Type-2 fuzzy thresholding using GLSC histogram of human visual nonlinearity
characteristics
Yang Xiao, Zhiguo Cao,* Wen Zhuo
National Key Laboratory of Science and Technology on Multi-spectral Information Processing, Institute for Pattern Recognition and Artificial Intelligence, Huazhong University of Science and Technology,1037 Luoyu Road, Wuhan
Abstract: Image thresholding is one of the most important approaches for image segmentation and it has been extensively used in many image processing or computer vision applications. In this paper, a new image thresholding method is presented using type-2 fuzzy sets based on GLSC histogram of human visual nonlinearity characteristics (HVNC).The traditional GLSC histogram takes the image spatial information into account in a different way from two-dimensional histogram. This work refines the GLSC histogram by embedding HVNC into GLSC histogram. To select threshold based on the redefined GLSC histogram, we employ the type-2 fuzzy set, whose membership function integrates the effect of pixel gray value and local spatial information to membership value. The type-2 fuzzy set is subsequently transformed into a type-1 fuzzy set for fuzziness measure computation via type reduction. Finally, the optimal threshold is obtained by minimizing the fuzziness of the type-1 fuzzy set after an exhaustive search. The experiment on different types of images demonstrates the effectiveness and the robustness of our proposed thresholding technique.
OCIS codes: (100.0100) Image processing; (100.2000) Digital image processing; (100.3008) Image recognition, algorithms and filters.
References and links
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1. Introduction
Image thresholding, which extracts the object from the background in an input image, is often required for many image processing or computer vision applications, such as shape recognition, handwritten texts recognition and image enhancement [1–6]. Furthermore, the generation of binary image by image thresholding is employed for feature extraction and object recognition. Image thresholding could be regarded as the simplest form of image segmentation, or more generally, as a two-class clustering procedure [1–5].
Over the past decades, various approaches for automatic threshold selection have been reported, Weszka [1], Sahoo et.al [2] and Sankura et.al [3] surveyed on image thresholding at different times. In recent years, fuzzy set theory has been widely applied to thresholding [4–17]. By incorporating human perception and linguistic concepts such as similarity, fuzzy technique as a nonlinear knowledge-based method, can remove grayness ambiguities significantly. Most fuzzy thresholding methods [4,5,9–17] are based on the regular fuzzy sets (also referred as type-1 fuzzy sets [18]). In [20], Mendel et.al pointed out that there are at least
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10657
four sources of uncertainties in fuzzy logic system. However, type-1 fuzzy sets are not able to directly model such uncertainties. In order to handle this problem, the concept of type-2 fuzzy sets, whose membership functions are fuzzy themselves, was proposed by Zadeh [19] in 1975. Unfortunately, type-2 fuzzy sets are not utilized widely because they are hard to understand compared with type-1 fuzzy sets. During the last decade, Mendel et.al [20–22], Wu et.al [23], Zhai et.al [24], Lucas et.al [25] and Aisbett et.al [26] discussed representation form and characteristics of type-2 fuzzy sets theoretically to motivate their applications. In 2005, Tizhoosh demonstrated the first use of type-2 fuzzy sets for thresholding [6] based on Mendel‟s work [20]. In Tizhoosh‟s paper, he addressed a new membership function and utilized the entropy in interval valued fuzzy sets [27] as ultrafuzziness measure to capture the uncertainties within type-2 fuzzy systems. Vlachos et.al [7] and Bustince et.al [8] gave comments on Tizhoosh‟s proposition recently to demonstrate the importance of type-2 fuzzy thresholding approach.
The existing thresholding methods that only depend on the first-order gray-level histogram (1-D histogram) have one common drawback that the spatial correlation between different gray levels is ignored. However, more information contained in the image could be used for better segmentation. Originating from the work of Kapur et.al [28] and that of Kirby and Rosenfeld [29], Abutaleb [30] extended the entropic thresholding technique by using two-dimensional histogram (2-D histogram) determined by the gray value and the local average gray value of the pixels. As following works, Brink [31], Pal et.al [32] and Sahoo et.al [33,34] refined Abutaleb‟s method, and Liu et.al [35] suggested 2-D Otsu [36] technique. Moreover, Wang et.al was the first to employ 2-D histogram in fuzzy thresholding [5]. However, 2-D thresholding methods suffer from the drawbacks of large time consumption and information loss during the optimal threshold search. Recently, a new entropic approach using gray level spatial correlation histogram (GLSC histogram) was proposed by Xiao et.al [37,38]. GLSC histogram, which takes image spatial correlation into account in a different way from 2-D histogram, is determined by using the gray value of the pixels and the number of neighboring pixels with similar gray value in the corresponding neighborhood. According to Xiao‟s research [37,38], entropic thresholding method using GLSC histogram achieved equivalent or even better segmentation performance compared with 2-D histogram based ones, while saving time remarkably.
In this paper, we combine type-2 fuzzy sets with GLSC histogram to utilize their superiority addressed above for image thresholding. A new type-2 fuzzy thresholding approach based on GLSC histogram of human visual nonlinearity characteristics (HVNC) [39] is proposed. The original GLSC histogram is refined by embedding HVNC to enable it with human perception. Based on the redefined GLSC histogram, the type-2 fuzzy set for fuzzy thresholding is suggested. In order to obtain crisp fuzziness measure, the type-2 fuzzy set is subsequently transformed into a type-1 fuzzy set by vertical slice centroid type reduction (VSCTR) [25]. Finally, the optimal threshold is obtained by minimizing the fuzziness of the type-1 fuzzy set after an exhaustive search. The overall flowchart of our thresholding method is shown in Fig. 1. Meanwhile, the main difference between our work and Xiao‟s proposition [37,38] is how to construct the criterion function for threshold selection. The criterion function in this paper is proposed based on fuzzy set theory, however Xiao addressed it by using the concept of Shannon entropy.
The remainder of this paper is organized as follows. Section 2 describes the GLSC histogram of HVNC. In section 3, type-2 fuzzy sets for thresholding is defined. Section 4 presents the fuzziness measure for type-2 fuzzy sets. Here, the optimal threshold is selected by minimizing the fuzziness. Section 5 shows comparative results to demonstrate the effectiveness and the robustness of the proposed approach and Section 6 draws the final conclusions.
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10658
Fig. 1. Overall flowchart of the proposed algorithm.
2. GLSC histogram of HVNC
In this section, we first introduce the concept of 2-D histogram and GLSC histogram, while discussing their intrinsic drawbacks. Secondly, we illustrate in details how to refine GLSC histogram by embedding HVNC into it.
2.1 2-D histogram and its drawbacks
For an image ( , ) | {1,2,..., }, {1,2,..., }F f x y x Q y R of size Q R with gray level
ranging in 0, L , 2-D histogram [5,30–35] is defined as
2
number of bin , , ,( , ) , [0, ]; [0, ].d
f x y s g x y tH s t s L t L
Q R
(1)
where ,g x y represents the local average in square window of size N N centered at
,x y , and we define it as
1 2 1 2
21 2 1 2
1, , , 1, ; 1, .
N N
i N j N
g x y f x i y j x Q y RN
(2)
Figure 2 shows the 2-D histogram plane where T axis represents the gray value of pixels,
S axis represents the local average gray value of the pixels and L is the number of gray
levels. 2-D histogram is defined by assuming that pixels corresponding to object or background
make more contributions to the diagonal quadrants, region A and D than edge pixels or the noise, as the gray level distribution is relatively more homogeneous in object or background.
According to the optimal threshold vector ,s t selection in 2-D thresholding techniques
[5,30–35], quadrants B and C are ignored. 2-D histogram provides an effective way for
applying local neighborhood information to thresholding, which results in better segmentation. However, it still suffers from the drawbacks listed below:
First, the bin - , , ,f x y g x y for object or background is occasionally farther
away from region A and D than edge or noise, which conflicts with the assumption
of 2-D histogram. For instance, as the two local neighborhoods of size 3 3 shown
in Fig. 3, Fig. 3(a) of more homogenous gray level distribution corresponds to object or background with greater probability than Fig. 3(b) and it should locate closer to the diagonal line of 2-D histogram. However, the bins for Figs. 3(a) and 3(b) are
2, 58 and 128,127.5556 , obviously (a) is farther away from the diagonal
quadrants.
Secondly, as discussed above, some object or background bins may be discarded in 2-
D thresholding for their location in region B and C , which is unreasonable.
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10659
Additionally, in our opinion, ignorance of edge pixels would lead to serious information loss in 2-D thresholding.
Lastly, the extension of 1-D approaches to 2-D histogram gives rise to the exponential increment of computation time.
In order to overcome the drawbacks in 2-D histogram, Xiao et.al [37,38] proposed the GLSC histogram and entropic thresholding method based on the GLSC histogram.
Fig. 2. 2-D histogram plane.
Fig. 3. Drawbacks of 2-D histograms. (a) Local neighborhood 1. (b) Local neighborhood 2.
2.2 GLSC histogram and its drawbacks
Xiao et.al [37,38] defined GLSC histogram which describes local spatial information in a different way from 2-D histogram as follows
2
number of bin , , ,( , ) , [0, ]; [1, ].GLSC
f x y k d x y mH k m k L m N
Q R
(3)
where ,d x y represents the number of neighboring pixels whose gray value is close to
,f x y in square window of size N N centered at ,x y , and it is defined as
1 2 1 2
1 2 1 2
( , ) ( ( , ) ( , )).
N N
i N j N
d x y T f x i y j f x y
(4)
where
, if | ( , ) ( , ) |
( ( , ) ( , )) ., if | ( , ) ( , ) |
f x i y j f x yT f x i y j f x y
f x i y j f x y
(5)
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10660
Object or background pixels have similar gray value as neighboring pixels more probably than edge or noise, for their gray level distribution is relatively more homogeneous. As a consequence, Xiao suggested that object or background make more contribution to the regions of high m value in GLSC histogram. Setting to 4 as [37,38], the bins
, , ,f x y d x y for Figs. 3(a) and 3(b) are 2,7 and 128,1 respectively, which is
consistent with Xiao‟s suggestion and the actual situation. In GLSC histogram based entropic thresholding approach [37,38], all the pixels in image are taken into consideration by assigning a weight correlated with m in entropy criterion function computation to avoid
information loss. As proved by [38], entropic thresholding using GLSC histogram yielded equivalent or even better segmentation than 2-D histogram while saving time remarkably, for
the computation complexity of Xiao‟s method is 2( )O N L while the original 2-D ones is
4( )O L .
As discussed above, GLSC histogram possesses some advantages over 2-D histogram. However, it also suffers from some drawbacks as referred in [38]. An important affair is how to fix to use GLSC histogram. In [37,38], is an empirical constant for all pixels, which
seems unreasonable. In the next subsection, we shall illustrate how to fix in a statistical
way by embedding HVNC into GLSC histogram.
2.3 The refined GLSC histogram
Research on vision physiology and psychology shows that under different background brightness - I , visual system does not yield the same perception response to absolute brightness difference - I [39]. Its discriminative ability decreases as background brightness
is increased. Denote TI as a just noticeable brightness difference for the visual system, then
TI will increase as I is enlarged. Apart from the scotopic region, the relationship between
TI and I obeys the DeVries-Rose or Weber law [39]. The relation curve in Fig. 4 serves to
illustrate this nonlinearity principle. At low light intensities, TI is a constant and then
converges asymptotically to DeVries-Rose or Weber behavior with I increasing as
1
2
" ".
, " "T
I DeVries Rose regionI
I Weber region
(6)
Next, the human visual nonlinearity characteristics demonstrated above will be applied to
GLSC histogram in this paper. In Fig. 5, 1 ,NE x y and 2 ,NE x y are two square window
centered at ,x y of size 1 1N N and
2 2N N (2 1N N ) respectively, and ,f x y is the
central pixel. When we consider the relationship between ,f x y and its neighboring pixels
in 1 ,NE x y for GLSC histogram computation, is regarded as TI and the gray value
mean of 2 ,NE x y - 2 ( , )Mean x y is regarded as I in Eq. (6). According to HVNC, could
be calculated approximately as
1 2 1
2 2 2
1 2 1 2
2 2 2 2
, " "
, " "
, , _.
, _ ,
Mean x y DeVries Rose region
Mean x y Weber region
Mean x y Mean x y Div R
Mean x y Div R Mean x y L
(7)
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10661
where _Div R is a threshold; 1 0 ,
2 0 ; 1 and
2 are two constants. Then, it can be
seen that is no longer a constant and it adjusts with local property of different image
regions obeying human visual nonlinearity law. In practice, parameters in Eq. (7) are set as
1 0.039592 , 1 3 ,
2 0.0392 , 2 2.8595 and _ 100Div R respectively.
Fig. 4. Relationship between TI and I .
Fig. 5. Relationship between ,f x y , 1 ,NE x y and 2 ,NE x y .
Fig. 6. Cameraman image and its GLSC histogram of HVNC. (a) Cameraman image. (b) GLSC histogram of HVNC.
Figure 6 shows the cameraman image and its GLSC histogram of HVNC. Figure 7 shows
the pixel mapping images corresponding to different m values ranging from 1 to 9 of the
cameraman image, under the conditions that 1 3N and
2 5N . In each mapping image,
pixels of corresponding m value remain their original gray value while the others are labeled
as green color. From Fig. 7, we can see that the mapping images with high m value
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10662
correspond to object or background more probably while the ones with low m value
correspond to edge or noise oppositely. In the next section, type-2 fuzzy sets based on the refined GLSC histogram are presented
for thresholding.
Fig. 7. Mapping images corresponding to different m .
3. Type-2 fuzzy sets based on GLSC histogram of HVNC
In this section, some basic concepts on type-2 fuzzy sets are introduced and then the proposed type-2 fuzzy sets based on GLSC histogram of HVNC are defined.
3.1 Basic concepts on type-2 fuzzy sets
In [20], Mendel et.al pointed out that there are at least four sources of uncertainty in type-1 fuzzy logic system [18]: (1) the meaning of the used words is uncertain; (2) consequents of rules have a histogram of values, especially when knowledge is extracted from a group of experts who do not all agree; (3) measurements may be noisy; (4) the data used to tune the parameters of type-1 fuzzy logic system may also be noisy. All these factors listed above result in the assignment of a membership degree to an element in type-1 fuzzy sets being not certain. The main problem of type-1 fuzzy sets is that they are not able to handle such uncertainties because their membership functions are totally crisp. Oppositely, type-2 fuzzy sets [19] could model such uncertainties since their membership functions are fuzzy themselves [20].
Definition 1: A type-2 fuzzy set denoted as A is characterized by a type-2 membership
function ,A
x u as [20–26]
, / 0,1 , , 0,1 .x
xA Ax X u JA x u u x J x u
(8)
where x is the primary variable, xJ , representing an interval in 0,1 , is the primary
membership of x , u is the secondary variable, ,A
x u is also noted as the secondary grade
and , /x
Au Jx u u
is the secondary membership function at x also called the vertical slice.
If all secondary grades equal 1, A is regarded as the interval type-2 fuzzy set.
As defined in Eq. (8), the membership value of x in A is no longer a crisp number but the
type-1 fuzzy set , /x
x Au JA x u u
. Figure 8 depicts an example of type-2 fuzzy set in
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10663
[20] and the shaded area is the footprint of uncertainty (FOU) used to verbalize the shape of
type-2 fuzzy sets. In particular, 1,2,3,4,5X , 0,0.2,0.4,0.6,0.8U and the secondary
membership function at 2x is 2 0.5 0 0.35 0.2 0.35 0.4 0.2 0.6 0.5 0.8xA .
Fig. 8. Example of a type-2 fuzzy set.
3.2 Type-2 fuzzy sets utilized in this paper
Tizhoosh [6] first applied type-2 fuzzy sets to image thresholding to eliminate the uncertainties in membership function selection. However, regardless of membership function shape, type-2 fuzzy sets are employed to remove the vagueness of image data in this paper. In our opinion, the membership value assigned to a pixel should not be only determined by its gray value as [4,6,9–17] but also the spatial correlation with neighboring pixels. As discussed in Section 2, the degree of a pixel belonging to object or background could be measured by its m value in GLSC histogram of HVNC, the greater the value of m , the more probably it
belongs to object or background. Therefore, in fuzzy thresholding, pixels of greater m should
be assigned greater membership value than the ones with the same gray value but of lower m .
As a consequence, each gray level k would have a union of membership grades, which could
be regarded as the primary membership of k denoted as kJ . The secondary variable ,u k m
in kJ is determined both by k and ... Additionally, the secondary grade of ,u k m could be
approximated as the normalized occurrence probability of ,u k m in kJ . Next, we shall
illustrate how to define type-2 fuzzy sets based on GLSC histogram of HVNC in details.
For one image, let _ ,GLSC HVNCH k m denote its GLSC histogram of HVNC and
_GLSC HVNCA denote the type-2 fuzzy sets based on _ ,GLSC HVNCH k m , where 0,k L and
2
11,m N . We have
_
_0, ,
, , , .GLSC HVNC
k
GLSC HVNCAk L u k m J
A k u k m u k m k
(9)
where 0,1kJ and _
, , 0,1GLSC HVNCA
k u k m is the secondary grade. In Eq. (9), kJ ,
,u k m and _
, ,GLSC HVNCA
k u k m are unknown and how to obtain them is the key point to
construct _GLSC HVNCA . Next, their definition is illustrated in details.
First, the definition of primary membership - kJ is given as
211,
, .km N
J u k m
(10)
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10664
Secondly, the secondary variable - ,u k m is obtained as
, .u k m k m (11)
where k and m represent the impact factor of gray value and local spatial
information to ,u k m respectively. How to define them is listed below.
Fig. 9. 0 t and 1 t computation.
Motivated by what Huang [4] has addressed, k is defined similarly to the membership
function applied in the 1-D histogram based fuzzy method [4]. Let 0,1H k denote the 1-
D histogram. As shown in Fig. 9, given a certain threshold t , the average gray value of
background 0 t and object 1 t could be obtained as follows
0
0
.t
k
t kH k
(12)
and
1
1
.L
k t
t kH k
(13)
The membership function - mu k in [4] is given as
0
1
1, 0,
1, 0.5,1 .
1,
1,1
k tk t C
mu k mu k
k t Lk t C
(14)
However, the parameter - C in Eq. (14) is not easy to fix and we find that it affects the
thresholding performance to some degree by experiment. To overcome this defect, k is
defined by making some changes on mu k as
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10665
0
1
1, 0,
1 sin _ 2
, 0.5,1 .1
,1,1 sin _ 2
k tk t Diff max
k k
k t Lk t Diff max
(15)
where
0 1_ the maximum value in , 0, 1, 1 .Diff max k t k t k L t L (16)
From Eq. (15) and Eq. (16), it can be seen clearly that now no parameter needs to be set
empirically in k .
As discussed above, pixels of greater m should be assigned greater membership value
than the ones with the same gray value but of lower m . So, m should be obviously
defined as a monotonically increasing function of m . In practical, m is suggested as a
nonlinear function below
21
21
9 9
99
1 1, 0,1 .
11
m N
m N
e em m
ee
(17)
where 1,9 and 1, are two constants. Figure 10 shows an example of m . It
can be observed that m monotonically increases with m . As a consequence, ,u k m is
also a monotonically increasing function of m , which agrees with the fact that object or
background pixels should be assigned greater membership value than edge or noise of the same gray value. In practice, parameters in Eq. (17) are set as 1 and 1 respectively.
Fig. 10. Impact factor of local spatial information - m .
In image thresholding, for a given threshold t , any gray level in the input image should
belong to either object or background. Hence, Huang [4] pointed out that the membership
value of any gray level should be no less than 0.5. In ,u k m , k is the primary function
and m can be regarded as the tuning function for k . Since k is derived from [4],
we expect that ,u k m of any gray level should be also no less than 0.5. Here, a simple
constraint is made to ensure ,u k m being in the range 0.5,1 .
, 0.5, , 0.5u k m if u k m (18)
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10666
Finally, the secondary grade - _
, ,GLSC HVNCA
k u k m is thought of as the possibility of
,u k m to be as the membership grade of gray level k . Here, _
, ,GLSC HVNCA
k u k m
is approximated as the normalized occurrence probability of .. in kJ , and we have
21
_ _ _1, , , , .
GLSC HVNC
N
GLSC HVNC GLSC HVNCA ik u k m H k m H k i
(19)
Using the type-2 fuzzy sets _GLSC HVNCA proposed in this section. How to choose the
optimal threshold *t is presented next.
4. Optimal threshold selection
Being as a special case of type-2 fuzzy sets, interval type-2 fuzzy sets were employed in [6]
and Tizhoosh selected the optimal threshold *t by maximizing the entropy of interval valued
fuzzy sets addressed in [27], which seems a little unreasonable as what Bustince discussed in
[8]. However, _GLSC HVNCA in this paper is a general type-2 fuzzy set, and Tizhoosh‟s
suggestion on *t selection does not work under this condition. Here, *t is chosen in a
different way illustrated below.
In order to obtain *t , the fuzziness measure of _GLSC HVNCA should be computed firstly as
the criterion. Most recently, Zhai [24] has addressed how to compute the fuzziness for general type-2 fuzzy sets. While, the proposition in [24] is not suitable for practical image thresholding for two reasons: (i) the fuzziness measure suggested in [24] is a type-1 fuzzy set but not a crisp number and (ii) computational expense for fuzziness is very high.
For the sake of practical use and convenience in computation, _GLSC HVNCA is transformed
into a type-1 fuzzy set 1TA via vertical slice centroid type reduction (VSCTR) [25] in our
paper. And then *t is selected by minimizing the fuzziness proposed by Yager [40] of 1TA .
As illustrated in Section 3, the membership grade for gray level k is not a crisp number in
_GLSC HVNCA but a type-1 fuzzy set kA also called the vertical slice defined as
_,
, , / , .GLSC HVNC
kk Au k m J
A k u k m u k m
(20)
According to [23], the centroid kc A of kA is given as
2 21 1
_ _1 1, , , , , .
GLSC HVNC GLSC HVNC
N N
k A Ai ic A u k i k u k i k u k i
(21)
So, if regarding kc A as the membership grade of k , _GLSC HVNCA could be transformed into
a type-1 fuzzy set 1TA via VSCTR as Lucas [25] suggested.
1[0, ]
.T kk L
A c A k
(22)
Based on Yager‟s proposition [40], the fuzziness pF t of 1TA could be defined as
1
01 2 1 , 1,2,3 .
ppL
p kkF t c A H k p
(23)
where H k is the 1-D histogram. Then the optimal threshold *t is chosen by minimizing
pF t as
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10667
* min 0, .pt Arg F t t L (24)
5. Experiment
In order to demonstrate the performance of our proposed technique, 14 images of different histogram types (including unimodal, bimodal and multimodal) and sizes are chosen for test. The test images which are noisy or smooth contain small and large objects of clear and fuzzy boundaries, and those images have both simple and complex relationships between object and
background. All the images are listed as ant ( 370 357 ), bear ( 315 450 ), block ( 203 203 ),
( 203 203 ), shadow2 ( 244 244 ), stele ( 244 244 ) and stone ( 244 244 ). Most of the test
images and the corresponding ground-truth images employed as gold standard are obtained form references [6,9,17], and the rest are downloaded from internet. Figure 12 shows the original test images, their gold standard and histograms.
Four well established algorithms are utilized to make comparison with the proposed approach (T2F2), they are Liu‟s 2-D Otsu method (2DO) [35], Sahoo‟s 2-D entropic technique (2DE) [33], Wang‟s 2-D type-1 fuzzy means (2DT1F) [5] and Tizhoosh‟s type-2 fuzzy approach (T2F1) [6]. Among them, the former two are non-fuzzy ways while the others are fuzzy ones. The binary images yielded by the five algorithms are shown in Fig. 13 and Fig. 14 as PART I and II respectively.
A performance measure [1,6,9] based on the misclassification error for different
techniques is applied. is defined as
100 .O T O T
O O
B B F F
B F
(25)
where OB and
OF represent the background and object of the original (ground-truth) image,
TB and TF are the background and object in resulting image, and . denotes the cardinality
of the set. The performance measure for all the algorithms is listed in Table 1. Mean - and
standard deviation - employed to measure effectiveness and robustness of different
approaches are also included. Figure 11 shows the performance comparison between our method and other techniques by graph.
As listed in Table 1, our method yields the highest performance of 94.41% with the lowest standard deviation of 5.96%, which is superior to other techniques apparently.
According to Fig. 11, a detailed comparison between T2F2 and other algorithms is made as follows.
2DO is famous for its good adaptability to different types of images. As seen from Fig. 13, Fig. 14 and Fig. 11(a), the segmentation yielded by 2DO for all the test images is acceptable, especially when the 1-D histogram is bimodal. However, for most cases 2DO is inferior to T2F2 significantly and its performance is not so satisfying when the image histogram is unimodal or multimodal, e.g., image 3, 5, 8, and 9.
2DE makes use of 2-D histogram to improve performance, but we find that it tends to yield over segmentation, e.g., image 2, 5, 7 and 12. This situation may be caused by the intrinsic characteristics of entropy. At the same time, there is also a great performance gap between 2DE and our method.
2DT1F integrates the concept of 2-D histogram and type-1 fuzzy sets together, generally its performance is close to T2F2 and sometimes even better as shown in Fig. 11(c). But under some condition, 2DT1F does not work, such as image 2 and 5. So, its robustness seems doubtful.
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10668
As the first type-2 fuzzy thresholding algorithm, T2F1 gives the best segmentation to some images, which can demonstrate the usefulness of type-2 fuzzy sets in thresholding. Unfortunately, its robustness is still not good enough as its is the
second greatest one of all the five approaches, while of T2F2 is the lowest.
From the experiment, it can be observed clearly that no method is the best choice for all the images. As a stable technique, our algorithm yields good segmentation for most test images with the lowest and averagely the greatest . But when the image content is
simple, T2F2 seems not so outstanding, which needs to be improved.
Fig. 11. Thresholding performance comparison between different algorithms. (a) 2DO and T2F2. (b) 2DE and T2F2. (c) 2DT1F and T2F2. (d) T2F1 and T2F2.
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10669
6. Conclusion
Image thresholding is a fundamental and difficult task in image processing. Till now, no superior technique has been proposed for all types of images. So, it is necessary to develop new methods which are effective and robust. In this paper, a new thresholding approach using type-2 fuzzy sets based on GLSC histogram of HVNC is suggested. Our work focuses on how to make use of both gray level and local spatial information in fuzzy thresholding to improve segmentation. By embedding HVNC into GLSC histogram, we make it of human perception characteristics. On the basis of the refined GLSC histogram, type-2 fuzzy set is defined to model the vagueness in image data for thresholding, which type-1 fuzzy set is incapable of. Comparing with other four fuzzy and non-fuzzy algorithms, experiment on 14 images demonstrates the advantage of our method. As future work, more adaptive mathematical model for HVNC is desired. Additionally, we are also interested in how to define the ultrafuzziness of type-2 fuzzy set directly for thresholding instead of transforming it into type-1 fuzzy set.
Fig. 12. Test images, their corresponding ground-truth images and histograms.
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10670
Fig. 13. Thresholding results of five algorithms - PART I. For each test image, from left to right: Liu‟s 2-D Otsu method, Sahoo‟s 2-D entropic technique, Wang‟s 2-D fuzzy means, Tizhoosh‟s type-2 fuzzy approach and our proposed algorithm.
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10671
Fig. 14. Thresholding results of five algorithms - PART II. For each test image, from left to right: Liu‟s 2-D Otsu method, Sahoo‟s 2-D entropic technique, Wang‟s 2-D fuzzy means, Tizhoosh‟s type-2 fuzzy approach and our proposed algorithm.
Acknowledgments
This work was supported by the Project of the National Fundamental Research of China under Grant No. A1420061266, the Project of the National Natural Science Foundation of China under Grant No. 60736010 and the Project of Nonprofit Sector of China under Grant No. GYHY200906032.
#143562 - $15.00 USD Received 3 Mar 2011; revised 7 May 2011; accepted 10 May 2011; published 16 May 2011(C) 2011 OSA 23 May 2011 / Vol. 19, No. 11 / OPTICS EXPRESS 10672
