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Novel Automatic Exact Histogram Specification for Contrast Enhancement in Images Debashis Sen and Sankar K. Pal Center for Soft Computing Research, Indian Statistical Institute 203 B. T. Road, Kolkata, India 700108. E-mail: {dsen_t.sankar}@isical.ac.in.

- increase in the overall discriminablity among the sam­ ples in the histogram. - maximization of a measure that represents increase in information entropy and decrease in average image ambiguity.

Abstract-Histogram equalization, which aims at information maximization, is widely

used in different ways to perform

contrast enhancement in images. In this paper, an automatic exact histogram specification technique is proposed, and used for global and local contrast enhancement of images. The desired histogram is obtained by first subjecting the image histogram to a modification process and then by maximizing a measure that represents increase in information and decrease in ambiguity. In comparison to a few existing methods, the effectiveness of the proposed automatic exact histogram specification technique in enhancing contrasts of grayscale images is demonstrated through visual assessment of results.

Index Terms-Average image ambiguity, beam theory, contrast enhancement, exact histogram specification, fuzzy sets, informa­ tion entropy

I.

INTRODUCTION

Contrast enhancement is an important image processing technique that makes various contents of images easily dis­ tinguishable through suitable increase in contrast. Contrast enhancement through histogram specification is obtained by suitably changing the underlying histogram into a desired one. Histogram specification can be used to perform global, mean brightness preserving, dynamic and local contrast enhance­ ment in images. Exact histogram specification [1] guarantees that the histogram obtained after the processing is almost exactly the desired one. However, there does not exist any obvious choice for the desired histogram [1]. Mostly, the desired histogram has been considered as uniform, where the specification process is referred to as histogram equalization. A few times the desired histogram has been considered as the one which makes the histogram of perceived brightness uniform (histogram hyperbolization [2]). In certain cases, the desired histogram has been considered as unimodal I bimodal Gaussian or exponential depending on the underlying application. It is well known that histogram equalization aims at max­ imization of information entropy, which gives the amount of available information. Therefore, exact histogram equalization, which transforms the original histogram to a nearly uniform or an exactly uniform histogram, could be inferred as an exact histogram specification method, where the desired histogram is obtained by maximizing information entropy measure. We propose that an automatic exact histogram specification tech­ nique more suitable for contrast enhancement could be devised considering:

We consider the first aforesaid aspect, as it is demonstrated in [3] that increasing the overall discriminablity among samples in the histogram through a histogram modification process aids operations such as image segmentation that require distinction of image contents. Average image ambiguity is considered in the second aforesaid aspect, as in [4], it is pointed out that average image ambiguity reduces with increase in the distinguishablity of various image parts. In this paper, a novel automatic exact histogram specifica­ tion technique is presented. The desired histogram is obtained by first subjecting the image histogram to a modification process that increases the overall discriminablity among sam­ ples in the histogram, and then maximizing a measure that represents increase in information entropy [5] and decrease in average image ambiguity [4]. The aforesaid approach is based on the finding that increase in information entropy and decrease in average image ambiguity, which are indicators of contrast enhancement, are contradictory. The proposed exact histogram specification technique is used for global and local contrast enhancement of grayscale images. These enhance­ ment techniques can be directly extended to color images using the approach given in [6]. Global and local contrast enhancement performed using the proposed automatic exact histogram specification technique is compared to the usage of exact histogram equalization [1] and to a few state of the art existing techniques. The effectiveness of the proposed exact histogram specification technique is demonstrated through visual assessment of contrast enhancement performances. II.

DETERMINATION OF THE DESIRED HISTOGR AM FROM

THE ORIGINAL HISTOGRAM: IN CRE ASING THE OVER ALL DIS CRIMINABILITY AMONG SAMPLES IN THE HISTOGR AM

In the proposed approach of determining the desired his­ togram from the original histogram, the first step is a histogram modification process. We increase the overall discriminablity among samples in the histogram that correspond to pixels in the underlying image through the modification process. The histogram modification process, which is based on an altered version of the density modification approach given in [3], is

978-1-4244-9799-7/11/$26.00 ©2011 IEEE

57

described here. Without loss of generality, we shall describe the process assuming a gray-level histogram of a grayscale image. Let 0 be the gray-level histogram of the grayscale image under consideration. We first normalize 0 as follows:

O(i) . E G L ZEG O(z)' z

- . O(z) =

(1)

where G is the universal set of gray levels. The quantity D(i) gives the probability of occurrence of gray value i in the image under consideration. We shall now consider some concepts from beam theory of solid mechanics. Consider a solid whose shape is given by D. Let the solid be placed over a beam of uniform (along the length) height and width. Let us assume that the entire uniform beam and the entire solid are made out of the same material. Consider that the beam is rested upon two pivots at the two ends. Let the two ends refer to the minimum (9min = minzEG z) and maximum (9max = maxzEG z) gray values. Let us consider that a uniform (along the length) force ("ip), say due to gravity, is always acting upon the uniform beam. Figure 1 gives the pictorial representation of the aforesaid setup. The total force acting on the beam is P(i) = D(i) + "iP and we consider "iP as ( L ZE G O(Z))-l, which represents one unit (force) with respect to O. !

Fig. 1.

!

!

!

!

I

!

!

!

!

!

!

!

I

I

!

!

---+y,

The setup for the proposed histogram modification

Once D(i) is obtained from O(i) using (1), we calculate the bending moment B(i) due to the force P(i) at a gray value i E G (refer Figure 1) using Euler-Bernoulli beam theory [7] as follows:

B(i) =

(RXI) - (CP(i)

x

(I-CG(i)) ) , where 1= i-9min

(2) In the above, CP(i1) stands for the total (cumulative) force between the gray values 9min and i, and it is given as i i CP(i) = P(z) = (D(z) + "ip) (3)

2::

2::

Z=gmin

Z=gmin

The symbol CG(i) stands for the center of gravity between the gray values 9min and i, and it is given as

CG(i) =

1 CP(i)

i

2::

(z - 9min)P(Z)

(4)

Z=gmin

In (2), R is the reactive force at the pivot at 9min and it is calculated as

R = CP(9max) x

[ L- CG(9max) ] L

' where L= 9max - 9min (5)

Now, consider the following function obtained by normalizing the bending moment B:

B(i) . . J1(z) = Z E G maxzEG B(z)'

(6)

Let us now consider a few attributes of J1. AI. Nonnegativity- It is evident from (2) that B(i) 2: 0 for all i E G and hence from (6) we get J1(i) 2: 0 for all i E G. A2. Range- From attribute Al and the observation that supz J1(z) == maxzEG J1(z) = 1 (see (6)), we get that 0:::; J1(i):::; 1 for all i E G. A3. Vanish Identically- It is easily deducible from (2) that B(i) = 0 only when i = 9min or i = 9max and hence from (6) we get J1(i) = 0 only when i = 9min or i =

9max· A4. Concavity- It is known that the relation between the bending moment B at a position (gray value) and the force P at that position (gray value) is as follows [7]:

B"

ex:

-P

(7)

Now, when the interval [9min, 9max] is considered as an interval in the real line, it becomes evident from (7) that the bending moment B would have a second derivative B" in [9min, 9max] and always B" :::; 0 as P 2: O. This condition B" :::; 0 is a necessary and sufficient condition for B to be a concave function. Hence, from (6) we get that J1 is a concave function in G ([9min, 9max]). From attribute A2, it is evident that J1 can be considered as a membership function corresponding to a fuzzy set [8] defined in G. From attributes A3 and A4, we can say that the value of the membership function J1 at a gray value i represents a property 'farness of the gray value from the nearest among 9min and 9max . Note that, the aforesaid terms 'farness' and 'nearest' are inherently defined in J1. The gray value where J1 takes a value of unity, is the gray value that is equally far from both 9min and 9max. A gray value smaller (larger) than that gray value is nearer to 9min (9max) compared to 9max (9min). '

As the groups of samples in the histogram associated with

9min and 9max are the most discriminable ones from each other, the aforesaid property J1 at a gray value gives the farness of the samples in the histogram associated with that gray value from the nearest among the mutually most discriminable groups of samples in the histogram. Therefore, the property J1 can be used to modify the histogram 0 such that the overall discriminability among samples in the histogram is increased. It is evident from (2) and (6) that J1 is dependent on the histogram O. However, one might like to have the property 'farness of a gray value from the nearest among 9min and 9max' such that it depends only on the gray value (i E G) and not on the histogram (0). Note that, if the value of "iP is considered such that "iP > > maxzEG D(z), the member­ ship function approximately becomes independent of 0 and depends only on the underlying gray value.

Now, we know that the bending moment B refers to a membership function J1, which represents a useful property

58

'famess of a gray value from the nearest among 9min and 9max '. We shall now modify the histogram 0 using concepts from beam theory, where the bending moment B will be appropriately considered. The curvature due to the bending moment B(i) at a gray value i E G is as follows:

p t.

B(i)

( ) - t(i) _

In order to ascertain that considering the aforesaid quantity as the modified histogram is appropriate, consider the following analysis. From the calculation of B (9)-(13) and its relation to J..L and 0, we infer that the modified histogram

OM

OM

at a gray value is such that: 1. The number of samples in

(8)

t(

where i) is the moment of inertia (opposing the bending) at i, which is calculated as o (i)+"YL

t(i)

L

=

(k - C(i))2

where c ( i) denotes the centroid of the solid and the beam taken together at the gray value i and it is given by 0.5 x (O(i) + 1,). We consider that the height of the beam 1, (see Figure 1) equals ( zEG which represents one unit with respect to O. However, it is observed that some 0 may correspond to solids which when considered would result in curvatures (p( i) that are very large at a few gray values, making the curvatures at other gray values insignificant. We find that such situations are unfavorable for increasing the overall discriminability among samples in the histogram, as many gray values would be ignored during the modification process. Therefore, we consider the following measure instead of p at a gray value i E G:

L

P'

O(z))-l,

( .) t(i) t

=

B(i)

+ maxzEG

(10)

t(z)

( ) L.JzEGi) (z) i p(i) i O(i) max ( [maxp(z) -p(i) (z), 0]) maxp zEG zEG p(

= "'"

'

P

EG

'

(11)

Once the value of for all E G has been obtained, we perform the following operation: +D x

=

(12)

where D is a real value in the interval (-1,00). We then normalize 0 as follows:

O t.

()

=

( )O , i L ZEG (z) Oi

EG

(13)

We consider the quantity B( i) as the probability of occurrence of gray value i in the image having the modified histogram. The modified histogram, say is then determined from B such that the number of samples in the modified histogram equals that in the original histogram, as follows:

OM

OM(i)

=

lB x

L

L O(z)J,

zEG

O(z) OM.

From the first aforesaid aspect, we see that the modified histogram contains a signature of the original histogram From the earlier discussion on the bending moment B and J..L, we infer that a larger value of J..L at a gray value means that the gray value is not nearer to any of the two mutually most discriminable groups of samples in the histogram, which are at the 9min and 9max values. Now, in order to have a increased overall discriminablity among samples, the samples in the modified histogram at a gray value should be larger (smaller) when the value of J..L at that gray value is smaller (larger), so that, more samples are nearer to any of the two mutually most discriminable groups of samples. From the second aforesaid aspect, we see that the modified histogram has more samples nearer to any of the two mutually most discriminable groups of samples compared to the original histogram. Therefore, overall discriminablity among samples

OM

O.

is increased by the histogram modification process and it is appropriate to consider as the modified histogram.

OM

and then normalize p as follows:

p' t.

OM

(9)

k=O

OM

is larger (smaller) at a gray value when the number of samples in 0 at that gray value is larger (smaller ), for a fixed value of J..L at that gray value. 2. The number of samples in is larger (smaller) at a gray value when the value of J..L at that gray value is smaller (larger), for a fixed number of samples in 0 at that gray value.

L

i EG

(14)

OM(z)

- ZEG The discrepancy € = ZEG omitted by adding one sample to each of the (number of occurrences) in

€

is then largest bins

III. DETERMINATION OF THE DESIRED HISTOGRAM FROM THE ORIGINAL HISTOGRAM: DETERMINATION OF THE PARAMETER D Here, we consider a trade-off between increase in informa­ tion entropy and decrease in average image ambiguity in order to determine a parameter required to get the desired histogram from the original histogram. Consider the expression of O(i) from (12), which we rewrite below.

( ) max ( [maxp(z) -p(i) zEG

Oi

=

+D x

(z), 0]) , i maxp zEG

EG

(15) As mentioned earlier, D is a real value in the interval ( -1,00 ). From Section II, it can be easily deduced that i) = 0 only when i = 9min or i = 9max. Therefore, it is evident from (15) that O(9min) = O(9max) and when D is close enough to -1, O(i) is non-zero only at i = 9min and i = 9max. In such a case, all the samples in the modified histogram would be at 9min and 9max. Now as D --+ 00, it is evident from (15) that the values of O(i) Vi tends to be almost the same. In such a case, all the samples in the modified histogram tend to be more or less equally distributed (uniform) at all gray values. From the above discussion, it is evident that the parameter D is a very significant one. We devise an approach

p(

OM

OM

59

here to determine the parameter D in an optimal sense. The modified histogram OM that uses the optimal value of D is the

desired histogram in our novel exact histogram specification technique. As mentioned earlier, when D --+ 00, OM tends to be uniform. It is well understood and also empirically observed that information entropy [5] calculated from OM increases as OM tends to be uniform [9]. Therefore, information entropy due to OM increases as D --+ 00. From the theory of histogram equalization [9] we know that information entropy maximiza­ tion is a potent way of contrast enhancement. However, it is also well known that information entropy maximization may cause artifacts such as washed-out effect and clutter intensification in the underlying image [10]. On the otherhand, when D --+ -1, the number of samples in OM tends to be non-zero only at 9min and 9max. Therefore, as also evident from the earlier discussion, information entropy due to OM decreases as the value of D decreases. However, we find that the average image ambiguity measure [4] based on grayness ambiguity [4] calculated from OM also decreases as D --+ -1. It is known that minimization of average image ambiguity or other measures of image ambiguity is a potent way of contrast enhancement [4]. The effect of the aforesaid minimization process in the underlying histogram is such that the samples in it get concentrated near two highly distinct gray values. Therefore, although the overall contrast of the underlying image would be intensified, the process may lead to loss of details. From the above explanation, we see that increase in in­ formation entropy and decrease in average image ambiguity calculated from OM are contradictory in the context of change in the parameter D. However, both increase in information entropy and decrease in average image ambiguity are de­ sirable traits in contrast enhancement of images. Therefore, the requirement is to perform a trade-off between increase in information entropy and decrease in average image ambiguity in order to determine the parameter D. Such a trade-off would constrain both increase of information entropy and decrease of average image ambiguity, which would diminish the aforesaid disadvantages of both. We consider the maximization of the following measure to determine D:

F

=

(1 - AIA)

x

II

(16)

In the above, II represents the Shannon's information entropy H [5] normalized such that II E [0,1] and H is calculated from OM. Average image ambiguity is represented by AIA, where AlA E [0,1]. The computation of AlA is based on grayness ambiguity calculated from OM, which has been elaborately explained in [4]. As mentioned earlier, we intend to obtain the optimal value of D as Dopt = arg max F (17) DE(-I,oo)

We consider that OM (see (14)) as the desired histogram in our exact histogram specification technique, which is obtained considering D = Dopt in (12). We have carried out the

(a) Original His- (b) Membership togram function 11

(c) F(D)

Desired (d) Histogram (OM, D Dopt ) =

(e) Modified Histograms OM with D

=

-

.

99 , -.1, .05, .1, 1, 10, 1000

Fig. 2. Graphical representation of the determination of the desired histogram from the original histogram

process of determination of desired histogram considering several original histograms and we have made the following observations: - Both AIA and II increase with increase in D. - Rate of increase of both AIA and II decrease with increase in D. - With respect to increase in D, initially, the rate of increase of II is more than that of AIA. Later, the rate of increase of II becomes less than that of AIA. - AIA and II as functions of the parameter D fit the expressions C1(-exp(-KD) + C2) and C3(-exp(-rD) +C4), respectively, where r > K and C1, C2, C3 and C4 are arbitrary constants. From the expression of F in (16) and the first three aforesaid observations, it can be easily inferred that F as a function of D E (-1, 00 ) would have a unique maximum. The same inference can be mathematically deduced considering the fourth observation. The value of D at the solitary maximum is Dopt and hence, searching for Dopt in the entire interval ' ( -1, 00 ) is not required. The nature of dependency of OM on the value of D and the behavior of F as a function of D, which are explained in Section III, are shown graphically in Figure 2. Note the unique maximum of F as a function of D in Figure 2(c), and the near extreme cases of OM when D is very near to -1 and when D takes a very large value in Figure 2(e). Observe in Figure 2 that as explained in Section II, the overall discriminability among samples in OM has increased compared to the original histogram as more samples in OM are nearer to any of the two mutually most discriminable groups of samples, which are at the minimum and maximum gray values. As pointed out in Section II, notice (see Figure 2(a) and (d)) the signature of the original histogram OM in the form of valleys and peaks. The attributes of the membership function /1 stated in Section II is evident from Figure 2(b). IV.

IMPLEMENTATION AND EXPERIMENTAL RESULTS

Once the desired histogram is determined using the pro­ posed approach, we implement exact histogram specification as proposed in [1]. The scheme given in [1] is based on strict ordering among image pixels via calculation of local mean values.

60

(a) image

(b) proposed EHS

(c) EHE

(d) GLG

Fig. 3. Global contrast enhancement of grayscale images using the proposed EHS,EHE and GLG techniques

(a) image

(b) Prop. (c) CLAEHS CLAEHE

(d) MUM

(e) RACE

Fig. 4. Local contrast enhancement of a grayscale image using the proposed CLAEHS,CLAEHE,MUM and RACE techniques

In this section, we perform global and local contrast en­ hancement of grayscale images by considering the proposed automatic exact histogram specification technique for global and local histograms. We provide experimental results in order to demonstrate the effectiveness of the proposed automatic exact histogram specification technique in comparison to a few existing methods. Due to space constraint, we have only shown application on grayscale images. The extension to color images can be directly performed using the approach given in [6]. Global contrast enhancement is aimed at increasing the overall contrast of an image. Consider the grayscale image given in Figure 3(a). The proposed exact histogram speci­ fication (proposed EHS) technique is applied on the global gray-level histogram of the image in order to perform global contrast enhancement. The performance of the proposed tech­ nique is compared to that of exact histogram equalization (EHE) [1] and gray-level grouping (GLG) [10]. The images in Figures 3(b), (c) and (d) are obtained by performing global contrast enhancement using the proposed EHS, EHE and GLG techniques, respectively. As can be seen the overall contrast is higher in the image in Figure 3(b), whereas, the brightness seems to be higher in the image in Figure 3(d). Local contrast enhancement is aimed at increasing con­ trast in local neighborhoods in images in order to reveal minute details. We consider the contrast limited adaptive (local) enhancement approach pioneered in [11], in order to carry out local contrast enhancement using the EHE and the proposed EHS techniques, which are applied to histograms calculated within local neighborhoods in images, and we call them contrast limited adaptive EHE (CLAEHE) and contrast limited adaptive EHS (proposed CLAEHS). The performance of the proposed CLAEHS is compared to that of CLAEHE, local standard deviation distribution modeling based unsharp masking (MUM) proposed in [12] and human visual properties based algorithm RACE given in [13], as all of them are adaptive (local) contrast enhancement techniques. Consider the grayscale image given in Figure 4(a). The images in Figures 4(b), (c), (d) and (e) are obtained by performing local contrast enhancement using the proposed CLAEHS, CLAEHE, MUM and RACE techniques, respectively. As can be seen, details such as that in the top-right of the image has been revealed better in the image in Figure 4(b). V. CONCLUSION

An automatic exact histogram specification technique has been proposed in this paper, and it has been used for global and

local contrast enhancement of images. The desired histogram has been obtained by first subjecting the original histogram to a modification process that increases the overall discriminablity among samples in the histogram and then by maximizing a measure that represents increase in information and decrease in ambiguity, which are contradictory indicators of contrast enhancement. Based on visual assessment of results, the proposed exact histogram specification technique has been found effective in enhancing contrasts of grayscale images in comparison to a few existing methods. It has been also found that the use of the new exact histogram specification method for grayscale images can be directly extended to color images. REFERENCES [1] D. Coltuc,P. Bolon,and 1. M. Chassery,"Exact histogram specification;' IEEE Trans. Image Process., vol. 15,no. 5,pp. 1143-1152,2006. [2] D. T. Cobra, "A generalization of the method of quadratic hyper­ bolization of image histograms," in Proceedings of the 38th Midwest Symposium on Circuits and Systems, vol. I, August 1995,pp. 141-144. [3] D. Sen and S. K. Pal, "Feature space based image segmentation via density modification," in Proceedings of the IEEE International Conference on Image Processing, 2009,pp. 4017-4020. [4] ,"Generalized rough sets,entropy,and image ambiguity measures," IEEE Trans. Syst.. Man. Cybern. B, vol. 39,no. 1,pp. 117-128,2009. [5] c. E. Shannon,"A mathematical theory of communication," Bell System Technical Journal, vol. 27,pp. 379-423,1948. [6] S. K. Naik and C. A. Murthy,"Hue preserving color image enhancement without gamut problem," IEEE Trans. Image Process., vol. 12,no. 12, pp. 1591-1598, 2003. [7] I. B. Prasad, Applied Mechanics and Strength of Materials, 5th ed. Delhi,India: Khanna Publishers,1983. [8] G. K1ir and B. Yuan,Fuzzy Sets and Fuzzy Logic: Theory and Applica­ tions. New Delhi,India: Prentice Hall,2005. [9] A. K. Jain, Fundamentals of Digital Image Processing. New Delhi, India: Prentice Hall of India,2001. [10] Z. Y. Chen, B. R. Abidi, D. L. Page, and M. A. Abidi, "Gray-level grouping (GLG): an automatic method for optimized image contrast enhancement-part I: the basic method," IEEE Trans. Image Process., vol. 15,no. 8,pp. 2290-2302,2006. [11] S. M. Pizer,E. P. Amburn,1. D. Austin,R. Cromartie, A. Geselowitz, T. Greer,B. T. H. Romeny,and 1. B. Zimmerman,"Adaptive histogram equalization and its variations," Computer Vision. Graphics and Image Processing, vol. 39,no. 3,pp. 355-368,1987. [12] D. C. Chang and W. R. Wu,"Image contrast enhancement based on a histogram transformation of local standard deviation," IEEE Trans. Med. Imag., vol. 17,no. 4,pp. 518-531,1998. [13] E. Provenzi,C. Gatta,M. Fierro,and A. Rizzi,"A spatially variant white­ patch and gray-world method for color image enhancement driven by local contrast;' IEEE Trans. Pattern Anal. Mach. Intell., vol. 30,no. 10, pp. 1757-1770,2008. --