u
, ..., t], and N
2
the
number of pixels with levels [t +1,... , T
l
]. The optimal threshold t
*
can be determined by RCramesh,
T
¼ Arg min
T
u
6t6T
l
fEðtÞg: ð20Þ
3.4. Method 2: Range-Constrained Tsallis method (RCtsallis)
For a transformed image, let p
i
be the probability of gray level i
appeared in the image, where T
u
6 i 6 T
l
. Assuming that the pixels
in the image are classified into two classes, A and B, by a gray level
t, where T
u
6 t 6 T
l
. Class A corresponds to the foreground and class
B to the background, or vice versa. Cumulative probability of each
class can be defined as
x
A
¼
X
t
i¼T
u
p
i
; ð21Þ
x
B
¼
X
T
l
i¼tþ1
p
i
: ð22Þ
A priori Tsallis entropy for each class is defined as
S
A
q
ðtÞ¼
1
P
t
i¼T
u
p
i
=
x
A
ðÞ
q
q 1
; ð23Þ
S
B
q
ðtÞ¼
1
P
T
l
i¼tþ1
p
i
=
x
B
ðÞ
q
q 1
: ð24Þ
And the optimal threshold t
*
can be determined by RCtsallis,
T
¼ Arg max
T
u
6t6T
l
fS
A
q
ðtÞþS
B
q
ðtÞþð1 qÞS
A
q
ðtÞS
B
q
ðtÞg: ð25Þ
3.5. Method 3: Range-Constrained Wang’s method (RCwang)
For a transformed image with gray levels [T
u
, T
u
+1,..., T
l
], the
number of pixels with level i (T
u
6 i 6 T
l
) is denoted by n
i
and the
total number of pixels by N ¼ N
T
u
þ N
T
u
þ1
þþN
T
l
. Suppose that
the pixels in the transformed image are divided into two classes, A
and B, by a level t (T
u
6 t 6 T
l
). A is the set of pixels with levels
[T
u
, ... , t], and B the set of pixels with levels [t +1,... , T
l
]. The
two sets constitute the corresponding probability spaces, and their
probability density function, p
A
(x, y) and p
B
(x, y), can be formulated
as follows by using Parzen window estimation:
p
A
ðx; yÞ¼
1
N
X
t
i¼T
u
X
N
i
j¼1
1
h
2
N
i
u
ðx; y; x
j
; y
j
; h
2
N
i
Þ; ð26Þ
p
B
ðx; yÞ¼
1
N
X
T
l
i¼tþ1
X
N
i
j¼1
1
h
2
N
i
u
ðx; y; x
j
; y
j
; h
2
N
i
Þ; ð27Þ
where
u
ðx; y; x
j
; y
j
; h
2
N
i
Þ¼
1
2
p
exp
ðx x
j
Þ
2
þðy y
j
Þ
2
2h
2
N
i
!
: ð28Þ
In Eqs. (26)–(28), h
N
i
denotes the window width of the set com-
posed of those pixels with gray level i,(x
j
, y
j
) is the coordinate of
the jth pixel in the set. And the optimal threshold t
*
can be deter-
mined by RCwang,
t
¼ Arg min
T
u
6t6T
i
ZZ
X
p
2
A
ðx; yÞdx dy þ
ZZ
X
p
2
B
ðx; yÞdx dy
2
ZZ
X
p
A
ðx; yÞp
B
ðx; yÞdx dy
; 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
p
2
A
ðx; yÞdx dy ¼
1
N
2
X
t
i¼T
u
X
N
i
j¼1
X
t
k¼T
u
X
N
k
r¼1
Gðx
j
; y
j
; x
r
; y
r
; h
2
N
i
þ h
2
N
k
Þ;
ð30Þ
ZZ
X
p
2
B
ðx; yÞdx dy ¼
1
N
2
X
T
l
i¼tþ1
X
N
i
j¼1
X
T
l
k¼tþ1
X
N
k
r¼1
Gðx
j
; y
j
; x
r
; y
r
; h
2
N
i
þ h
2
N
k
Þ;
ð31Þ
ZZ
X
p
A
ðx;yÞp
B
ðx;yÞdxdy ¼
1
N
2
X
t
i¼T
u
X
N
i
j¼1
X
T
l
k¼tþ1
X
N
k
r¼1
Gðx
j
;y
j
; x
r
;y
r
; h
2
N
i
þ h
2
N
k
Þ;
ð32Þ
where
Gðx
j
; y
j
; x
r
; y
r
; h
2
N
i
þ h
2
N
k
Þ¼
1
h
2
N
i
þ h
2
N
k
u
ðx
j
; y
j
; x
r
; y
r
; h
2
N
i
þ h
2
N
k
Þ; 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
Table 1
Thresholds, numbers of misclassified pixels, ME values and running times obtained by applying various methods to the NDT images.
NDT images Thresholding methods
Ramesh’s method RCramesh Tsallis RCtsallis Wang’s method RCwang
Cell
Threshold 241 222 171 226 171 225
Misclassified pixels 53,861 2634 7375 4172 7375 3780
ME 0.82185 0.040192 0.11253 0.06366 0.11253 0.057678
Running times (s) 1.641 1.844 3.469 3.672 281.27 281.47
Tile
Threshold 213 197 149 200 149 199
Misclassified pixels 46,719 1711 10229 2703 10229 2257
ME 0.71288 0.026108 0.15608 0.041245 0.15608 0.034439
Running times (s) 1.375 1.563 3.406 3.594 265.69 265.88
PCB
Threshold 173 79 161 80 161 80
Misclassified pixels 15,707 2655 14,426 2501 14,426 2501
ME 0.23967 0.040512 0.22012 0.038162 0.22012 0.038162
Running times (s) 0.359 0.625 0.375 0.641 203.59 203.86
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
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 RCramesh Tsallis RCtsallis Wang’s method RCwang
Gearwheel
Threshold 228 84 13 85 202 85
Misclassified pixels 26,829 106 2247 131 12,664 131
ME 0.40938 0.0016174 0.034286 0.0019989 0.19324 0.0019989
Running times (s) 0.5 0.672 1.094 1.266 262.34 262.52
Potatoes
Threshold 143 97 64 98 65 98
Misclassified pixels 1805 0 1823 38 1593 38
ME 0.027542 0 0.027817 0.00057983 0.024307 0.00057983
Running times (s) 0.609 0.812 0.734 0.937 189.13 189.33
Block
Threshold 126 52 17 53 128 53
Misclassified pixels 12,554 428 2589 378 13,910 378
ME 0.19156 0.0065308 0.039505 0.0057678 0.21225 0.0057678
Running times (s) 0.437 0.593 0.781 0.937 232.56 232.72
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
range of threshold selection from [0 L 1] to [T
u
T
l
]. 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
jB
O
\ B
T
jþjF
O
\ F
T
j
jB
O
jþjF
O
j
; ð34Þ
where B
O
and F
O
are the background and foreground of the ground
truth image, B
T
and F
T
the 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
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 [T
u
T
l
] 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
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.
Images Range of object Range of background Images Range of object Range of background
Cell (227 255] [0 222) Block (67 255] [0 52)
Tile (201 255] [0 197) Lena (128 255] [0 119)
PCB (90 255] [0 79) Peppers (135 255] [0 124)
Gearwheel (104 255] [0 84) Flower (63 255] [0 54)
Potatoes (113 255] [0 97) Corn (112 255] [0 100)
400 Z. Li et al. / Pattern Recognition Letters 32 (2011) 392–402
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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RCwang
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Gearwheel 0.00200 0.00200 0.00200 0.00200 0.00200 0.00200 0.03253 0.03844 0.03844 0.03844
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Block 0.00577 0.00577 0.00577 0.00577 0.00577 0.00577 0.02237 0.09178 0.09178 0.09178
MME 0.02310 0.02310 0.02310 0.02310 0.03456 0.11848 0.13342 0.14597 0.14597 0.14597
Average MME for three methods
0.02250 0.02250 0.02250 0.02250 0.06735 0.22325 0.24778 0.30912 0.30912 0.30912
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402 Z. Li et al. / Pattern Recognition Letters 32 (2011) 392–402
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