Table Of ContentPattern Recoynition. Vol. 27, No. 9. pp. 1275 1289, 1994
nomagreP Elsevier Science Lid
Copyright )~( 1994 Pattern Recognition Society
Printed in Great Britain. All rights reserved
0031 320394 $7.0()+.00
0031-3203 (94) E0026-H
A RELATIVE ENTROPY-BASED APPROACH TO
IMAGE THRESHOLDING
CHEIN-I CHANG, t KEBO CHEN, t JIANWEI WANG+ and MARK L. G. ~ESUOHTLA
t Department of Electrical Engineering, University of Maryland, Baltimore County Campus, Baltimore,
MD 21228-5398, U.S.A.
+ Edgewood Research, Development and Engineering Center, Aberdeen Proving Ground, MD 21010-5432,
U.S.A.
devieceR( 4 Auqust 1993; deviecer roJ noitacilbup 24 February 1994l
tcartsbA In this paper, we present a new image thresholding technique which uses the relative entropy
(also known as the Kullback-Leiber discrimination distance function) as a criterion of thresholding an
image. As a result, a gray level minimizing the relative entropy will be the desired threshold. The proposed
relative entropy approach is different from two known entropy-based thresholding techniques, the local
entropy and joint entropy methods developed by N. .R Pal and .S K. Pal in the sense that the former is
focused on the matching between two images while the latter only emphasized the entropy of the co-
occurrence matrix of one image. The experimental results show that these three techniques are image
dependent and the local entropy and relative entropy seem to perform better than does the joint entropy.
in addition, the relative entropy can complement the local entropy and joint entropy in terms of providing
different details which the others cannot. As far as computing saving is concerned, the relative entropy
approach also provides the least computational complexity.
Thresholding Relative entropy Local entropy Joint entropy Co-occurrence matrix
l. INTRODUCTION transitions from background to background (BB),
background to objects (BO), objects to background
Image thresholding often represents a first step in (OB) and objects to objects (OO). The local entropy is
image understanding. In an ideal image where objects defined only on two quadrants, BB and OO, whereas
are clearly distinguishable from the background, the the joint entropy is defined only on the other two
grey-level histogram of the image is generally bimodal. quadrants, BO and OB. Based on these two definitions,
In this case, a best threshold segmenting objects from Pal and Pal developed two algorithms, each of which
the background is one placed right in the valley of two maximizes local entropy and joint entropy, respectively.
peaks of the histogram. However, in most cases, the In this paper, we present an alternative entropy-
grey-level histograms of images to be segmented are based approach which is different from those in refer-
always multimodal. Therefore, finding an appropriate ences (1-61. Rather than looking into entropies of
threshold for images is not straightforward. Various background or object individually, we introduce the
thresholding techniques have been proposed to resolve concept of the relative entropy 1vI (also known as cross
this problem. entropy, Kullback-Leiber's discrimination distance and
In recent years, information theoretic approaches directed divergence), which has been widely used in
based on Shannon's entropy concept have received source coding for the purpose of measuring the mis-
considerable interest. ~6-1~ Of particular interest are matching between two sources. Since a source is gene-
two methods proposed by N. R. Pal and S. K. Pal ul rally characterized by a probability distribution, the
which use a co-occurrence matrix to define second- relative entropy can be also interpreted as a distance
order local and joint entropies. The co-occurrence measure between two sources. This suggests that the
matrix is a transition matrix generated by changes in relative entropy can be used for a criterion to measure
pixel intensities. For any two arbitrary grey levels i and the mismatching between an image and a thresholded
j (i, j are not necessarily distinct), the co-occurrence bilevel image. One method to apply the relative entropy
matrix describes all intensity transitions from grey concept to image thresholding is to calculate the gray-
level i to grey level j. Suppose that t is the desired level transition probability distributions of the co-
threshold. The t then segments an image into the occurrence matrices for an image and a thresholded
background which contains pixels with grey levels bilevel image, respectively, then find a threshold which
below or equal to t and the foreground which cor- minimizes the discrepancy between these two transition
responds to objects having pixels with grey levels above probability distributions, i.e. their relative entropy.
t. This t further divides the co-occurrence matrix into The threshold rendering the smallest relative entropy
four quadrants which correspond, respectively, to will be selected to segment the image. As a result, the
5721
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
A Relative Entropy-Based Approach to Image Thresholding
5b. GRANT NUMBER
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6. AUTHOR(S) 5d. PROJECT NUMBER
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13. SUPPLEMENTARY NOTES
14. ABSTRACT
In this paper, we present a new image thresholding technique which uses the relative entropy (also known
as the Kullback-Leiber discrimination distance function) as a criterion of thresholding an image. As a
result, a gray level minimizing the relative entropy will be the desired threshold. The proposed relative
entropy approach is different from two known entropy-based thresholding techniques, the local entropy
and joint entropy methods developed by N. R. Pal and S. K. Pal in the sense that the former is focused on
the matching between two images while the latter only emphasized the entropy of the cooccurrence matrix
of one image. The experimental results show that these three techniques are image dependent and the local
entropy and relative entropy seem to perform better than does the joint entropy. In addition, the relative
entropy can complement the local entropy and joint entropy in terms of providing different details which
the others cannot. As far as computing saving is concerned, the relative entropy approach also provides the
least computational complexity.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON
a. REPORT b. ABSTRACT c. THIS PAGE Same as 15
unclassified unclassified unclassified Report (SAR)
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
6721 C.-I GNAHC te .la
thresholded bilevel image will be the best approxima- (f(l,k)=i, f(l,k+l)=j
tion to the original image. Since transitions of OB and 6(l,k) = ,1 if ~ . and/or
BO generally represent edge changes in boundaries
Lf(l,k)=i, f(l+ 1,k)=j
and transitions of BB and OO indicate local changes in
6(l,k)= 0, otherwise.
regions, we can anticipate that a thresholded bilevel
image produced by the proposed relative entropy ap- One may like to make the co-occurrence matrix
proach will best match the co-occurrence matrix of the symmetric by considering horizontally right and left,
original image. This observation is demonstrated and vertically above and below transitions. It has,
experimentally by several test images. In general, the however, been found )1~ that including horizontally left
performance of all the three methods is image depen- and vertically above transitions does not provide more
dent. Although there is no evidence that one is generally information about the matrix or significant improve-
better than the others, according to the experiments ment. Therefore, it is sufficient to consider adjacent
conducted in this paper, the joint entropy did not work pixels which are horizontally right and vertically below
as well as did the local entropy and relative entropy. so that the required computation can be reduced.
Interestingly, among all images tested the relative Normalizing the total number of transitions in the
entropy approach seems to be better than the others co-occurrence matrix, we obtain the desired transition
at finding edges. In addition, our experiments show probability from grey level i to j~) as follows.
that the relative entropy seems to be a good comple-
ment to the local entropy and joint entropy methods
in terms of providing different image details and des-
criptions from those provided by the local entropy
and joint entropy. Finally, an advantage of the relative 2.2. Quadrants of the co-occurrence matrix
entropy approach is the computational saving based
Let teG be a threshold of two groups (foreground
on arithmetic operations required for calculating
and background) in an image. The co-occurrence
entropies compared to the local and joint entropy
matrix ,T defined by (1), partitions the matrix into four
approaches.
quadrants, namely, A, B, C, and D, shown in Fig. .1
This paper is organized as follows. Section 2 de-
These four quadrants may be separated into two
scribes previous work on entropy-based thresholding
types. If we assume that pixels with grey levels above
approaches. Section 3 introduces the concept of relative
the threshold be assigned to the foreground (objects),
entropy and presents a relative entropy-based thres-
and those below, assigned to the background, then, the
holding algorithm. In Section 4, experiments are con-
quadrants A and C correspond to local transitions
ducted based on various test images in comparison to
within background and foreground, respectively;
the local entropy and joint entropy methods described
whereas quadrants B and D represent transitions across
in reference (1). Finally a brief conclusion is given in
the boundaries of background and foreground. The
Section .5
probabilities associated with each quadrant are then
defined by
2. PRELIMINARIES PA(t)= ~ ~ Po
i=O.j=O
2.1. Co-occurrence matrix ~ L-I
Pn(t) = ~ jiP
Given a digitized image of size M × N with L
i=0 j=t+l
gray levels G = {0, 1,2,...,L-1}, we denote F = L-I L 1
If(x, Y)M N× to represent an image, where f(x,y)e G Pc(t) = ~ ~ j,P
is the grey level of the pixel at the spatial location (x, y). i=t+l j=t+l
A co-occurrence matrix of an image is an L x L dimen- L-I ~
sional matrix, T= tJL × ,L which contains information Po(t)= ~, .jiP )3(
i=t+l j=O
regarding spatial dependency of grey levels in image
F as well as the information about the number of The probabilities in each quadrant can be further
transitions between two grey levels specified in a parti- defined by the "cell probabilities" and obtained as
cular way. A widely used co-occurrence matrix is an
asymmetric matrix which only considers the grey level 0 | L-I
transitions between two adjacent pixels, horizontally
A B
right and vertically below. )l~ More specifically, let tij
be the (i,j)th entry of the co-occurrence matrix .T
Following the definition in reference ,)1(
D C
M N
tij= ~ ~ 6(I,k), )1( L-I
I-1 k=l
where .giF .1 Quadrants of a co-occurrence matrix.
evitaleR entropy approach 7721
follows by normalization.
object (BO) and object to background (OB). In analogy
with the local entropy defined above, another second-
order joint entropy of the background and the ob-
Ap = pUpA i=0 j=0 ject was also defined in reference )1( and given as
__ , 1 l
follows by averaging the entropy H(B;O) resulting
from quadrant B, and the entropy H(O; B) from quad-
i=0 j i=0 j=O
rant D.
lij
)t(t,~oZlH = (H(B; O) + H(O; B))/2
i-O j=O
-- 2 pologpuB B
for O<i<_t,O<j<t )4( i=0 j=|+l
'o=
~P tij + ~ ~ p~logp .2 111(
= PiJ/PB -- L 1 '
i=t+l j
i=0 j t+l The algorithm maximizing (11) si called the joint
for O<i<t,t+l<jgL-I )5( entropy-based algorithm, which si the second algorithm
developed by Pal and Pal/~)
tq
pC=Pij/Pc= L-I L 1 ,
2 Z ti,
3. RELATIVE ENTROPY-BASED
i t+l j-t+l
THRESHOLDING TECHNIQUE
for t+l<i<_L-l,t+l<j<L-I )6(
3.1. Definition of relative entropy
pO=pijp °= L-I tij -'
Let S be an L-symbol source and pj and p) be two
probability distributions defined on .S The relative
i-t+l j--O
entropy between p and p' (or equivalently, the entropy
for t+l<_i<L-l,O<_j<t. )7( ofp relative to p') is defined by
L-I
L(p;p')= ~ pjlogQ. )21(
2.3. Algorithms of Pal and Pal )l<
j=O pj
The algorithms suggested by Pal and Pal m attempted The definition given by Equation (12) was first
to take advantage of spatial correlation in an image. introduced by Kullback ~vI as a distance measure between
By doing so, they introduced two concepts of second- two probability distributions, and later was found to
order entropy based on Equations (4)-(7), which are be very useful in many applications, 8t ~2~ Since the
called local entropy and joint entropy. information contained in an image source can be de-
Since quadrant A and quadrant C reflect the local scribed by its entropy, which in turn can be completely
transitions from background to background (BB), and characterized by source symbol probabilities, the relative
object to object (OO), they defined local entropy of entropy basically provides a criterion to measure the
background and local entropy of object by HB(t ) and discrepancy between two images determined by prob-
Ho(t), respectively, as follows. ability distributions pj and p), respectively. The smaller
'iiA
the relative entropy, the less the discrepancy. It is
)t(~2BH = -- ~ OP log p 0 )8( natural to use relative entropy as a measure of difference
i=o j=o
between an image and its segmented image; in our case,
l ~"1 ~ ~' a bilevel thresholded image. There are several synonyms
c c
H~)(t) = --2i=,+1 j=,+a p°I°gp0" )9( of relative entropy, e.g. cross entropy, Kullback-
Leiber's discrimination distance function and directed
It should be noted that )8( and )9( are determined by divergence.
the threshold t, thus they are a function of t. In order to obtain a bilevel image of good quality,
By summing up the local entropies of the object and our aim is to find a threshold to segment an image such
the background, the second-order local entropy can be that the resulting thresholded bilevel image will best
obtained by match the original image. Using the measure of relative
)t(la~o,H (2) = H~2)(t) + H~'(t). )01( entropy, one can choose the threshold t in such a
manner that the grey-level probability distribution p)
The algorithm proposed by Pal and Pal m is one to of the thresholded image has minimum relative entropy
select a threshold which maximizes the ~,)~2olH over t. In L(p;p') with respect to that of the original image, p.
this paper, it will be called the local entropy-based More specifically, the desired threshold t minimizes the
algorithm. discrepancy between p and p', where p and p' are the
Alternatively, quadrant B and quadrant D provide grey-level probability distributions of the original
edge information on transitions from background to image and the resulting thresholded image, respectively.
8721 C.-I GNAHC et .la
3.2. Joint relative entropy-based approach PB(t)
pi~m(t) = qn(t)=
As indicated previously, a thresholding method based (t + 1) × (L- t - li
on first-order statistics of an image does not consider for O<i<t,t+l<j<L-1 (15)
spatial correlation of an image. Therefore, exploiting
Pc(t)
the spatial dependency of the pixel values in the image )t()C~:p = qc(t) =
can help to determine a good threshold. It seems (L- t- )1 x (L- t- )1
reasonable to extend the first-order relative entropy
for t+l<i<L-l,t+l<j<L-1 (16)
defined in the previous subsection to a second-order
joint relative entropy between jiP and p'ij, where j~p and Po(t)
)t()O~ip = qo(t) -
j~'p are the transition probability distributions of the
(L- t - 1) × (t + 1)
co-occurrence matrices defined by Equation (1) and (2)
generated by the original image and the thresholded for t+l<i<_L--l,O<_j<t. (17)
image, respectively. Since transition probability distri-
butions defined by the co-occurrence matrix contain where PA(t), PB(t), Pc(t), and Po(t) are defined by
the spatial information which reflects homogeneity Equation (3). For each selected t, pi~ )A(, (t), Pij )n(, (t), Pij )C(, (t),
within groups (quadrants A and C in Fig. ,)1 and and p'~°)(t) are constants in each individual quadrant
changes across boundaries (quadrants B and D in and only depend upon the quadrant to which they
Fig. ,)1 one can envision that a better result may be belong. Therefore, we can denote them by qA(t), qn(t),
obtained if we choose the thresholded bilevel image to qc(t), and qo(t), respectively.
be the one which has the best transition match to that
of the original image in terms of relative entropy.
3.4. Relative entropy-based algorithm
Let the joint relative entropy of the probability dis-
tributions j~p and j~'p be defined by: By expanding Equation (13), we have:
L(p;p')= L-I "~ L-I ~ pijlog pij, (13) L(p;p')= L-I ~ L Z 1 ,i~Pg°ljip j
i=o j=o j~p i=0 j=O Pij
where j~p and j~'p are the transition probabilities from L-1 L-1 L 1 L-1
= E E Pijlogpij- E E Pijlogp'ij.
grey level i to grey levelj of the original image and the
i~O j=O i-O j 0
bilevel image, respectively. Minimizing L(p;p') over t )81(
generally renders a bilevel image which best matches
the original image. Because the first term in Equation (18) is independent
It should be noted that when we threshold an image, of the threshold t, minimizing the relative entropy
we basically assign all gray levels in an original image described by Equation (13) is equivalent to maximizing
to either 0 or 1 which corresponds to background or the second term of Equation 08).
objects. As a result, there are only two grey levels in We can simplify even further the second term of the
the thresholded image. The subscript ij used in the not- right side of the Equation (18) as follows:
ation of the transition probability j~p still refers to the L-1 L 1
~ pijlogp'ij = ~pijlOgqA(t)+ ~pijlogq,(t)
grey levels of the original image. In addition, the stat-
istics of pixels not adjacent to one another could also i-0 j=0 A B
be considered, but the estimation of probabilities for + ~ jiP log qc(t) + ~ jiP log qo(t)
such cases would be very difficult. In this paper we only C D
consider the 'asymmetric co-occurrence matrix defined = Pa(t)logqA(t) + Pn(t)logqn(t)
in Section 1.2 for joint relative entropy.
+ Pc(t) log qc(t) + Po(t) log qo(t).
(19)
3.3. Co-occurrence matrix of a thresholded
bilevel image This implies that in order to obtain a desirable
threshold for classifying the object from the background,
Let us assume that t is the selected threshold. By
we need only maximize the last expression in Equation
assigning 1 to all grey levels above threshold t, G 1 =
(19) over t. The expression consists of four terms only,
{t + 1 .... , L- }1 and 0 to all grey levels below t, 6 2 = each of which is a product of Pi and log qi(t) for i = A,
,0{ 1 ..... t}, we obtain a binary image. It should be
B, C, D. In comparison with Equations (8) and (9)
noted that the grey levels in G 1 will be treated equally
required for the local entropy and Equation 1( l) for
likely in probability, as will be the grey levels in .2G
the joint entropy, the computational load for the relative
Consequently, the P'ij can be found as follows (see
entropy is significantly reduced. From Equations (8)
Fig. :)1
and (9), (t + 1) 2 -k- (L- t) 2 multiplications are required
P,4(t) to calculate j~p log j~p for finding the local entropy, and
)t()A~;P = qa(t) =
from Equation (11), 2(t + I)(L-- t) multiplications for
(t + 1) × (t + I i
the joint entropy. However, only four multiplications
for O<_i<t,O<j<t (14) and four divisions are needed in Equations (19) for the
Relative entropy approach 9721
relative entropy. As a result, the computational saving grey-level histogram of the building image given by
can be tremendous when the size of an image is very Fig. 5(e). It should be noted that the histogram of
large. Fig. 5(e) is very different from that of previous images,
Figs 2(e), 3(e) and 4(e).
.4 LATNEMIREPXE STLUSER
tnemirepxE :5 Coffee puc ,egami Fig. 6(a)
In order to see the performance of the relative entropy-
Compared to the grey-level histogram of the building
based thresholding method we conducted tests for a
image, Fig. 5(e), the coffee cup image has a very similar
set of various images. As shown in experiments, the
grey-level histogram distribution, Fig. 6(e). Coinciden-
relative entropy approach provides an alternative effi-
tally, the relative entropy produced the same threshold
cient and effective image thresholding tool. All test
t = 237 which was used for the building image. The
images have 256 grey levels. In all experiments, the
local entropy and joint entropy produced Fig. 6(b)
images labelled (a) are original images; the images
and (c) with t = 130 and t = 156, respectively. As shown
labelled (b), (c) and (d) are generated by the local
in Fig. 6(b)-(d), Fig. 6(d) picks up the open edge of the
entropy, joint entropy and relative entropy, respectively.
cup while Fig. 6(b) and (c) show the side edges of the
All the figures labelled (e) represent the corresponding
cup.
grey-level histograms of the original images.
tnemirepxE 6: Vase ,egami Fig. 7(a)
Experiment :1 Peppers ,egami Fig. 2(a)
The grey-level histogram of the vase image is very
From Fig. 2(b-d), it is obvious that the joint entropy
different from that of other images, where its grey
produced the worst image with threshold t = 90, while
levels are distributed more uniformly than others. The
the local entropy and relative entropy produced an
image Fig. 7(c) produced by t = 361 for the joint entropy
identical image since both generated the same threshold
does not seem as good as Fig. 7(b) and (d) with t = 521
t= 127.
for the local entropy and t = 231 for the relative entropy,
respectively. In addition, Fig. (d) looks a little better
tnemirepxE 2: F-16 jet ,egami Fig. 3(a)
than Fig. 7(b), since there is blurring over the top of
In this image, three methods generated different the vase in Fig. 7(b).
details, shown in Fig. 3(b-d). For instance, the local
entropy with threshold t = 511 gave the best description tnemirepxE :7 Lena ,egami Fig. 8(a)
of the lettering "F-16" on the tail, while the relative
Figure 8(b)-(d) shows that the quality of images
entropy with threshold t = 571 shows more clearly the
produced by t = 951 for the local entropy, t = 124 for
cockpit, the insignia, and the lettering "US AIR FORCE"
the joint entropy and t = 170 for the relative entropy
on the fuselage. The joint entropy with t = 731 produced
is nearly the same except that they pick up different
an image between the quality of the other methods.
tiny descriptions and details. For instance, ~ relative
entropy shows Lena's mouth at the expense of some
Experiment :3 Couple ,egami Fig. 4(a)
details of the feather on Lena's hat. In contrast, the
Evidently, the couple image thresholded by the rela- local entropy and the joint entropy give a little more
tive entropy produced the best image, Fig. 4(d), com- detail on the feather while missing Lena's mouth and
pared to those in Figs 4(b) and (c) generated by the some details of Lena's hat.
local entropy and joint entropy. The threshold used
for the relative entropy was 111, whereas both the local tnemirepxE :8 City ,egami Fig. 9(a)
entropy and joint entropy used the same threshold
Like the Lena image, the city images produced by
.171
t = 321 for the local entropy, t = 821 for the joint
entropy and t = 211 for the relative entropy are very
Experiment 4: Building ,egami Fig. 5(a)
close. It is interesting to compare the grey-level histo-
The building image is interesting. Both the local grams of these two images. If one of them is flipped
entropy and joint entropy generated close thresholds over, it is found that their distributions turned out very
t = 166 and t = 172, respectively. As a result, the cor- similar. Therefore, these two experiments should be
responding thresholded images, Figs 5(b) and (c) are expected to have similar results.
close. However, Fig. 5(d) produced by the relative Based on the experiments conducted above it seems
entropy using threshold t = 237 is quite different from that the grey-level histograms of these eight images can
Figs 5(b) and (c). The local entropy and joint entropy be roughly classified into four categories. The first
seem to give a better description of the building while three experiments (peppers, F-16 jet and couple images)
failing to pick up the middle edges of the building and are grouped together into Category 1 since their histo-
the outside stairs, which are shown in Fig. 5(d). The grams have many saw-like sharp peaks with short
reason for this is probably that the relative entropy can durations. The next two experiments (building and cup
best match all possible transitions made from one grey images) are in Category 2 because they share the same
level to another. This seems to be justified from the characteristics of the histograms which are Gaussian-
1280 C.-I CHANG et al.
(a) original image
.p
(c) joint entropy (t=90) (d) relative entropy (t= 127)
1.0
0.09
I
0,08
0,07
0.06
0.05
0.04
0.03
0.02
10.0
.A .^ A.^A_,
00 50 001 051 200 250 3~
(e) grey-level histogram
Fig. 2. Peppers image: (a) original image, (b) image obtained by local entropy, t = 127, (c) image obtained
by joint entropy, t = 90, (d) image obtained by relative entropy, t = 127, (e) grey-level histogram.
Relative entropy approach 1821
~...---,~.,.~,,,~,," ..., ...,
d~
(a) origina ! image (b) local entropy (t= 115)
- ~
-w,
(c) joint entropy (t= 137) (d) relative entropy (t= 175)
2.0
81.0
61.0
41.0
21.0
1.0
80.0
I
60.0
0'04 t
0.02
o, ^- -^,^ ~'-
0 50 00I 051 002 052 003
(e) grey-level histogram
Fig. .3 F-16 jet image: )a( original image, (b) image obtained by local entropy, t = 115, )c( image obtained
by joint entropy, t = 137, )d( image obtained by relative entropy, t = 175, )e( grey-level histogram.
1282 C.-I CHANG et al.
I
(a) original image i"
:n::t
~, ~. ,,--
(c) joint entropy (t= 171) (d) relative entropy (t= 111)
0.25
0.2
0.15
I.0
0.05 IAAltA ..... A
00 50 ' 001 051 200 250 300
(e) grey-level margotsih
Fig. 4. Image of couple: (a) original image, (b) image obtained by local entropy, t = 171, )c{ image obtained
by joint entropy, t = 171, (d) image obtained by relative entropy, t = 111, (e) grey-level histogram.
Relative entropy approach 1283
m i i (cid:127) K g (cid:127)
(a) original image (b) local entropy (t=166)
N
(c) joint entropy (t= 172) (d) relative entropy (t=237)
52.0
0.2
51.0
0.I
50.0
0 - -
0 50 001 051 200 250 300
(e) grey-level histogram
Fig. .5 Building image: (a) original image, (b) image obtained by local entropy, t = 166, (c) image obtained
by joint entropy, t = 172, (d) image obtained by relative entropy, t = 237, (e) grey-level histogram,
