Brightness–preserving weighted subimages for contrast enhancement of gray–level images

Zongwei Lu

doi:10.1117/1.JEI.21.3.033001

Abstract

Abstract. Although many histogram equalization (HE) based methods have been developed to overcome the problem of mean-shift, they often suffer from over-enhancement caused by an abrupt jump of gray levels. Consequently, mean brightness cannot be preserved with higher accuracy. In this paper, one simple and efficient method, brightness-preserving weighted subimages (BPWSI), for brightness preservation and contrast enhancement of gray level images is proposed. First, based on brightness preserving bi-histogram equalization, two new subimages are defined. Then one brightness-preserving way to calculate two weight coefficients is given. Finally, the output image is defined as the weighted sum of two defined subimages. Extensive simulations show that BPWSI can preserve the mean brightness with higher accuracy than many other HE-based methods while enhancing the contrast efficiently. Furthermore, due to its simplicity, BPWSI can be implemented on real-time systems.

Figures in this Article

Introduction

Contrast enhancement plays an important role in many image-processing applications. Its aim is to improve the visual quality of an image such that the resulting image is more visually pleasing than the original. Current techniques for image contrast enhancement can be broadly categorized into two groups: transform domain and spatial domain.

Transform-domain-based techniques map the image into a transform domain by using two-dimensional (2-D) discrete Fourier transform (DFT) and discrete cosine transform (DCT) etc., such as in Refs. ¹ and ². Although transform-domain-based techniques are effective for contrast enhancement, they have such limitations as pointed out in ². In addition, many transform-domain-based techniques are time-consuming, and they cannot meet the requirements of real-time. On the other hand, spatial-domain-based techniques manipulate the image’s intensity values directly such that the histogram of the processed image is more spread and uniform than the original.³^–¹³ One of the most popular techniques is histogram equalization (HE).

Based on the cumulative distribution function, HE flattens and stretches the dynamic range of the image’s histogram, thereby improving the image contrast. However, because HE may shift the mean brightness of the image greatly, and the excessive change in mean brightness may lead to annoying artifacts and unnatural enhancement, it is not well suited for consumer electronics, such as TV, where brightness preserving is necessary to avoid degraded images.

The idea of keeping the mean brightness for better contrast is first proposed by Kim in ⁴. Brightness preserving bi-histogram equalization (BBHE) separates the histogram into two subhistograms based on the mean of the image, one ranging from minimum gray level to mean and the other ranging from mean to maximum gray level. Then two subhistograms are equalized independently. It has been analyzed mathematically and simulated that BBHE can preserve brightness better than HE.

Similarly, dualistic subimage HE (DSIHE) subimage was proposed by Wan et al.⁵ DSIHE chooses the median (intensity with cumulative probability density equal to 0.5) to separate the histogram into two subhistograms and will yield maximum entropy for the image. Hence DSIHE claimed to outperform BBHE both in brightness preservation and contrast enhancement.

In order to preserve the mean brightness of the input image optimally, Chen and Ramli proposed minimum mean brightness error bi-HE (MMBEBHE), which is actually one enhanced version of BBHE.⁶ Each possible gray value, from 0 to $L - 1$ , was used as one separating point in BBHE, and the value that would produce minimum difference between input and output means was chosen as the separating point.

In addition, Chen and Ramli also proposed recursive mean separate HE (RMSHE), which is actually a generalization of BBHE.⁷ First it uses mean to separate the histogram into two subhistograms. Then the mean for each subhistogram is calculated, and each subhistogram is further divided into two parts by the mean value. The process can be repeated for $r$ times and will produce $2^{r}$ subhistograms. Each subhistogram is equalized independently. The larger the value of $r$ , the closer the brightness of the output image is to the brightness of the input image. Similar to RMSHE, Sim proposed recursive subimage HE (RSIHE), which uses median rather than mean to separate the histogram recursively.⁸ However, when the value of $r$ is too large, the output image for RMSHE and RSIHE is exactly a copy of the input image, and contrast cannot be enhanced at all.⁹ So there is a trade-off between brightness preservation and contrast enhancement.

Instead of using mean or median as the separating point, Wongsritong proposed multipeak HE with brightness preserving (MBPHEBP), which separates the histogram into multi-subhistograms based on the shape of the histogram.¹⁰ First the histogram is smoothed with one-dimensional smoothing filter. Then the histogram is divided into multi-subhistograms based on the local maxima of the smoothed histogram. Last, each subhistogram is equalized independently. It is claimed that MBPHEBP can preserve the brightness better than BBHE.

To enhance the contrast efficiently, Wadud proposed dynamic HE (DHE).¹¹ Similar to MBPHEBP, the method smooths the histogram by using a one-dimensional smoothing filter. In contrast with MBPHEBP, the histogram is separated into multi-subhistograms based on local minima. Then each subhistogram is assigned to a new dynamic range before HE is implemented. The subsections with a large number of pixels will occupy bigger dynamic ranges. DHE can enhance the contrast better than other HE-based methods. However, DHE does not put any constraints on the preservation of brightness. Thus some artifacts, such as over-enhancement, may be observed.

Based on DHE, Ibrahim and Kang proposed brightness-preserving dynamic HE (BPDHE).¹² This method uses the local maxima, instead of local minima, to separate the histogram into multi-subhistograms, as Ibrahim and Kang claimed that local maxima are better for brightness preservation. Then one new dynamic range is assigned for each subhistogram, and HE is implemented. Last, brightness normalization is used for the output image in order to preserve the mean brightness. Although BPDHE can preserve the brightness well, the contrast enhancement is determined by brightness normalization where a smaller ratio will lead to insignificant enhancement and a bigger ratio will cause the intensity saturation.

Recently, Chen and Isa proposed bi-HE median plateau limit (BHEPL-D).¹³ Similar to BBHE, this method separates the histogram into two subhistograms based on the mean of the input image. Then clipping process is used for each subhistogram with the median for occupied intensities in each subsection as the plateau limit. Finally, HE is implemented for each subhistogram independently. The purpose of the clipping process in the BHEPL-D is to avoid over-enhancement and noise amplifying, which often appear in many other HE-based methods. It is claimed that BHEPL-D outperforms other HE-based methods in terms of brightness preservation and contrast enhancement.

However, all the above HE-based methods cannot preserve the mean brightness with higher accuracy and easily produce over-enhancement when some gray levels with high frequencies dominate other gray levels with lower frequencies. This condition leads to abrupt jump of gray levels in the transformation function of image and fails to enhance the contrast.

To cope with abrupt jump of gray levels, we propose one simple and efficient method in this paper. The proposed method, brightness-preserving weighted subimages (BPWSI), can preserve mean brightness with higher accuracy than almost all contemporary methods without introducing serious side-effects, such as over-enhancement, level saturation, and noise amplifying, etc.

In addition, it is well known that contrast enhancement is very much problem-oriented. In other words, a method that is quite useful for enhancing x-ray images may not necessarily be the best approach for enhancing pictures of Mars transmitted by a space probe.¹⁴ For example, contrast limited adaptive HE (CLAHE) is effective in enhancing the medical images by considering local information.¹⁵ However, CLAHE has relatively higher computational cost and thus it is not suitable to be implemented in consumer electronics. So, in this paper, we only focus on developing efficient method for consumer electronics where it is necessary to preserve the brightness to avoid annoying artifacts, which will degrade the quality of the image.

The rest of the paper is organized as follows. Section 2 briefly introduces the BBHE method. In Sec. 3, the proposed method, BPWSI, is described in detail. Section 4 presents simulation results to demonstrate the effectiveness of the proposed method. In Sec. 5, the effect of one parameter $δ$ on contrast enhancement and brightness preservation is described. Then in Sec. 6 we discuss some limitations of brightness-preserving methods. The conclusion of the work is given in Section 7.

Brightness-Preserving Bi-Histogram Equalization

The following is basically a reprint of ⁴.

Denote by $X_{m}$ the mean of the image $X$ and assumes $X_{m} \in {X_{0}, X_{1}, \dots, X_{L - 1}}$ . Based on the mean, the input image is decomposed into two subimagesubimages, $X_{L}$ and $X_{U}$ , asDisplay Formula

X = X_{L} \cup X_{U},

(1)

where Display Formula

X_{L} = {X (i, j) | X (i, j) \leq X_{m}, \forall X (i, j) \in X}

(2)

and Display Formula

X_{U} = {X (i, j) | X (i, j) > X_{m}, \forall X (i, j) \in X} .

(3)

Note that the subimage

X_{L}

is composed of

{X_{0}, X_{1}, \dots, X_{m}}

, and the other image

X_{U}

is composed of

{X_{m + 1}, X_{m + 2}, \dots, X_{L - 1}}

.

Next, define the respective probability density functions of the subimagesubimages $X_{L}$ and $X_{U}$ as Display Formula

p_{L} (X_{k}) = \frac{n_{L}^{k}}{n_{L}},

(4)

where

k = 0, 1, \dots m

, and Display Formula

p_{U} (X_{k}) = \frac{n_{U}^{k}}{n_{U}},

(5)

where

k = m + 1, m + 2, \dots L - 1

, in which

n_{L}^{k}

and

n_{U}^{k}

represent the respective numbers of

X_{k}

in

X_{L}

and

X_{U}

, and

n_{L}

and

n_{U}

are the total number of samples in

X_{L}

and

X_{U}

, respectively. Note that

n_{L} = \sum_{k = 0}^{m} n_{L}^{k}

,

n_{U} = \sum_{k = m + 1}^{L - 1} n_{U}^{k}

and

n = n_{L} + n_{U}

.

The respective cumulative density functions for $X_{L}$ and $X_{U}$ are defined as Display Formula

c_{L} (x) = \sum_{j = 0}^{k} p_{L} (X_{j})

(6)

and Display Formula

c_{U} (x) = \sum_{j = m + 1}^{k} p_{L} (X_{j}),

(7)

where

X_{k} = x

. Note that

c_{L} (X_{m}) = 1

and

c_{U} (X_{L - 1}) = 1

by definition.

Similar to HE where cumulative density function is used as a transformation function, let’s define the following transformation functions exploiting the cumulative density functions Display Formula

f_{L} (x) = X_{0} + (X_{m} - X_{0}) c_{L} (x)

(8)

and Display Formula

f_{U} (x) = X_{m + 1} + (X_{L - 1} - X_{m + 1}) c_{U} (x) .

(9)

Based on these transformation functions, the decomposed subimages are equalized independently, and the composition of the resulting equalized subimages constitutes the output of BBHE. That is, the output image of BBHE, $Y$ , is finally expressed asDisplay Formula

Y = {Y (i, j)} = f_{L} (X_{L}) \cup f_{U} (X_{U}),

(10)

where Display Formula

f_{L} (X_{L}) = {f_{L} (X (i, j)) | \forall X (i, j) \in X_{L}}

(11)

and Display Formula

f_{U} (X_{U}) = {f_{U} (X (i, j)) | \forall X (i, j) \in X_{U}} .

(12)

If one notes that

0 \leq c_{L} (x)

,

c_{U} (x) \leq 1

it is easy to see that

f_{L} (X_{L})

equalizes the subimagesubimage

X_{L}

over the range

(X_{0}, X_{m})

whereas

f_{U} (X_{U})

equalizes the subimagesubimage

X_{U}

over the range

(X_{m + 1}, X_{L - 1})

. As a consequence, the input image

X

is equalized over the entire dynamic range

(X_{0}, X_{L - 1})

with the constraint that the samples less than the input mean are mapped to

(X_{0}, X_{m})

and the samples greater than the mean are mapped to

(X_{m + 1}, X_{L - 1})

.

Brightness-Preserving Weighted Sub-Images

As stated above, BBHE separates the image into two subimages based on the mean of the input image, and HE is implemented for each subimage. Consequently, the mean brightness of each subimage may shift a lot from the brightness of the input image and artifacts may be observed in each subimage. Since two equalized subimages constitute the output image, the mean brightness of the output image may shift greatly, and artifacts appearing in each subimage will also be shown in the output image.

In contrast to the two subimages defined in Eqs. (11) and (12), we take a different way to define two subimages $Y_{L}$ and $Y_{U}$ as Display Formula

Y_{L} = f_{L} (X_{L}) \cup X_{U}

(13)

and Display Formula

Y_{U} = X_{L} \cup f_{U} (X_{U}) .

(14)

For subimage

Y_{L}

, the samples less than mean

X_{m}

are mapped from Eq. (8) and samples greater than

X_{m}

are mapped from Eq. (3); similarly, for subimage

Y_{U}

, the samples less than

X_{m}

are mapped from Eq. (2) and samples greater than mean

X_{m}

are mapped from Eq. (9).

Based on Eqs. (13) and (14), let’s define the output image $Y$ as the weighted sum of subimages $Y_{L}$ and $Y_{U}$ Display Formula

Y = λ_{L} Y_{L} + λ_{U} Y_{U},

(15)

where

λ_{L}

and

λ_{U}

are weight coefficients.

In the following, one way to calculate weight coefficients $λ_{L}$ and $λ_{U}$ is given.

First, in order to preserve the mean brightness of the input image, the weight coefficients $λ_{L}$ and $λ_{U}$ should meet the following constraint Display Formula

M_{X} = λ_{L} M_{Y_{L}} + λ_{U} M_{Y_{U}},

(16)

where

M_{X}

,

M_{Y_{L}}

, and

M_{Y_{U}}

denote the mean of the input image

X

, subimages

Y_{L}

and

Y_{U}

, respectively.

Second, the sum of the differences between the mean of each subimage $Y_{L}$ and $Y_{U}$ and the input image $X$ should satisfy Display Formula

λ_{L} (M_{Y_{L}} - M_{X}) + λ_{U} (M_{Y_{U}} - M_{X}) = 0.

(17)

Expanding the above equation and substituting the Eqs. (16) into (17) Display Formula

λ_{L} + λ_{U} = 1

(18)

Combining Eqs. (16) and (18), the solutions for

λ_{L}

and

λ_{U}

are Display Formula

λ_{L} = \frac{M_{X} - M_{Y_{U}}}{M_{Y_{L}} - M_{Y_{U}}}

(19)

Display Formula

λ_{U} = \frac{M_{Y_{L}} - M_{X}}{M_{Y_{L}} - M_{Y_{U}}} .

(20)

Now the output image

Y

can be expressed as Display Formula

Y = \frac{M_{X} - M_{Y_{U}}}{M_{Y_{L}} - M_{Y_{U}}} Y_{L} + \frac{M_{Y_{L}} - M_{X}}{M_{Y_{L}} - M_{Y_{U}}} Y_{U} .

(21)

It is clearly seen from the above steps that the output image

Y

can preserve the mean brightness of the input image

X

exactly.

Last, observing Eqs. (19) and (20), the values of $λ_{L}$ and $λ_{U}$ depend on $M_{X}$ , $M_{Y_{L}}$ and $M_{Y_{U}}$ .

Case 1: $λ_{L} < 0$ , $λ_{u} > 1$ when $M_{Y_{L}} < M_{Y_{L}} < M_{X}$ or $M_{X} < M_{Y_{U}} < M_{Y_{L}}$ ; Similarly, $λ_{L} > 1$ , $λ_{u} < 0$ when $M_{Y_{U}} < M_{Y_{U}} < M_{X}$ or $M_{X} < M_{Y_{L}} < M_{Y_{U}}$ . Because $λ_{u} > 1$ or $λ_{L} > 1$ , the subimage $Y_{U}$ or $Y_{L}$ may be overenhanced.

Case 2: $λ_{L} \approx 0$ , $λ_{u} \approx 1$ when $M_{X} \approx M_{Y_{U}}$ ; Or, $λ_{L} \approx 1$ , $λ_{u} \approx 0$ when $M_{X} \approx M_{Y_{L}}$ . The output image $Y \approx Y_{U}$ or $Y \approx Y_{L}$ . So, based on Eqs. (19) or (20), the mean of the output image $Y$ is almost the same as the mean of the input image $X$ . However, as shown in Eqs. (13) and (14), because only $f_{L} (X_{L})$ or $f_{U} (X_{U})$ is equalized, the contrast of the whole image $Y_{L}$ or $Y_{U}$ may not be enhanced well.

Case 3: For other cases, $0 < λ_{L} < 1$ , $0 < λ_{u} < 1$ . The mean brightness of the input image can be preserved accurately, and the contrast may be enhanced well.

Then it is necessary for us to find one efficient way to overcome the problems for Case 1 and 2.

By analyzing the Eqs. (19) and (20), it is found that $λ_{L}$ and $λ_{U}$ are determined by the values of $M_{X}$ , $M_{Y_{L}}$ and $M_{Y_{U}}$ . For one input image $X$ , the values of $M_{Y_{L}}$ and $M_{Y_{U}}$ depend on the separating point that separates the histogram into two subhistograms. So the values of $λ_{L}$ and $λ_{U}$ vary with different separating points, which suggests that Case 1 and 2 can be remedied by choosing one different separating point instead of the mean used for BBHE.

However, practically, it is not easy to find a good separating point for each image. Moreover, sometimes Case 3 does not meet at all, no matter what separating point is chosen for BBHE. So in order to overcome this drawback, an efficient approach is proposed in the following.

The constraints in Eqs. (16) and (17) are relaxed as follows, Display Formula

λ_{L} M_{Y_{L}} + λ_{u} M_{Y_{L}} = M_{x}^{'}

(22)

Display Formula

λ_{L} (M_{Y_{L}} - M_{X}^{'}) + λ_{u} (M_{Y_{U}} - M_{X}^{'}) = δ,

(23)

where

M_{X}^{'} = \min {M_{Y_{L}}, M_{Y_{U}}} + δ

and

δ > 0

.

Now the relaxed mean brightness $M_{X}^{'}$ lies between $M_{Y_{L}}$ and $M_{y_{U}}$ and Case 3 is met. Although the relaxed mean brightness $M_{X}^{'}$ is somewhat different from the mean brightness $M_{X}$ of the input image, this shortcoming can be compensated by the enhanced contrast.

The solutions for Eqs. (22) and (23) areDisplay Formula

λ_{L} = \frac{M_{X}^{'} - (1 - \frac{δ}{M_{X}^{'}}) M_{Y_{U}}}{M_{Y_{L}} - M_{Y_{U}}}

(24)

Display Formula

λ_{u} = \frac{M_{X}^{'} - (1 - \frac{δ}{M_{X}^{'}}) M_{Y_{L}}}{M_{Y_{U}} - M_{Y_{U}}}

(25)

The output image

Y

is Display Formula

Y = \frac{M_{X}^{'} - (1 - \frac{δ}{M_{X}^{'}}) M_{Y_{U}}}{M_{Y_{L}} - M_{Y_{U}}} Y_{L} + \frac{M_{X}^{'} - (1 - \frac{δ}{M_{X}^{'}}) M_{Y_{L}}}{M_{Y_{U}} - M_{Y_{L}}} Y_{U} .

(26)

Based on the above relax equations, although the mean brightness of the input image may shift a little, contrast can be enhanced efficiently as Case 3 stated in the above. Furthermore, a lot of CPU time to calculate a good separating point for BBHE is saved.

Now, considering Eqs. (24) and (25), it is necessary for us to calculate the range for $δ$ in order to avoid $λ_{L} < 0$ ( $λ_{u} < 0$ ) or $λ_{L} > 1$ ( $λ_{u} > 1$ ).

Sum Eqs. (24) and (25), Display Formula

λ_{L} + λ_{u} = 1 - \frac{δ}{M_{X}^{'}} .

(27)

Since

0 < \frac{δ}{M_{X}^{'}} < 1

, Eq. (27) means Display Formula

0 < λ_{L} + λ_{u} < 1.

(28)

So, based on the above equation,

λ_{L} < 0

(

λ_{u} < 0

) or

λ_{L} > 1

(

λ_{u} > 1

) can be avoided when

λ_{L} > 0

and

λ_{u} > 0

.

To calculate the range for $δ$ , two cases are considered as follows.

$\min {M_{Y_{L}}, M_{Y_{U}}} = M_{Y_{L}}$ , $M_{X}^{'} = M_{Y_{L}} + δ$ .Equation (24) must meet the following constraint, Display Formula
$λ_{L} = \frac{M_{X}^{'} - (1 - \frac{δ}{M_{X}^{'}}) M_{Y_{L}}}{M_{Y_{L}} - M_{Y_{U}}} > 0.$ (29)
Solving the above inequality, Display Formula
$δ < \sqrt{M_{Y_{L}} M_{Y_{U}}} - M_{Y_{L}} .$ (30)
Also, Eq. (25) must meet the following constraint, Display Formula
$λ_{u} = \frac{M_{X}^{'} - (1 - \frac{δ}{M_{X}^{'}}) M_{Y_{L}}}{M_{Y_{U}} - M_{Y_{L}}} > 0.$ (31)
Solving the above inequality, Display Formula
$δ + \frac{δ}{M_{Y_{L}} + δ} > 0.$ (32)
Obviously, the above inequality holds when $δ > 0$ .
$\min {M_{Y_{L}}, M_{Y_{U}}} = M_{Y_{U}}$ , $M_{X}^{'} = M_{Y_{U}} + δ$ . Similarly, we can get the following constraint for $δ$ , Display Formula
$δ < \sqrt{M_{Y_{L}} M_{Y_{U}}} - M_{Y_{U}} .$ (33)
Consequently, combining the above discussions in Eqs. (1) and (2), in order to avoid $λ_{L} < 0$ ( $λ_{u} < 0$ ) or $λ_{L} > 1$ ( $λ_{u} > 1$ ), $δ$ must meet the following constraints. Display Formula
${\begin{cases} 0 < δ < \sqrt{M_{Y_{L}} M_{Y_{U}}} - M_{Y_{L}}, & if \min {M_{y_{L}}, M_{y_{L}}} = M_{y_{L}} \\ 0 < δ < \sqrt{M_{Y_{L}} M_{Y_{U}}} - M_{Y_{U}}, & if \min {M_{y_{L}}, M_{y_{L}}} = M_{y_{U}} \end{cases} .$ (34)

Simulation Results

In order to demonstrate the performance of BPWSI, we test it on 80 various images by comparing it with HE, BBHE, DSIHE, RSIHE, RSMHE, CLAHE, and BHEPL-D. The RSIHE and RMSHE separate the histogram into four subhistograms by setting $r = 2$ .

Three metrics—average absolute mean brightness error (AAMBE), average peak signal noise ratio (APSNR), and average discrete entropy (ADE)—are used to evaluate the effectiveness of these methods.¹³

AAMBE is defined as Display Formula

AAMBE = \frac{1}{N} \sum_{i = 1}^{N} | M_{i} (X) - M_{i} (Y) |,

(35)

where

N

is the total number of test images,

M_{i} (X)

and

M_{i} (Y)

are the mean brightness of input and output images, respectively. A lower value of AAMBE indicates that the output image preserves the mean brightness of the input image better.

APSNR is defined as Display Formula

APSNR = \frac{1}{N} 10 \log_{10}^{\frac{255^{2}}{\frac{1}{HW} \sum_{i = 1}^{H} \sum_{j = 1}^{W} {(X_{i, j} - Y_{i, j})}^{2}}},

(36)

where

X_{i, j}

and

Y_{i, j}

denote the intensities of the input and output images at the spatial location

(i, j)

.

H

and

W

are the image size. The higher the value of APSNR, the less the noise level in the output image.

ADE is defined as Display Formula

ADE = - \frac{1}{N} \sum_{i = 0}^{255} p_{Y} (X_{i}) \times \log_{2}^{p_{Y} (X_{i})},

(37)

where

p_{Y} (X_{i})

is the probability density function for the output image

Y

at intensity level

X_{i}

. Often, the higher the value of ADE, the richer details and information the output image holds.

The results for three metrics defined above on 80 various images are shown in Table 1.

Table 1Simulation results for three metrics.

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Method	AAMBE	APSNR	ADE
HE	41.4557	12.9419	6.1070
BBHE	16.6857	16.0426	6.1052
DSIHE	15.5231	15.2839	6.1361
RSIHE	6.9589	19.8704	6.0950
RMSHE	4.2493	22.6584	6.0955
CLAHE	16.4150	17.1618	7.3522
BHEPL-D	3.9764	24.2088	6.2853
BPWSI	0.2191	24.6564	6.2159

As shown in the second column of Table 1, the proposed method BPWSI has the least AAMBE value, which indicates that BPWSI can preserve mean brightness more accurately than other methods. In addition, it is noted that the value of AAMBE for BPWSI is less than half of one gray level. So BPWSI performs excellently in brightness preservation.

The third column in Table 1 indicates that BHEPL-D and BPWSI have larger values of APSNR where BPWSI ranks first. So, in terms of less noise level, BHEPL-D and BPWSI outperform other methods.

Among eight methods, CLAHE has the largest ADE value, and BHEPL-D and BPWSI rank next largest for similar ADE values, as shown in the fourth column in Table 1. However, the output images of CLAHE often look unnaturally and have some artifacts, as will be demonstrated in the following examples.

Consequently, it is obviously seen from Table 1 that the proposed method, BPWSI, performs best among the eight methods in terms of brightness preservation and contrast enhancement.

In addition to the quantitative evaluation of brightness preservation and contrast enhancement by using three metrics—AAMBE, APSNR, and ADE—it is also necessary for us to inspect the visual quality of the output images enhanced by all the methods mentioned above.

We choose three images with high and low contrast to demonstrate the performance of BPWSI, shown in Figs. 1 2–3, respectively. Notice the number in the brackets denotes the mean brightness of each image.

The original image shown in Fig. 1(a) has good contrast. The qualities of the images enhanced by BPWSI, HE, BBHE, DSIHE, RSIHE, RMSHE, and BHEPL-D, shown in Fig. 1(b) through 1(h), respectively, are acceptable. However, it is seen clearly from Fig. 1(i) that the quality of the image enhanced by CLAHE is degraded and it looks unnatural. At the same time, the image enhanced by BPWSI, shown in Fig. 1(b), has almost the same mean brightness as that of the input image shown in Fig. 1(a). So BPWSI will not deteriorate the quality of the image with good contrast, which is important for the contrast enhancement methods used in consumer electronics.

Enhanced results for Image 1: (a) Original (120.2164), (b) BPWSI (120.2103), (c) HE (128.2109), (d) BBHE (124.9352), (e) DSIHE (105.7611), (f) RSIHE (125.2669), (g) RMSHE (125.9255), (h) BHEPL-D (128.4071), (i) CLAHE (129.3829).

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Enhanced results for Image 2: (a) Original (107.7522), (b) BPWSI (107.7239), (c) HE (128.8276), (d) BBHE (125.2168), (e) DSIHE (119.7769), (f) RSIHE (116.9218), (g) RMSHE (116.9452), (h) BHEPL-D (112.7132), (i) CLAHE (124.3001).

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Enhanced results for Image 3: (a) Original (69.7949), (b) BPWSI (79.5240), (c) HE (127.6524), (d) BBHE (115.0546), (e) DSIHE (103.2182), (f) RSIHE (86.5241), (g) RMSHE (84.6074), (h) BHEPL-D (96.0357), (i) CLAHE (113.2089).

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Figure 2(a) shows one image with low contrast. Obviously, over-enhancement can be observed for the face and background in the images enhanced by HE, BBHE, DSIHE, RSIHE, RMSHE, and BHEPL-D, shown in Fig. 2(c) through 2(h), respectively. In Fig. 2(i), many annoying artifacts can be seen easily in the face and hair of the image enhanced by CLAHE. In Fig. 2(b), the image enhanced by BPWSI cannot only preserve the brightness accurately but also enhance the contrast efficiently. Moreover, over-enhancement and artifacts do not show in Fig. 2(b), and it looks more natural than all other enhanced images.

Figures 1(a) and 2(a) meet Case 3 mentioned above in the Sec. 3. In the following, we will show another image in Fig. 3(a), which doesn’t meet Case 3, and relaxation approach, as shown in Eq. (26), is taken to get the enhanced image.

Figure 3 shows the enhanced results for Image 3, which meets Case 2 described in Sec. 3 ( $λ_{L} \approx 1$ and $λ_{U} \approx 0$ ). The relaxed output image based on Eq. (30) is shown in Fig. 3(b). Here, $M_{X} = 69.7949$ , $M_{Y_{L}} = 69.5280$ , and $M_{Y_{U}} = 116.5653$ . So let $δ = 10$ . Obviously, the contrast of Fig. 3(b) is enhanced a lot, compared with Fig. 3(a). At the same time, over-enhancement is observed for clouds and buildings of the images enhanced by HE, BBHE, DSIHE, RSIHE, RMSHE, BHEPL-D, and CLAHE, shown in Fig. 3(c) through 3(i), respectively.

By inspecting the visual qualities of the three images enhanced by eight methods, it is obvious that the images enhanced by the proposed method, BPWSI, are the best. Furthermore, it is found by the experimental results on 80 different images that the performance of BPWSI is quite stable while the qualities of some images enhanced by other methods, such as BHEPL-D and CLAHE, are often degraded greatly.

In summary, combining the three quantitative metrics and the qualitative visual qualities of the enhanced images, the proposed method, BPWSI, outperforms other HE-based methods in terms of brightness preservation and contrast enhancement. Furthermore, it is noted that BPWSI can preserve mean brightness with higher accuracy than other methods.

In order to demonstrate the excellent performance in preserving brightness for BPWSI, the transformation functions for Images 1, 2, and 3 are shown in Fig. 4(a)4–4(c), respectively (for clarity, we show transformation functions for HE, BBHE, BHEPL-D, and BPWSI). Here, the x-axis denotes input gray levels and the y-axis denotes the output gray levels.

Transformation functions for (a) Image 1, (b) Image 2, and (c) Image 3.

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It is seen clearly from Fig. 4 that the transformation function for BPWSI has the least gradient value in the whole dynamic levels among four methods. In contrast, the transformation function for HE has the greatest gradient value. That is the reason why HE often shifts the mean brightness greatly, and BPWSI can preserve the mean brightness with higher accuracy.

The Effect of δ on Contrast Enhancement and Brightness Preservation

Although the range for $δ$ is given in Eq. (34), it is difficult to have one general rule to find one optimal value for each image. Generally, when the value of $δ$ is greater, the mean brightness of output image will shift a lot from the mean brightness of the input image. In addition, annoying artifacts may be shown in the output image.

Here Image 3 is used to illustrate the phenomenon. Based on Eq. (34), for image 3, $δ$ should meet the following constraint, Display Formula

0 < δ < 20.11.

In Fig. 5, images for $δ$ with different values are shown.

The effect of δ on Image 3: (a) Original (69.7949), (b) δ=2(71.5630), (c) δ=5(74.5590), (d) δ=10(79.5240), (e) δ=16(85.6489).

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Obviously, mean brightness is preserved well with smaller $δ$ . However, contrast cannot be enhanced much. In contrast, with greater $δ$ , contrast can be enhanced a lot. At the same time, annoying artifacts appear for the clouds and building, as shown in Fig. 5(e). Consequently, there is a trade-off between brightness preservation and contrast enhancement for different values of $δ$ .

To balance brightness preservation and contrast enhancement, $δ$ depends on the absolute mean error between $M_{X}$ and $\min {M_{Y_{L}}, M_{Y_{U}}}$ . In other words, it is suggested that $δ$ is smaller when the absolute mean error is large and $δ$ is bigger when the absolute mean error is small.

Limitations of Brightness-Preserving Methods

In this section, we discuss some limitations of brightness-preserving methods.

First, many brightness-preserving methods, such as in Refs. ⁴ through ¹³ and BPWSI, are based on spatial domain. In contrast with those methods based on transform domain, spatial-domain-based brightness-preserving methods cannot enhance the local details effectively since they have to satisfy the constraint of brightness preservation. For example, in ², three transform-based methods—logarithmic transform histogram matching, logarithmic transform histogram shifting, and logarithmic transform histogram shaping—are introduced for effective contrast enhancement. In Fig. 6(b) through 6(d), enhanced results by the three methods are shown. It is easily seen that all the three methods in ² enhance the image better than BPWSI shown in Fig. 6(e). However, we also notice that over-enhancement and noise are shown in the background of Fig. 6(b) through 6(d).

Enhanced results by ² and BPWSI: (a) Original (219.8437), (b) logarithmic transform histogram matching (212.1257), (c) logarithmic transform histogram shifting (204.8763), (d) logarithmic transform histogram shaping (215.8780), (e) BPWSI (219.8576).

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In addition, it should be pointed out that most brightness-preserving cannot enhance the dark image efficiently because they have to preserve the mean brightness of the dark image. In ³, human visual system based multi-HE (HVSMHE) is proposed to enhance such dark images.

Figure 7(a) shows one dark image from ³. The results enhanced by HE, RMSHE, HVSMHE, and BPWSI are shown in Fig. 7(b)–7(e), respectively.

Obviously, brightness-preserving methods, such as RMSHE and BPWSI, cannot enhance the dark image, as shown in Fig. 7(c) and 7(e). The visual quality of the image enhanced by HVSMHE is the best. However, the mean brightness of the image shown in Fig. 7(d) is far different from the brightness of the original image shown in Fig. 7(a). In other words, HVSMHE cannot preserve the brightness accurately. In addition, although HE is usually effective in enhancing the dark image, pronounced noise is observed in the smooth area, as shown in Fig. 7(b).

Enhanced results for dark image: (a) Original (34.7131), (b) HE (130.7999), (c) RMSHE (37.4289), (d) HVSMHE (84.5970), (e) BPWSI (38.5952).

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As stated above, although brightness-preserving methods have some limitations, they are still wildly used in the area of consumer electronics because of their simplicity and ease of implementation.

Conclusions

BBHE is a well-known contrast-enhancement method for consumer electronics. However, in many cases, it tends to change significantly the brightness of the input image. Thus the quality of the output image will be degraded because of the annoying artifacts, such as over-enhancement, etc. In order to preserve the mean brightness of the input image and enhance the contrast efficiently, a novel method, BPWSI, is proposed in this paper. The basic idea of the proposed method is to combine two subimages in one brightness-preserving way. Experimental results on many images with low and high contrast show that BPWSI can preserve the mean brightness with higher accuracy than BBHE, DSIHE, RSIHE, RMSHE, CLAHE, and BHEPL-D. At the same time, contrast can be enhanced with less annoying artifacts than the above-mentioned methods. Analysis on Eqs. (21) and (26) indicates that the principle of BPWSI is quite simple, which makes it suitable for real-time systems.

Finally, it should be pointed out that the principle of BPWSI can also be extended for many other methods, such as DSIHE, RSIHE, RMSHE, and BHEPL-D, etc. In other words, we can combine multi-subimages in one brightness-preserving way. For the sake of space limits, it is not discussed in detail in this paper.

Acknowledgments

This work is supported by the Fundamental Research Funds for Central University, China (No. CDJZR10150015).

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Zongwei Lu received his BSc and MSc degrees in electrical engineering from Tianjin University, China, in 1995 and 1999, respectively. He received his PhD in electrical engineering from Shanghai Jiaotong University in 2002. From 2002 to 2005, he was with the department of electrical engineering, Shanghai Jiaotong University, as an instructor. From 2005 to 2008, he worked as a senior engineer at Synopsys Inc., Mount View, USA. Now he is an associate professor at the department of electrical engineering, Chongqing University, China. His research interests include image enhancement, image segmentation, and signal processing.

Brightness–preserving weighted subimages for contrast enhancement of gray–level images

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Abstract

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Introduction

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Brightness-Preserving Bi-Histogram Equalization

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Brightness-Preserving Weighted Sub-Images

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Simulation Results

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