Visual Contrast Enhancement by Histogram Modification Based on Generalized Extreme Value Distribution

Contrast enhancement is a crucial method for improving the quality of an image. This paper proposes a visual contrast enhancement of a color image by using histogram modification. Transfer function of the modified histogram was designed with generalized extreme value distribution. To automatically enhance image contrast and tone meanwhile improving color balance, three parameters of the distribution that consists of shape, scale, and location were estimated by probability weighted moments. The location parameter was employed to improve brightness and color balance. The scale and shape was provided to enhance contrast and to adjust tone of the images. The proposed algorithm was evaluated with a low dynamic range (true-color images) and a high dynamic range images.


Introduction
Due to the widespread applications of digital cameras, many digital color images have been taken under improper lighting conditions as a result of low visual quality.To solve this problem, recently many proposed methods have been proposed to produce the better quality images.Contrast enhancement based on histogram modification (HM) (1)(2)(3) is a widely used technique for improving visual quality of the low contrast pictures in an image processing.
Histogram equalization (HE) is normally used to enhance the brightness and contrast of an image by using cumulative distribution function for stretching its dynamic range.HE is often questioned for excessive enhancement; because, it may shift the mean intensity values to the middle gray level of the intensity range.To overcome this mean-shift problem, many researchers (4)(5)(6)(7)(8)(9) have proposed methods to solve the mean shift problem of HE.Fundamental objective of the solving mean shift techniques provides the output images with the mean of brightness close to the original.Those techniques are not suit for underexpose or overexpose images; because, they will lead to human visual perception loss problem.
Recently, visual contrast enhancement algorithm (VCEA) based on histogram equalization was introduced by Meng-Liang Chung et al. (10) .The image results of VCEA have much more visual quality than other HE-based methods; however, the space adjustment mechanism cannot control the brightness mean.This mechanism will affect to improper appearance of color balance.
In this paper, a new visual contrast enhancement is proposed.Our model employs histogram modification (2) with the generalized extreme value (GEV) distribution (11) to characterize the transfer functions.Three parameters of GEV, which consist of shape, scale, and location, are estimated by probability weighted moments (PWMs) (12) .
By setting optimal values of the GEV parameters can use to enhance the retinal images (13) ; however, our proposed algorithm uses the standard white point, D55, to specify location parameters for controlling the brightness and color balance.The scale and shape parameters are employed to improve contrast and to adjust tone of color images, respectively.Our algorithm employs Kullback-Leibler divergence (KLD) (14) to select estimated contrast and tone parameter values.These parameters are manipulated by scale and shape parameters.Optimal tuning process of the three parameters is employed to formulate the transfer functions of HM and thus the algorithm can produce good quality color images.
The performance of the proposed scheme is evaluated with public available datasets True-color Kodak Dataset (15) (for low dynamic range images) and ESPL-LIVE HDR Image Quality Dataset (16) .
The paper proceeds as follows: Section 2 describes designing transfer functions, which consists of histogram modification, GEV distribution, and PWMs estimation; Section 3 describes the proposed algorithm; Section 4 presents the experimental results and the paper concludes in Section 5.

Transfer Functions
The proposed method relates to three topics for fabricating our algorithm.The theme is the transfer functions of histogram modification.The either two topics are provided to support the HM that is a GEV distribution and its parameter estimation method by PWMs.

Histogram Modification
Histogram analysis dates back to the early era of digital image processing (1)(2)(3) .It provides to analyze an intensity distribution for characterizing and designing to enhance pictures.As it includes a process for analyzing characteristics and design, the histogram has been named a histogram equalization, histogram matching, histogram specification, or histogram modification.Histogram processing employs mathematical models known as probability distribution functions (PDF).A simple version of histogram equalization uses a uniform distribution.The advance processes need to specify the shape of frequency distributions.The PDF has to support the advance processes.
Let x be the intensity levels of an image, X, in the range [0, L-1] (where L denotes the maximum intensity value).The number of pixels in each intensity level, h(x), is histogram of the image, X. h(x) could also be expressed as a percentage of the pixel numbers against the total number of pixels in the image, X; that is, ⁄ , where the size of image is MN pixels.In statistical terms, is the PDF.
Histogram matching based on HM (2) could be regarded as a monotonic point transformation, g d =T{f c }.The input variable, f c [f 1 , f C ], was mapped into an output variable, g d [g 1 , g D ], such that the output probability distribution, P R {g d =b d }, follows some desired form for a given input probability distribution, P R {f c =a c }. a c and b d represent reconstruction values between the c th and d th intensity levels.C and D were the maximum intensity values of the histogram; thus, the sum of input and output probability distribution must be equal to unity: The probability that pixels of the input image had an amplitude less than or equal to a c must be equal to the probability that pixels of the output image have amplitude less than or equal to b d , where b d = T{a c } because the transformation is monotonic.Hence The histogram modification in (3) can be obtained in approximate form by replacing the discrete probability distributions of (2) by continuous probability densities.
where and are the probability densities of f and g, respectively.From the given image, X, the integral on the right was replaced by the cumulative distribution function (CDF), ∑ .Thus, equation (3) was in the form The histogram was modified by many probability functions such as uniform, hyperbolic, exponential, and Rayleigh distribution, etc. (2) .Those distributions usually control only the range (uniform and hyperbolic) and some of them can adjust the shape parameters (exponential and Rayleigh).In a color image enhancement, the process needs a location parameter to control the color balance.In this paper, GEV distribution had three parameters consisting of shape, scale, and location, which were provided to modify the histogram of color fundus images (13) .

Generalized Extreme Value Distribution
Frequency analysis was an interesting topic for us because one important topic in a digital image processing has to analyze intensity values; especially, in HM (2) .For a color image, histogram transfer functions need more parameters to enhance the image; therefore, three parameters of GEV could be provided to adjust contrast, brightness, and color balance of the color images.The CDF of a GEV (11) was given by exp exp , , for 0.
where , , and , denote the location, scale, and shape parameters, respectively.When  = 0, CDF is Gumble's type I (-∞<x<∞).When  ≠ 0, CDF includes two types.When < 0 it becomes Frechet's type II, but when >0 it is the Weibull or type III distribution.
The PDF corresponding to ( 5 , for 0; The transformed intensity, g, could be solved by replacing the density function, p g (g), of (4) with the p(x) of ( 6) and integral the left, which becomes to the CDF of (5).Thus, the transfer function of HM by GEV distribution becomes ln ln for 0.
The modified intensity, g, of each color band comes from the CDF of the input image and the GEV parameters, which might be estimated by the method of PWMs.

Probability Weighted Moments
Three parameters of GEV distribution could be estimated by PWMs (11,12) as follows: The unbiased estimator of r  is where denotes the ordered observations from a sample of size n, that is ⋯ .The transfer function of HM by using GEV distribution as a transfer function was reviewed.The next section will describe the algorithm to enhance the color images by employing HM for adjusting the color balance, brightness, and contrast.

Color Enhancement Algorithm
The proposed method employs GEV parameters to adjust brightness, contrast, and color balance of natural images by HM. "Aster" image (17) in Fig. 1(a) was provided to demonstrate how our algorithm operates.The algorithm consists of four steps as shown in Fig. 2. The details of each step are described in the following subsections.

Initial Values Setting
The first step initializes the specified parameters of brightness, contrast, and color balance.Input image was resampled by reducing the size following the eyes adaptation for the best view in the fovea, which could be approximated each luminance over a 1° diameter solid angle corresponding to a potential foveal fixation point in the scene (18) .The brightness parameters, Bright Red , Bright Green , and Bright Blue , were approximately specified approaching to white point proportion of D55; thus, The shape a p for manipula namic range a e, the scale pa shape parame Let y denot und (UB) of th nd out of ra dom variable uating x to ameter (11) .In o .rom (7) and h PDFs were e D) (14) as in the , respective incremental va ing the shape ed by the tole ynamic range rative evaluati n in (7) were which was u denotes PDF represents evaluated by K following   
function.The values, b 0 , b 1 , and b 2 , were calculated by employing an unbiased estimator of the first three PWMs that were given by in Fig 1(a), w ue of the red b parameter, ̂ , replaces the lo he data; when ange.It was e, x, in (6), w ⁄ and r our case, the s Fig the