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Procedia Computer Science 46 (2015) 1786 - 1793

International Conference on Information and Communication Technologies (ICICT 2014)

A New Mathematical Model in Image Enhancement Problem

Sasi Gopalana, Arathy Sb *

aAssociate professor, School of Engineering,Cusat, Kochi-682022, India bJRFfellow, School of Engineering, Cusat, Kochi-682022, India

Abstract

In this problem a new mathematical model for image enhancement is developed in fuzzy domain. The triangular membership function is used for fuzzification which reduces the number of parameters for optimisation of the model. Image enhancement has been done using sigmoid operators whose parameters are found by optimisation of entropy and visual factor using simulated annealing. The cross over point is chosen as the membership function corresponding to average intensity level. Scaling is an important part of this model to avoid the discontinuity arising from division of intensity levels.

© 2015Published by ElsevierB.V.Thisis an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-reviewunderresponsibility of organizing committee of the International Conference on Information and Communication Technologies (ICICT 2014)

Keywords:triangular membership function;sigmoid function;simulated annealing

1. Introduction

Image enhancement is the process of increasing the quality of image such that more details can be perceived from the image. Image may be degraded due to many reasons like noise, lighting conditions, poor techniques of photography. These may result in poor contrast, overexposed and under exposed regions of image.

Image enhancement methods can be broadly divided into spatial domain methods and frequency domain methods. Spatial domain methods included the modification of histogram, application of spatial filters. Frequency domain methods included the Fourier transforms based image sharpening, image smoothing etc. Earlier works

* Corresponding author. Tel.:0474 2744840. E-mail address:arathys2010@gmail.com

1877-0509 © 2015 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of organizing committee of the International Conference on Information and Communication Technologies (ICICT 2014) doi:10.1016/j.procs.2015.02.134

mainly concentrated on gray scale images. Later enhancements were done in colour images. Using RGB colour space for image enhancement is inadequate since it produces artificial colours. Hence several works have used HSV models, LSV models, HIS models etc. In this paper we use the HSV model as it separates the chromatic information from achromatic information. Each image in the model is represented by three matrices of hue, saturation and intensity. Hue represents the pure colour of pixels and hence it is preserved. Saturation represents the amount of dilution done to the image by white light, saturation value of '1' indicates no dilution has taken place and hence pure colour. A saturation value of 0 indicates maximum dilution and hence white pixel. Intensity component represents the intensity or brightness of the pixel. A value of 0 indicates dark pixel and 1 indicates white pixel. Enhancement operations are generally performed on Intensity (V) component.

Since uncertainty and vagueness occur in terms like contrast, high quality etc it is found that fuzzy domain is applicable in image processing. The initial works on fuzzy image enhancement uses fuzzy rules and fuzzy filter classes 1,2 Hanmandlu et al. proposed an image enhancement method using contrast intensification operator defined by zadeh3,4. They applied iterative process for the desired enhancement. Later contrast intensification operator was modified to NINT and GINT operators and these were used for enhancement of underexposed images5, 6. Another approach of image enhancement used 'S' function to model the intensity level and the parameters of 'S' function was found by maximizing entropy7. This approach does not preserve the shape of the original histogram. A simple and computationally fast approach for enhancement of low contrast and low bright images was proposed by Raju et.al8. Yu et al. tried to understand the relationship between intensity and saturation and proposed a new method called SI correction for removing color distortion in the HSV space9. An analysis of image enhancement based on lattice theory and partially ordered sets was proposed in 2009 by Gopalan et.al10. All the above works concentrated on low contrast images. Hanmandalu et.al has proposed the enhancement of underexposed and overexposed regions of image by entropy optimization of four parameters using bacterial foraging algorithm11. Better enhancement could be provided by different fuzzification curves and transformation operators12, 13. But the number of parameters to be optimized increases in these algorithms. Section 2 describes the various stages of mathematical model which includes fuzzification, image enhancement, optimization, and defuzzification. Section 3 discusses the parameters for image evaluation. Section 4 discusses the results to the model.

2. Mathematical modeling

2.1. Fuzzification

Image fuzzification is the first step of this model. Image properties like intensity, contrast, number of gray levels etc can be converted to fuzzy sets by defining corresponding membership values. The membership values are defined over the domain of intensity levels from 0 to 255. Two non overlapping membership functions are defined for underexposed and overexposed regions. So that different characteristics are transformed into enhancement region of the model.

In image enhancement problem exposure is a modified parameter used to divide the underexposed and overexposed regions is done. For the new mathematical model, a modified version of exposure is defined by

Where p(x) is the probability of gray levels having the intensity level x. Exposure gives an indication of the ratio of overexposed region of image to the underexposed region of image. If the exposure value is positive then the ratio is higher and if negative the ratio is lower. An exposure value of zero indicates that the overexposed and underexposed regions of the image are equal. The maximum value of exposure is one and minimum value is -1. The intensity levels for underexposed and overexposed region is divided into [0, d-1] and [d, L] respectively where'd' is given by

2.2. Exposure

d = (L / 2) * (1 - e) (2)

2.3. Membership function

Any monotonically increasing or decreasing function can be used as the membership function for the two regions. So triangular membership function is used in the model and it is defined as below.

For overexposed region

¡Uu = (x / d) +1 (3)

And for the overexposed region is given by

£ x-255 I / (255 - d) +1 (4)

V x=d J

From (3) it can be seen that slope of the membership function is higher for underexposed region of the underexposed image. Similarly from (4), it can be seen that slope is lower. That is the defuzzification process is helping to map the gray levels in the exposed region to a longer range of gray levels.

2.4. Image enhancement

In image enhancement the contrast of an image can be increased or decreased by adjusting the intensity level. This is done by using membership functions. The sigmoid membership function with two asymptotic curves is used for lower intensity levels and higher intensity levels respectively.

The corresponding sigmoid function is defined as i

g (t) =

1 + e A(( B) (5)

The parameters A and B in the function determine its shape. In this problem value of A decides the slope of the functions and B is the value of t at which g(t) becomes 0.5.Clearly the intensification depends on the crossover point and the slope of the sigmoid curve. When the sigmoid function defined in (5) is applied on the original membership function defined in (3, 4), it can be seen that as slope increases, the membership value above the crossover point decreases. Similarly increases below the crossover point. The cross over point is set as the membership function of statistical mean in each of the regions. This is done in order to maintain the shape of the histogram. In previous works crossover point was set as 0.512.

For the underexposed region the sigmoid function is defined as in12

^ (X) = 1 + e-*iMu (x) (6)

d-\ d-\

Vu = Mu ("Z XP(X) / Z P(X)) (7)

For the overexposed region

^(x) = 1+e/<X (x)) (8)

255 255

Mao = Mo ("Z XP(X) / Z P(X)) (9)

x=d x=d

Where k and l are the slopes and ^au and nao are the crossover points of the underexposed and overexposed images respectively. The maximum contrast is obtained when the slope of the sigmoid function is <x> and this forms a black and white image. To overcome this following optimisation problem is considered.

2.5. Optimization problem

The objective function is to minimize

J = E + 2*exp(l(| vf - vsf |)) (10)

Subject to constraints

vf = vsf, k >0,l >0, >1 = 1 The fuzzy entropy defines the amount of uncertainty in the image. Hence the entropy must be minimized. While minimizing the entropy the image quality has to be maintained. Hence optimization has to be done to meet the constraint of visual quality. The fuzzy entropy is given by12

L *ln(2)

Ë( u ^(u )Mi-o/(k)))*in(i-(/Uk )) ))+Ë (/<(k )*in(//;(k))+(i-(//;(k)^in(i-(//;(k ))))

'E' becomes zero when all the membership values are zero or one. This indicates a sharpened image. The entropy has maximum value when all the membership values is 0.5 which indicates a low contrast image.

2.6. Quality factor

The quality factor of an image is defined as the absolute ratio of the average fuzzy contrast to the fuzzy contrast. i.e the quality factor for underexposed region

Q = auavg / au (12)

Where 'auavg' and 'au' are the average fuzzy contrast and fuzzy contrast of underexposed region respectively. Similarly the quality factor for overexposed regions of image can be defined.

2.7. Average fuzzy contrast and fuzzy contrast

For underexposed region average fuzzy contrast is defined as in12

c;avg = (d (M;(x) - 05)

Similarly average fuzzy contrast can be defined for overexposed region. Fuzzy contrast is defined for underexposed region as 12

cu = (Mu (x) - 0.5)

2.8. Visual factor

Visual factor is the normalized contrast factor. It measures the improvement in visual appearance due to enhancement. Hence the overall visual factor is the combination of underexposed and overexposed visual factors with weights proportional to the threshold of the regions. It is defined as12

vf = \* qu + | 1 I * —

^255J I 255) qo

The empirical visual factor is defined as

vsf = 1.2 -

2.9. Algorithm for the optimization problem

Fig. 1. Flow chart for simulated annealing

2.10. Defuzzification

In general defuzzification is done by applying inverse operator to the corresponding membership functions. Hence the intensity levels obtained is defined as

Xe =^V( ^)] (17)

While combining the intensity levels, it can be seen that there exists some discontinuity in it at the division of underexposed and overexposed regions. This can be overcome by scaling the intensity level of the shorter region to the maximum or minimum of longer region. This process is done by using the formula

x0e = ((xoe -min(xoe))*(max)-min(x^)) / mwc(xue)-min(xue)) + max(xue) (18)

Where xoe and xue are the modified intensity levels. Further scaling is done to achieve the gray level range between zero and 255. In some situations a portion of the images are highly underexposed or overexposed then image enhancement is not sufficient on the intensity component. In such situations image enhancement can be done on the saturation level. For the overexposed region saturation has to be increased to maintain the original colour and in the underexposed regions saturation has to be decreased.

3. Evaluation measures

3.1. Contrast information index

The contrast information factor is defined to check the quality of image. For a good quality image the contrast information factor should be higher. It can be defined as9

Cavg = Cproposed / Coriginal (19)

Where local contrast C can be measured with 3 x 3 window as.

C = (max - min) / {max + min) (20)

3.2. Index of fuzziness

This is used to check whether there is fuzziness in a high quality image. The index of the model developed is 6

y = jr min(M(k), (1 - M(k)))^(k)

k=0 (21)

The result obtained is compared with other related works and this model has low index of fuzziness.

3.3. Exposure

Exposure is also considered as an evaluation measure in the image enhancement problem. The exposure for pleasing image is close to zero. Using the newly developed model exposure is estimated and it is shown that exposure is closer to the target zero compared to previous works.

4. Results and discussion

The algorithm for the newly developed mathematical model was performed in Matlab version 7.9. An overexposed and underexposed image is taken for analysis. The Fig. 2-3 present a comparison between original image and enhanced images. In the overexposed image of fig. 2 the optimum value of parameters obtained are k=5, l=6 and difference of visual factors is 0.5651. The enhanced images subjectively look better than the original image.Table1-2 shows the comparison of the evaluation parameters for existing algorithms and the algorithm presented in this paper. The comparison shows that the model presented provides a better mathematical framework for image enhancement.

Fig. 2. (a) Original image; (b) Enhanced image of 'village'

Fig. 3. (a) Original image; (b) Enhanced image of 'Boy'

Table 1 .Comparison of parameters for the image 'village'

Evaluation Parameters Original image Algorithm proposed in [12] Algorithm proposed here

Contrast information index 1 1.0913 1.1442

Exposure 0.8052 0.8381 0.5946

Entropy 0.693 0.5946 0.4903

Table 2. Comparison of parameters for the image 'Boy'

Evaluation Parameters Original image Algorithm proposed in [12] Algorithm proposed here

Contrast information Factor 1 1.023 1.23

Index of fuzziness 1 2.17 2

Exposure -0.9 -0.608 -0.406

Entropy 0.6678 0.5565 0.5431

5. Conclusion

In this work a new mathematical model is developed using mathematical and statistical concepts, fuzzy concepts. Its equations and functions are considered in the fuzzy domain. Triangular membership function, sigmoid operators are used to move from real world into the abstract world of mathematical concepts. Statistical mean of intensity levels is considered while selecting crossover point. Slope of the sigmoid operators were found by optimising the objective function using simulated annealing. Only two parameters are optimised in the algorithm. Scaling of the output membership function helped to avoid discontinuity in the histogram of the image. With the help of the algorithm it is verified that the received image is better than original image. In a better way the solution to the mathematical problem is translated to a useful solution to the real world of images.

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