Study of Various Histogram Equalization Techniques

IOSR Journal of Electronics and Communication Engineering (IOSR-JECE)
e-ISSN: 2278-2834,p- ISSN: 2278-8735.Volume 8, Issue 1 (Sep. - Oct. 2013), PP 12-18
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Study of Various Histogram Equalization Techniques
Prasanth C. R.
Pursing Mtech in Digital signal processing Department of Electronics College of engineering cherthala
Abstract: Histogram equalization (HE) works well on single channel images for contrast enhancement.
However, the technique used is ineffective on multiple channel images. So, it is not suitable for consumer
electronic products, where preserving the original brightness is necessary in order not to introduce unnecessary
visual deterioration. Bi-histogram equalization (BHE) has been developed and it is analyzed
mathematically.BHE separates the input image’s histogram into two, based on its mean before equalizing them
independently so that it can preserve the original brightness up to certain extends. Recursive Mean-Separate
Histogram Equalization (RMSHE) is another technique to provide better and scalable brightness preservation
for gray scale and color images. While the separation is done only once in BHE, RMSHE performs the
separation recursively based on their respective mean. It is analyzed mathematically that the output images
mean brightness will converge to the input images mean brightness as the number of recursive mean separation
increases. The recursive nature of RMSHE also allows scalable brightness preservation, which is very useful in
consumer electronics. Finally a comparative study was made to analyze all the above methods using gray scale
and color images.
Keywords: Bi-histogram equalization, histogram equalization, scalable brightness preservation, recursive
mean-separate
I. INTRODUCTION
Histogram equalization (HE) is a very popular technique for enhancing the contrast of an image. It
basic idea lies on mapping the gray levels based on the probability distribution of the input gray levels. It
flattens and stretches the dynamics range of the images histogram and resulting in overall contrast improvement,
HE has been applied in various fields such as medical image processing and radar image processing. But, HE is
not commonly used in consumer electronics such as TV because it may significantly change the brightness of an
input image and cause undesirable artifacts. In theory, it can be shown that the mean brightness of the histogram
equalized image is always the middle gray level regardless of the input mean. This is not a desirable property in
some applications where brightness preservation is necessary. Mean preserving Bi-histogram equalization
(BHE) has-been proposed to overcome the above mentioned problems.BHE firstly separate the input images
histogram into two based on its mean; one having range from minimum gray level to mean and the other ranges
from mean to the maximum gray level. Next, it equalizes the two histograms independently. It has been
analyzed both mathematically and experimentally. This technique is capable to preserve the original brightness
to a certain extends .There are still cases that are not handled well by BHE.These images require higher degree
of brightness preservation to avoid annoying artifacts. Therefore, this paper proposes generalization of BHE to
overcome such limitation and provide not only better but also scalable brightness preservation. BHE separates
the input images histogram into two based on its mean before equalizing them independently. While the
separation is done only once in BHE, this paper proposes to perform the separation recursively; separate each
new histogram further based on their respective means. It has-been analyzed mathematically that the output
images mean brightness would converge to the input images mean brightness as the number of recursive mean
separations increases. Besides, its recursive nature also implies scalable preservation, which is very useful in
consumer electronic products. The generalization of BHE, namely – Recursive Mean Separate Histogram
Equalization (RMSHE) will be presented with mathematical analysis.
.
II. TYPICAL HISTOGRAM EQUALIZATION
Histogram equalization (HE) has been widely used for contrast enhancement of images by uniformly
distributing the probability of intensity values. Histogram equalization is widely used for contrast enhancement
in a variety of applications due to its simple function and effectiveness. Examples include medical image
processing and radar signal processing. One drawback of the histogram equalization can be found on the fact
that the brightness of an image can be changed after the histogram equalization, which is mainly due to the
flattening property of the histogram equalization.
Let X = {X (I, j)} denote a given image composed of L discrete gray levels denoted as {X0,X1,………XL-1} where
X ( i , j ) represents an intensity of the image at the spatial location ( i , j ) and X ( i , j ) € {X0,X1,………XL-1}.For a
given image X, the probability density function p ( X k ) is defined as

for K = 0 , 1 , . . . , L - 1, where nK
represents the number of times that the level XK appears in the input image X
and n is the total number of samples in the input image. Note that p (X k) is associated with the histogram of the
input image which represents the number of pixels that have a specific intensity Xk. In fact, a plot of nk
vs. Xk is
known as the histogram of X. Based on the probability density function, we define the cumulative density
function as
Where k=O, 1... L-1, and it is obvious that c (XL-1) =l.Thus the transform function of histogram equalization can
be defined as
Then the output image of the histogram equalization, Y = {Y (i, j,)}, can be expressed as
The disadvantages of typical Histogram equalization method is that it cannot however maintain
average brightness level which may result in either under or over saturation in the processed image and Overall
contrast of the input image is degraded after the histogram equalization. More fundamental reason behind the
limitations of the histogram equalization is that the histogram equalization does not take the mean brightness of
an image into account. In the subsequent sections, a new contrast enhancement algorithm is formally proposed
based on the histogram equalization. The new algorithm utilizes the mean brightness which is the one of the
important statistics of an image. This is known as brightness preserving bi histogram equalization (BBHE).
III. Brightness Preserving Bi Histogram Equalization (BBHE)
The ultimate goal of the proposed algorithm is to preserve the mean brightness of a given image while
the contrast is enhanced. The BBHE firstly decomposes an input image into two sub images based on the mean
of the input image. One of the sub images is the set of samples less than or equal to the mean whereas the other
one is the samples greater than the mean. Then the BBHEequalizes the sub images independently based on the
respective histograms with the constraint that the samples in the formal set are mapped into the range from the
minimum gray level to the input mean and the samples in the latter set are mapped into the range from the mean
to the maximum gray level. In other words, one of the sub images are equalized over the range up to the mean
and the other sub images equalized over the range from the mean based on the respective histograms. Thus, the
resulting equalized sub images are bounded by each other around the input mean, which has an effect of
preserving mean brightness.
Denote by Xm the mean of the image X and assume that Xm{X0, X1… XL-1}. Based on the mean, the input
image is decomposed into two sub-images XL and XU as
Where
Note that the sub-image XL is composed of {X0, X1… Xm} and the other image XU is composed of {Xm+1,
Xm+2…XL-1}.Next, define the respective probability density functions of the sub-images XL and XU as

Where k = 0, 1… m, and
Where k = m+1, m+2… L-1, in which nL
k
and nu
k
represent the respective numbers of Xk in XL and XU, and nL
and nU are the total number of samples in XL and XU respectively. The respective cumulative density functions
for XL and XU are then defined as
Similar to the case of HE where a cumulative density function is used as a transform function, let’s define the
following transform functions exploiting the cumulative density functions
Based on these transform functions, the decomposed sub images are equalized independently and the
composition of the resulting equalized sub-images constitute the output of BBHE.That is, the output image of
BBHE, Y, is finally expressed as
Where
it is easy to see that fL(XL)equalizes the sub-image XL over the range (X0, X m) whereas fU(Xu)equalizes the
sub-image XU over the range (X m+1, X L-1).As a consequence, the input image X is equalized over the entire
dynamic range (X0, X L-1) with the constraint that the sample less than the input mean are mapped to (X0, X m)
and the samples greater than the mean are mapped to (X m+1, X L-1).
3.1 Analysis on the brightness change by BBHE
Suppose that X is a continuous random variable, i.e., L = ∞, then the output of the HE, Y is also
regarded as a random variable. It is well known that the HE produces an image, whose gray levels have a
uniform density, i.e
For X0 ≤x ≤X L-1. Thus, it is easy to show that the mean brightness of the output image of the HE is the middle
gray level since

Where E (.) denotes a statistical expectation. It should be emphasized here that the output mean of the HE has
nothing to-do with the input image. That is, it is always the middle gray level no matter how much the input
image is bright/dark. Clearly, this property is not desirable in many applications. Turning the attention to the
man change by the BBHE, suppose that X is a random variable, which has symmetric distribution around its
mean, Xm. When the sub-images are equalized independently, the mean brightness of the output of the BBHE
can be expressed as
Where Pr (X ≤Xm) = Pr (X > Xm) = ½ is used since X is assumed to have asymmetric distribution around
Xm. From above equations we can write
Substituting (24) and (25) in equation (23) we get
Where
Is the middle gray level, which implies that the mean brightness of the equalized image by BBHE locates in the
middle of the input mean and the middle gray level. Note that the output mean of the BBHE is a function of the
input mean brightness Xm. This fact clearly indicates that the BBHE preserves the brightness compared to the
case of typical HE where output mean is always the middle gray level.
In typical HE, no mean-separation is performed and thus, no brightness preservation. In BBHE, the
mean-separation is done once and thus, achieve certain extends of brightness preservation. In cases where more
brightness preservation is required, perform the mean separation recursively; separate the resulting histograms
again based on their respective means. This is known as Recursive Mean-Separate Histogram Equalization
(RMSHE). This is basically a generalization of HE and BBHE in the aspect of brightness preservation
IV. Recursive Mean Seperparate Histogram Equalization (RMSHE)
the output mean E (Y) of typical HE is given as follows:
No mean-separation is performed before equalization and hence, no brightness preservation. Where output
mean is always the middle gray level, XG and has nothing to do with input mean. In fact, HE is equivalent to
RMSHE with recursion level, r=0.Fig. 4.1 shows histogram before and after BBHE. As the output mean, E (Y)
after BBHE is as follows
One mean-separation is performed before equalization and hence, results in some level of brightness
preservation. This is indicated by equation (29) where input mean, Xm has equal weight as the middle gray
level, XG in the output mean. In fact, BBHE is equivalent to RMSHE with recursion level r = 1. In order to

achieve higher brightness preservation, this paper proposed to perform the mean separation recursively; separate
the resulting histograms again based on their respective means. Supposed that X is further separated into 4
portions based on the mean of the two new histograms, Xml and Xmu as shown in Fig. 4.3
Fig. 4.1 Histogram before and after HE or
equivalently,RMSHE, r = 0 Fig. 4.2 Histogram before and after HE or
Equivalently RMSHE.,r=1
Let’s define Xml and Xmu
Fig. 4.3 Histogram before and after , RMSHE, r = 2

Because X is assumed to have a symmetric distribution around Xm. Fig. 4.3 shows the histogram before and after
equalizing the four portions of the histogram independently. This is the result of RMSHE with recursion level, r
= 2. The following shows the formulation of the output mean.
where Pr(X ≤Xml) = Pr(Xml < X ≤Xm) = Pr(Xm < X ≤Xmu) =Pr(X > Xmu) = ¼ is used since X is
assumed to have a symmetric distribution around Xm
From (29) and (30)
Two mean-separations before equalization results in higher-level of brightness preservation compare to zero and
one mean-separations as in the case of HE and BBHE respectively. This is indicated by equation (33) where the
weight of input mean, Xm has increased to three times as much as the weight of middle gray level, XG.
Following the above discussion, it is very reasonable to expect that the brightness preservation will increase as
the number of recursive mean-separations increases. Base on the observations on the output mean for RMSHE
recursion level r = 0, 1 and 2 shown above, it is not difficult to generalize the output mean E (Y) for RMSHE
recursion level r = n as follows:
Equation indicates that as the recursion level, n grows larger, E (Y) will eventually converge to the input mean,
Xm.In other words, it allows scalable degree of brightness preservation range from 0% (output of HE) - 100%
(getting back the original image). In applications of consumer electronics, the variety of image involve are too
wide to be covered with only a specific level of brightness preservation. Therefore, the scalability in this
algorithm is the most desirable property to allow adjustment of level of brightness preservation base on
individual image’s requirement. Equation indicates that as the recursion level, n grows larger, E (Y) will
eventually converge to the input mean, Xm.In other words, it allows scalable degree of brightness preservation
range from 0% (output of HE) - 100% (getting back the original image). In applications of consumer electronics,
the variety of image involve are too wide to be covered with only a specific level of brightness preservation.
Therefore, the scalability in this algorithm is the most desirable property to allow adjustment of level of
brightness preservation base on individual image’s requirement

V. Conclusion
In this paper, a new contrast enhancement algorithm referred as the recursive mean-separate histogram
equalization (RMSHE) with scalable brightness preservation is proposed. The RMSHE is a generalization of HE
and BHE in term of brightness preservation. The main idea lies on recursively separating the input histogram
based on the mean. The ultimate goal behind in the RMSHE is to allow higher-level of brightness preservation
to avoid unwanted noises. The analysis shows that the output mean will converge into the input mean as the
number of recursive mean-separation increases. Therefore, the scalability in this algorithm is the most desirable
property especially in consumer electronics to allow scaling of brightness preservation, suited to individual
image.
REFERENCES
[1] R. Gonzalez and R. Woods, Digital Image Processing, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 2007
[2] Y. Kim, “Contrast enhancement using brightness preserving bi-histogram Equalization,” IEEE Transactions
[3] S. Chen and A. Ramli, “Contrast enhancement using recursive mean separate Histogram equalization for scalable brightness
preservation, IEEE Transactions
AUTHORS PROFILE
PrasanthC.R Received Diploma in Electronics and Instrumentation from govt. Polytechnic
cherthala (first class with distinction, third rank and gold medallist). He passed Btech in
Electronics and Communication Engg from M.G university kerala, india.he currently
pursing Mtech in Digital signal processing from Govt. Engg college cherthala under Cochin
University of science and technology

Study of Various Histogram Equalization Techniques

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