Image Contrast Enhancement by Constrained Local Histogram Equalization

Computer Vision and Image Understanding Vol. 73, No. 2, February, pp. 281–290, 1999 Article ID cviu.1998.0723, available online at http://www.idealibrary.com on

NOTE Image Contrast Enhancement by Constrained Local Histogram Equalization Hui Zhu, Francis H. Y. Chan, and F. K. Lam Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong Received November 13, 1997; accepted July 17, 1998

gray level range of the original image. Another very popular method is histogram equalization [4]. This method assumes the information carried by an image is related to the probability of the occurrence of each gray level. To maximize the information, the transformation f should redistribute the probabilities of gray level occurrences to be uniform. In this way, the contrast at each gray level is proportional to the height of the image histogram. Frei [5] suggested the redistribution be hyperbolic, considering that the human visual system has a logarithmic response to stimuli. Generally, other distributions can be specified in different situations and this is called histogram specification [2]. Actually, any intensity-based enhancement method is a gray level transformation. It tries to rearrange the gray levels so that an image appears more distinct. It does not consider the contents of an image except its gray level distribution. To keep the appearance of the enhanced image similar to the original one, the order of the gray levels in the original image must be kept invariant; that is, the transformation function f must be monotonically increasing. The basic idea of feature-based methods is to enhance the small-scale components, which represent some specified features, of an image. These methods can be expressed as

Histogram equalization is a widely used image contrast enhancement method. While global histogram equalization enhances the contrast of the whole image, local histogram equalization can enhance many image details by taking different transformation of the same gray level at different places in the original image. However, the local histogram equalization process often results in unacceptable modification of the original image appearance. In this paper, a constrained local histogram equalization method is proposed to balance the conflicting requirements: enhancement of the image details and the maintenance of the overall image appearance. Our method uses the variational form of histogram equalization so that a constraint condition, which forces the local gray level transformations to change continuously in the spatial domain, can be introduced into the equalization process. Experimental results of different kinds of images show the effect of our method. °c 1999 Academic Press

1. INTRODUCTION The purpose of image contrast enhancement is to increase the visibility of images. Many methods have been proposed emphasizing enhancing different properties or components of images [1–3]. Most of the contrast enhancement methods can be classified into two main categories: intensity-based techniques and feature-based techniques. Intensity-based methods can be expressed by the form I O (x, y) = f (I (x, y)),

(1)

where I (x, y) is the original image, I O (x, y) is the output image after enhancement, and f is a transformation function. In these methods, a transformation of the image gray levels is applied to the whole image. In other words, pixels with the same gray level at different places of the original image are still kept the same in the processed image. Contrast stretching is the most representative and the simplest method in this category [2]. Linear function and nonlinear functions, such as square, exponential, and logarithmic function, can be used to fully utilize a displaying device’s intensity and enhance the contrast of an interested

I O (x, y) = L I (x, y) + G(x, y)HI (x, y),

(2)

where L I represents the low frequency components which keep the basic appearance of the original image, HI represents the high frequency components which contain the concerned features, and G is the enhancement gain. Feature-based contrast enhancement can be done in spatial domain or frequency domain [2]. In the spatial domain, various unsharp masks are used to convolute with the original image [6]. Statistical methods are also adopted [7, 8]. Local means and variances are calculated and transformed to desired values. In the frequency domain methods [2], the Fourier transform is first applied to the image and then a range of frequency components is selected to be enhanced. Finally, the enhanced image is obtained by the inverse Fourier transform. Nowadays, spatial-frequency analysis has also been used for image contrast enhancement. Laine et al. proposed a

281 1077-3142/99 $30.00 c 1999 by Academic Press Copyright ° All rights of reproduction in any form reserved.

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multiscale method to enhance mammograhic features, such as calcification and breast tumors [9]. In most cases, spatial or spatial-frequency methods are preferred because different gains of enhancement can be determined according to local image characteristics. For example, Ji et al. proposed to adaptively enhance local contrast based on human visual properties [10]. Dhawan et al. calculated local contrast by two concentric square windows and increased the contrast by some reasonable functions [11, 12]. Morrow et al. improved this method by replacing the square window with an adaptive neighborhood [13]. Compared with the intensity-based methods, feature-based contrast enhancement methods emphasize the contents of images. The feature-based methods always aim at stressing some known objects in images. Generally, all feature-based methods have three steps. First, decompose the image and extract the feature components to be enhanced; second, enhance the extracted feature components; third, reconstruct the image. In these methods, the ways to extract the feature components to be enhanced must be based on the knowledge about these features. So feature-based methods are often used in special applications. On the other hand, the intensity-based methods are more general. The problem these methods try to solve is how to fully use the display device’s intensity to provide a substantial increase in the perceptibility of images. By stretching some ranges of gray levels, some invisible or ignored unknown objects or image features in the original image may be detected. So they are widely used in the preprocessing of various images. All the above mentioned intensity-based methods are global enhancement methods. They can enhance the overall visibility of an image, but may not increase, or even decrease, the local contrast at some local positions in the image. Then a lot of detailed information of the image may be ignored. For example, if an image has a dark region and a bright region both with some objects inside, it is hard to simultaneously enhance the contrast in both regions by one transformation. The human visual system can adaptively set the gain to this situation. To simulate this characteristic, local intensity-based contrast enhancement methods are used to deal with such problems. The local intensitybased methods can be expressed by the form I O (x, y) = f (I (x, y), x, y)

(3)

which means the transformation function changes spatially. Fahnestock and Schowengerdt proposed a method called local range modification (LRM) [14]. In their method, linear stretch was applied according to the local maximal and minimal gray level. Histogram equalization was also extended to adaptive (or local) histogram equalization (AHE or LHE) [15–17]. For each pixel, as the center, a window was set to calculate its local histogram. Then histogram equalization was applied to the local histogram to obtain the gray level transformation for the centric pixel. To reduce the computational burden, Pizer et al. proposed interpolated AHE [17]. They calculated the local transformations only at a sample grid of pixels and interpolated the trans-

formations between these sample locations. Another improvement in their work was clipped AHE which we will introduce in Section 3.1. Paranjape et al. suggested the use of an adaptive growing region to replace the arbitrary defined window [18] for local histogram calculation. While adaptive histogram equalization and its variations are very useful to enhance the details contrasting in local areas of an image, they share a common shortcoming with the feature-based methods—the pixels with the same gray level in the original image may be different and the gray level order may even reverse at different places after enhancement. These effects destroy the appearance of the original image and make the image appear very noisy. Rehm and Dallas found that the LHE method produced edge artifacts at sharp boundary points where the local transformation changes abruptly due to the rapid change of the local histogram [19]. To decrease such effects, they suggested controling the abrupt change in the local transformations by subtracting a smooth version of the image prior to applying the LHE. In fact, this was equivalent to performing histogram equalization to the high spatial frequency components of the image. Pizer et al. also found this problem and tried to solve it by developing a method called SHAHE which combined unsharp masking and contrast-limited local histogram equalization [20, 21]. In the above LHE methods, the local histogram equalization at each pixel is done independently. So if there are big changes between nearby local histograms, the local gray level transformation will change abruptly. To reduce this effect, we argue that the change of the local histogram transformation should be constrained to be continuous spatially. Perona and Tartagni proposed a contrast normalization method, which used a smoothness function to build a full local frame of reference for the gray level values of the image [22]. Although the proposed LHE methods may generate continuous local transformation under the proper conditions by assigning carefully the parameters to individual images, the continuity cannot be proved and is not guaranteed. Here, we make use of Perona and Tartagni’s idea and apply a smoothness condition to constrain the local transformation as a solution to the problem. In this paper, we will propose a constrained local histogram equalization (CLHE) method. In Section 2, we first introduce a variational formulation of histogram equalization proposed by Altas, Louis, and Belward [23]. By the variational approach, histogram equalization is expressed as a functional optimization problem. Then this formulation is extended to LHE directly, and a smoothness constraint term is introduced into the optimization function so that the enhancement result is a balance of the enhancement of details of interest and the maintenance of the original image’s appearance. In Section 3, the numerical algorithm of our method is presented. A modified local histogram construction method is proposed to make the algorithm easy to implement. Results of common images and X-ray medical images enhanced by our method are shown in Section 4, and they are compared with global histogram equalization (GHE) and LHE results. Section 5 is the conclusion.

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2. METHOD

that is,

To combine the constraint condition with the LHE method, we will first introduce a variational formulation of histogram equalization [23]. It should be pointed out that Sapiro and Caselles have used a partial differential equation (PDE) to represent histogram equalization [24]. They have showed that their PDE corresponded to another variational formulation, which is different from [23], of histogram equalization. They are in favor of such PDE or variational methods because such formulations permit users to combine algorithms. They demonstrated this advantage by proposing an algorithm performing simultaneous histogram equalization and denoising. This is also why we adopt variational approaches here to represent the histogram equalization.

Z

(4)

maps every gray level rk to a new value sk (Fig. 1). To keep the same intensity order from black to white in the output image, f must be monotonically increasing. By this mapping, h r (r ) is transformed to h s (s). Histogram equalization finds the mapping f that makes h s (s) uniform. Following from elementary probability theory, dr h s (s) = h r (r ) ds

¸ r = f −1 (s)

.

v[ f (r )] =

Z F(r, f, fr ) dr =

1 2 f dr, h(r ) r

(8)

where r represents gray level, f is the gray level transformation function, fr is the first derivative of f to r , and h is the normalized histogram. To find the function f which will minimize the functional v, the following Euler equation is first obtained: Ff −

d F f = 0. dr r

(9)

That is, d − dr

µ

1 2 fr h(r )

¶ = 0.

Integrating with respect to r , we have (5) fr = Ch(r ),

If h s (s) = 1, ds = h r (r ) dr ;

(7)

which means the transformation function f is the cumulative distribution function of the image gray level r . Altas, Louis, and Belward viewed the histogram equalization problem from another angle [23]. They found that the procedure of histogram equalization attempted to redistribute the histogram over the full range of gray levels in such a way as to minimize the cumulative interhistogram bar spacing, subjected to a weighting by the histogram bar height. So they reformulated the histogram equalization problem as a general variational minimization problem. Let v[ f (r )] be a functional of f ; we have Z

Let r, s represent the gray level of the original image and its enhanced output image. Here, we set 0 ≤ r, s ≤ 1. h r (r ) and h s (s) are their normalized histogram which also can be viewed as their gray level probability density functions (PDFs) of the original and output images, respectively. A transformation function

·

h r (w) dw, 0

2.1. Variational Formulation of Histogram Equalization

s = f (r )

r

s = f (r ) =

(6)

where C is an arbitrary constant. Integrating with respect to r a second time, Z r h(w) dw. f (r ) = f (0) + C 0

Because f (0) = 0 and f (1) = 1, C = 1 can be achieved. Then Z f (r ) =

r

h(w) dw.

(10)

0

We see that the solution of f is just (7). So histogram equalization can be achieved by two methods, from the integral of (7) or from solving the differential equation (9). 2.2. Constrained Local Histogram Equalization (CLHE) FIG. 1.

A gray level transformation function.

If we use a 3-D function f (x, y, r ) to represent the local gray level transformation, where (x, y) is any place in the image,

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according to the descriptions in Section 2.1, local histogram equalization can be solved independently at each position (x, y) by a variational approach to minimize the functional Z v[ f (x, y, r )] =

1 f 2 dr, h(x, y, r ) r

where h(x, y, r ) is the local histogram function. It is equivalent to minimize the functional ZZZ v[ f (x, y, r )] =

1 f 2 dr d x dy h(x, y, r ) r

(11)

(see Appendix). As we mentioned in Section 1, although local histogram equalization enhances the contrast according to the local situations of an image so that many details can be visualized, it will distort the image appearance because the same gray level pixels in different places may be transformed to different gray levels, and sometimes the difference is quite large. This can be viewed as the costs of local contrast enhancement. The problem we want to solve is how to balance the enhancement gain of the local contrast and the distortion of the global image appearance. So we introduce a smoothness term into the above minimization functional and get v[ f (x, y, r )] ZZZ · =

¸ ¡ ¢ 1 fr2 + λ f x2 + f y2 dr d x dy, h(x, y, r )

(12)

where the second term in the integral, which tries to keep a spatially continuous gray level transformation in the whole image, is used to represent the costs of the global image appearance distortion, λ is the balance parameter, and λ > 0. In this way, we want to find such a transformation function with two objectives: on one hand, it aims at the local histogram equalization; on the other hand, it attempts to keep the transformation relatively unchanged with the position. The corresponding Euler equation of the above minimization problem is f x x + f yy +

h r fr frr − = 0. λh λh 2

(13)

Now we have modeled a constrained local histogram equalization method by a 3-D PDE. In the next section, we will describe the implementation algorithm. 3. IMPLEMENTATION To facilitate implementation of the numerical algorithm for solving (12), a new local histogram construction method will be presented before the whole CLHE algorithm is proposed.

FIG. 2. Local window for local histogram calculation.

3.1. Local Histogram Formulation Because by histogram equalization, the difference between two neighboring gray levels is proportional to the height of the histogram, over-enhancement may occur if there are high peaks in the histogram. This phenomenon also exists in some local histogram equalization methods. To overcome this problem, Pizer et al. suggested a clipped histogram technique which could limit the slope of the transformation function [17]. They enforced a maximum on the counts in the histogram and redistributed these “overflow” pixels in all histogram bins uniformly. Local histogram construction is an important step in local histogram equalization. Usually, the local histogram of a pixel is the histogram of a window centered at this pixel (Fig. 2). Different window sizes are used in different applications for good performance of local histogram equalization. Paranjape et al. suggested an adaptive neighborhood to replace the square window for local histogram calculation [18]. All these local histogram construction methods only consider the pixels in the local area and do not consider the pixels in the remaining area of the image. According to a human’s visual properties, on one hand, the visual system can change adaptively with the region of interest (ROI); on the other hand, it is influenced by the surroundings of the ROI. So here, we give another way to construct the local histogram redistribution. We construct the local histogram by two parts: one is the histogram in the local window and the other is the histogram outside the window (Fig. 2). h L (r ) = αh W (r ) + (1 − α)h B (r ),

(14)

where h W is the normalized histogram of the window, h B is the normalized histogram of remaining of the image, and 0 ≤ α ≤ 1. Let A W and A B represent the area of regions W and B. If α = A W /A B , then h L (r ) = h(r ); that is, the local histogram is equal to the global histogram. If α > A W /A B , the local histogram emphasizes on the local information. In this way, the parameter α can be used to adjust the local histogram simulating the influence of surroundings to the ROI. By using this

IMAGE CONTRAST ENHANCEMENT

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FIG. 3. (a) A baby image; (b) GHE enhancement result; (c) LHE enhancement result; (d) CLHE enhancement result with λ = 1; (e) CLHE enhancement result with λ = 5.

method to construct the local histograms, all local histograms have the same gray levels as the global histogram, but with different amplitudes at each gray level at different positions. By our local histogram construction method, the local histograms all have the same gray levels but different heights, which emphasize the local features, with the image. This property also makes it convenient to construct the discrete form for Eq. (13)

in the numerical algorithm, because the discrete mesh grid is regular. 3.2. The Algorithm of CLHE To implement the histogram equalization algorithm by solving the Euler equation (13), the image histogram should be modified to only containing the gray levels occurring in the original

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FIG. 4. (a) The image “camera”; (b) GHE enhancement result; (c) LHE enhancement result; (d) CLHE enhancement result with λ = 1; (e) CLHE enhancement with λ = 5.

image. This is because h must not be zero in Eq. (13). The reconstructed normalized histogram can be used to represent the PDF of the gray level. As done by Pizer et al., we use a sample grid of points for calculating the local histogram equalization and interpolate the transformations between these sample locations. The main reason for using this sample grid is the memory limitation. We need

two 3-D arrays, one to keep the local histogram and the other for the local gray level transformation. The successive over-relaxation (SOR) method is used to solve Eq. (13) [25]. First f x x , f yy , frr , fr , and h r should be represented in their finite-difference forms: f yy = f i+1, j,k + f i−1, j,k − 2 f i, j,k

(15)

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FIG. 5. (a) A chest X-ray image; (b) GHE enhancement result; (c) LHE enhancement result; (d) CLHE enhancement result.

f x x = f i, j+1,k + f i, j−1,k − 2 f i, j,k

(16)

frr = f i, j,k+1 + f i, j,k+1 − 2 f i, j,k

(17)

fr = ( f i, j,k+1 − f i, j,k−1 )/2

(18)

h r = (h i, j,k+1 − h i, j,k−1 )/2.

(19)

Then the SOR method here can be written as n f i,n+1 j,k = f i, j,k +

ω n ξ , a i, j,k

(20)

where ξi, j,k = ( f x x + f yy + frr /λh − h r fr /λh 2 )i,n j,k , a = 4 + 2/λh i,n j,k , ω is the over-relaxation parameter and 1 < ω < 2. We use the maximum–minimum linear stretching transformation as all of the initial local gray level transformation f i,0 j,k . For the

P convergence decision, i, j,k ξi,n j,k is calculated for each iteration, and the iterations will stop after it is smaller than a constant or its change is very small. 4. EXPERIMENTAL RESULTS AND DISSCUSIONS In this section, we will present some results of our method and compare them with global histogram equalization and local histogram equalization. All these experiments were implemented on an SGI Indy PC workstation using C language. It took about 15–25 min for each of the experiments. The time depends on the image size, gray levels in the image, local window size, the sampling grid size, and the number of iteration for convergence. The first one was a baby image (Fig. 3a). Figure 3b is the global histogram equalization result of Fig. 3a. Although the

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FIG. 6. (a) A spine X-ray image; (b) GHE enhancement result; (c) LHE enhancement result; (d) CLHE enhancement result.

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GHE result shows higher contrast in the whole image, the contrast of the boy’s face is lower. Figure 3c is the LHE result. The sampling grid of LHE is 16 × 16, and the local window size is 33 × 33. In calculating the local histogram, α is set to 0.8. Comparing Fig. 3b with Fig. 3c, the baby’s facial features are clearer by LHE than by GHE, but the image is noisier in the relatively uniform regions, such as the background, and the boy’s hair becomes gray. This is because LHE enhanced the contrast locally. Figure 3d is the result by our CLHE method with λ = 1. The boy’s face features are still clear to see, and his hair is almost dark. Although the local contrast is not as high as in Fig. 3c, the whole enhancement effect is better. It is not as noisy as Fig. 3c. So our CLHE keeps the advantage of enhancing details as LHE on one hand, and overcomes its disadvantage of generating artifacts to a certain extent. Figure 3e is another result by our CLHE with λ = 5. A similar result can be seen in another experiment—an outdoor image of a man taking a picture (Fig. 4). The parameters used were the same as in Fig. 3 except α = 0.5. More details, such as the pocket style of the man’s coat, can be seen in the LHE result (Fig. 4c) compared with the GHE result (Fig. 4b). But the sky and the grass are quite noisy, and the man’s coat and his hair turn gray. By our CLHE (Fig. 4d, λ = 1, and Fig. 4e, λ = 5), the situation is much improved. The local contrast is enhanced much more than the GHE result and the original image appearance is kept quite well. Pizer et al. described the LHE as an excellent contrast enhancement method for medical images [17], and it has been used for X-ray chest image enhancement [26]. Now we will try our CLHE for two X-ray medical images. The first one is a chest image. The spine can be seen much more clearly in the LHE result (Fig. 5c) but is almost not visible in both the original image (Fig. 5a) and the GHE result (Fig. 5b). With our CLHE, the spine still can be seen, but the whole image is “cleaner.” More similar results of a spine image are shown in Fig. 6. The global enhancement effect of our CLHE (Fig. 6d) is somewhat like GHE (Fig. 6b), which keeps the global appearance of the original image (Fig. 6a), and its local enhancement effects is like LHE, which enhances many details of the original image, but avoids many artifacts. The basic idea of LHE and its variations is to equalize the local histograms so that more details are visible. Because the local histogram changes here and there, the local gray level transformation changes with it, sometimes abruptly. This property of LHE distorts the original appearance of an image and the image is always over enhanced. The idea of using the smoothness function to constrain image enhancement has appeared in other methods [19, 24, 27]. Here we have applied this idea to the LHE method. The smooth constraint condition we set can be viewed as the cost that should be paid for LHE. The balance parameter λ can be set to different values in different applications.

5. CONCLUSION In this paper, we have proposed a constrained local histogram equalization method for image contrast enhancement. In this method, by the variational approach to reformulate the local histogram equalization, a constrained condition, which is to keep the local gray level transformations varying continuously, is introduced into the minimization function. A new local histogram construction method is suggested to simulate a human’s visual system and simplify the numerical implementation. We have tried our method on common and medical images. By our method, the enhanced images have enough local enhancement for details as well as relatively “clean” appearances. APPENDIX R 1 f 2 dr . If v( f 0 (x, y, r )) = LetRRg(x, y, f (x, y, r )) = h(x,y,r ) r min f g(x, y, f (x, y, r )) d x d y, then at any place (x0 , y0 ), we have v( f 0 (x0 , y0 , r )) = min f g(x0 , y0 , f (x0 , y0 , r )). Here, h ≥ 0, f ≥ 0, and the integral intervals of x, y, and r are positive. Proof. (by contradiction) Apparently, g > 0. Assume at (x0 , y0 ), it is not f 0 (x0 , y0 , r ) but u(r ) makes min f g(x0 , y0 , f (x0 , y0 , r )), then we can construct another function ½ u(r ), at (x0 , y0 ); . f 1 (x, y, r ) = f 0 (x, y, r ), else. Therefore 0 < g(x, y, f 1 ) ( at (x0 , y0 ) g(x0 , y0 , u(r )), < g(x, y, f 0 ). = g(x, y, f 0 (x, y, r )), else Because are positive and g > 0, we RR all the integral intervals RR have g(x,RRy, f 1 ) d x d y < g(x, y, f 0 ) d x d y. But v(x, y, f 0 ) = min f g(x, y, f ) d x d y, therefore contradiction is reached. ACKNOWLEDGMENTS This work is partly supported by the research grants of the University of Hong Kong. We are grateful to Prof. Rangaraj M. Rangayyan, the University of Calgary, Canada for providing his papers for reference. The images of the first two examples are obtained from ftp://ftp.eedsp.gatech.edu/database/images/. The medical images are obtained from UMDS Radiology Teaching File by ftp (http://www-ipg.umds.ac.uk/IPG/ ).

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