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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 4, APRIL 2000
255 for 8 bits) as thresholds, it has been found better to use xmin and
xmax , defined as the minimum and maximum values respectively of
the input pixels. Then, the clipping operation is as follows. For each
computed pixel x
x=
xmin ; if x < xmin
xmax ; if x > xmax
x;
otherwise.
(22)
In video coding, blocks of 16 2 16 pixels can be lost in transmission.
For that case, expression (20) with M = N = 18 leads to
Thresholding Using Two-Dimensional Histogram and
Fuzzy Entropy Principle
H. D. Cheng, Y. H. Chen, and X. H. Jiang
Abstract—This paper presents a thresholding approach by performing
fuzzy partition on a two-dimensional (2-D) histogram based on fuzzy relation and maximum fuzzy entropy principle. The experiments with various gray level and color images have demonstrated that the proposed approach outperforms the 2-D nonfuzzy approach and the one-dimensional
(1-D) fuzzy partition approach.
Index Terms—Fuzzy region, fuzzy relations, maximum fuzzy entropy
principle, threshold, 2-D histogram.
I. INTRODUCTION
P
=Q
=
0[0:9848; 0:9549; 0:9111; 0:8549; 0:7879;
0:7121; 0:6299; 0:5437; 0:4563; 0:3701; 0:2879;
0:2121; 0:1451; 0:0889; 0:0451; 0:0152]
T:
(23)
Again, each calculated pixel requires a maximum of eight multiplications.
In practice, several consecutive square blocks can be lost, which results in a rectangular lost block. It may also happen that such a block
is located at the right border of the image and no right border of the
lost block is available. The approach presented can be readily applied
in that case and a simplified expression is obtained for the interpolation mask G, because the terms which contain JN 02 Q disappear in
expressions (20) and (24), shown at the bottom of the previous page.
The same simulations as above have been run with one border missing,
i.e., input pixels are available on three borders only instead of four. The
PSNR values obtained are 26.6, 25.4, and 24.2 dB for the three images
“Zelda,” “tennis,” and “hotel,” respectively. The loss in PSNR reflects
the reduction of the input information.
V. CONCLUSION
An efficient technique has been presented to interpolate a rectangular
block of pixels in an image, each missing pixel being calculated from
a set of eight border pixels. The performance has been illustrated with
three images and a specific grid of isolated lost blocks of 8 2 8 pixels.
The interpolation technique which has been described can be included as a post processing operation in a video communication receiver and contribute to improve the quality in the presence of transmission errors.
Image thresholding is an important technique for image processing
and pattern recognition, which is also regarded as the first step for
image understanding. Many methods have been proposed to select the
thresholds automatically [1]. Most bilevel thresholding techniques can
be extended to the case of multithresholding, therefore, we focus on the
bilevel thresholding techniques in this paper. The proposed approach
will automatically determine the fuzzy region and find the thresholds
based on the maximum fuzzy entropy principle. It involves a novel
fuzzy partition on a two-dimensional (2-D) histogram where a 2-D
fuzzy entropy is defined, and a genetic algorithm is employed to find
the optimal result.
II. PROPOSED APPROACH
In order to obtain a 2-D histogram of an image, we define the local
average of a pixel, f (x; y ), as the average intensity of its four neighbors
denoted by g (x; y )
g(x; y) = b 14 [f (x; y + 1) + f (x; y 0 1)
+ f (x + 1; y ) + f (x 0 1; y )] + 0:5c:
(1)
A 2-D histogram is an array (L 2 L) with the entries representing the
number of occurrences of the pair (f (x; y ); g (x; y )). A 2-D histogram
can be viewed as a full Cartesian product of two sets X and Y , where
X represents the gray levels and Y represents the local average gray
levels: X = Y = f0; 1; 2; 1 1 1 ; L01g. The pixels having the same intensity but different spatial features can be distinguished in the second
dimension (local average gray level) of a 2-D histogram.
The BlockB and BlockW are defined in Fig. 1(a) and (b), respectively. Four fuzzy sets, BrightX , DarkX , BrightY , and DarkY , are defined based on the S -function and the corresponding Z -function [2],
[3] as follows:
BrightX (x)
S (x; a; b; c)
=
x
x
x2X
x2X
DarkX (x)
Z (x; a; b; c)
DarkX =
=
x
x
x2X
x2X
BrightX =
REFERENCES
[1] L. T. Chia et al., “On the use of transform domain information for concealment of errors in JPEG images,” in Proc. Conf. EUSIPCO-94, Edinburgh, U.K., Sept. 1994, pp. 616–619.
[2] Y. Wang, Q.-F. Zhu, and L. Shaw, “Maximally smooth image recovery
in transform coding,” IEEE Trans. Commun., vol. 41, pp. 1544–1551,
Oct. 1993.
[3] J. W. Park, J. W. Kim, and S. U. Lee, “DCT coefficients recovery-based
error concealment technique and its application to the MPEG-2 bit
stream error,” IEEE Trans. Circuits Syst. Video Technol., vol. 7, pp.
845–854, Dec. 1997.
[4] M. C. Pease, Method of Matrix Algebra, New York: Academic, 1965.
Manuscript received August 25, 1997; revised July 27, 1999. The associate
editor coordinating the review of this manuscript and approving it for publication was Prof. Jeffrey J. Rodriguez.
The authors are with the Department of Computer Science, Utah State University, Logan, UT 84322-4205 USA (e-mail: cheng@hengda.cs.usu.edu).
Publisher Item Identifier S 1057-7149(00)03012-8.
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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 4, APRIL 2000
733
Hnonfuzzy (RB ) =
nxy
0
2
(x; y ) R
nxy
1 log
2
nxy
(x; y ) R
2
(7)
nxy
(x; y ) R
Hfuzzy (R2 ) =
0
nxy
Bright (x; y)
2
(x; y ) R
nxy
1 log
2
2
nxy
(x; y ) R
(8)
nxy
(x; y ) R
Hnonfuzzy (RW ) =
0
2
nxy
(x; y ) R
1 log
Fig. 1. (a) Block is divided into R and R , (b) Block is divided into R
and R , and (c) 2-D histogram is divided into four blocks.
BrightY (y )
S (y; a; b; c)
=
y
y
y2Y
y 2Y
DarkY (y)
Z (y; a; b; c)
DarkY =
=
y
y
y 2Y
y 2Y
BrightY =
where Z () = 1 0 S ().
The fuzzy relation Bright is a subset of the full Cartesian product
space X 2 Y , i.e., Bright = BrightX 2 BrightY X 2 Y
Bright (x; y) = BrightX 2BrightY (x; y)
(2)
= min (BrightX (x); BrightY (y )):
Similarly, Dark = DarkX 2 DarkY X 2 Y
Dark (x; y) = DarkX 2DarkY (x; y)
(3)
= min (DarkX (x); DarkY (y )):
Let A be a fuzzy set with membership function A (xi ), where
xi ; i = 1; 1 1 1 ; N , are the possible outputs from source A with the
probability P (xi ). The fuzzy entropy of set A is defined as [4]
Hfuzzy (A) = 0
N
A (xi )P (xi ) log P (xi ):
(4)
i=1
The total image entropy is defined as
H (image) = H (BlockB ) + H (BlockW ):
(5)
As shown in Fig. 1(a), the dark block BlockB can be divided into a
nonfuzzy region RB and a fuzzy region R1
[ R1
RB = f(x; y)jDark (x; y) = 1; (x; y) 2 BlockB g
R1 = f(x; y)jDark (x; y) < 1; (x; y) 2 BlockB g:
BlockB = RB
Similarly, the bright block BlockW is composed of a nonfuzzy region
The following four entropies can be computed:
Hfuzzy (R1 ) =
0
Dark (x; y)
2
(x; y ) R
1 log
nxy
2
(x; y ) R
nxy
2
nxy
(x; y ) R
nxy
(6)
nxy
nxy
2
nxy
(9)
(x; y ) R
where nxy is the element in the 2-D histogram which represents the
number of occurrences of the pair (x; y ). The membership functions
Bright (x; y) and Dark (x; y) are defined in (2) and (3), respectively.
nxy in
It should be noticed that the probability computations nxy =
the four regions are independent of each other.
To find the best set of a, b, and c is an optimization problem which
can be solved by: heuristic searching, simulated annealing, genetic algorithm, etc. In this paper, we use genetic algorithm [5] to search for
the optimal solution. The proposed method consists of the following
three major steps:
1) find the 2-D histogram of the image;
2) perform fuzzy partition on the 2-D histogram;
3) compute the fuzzy entropy.
Step 1) needs to be executed only once while steps 2) and 3) are performed iteratively for each set of (a; b; c). The optimum (a; b; c) determines the fuzzy region (i.e., interval [a; c]). The threshold is selected
as the crossover point of the membership function which has membership 0.5 implying the largest fuzziness.
Once the threshold vector (s; t) is obtained, it divides the 2-D
histogram into four blocks, i.e., a dark block Block0 , a bright block
Block1 , and two noise (edge) blocks, Block2 and Block3 , as shown in
Fig. 1(c). The bright extraction method is expressed as [6]
fs; t (x; y; bright) =
g1 ; f (x; y) t ^ g(x; y) s
g0 ; otherwise.
(10)
Conversely, the dark portion extraction is
fs; t (x; y; dark) =
RW and a fuzzy region R2 , as shown in Fig. 1(b)
BlockW = RW [ R1
RW = f(x; y)jBright (x; y) = 1; (x; y) 2 BlockW g
R2 = f(x; y)jBright (x; y) < 1; (x; y) 2 BlockW g:
2
(x; y ) R
g0 ; f (x; y) < t ^ g(x; y) < s
g1 ; otherwise.
(11)
III. EXPERIMENTAL RESULTS AND DISCUSSIONS
Most gray level image thresholding techniques can be extended to
color images by directly processing each component of a color space,
then combining the results in some way to obtain the final image. For
a color image, we apply the proposed approach to RGB components of
the color space, respectively, and then combine the three results into a
new RGB color image.
We test the proposed approach on many monochrome and color images. We only show three figures here. The gray level of monochrome
images and RGB components of color images ranges from 0 to 255.
For a monochrome image, the corresponding bilevel threshold image
is expressed in two intensities 0 and 255. For a color image, the bilevel
threshold image of each component (R, G, and B) is expressed in two
734
Fig. 2.
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 4, APRIL 2000
(a) Original image, (b) result using 1-D fuzzy approach, (c) result using 2-D nonfuzzy approach, and (d) result using proposed 2-D fuzzy approach.
Fig. 3. (a) The original image, (b) result using 1-D fuzzy approach, (c)
result using 2-D nonfuzzy approach, and (d) result using proposed 2-D fuzzy
approach.
intensities: g0 and g1 , where g0 represents the gray level with the maximal number of pixels which are less than the threshold, and g1 , the
gray level with the maximal number of pixels which are greater than
the threshold. Then these bilevel images are combined to a RGB color
image.
In order to show the importance of spatial information in threshold
selection, we compare the results of the proposed approach with the
ones of the 1-D fuzzy approach which uses fuzzy c-partition and maximum fuzzy entropy principle to select thresholds [6]. The results of the
proposed approach are also compared with those of the 2-D nonfuzzy
(crisp) approach to demonstrate the power of fuzzy set theory. The 2-D
nonfuzzy approach uses 2-D histogram and maximum entropy principle to select thresholds [7]. For Fig. 2, the threshold vectors are (119,
159) and (112, 112) by using 2-D nonfuzzy approach and 2-D fuzzy
approach, respectively. The threshold value is 127 by using 1-D fuzzy
approach. Fig. 2(d) has much clearer features of image than those in
Fig. 2(b) and (c). The sky and the tower are better segmented in Fig.
2(d) than that in Fig. 2(b) and (c). For Fig. 3(b), the threshold values
are 102, 113, and 112 for R, G, and B components, respectively. For
Fig. 3(c), the threshold vectors are (83, 82), (81, 75), and (66, 69) for R,
G, and B components, respectively. For Fig. 3(d), the threshold vectors
are (81, 81), (100, 100), and (154, 154) for R, G, and B components respectively. Fig. 3(d) is the only one discriminating the blue sky from the
cornfield, and the tractor in Fig. 3(d) is much better extracted than that
in Fig. 3(b) and (c). The upper-right corner of Fig. 3(d) was misclassified which was caused by bilevel thresholding. This also happened
in Fig. 3(c). However, Fig. 3(d) has the best result. For Fig. 4(b), the
threshold values are 252, 211, and 164 for R, G, and B components,
respectively. For Fig. 4(c), the threshold vectors are (138, 128), (164,
196), and (180, 182) for R, G, and B components, respectively. For Fig.
4(d), the threshold vectors are (215, 215), (196, 196), and (171,171) for
R, G, and B components, respectively. In Fig. 4(d), the eyes, nose, and
mouth, are well extracted, and the colors of the dress and hair are different. In Fig. 4(b), the colors of the dress,
IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 4, APRIL 2000
735
[2] S. K. Pal and D. K. D. Majumder, Fuzzy Mathematical Approach to
Pattern Recognition. New York: Wiley, 1986.
[3] C. A. Murthy and S. K. Pal, “Histogram thresholding by minimizing
graylevel fuzziness,” Inform. Sci., vol. 60, pp. 107–135, 1992.
[4] L. A. Zadeh, “Probability measures of fuzzy events,” J. Math. Anal. Applicat., vol. 23, pp. 421–427, 1968.
[5] Y. H. Chen, “Thresholding of gray-level pictures using two-dimensional
histograms and fuzzy entropy principle,” M.S. thesis, Dept. Comput.
Sci., Utah State Univ., Logan, 1997.
[6] H. D. Cheng, J. R. Chen, and J. Li, “Threshold selection based on fuzzy
c-partition entropy approach,” Pattern Recognit., vol. 31, pp. 857–870,
July 1998.
[7] A. D. Brink, “Thresholding of digital images using two-dimensional entropies,” Pattern Recognit., vol. 25, pp. 803–808, Aug. 1992.
Regions Adjacency Graph Applied to Color Image
Segmentation
Alain Trémeau and Philippe Colantoni
Fig. 4. (a) Original image, (b) result using 1-D fuzzy approach, (c) result using
2-D nonfuzzy approach, and (d) result using proposed 2-D fuzzy approach.
TABLE I
COMPARISONS OF
THE COMPUTATIONAL TIME
Abstract—The aim of this paper is to present different algorithms, based
on a combination of two structures of graph and of two color image processing methods, in order to segment color images. The structures used in
this study are the region adjacency graph and the line graph associated.
We will see how these structures can enhance segmentation processes such
as region growing or watershed transformation. The principal advantage
of these structures is that they give more weight to adjacency relationships
between regions than usual methods. Let us note nevertheless that this advantage leads in return to adjust more parameters than other methods to
best refine the result of the segmentation. We will show that this adjustment
is necessarily image dependent and observer dependent.
Index Terms—Color image segmentation, region adjacency graph, region
growing process, watershed process.
I. INTRODUCTION
hair, and face features are all the same. In Fig. 4(c), the dress has been
misclassified as the background, therefore, it cannot be distinguished
from the background. We use HP DCE/9000 to conduct the experiments, and the computation times are listed in Table I.
IV. CONCLUSIONS
A novel 2-D fuzzy partition characterized by parameters a, b, and
is proposed which divides a 2-D histogram into two fuzzy subsets
“dark” and “bright.” For each fuzzy subset, one fuzzy entropy and one
nonfuzzy entropy are defined based on the fuzziness of the regions. The
best fuzzy partition was found based on the maximum fuzzy entropy
principle, and the corresponding parameters a, b, and c determines the
fuzzy region [a; c]. The threshold is selected as the crossover point of
the fuzzy region. The experimental results show that the spatial information of pixels should be considered in the selection of thresholds
and the 2-D fuzzy approach outperforms the 2-D crisp approach and
the 1-D fuzzy approach.
c
REFERENCES
[1] P. K. Sahoo, S. Soltani, A. K. C. Wong, and Y. Chen, “A survey of thresholding techniques,” Comput. Vis., Graph., Image Process., vol. 41, pp.
233–260, 1988.
Even if many algorithms are available for color image segmentation
[1]–[7], the literature is not as rich as that for grey level images [7],
especially when we refer to segmentation algorithms based on region
growing processes. Yet, it has been demonstrated that, for several sets
of images, region growing processes best perform than clustering or
thresholding approaches because they deal with spatial repartition of
color information. Region growing algorithms typically start with seed
pixels, then iteratively add to regions unassigned neighboring pixels
which satisfy one or several homogeneity criteria. Thus, we can define a region as being a set of connected pixels which satisfy some homogeneity criteria. Several criteria linked to color similarity or spatiocolor similarity can be used to analyze if a pixel belongs or not to a region [8], [9]. These criteria can be defined according to local, regional
and global relationships. In a previous approach, we have shown that
three criteria of homogeneity must be used to perform a relevant segmentation [10]. These criteria are as follows:
Manuscript received April 8, 1998; revised September 7, 1999. This work
was supported in part by the Région Rhône-Alpes through the ACTIV program.
The associate editor coordinating the review of this manuscript and approving
it for publication was Prof. Robert J. Schalkoff.
The authors are with the Institut d’Ingnierie de la Vision-LIGIV, Universit
Jean Monnet, BP 505, 42007 Saint-Etienne Cedex 01, France (e-mail:
tremeau@vision.univ-st-etienne.fr; colanton@vision.univ-st-etienne.fr).
Publisher Item Identifier S 1057-7149(00)02684-1.
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