## 1 Introduction  
Over the last few decades, X-ray images have been widely used in helping physicians make their diagnostic decision. However, the raw X-ray image quality is not always good. Visual image quality has been actively improved using contrast enhancement techniques. The main purpose of image contrast enhancement is to improve the interpretability or perception of information contained in the image for human viewers. Image contrast enhancement is a process that produces an output image that subjectively looks better than the original image by changing the pixel's intensity of the input image. Image enhancement can be used as a preprocessing step in medical image processing, texture synthesis and many other image/video processing applications [1; 2; 3; 4; 5; 6; 7; 8]. These techniques also can be used to improve the visual appearance of an image or to convert an image to a form better suited to the subsequent processing, such as segmentation, feature extraction, detection and recognition. A considerable amount of research has focused on this subject, and numerous histogram modification techniques have been developed [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18]. Generally speaking, histogram modification techniques can be categorized into two classes: global histogram modification [9; 10] and local histogram modification [11; 12]. The global histogram equalization (GHE) method is one of the most commonly used methods for image contrast enhancement because of its simplicity and effectiveness. The GHE method employs the cumulative density function of the input image as its intensity mapping function. The basic idea of the GHE method is to re-map the intensities of the input image by using the intensity mapping function. Then, the GHE method can stretch the dynamic range of the image histogram to enhance the overall contrast of the original image. The resulting image may have a uniform distribution of intensity [13]. However, the GHE method suffers from some drawbacks, such as over enhancement, an increase in the noise level, loss in detail and a washed-out effect in some almost homogeneous areas [8]. Many researchers have proposed many useful algorithms to solve these problems involved in the global histogram equalization technique [14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. In [14], a brightness preserving bi-histogram equalization (BPBHE) method first separates the input image histogram into two sub-histograms based on the mean of the input image brightness. Then, these two sub-histograms are equalized independently. The mean brightness of the resultant image will lie between the mean brightness and middle grey level of the input image. The BPBHE technique is capable of preserving the original image brightness to a certain extent. In [15], the dualistic sub-image histogram equalization (DSIHE) method also separates the input image histogram into two sub-histograms, but the separation is based on the median value. The minimum mean brightness error bi-histogram equalization (MMBEBHE) is the extension of the brightness-preserving bi-histogram equalization (BPBHE) method. This method separates the image histogram by using the threshold level. The MMBEBHE method yields a minimum brightness difference between the input image and the output image [16]. These methods separate the input image's histograms into two sub-histograms by using one intensity value as their separating point. Although these methods are simple, they can preserve the mean brightness only to a certain extent [17]. Moreover, other solutions can be found in [18; 19; 20; 21]. The recursive mean-separate histogram equalization (RMSHE) method is another improvement scheme of the brightness-preserving bi-histogram equalization (BPBHE) method [22]. The recursive mean-separate histogram equalization (RMSHE) method recursively divides the image's histogramsinto several sub-histograms based on the local mean values.