1 INTRODUCTION

Contrast enhancement is a technique that makes the image features to be more clearly prominent by making optimum use of the gray levels on the display or output device. This improves the visual quality of an image that helps to analyze and interpret the image easily for machine vision applications. Traditional methods such as intensity transformation and histogram-based techniques are used for contrast enhancement [1]. Recently, numerous algorithms are developed to improve the contrast of the image based on the histogram. Bi-histogram equalization methods and two-dimensional histogram-based enhancement methods are performing well in the state of art algorithms. Though contrast enhancement is a subjective process, the efficacy of the algorithms is assessed by quantitative and qualitative analysis.

Qualitative analysis is based on visual examination and offers details about annoying artifacts, unusual enhancement, and over enhancement. It resolves the ability of the enhancement technique which is justified by human eyes. Mean opinion score (MOS) is used as a particular quality measure for many years. However, the MOS method is problematic, time-consuming, and expensive for realistic usage. This analysis provides various opinions for different users [2]. Additionally, subjective measures are more complex by numerous factors including viewing distance, lighting condition, and subject’s vision ability.

Because of the above-mentioned problems linked with the subjective measures of image quality, It is important to develop a mathematical model for assessing the quality of an image known as objective measures. Objective measure forecasts the quality of an image precisely and automatically. A perfect objective measure should have the ability to signify the quality predictions of a subjective measure. Based on the availability of the reference image the objective measures or quantitative metrics are classified into three types. They are full reference approaches that determine the similarity between the processed and reference images, reduced reference metrics where reference images are not always available, no reference metrics where the assessment is done without prior knowledge of the reference image. Many of the applications, reference images are not available, so no reference metrics are convenient in practice [3].

In this paper, various performance metrics that help to evaluate the quality of the contrast-enhanced images are discussed. Absolute mean brightness error measures the shifting of the mean brightness. Contrast, contrast improvement index (CII), contrast per pixel (CPP) and standard deviation (σ) quantifies the amount of contrast enhancement in the processed image. The structural information loss is measured by universal quality index (UQI), structural similarity index (SSIM), quality index based on local variance (QILV), and gradient magnitude similarity deviation (GMSD). The amount of contrast improvement can also be measured by Measure of enhancement (EME), Measure of enhancement by entropy (EMEE), Michelson-law measure of enhancement (AME), and Michelson-law measure of enhancement by entropy (AMEE). The information loss is identified by discrete entropy (DE) and two-dimensional entropy is calculated from the gray-level co-occurrence matrix which is known as Mutual information-based contrast measure (MICM). The objective measures like peak signal to noise ratio (PSNR), UQI, SSIM, and QILV are full reference metrics whereas DE, Absolute Mean Brightness Error (AMBE), CII, CPP, Standard Deviation, EME, EMEE, AME, AMEE, GMSD, and MICM are no reference metrics. By considering the pros and cons of the quantitative measures, two new performance metrics are suggested in this paper. To quantify the mean shifting and improvement of the contrast a metric called new contrast (New_cont) is developed and another metric difference in entropy (Diff_ E) is given to measure entropy preservation.

The rest of the paper is organized as follows. In Section 2, various measures used to quantify image enhancement are discussed and analyzed. Section 3 signifies the experimental results and comparisons to the existing metrics and a brief discussion of the algorithm which is used to obtain the enhanced image for analysis. The conclusions are discussed in Section 4.

2 TAXONOMY OF PERFORMANCE METRICS

The objective of the quantitative analysis is that it numerically measures the effectiveness of the enhancement algorithms. In this section, various full reference and no reference quality metrics are discussed. In the complete taxonomy of performance measures, \(I\) and \(J\) represent low contrast input and contrast-enhanced images, respectively with the size of \(M \times N\). \({{\mu }_{I}}\) and \(\sigma _{I}^{2}\) denote mean and variance of the low contrast input image, \({{\mu }_{J}}\) and \(\sigma _{J}^{2}\) denote the mean and variance of the contrast-enhanced image, respectively. \({{\sigma }_{{I,J}}}\) is the covariance of images \(I\) and \(J\).

2.1 Peak Signal to Noise Ratio (PSNR)

PSNR is used for evaluating image quality. It indicates the level of degradation of enhanced image when compared to the input image [4]. It quantifies the reconstruction ability of an algorithm and it measures the amount of deviation between enhanced and original images.

$$PSNR\, = \,10{{\log }_{{10}}}\left[ {\frac{{{{{\left( {L - 1} \right)}}^{2}}}}{{\frac{1}{{M \times N}}\sum\limits_x {\sum\limits_y {{{{\left| {I(x,y)\, - \,J(x,y)} \right|}}^{2}}} } }}} \right],$$
(1)

where \(\left( {L - 1} \right)\) indicates the maximum intensity of the image. The total number of pixels in the image is represented by \(M \times N\). For an 8-bit image, the PSNR value lies between 30–50 dB and it signifies the better enhancement.

2.2 Entropy

Image entropy stipulates the uncertainty in the image values. It measures histogram dispersion and averaged quantity of information to encode the image values [5], [6], [7]. A higher value of entropy specifies that the image has better-off details. Entropy is computed by:

$$E\left( I \right) = - \mathop \sum \limits_{l = 1}^P \,p\left( {{{i}_{l}}} \right){{\log }_{2}}p\left( {{{i}_{l}}} \right),$$
(2)

where \(p\left( {{{i}_{l}}} \right)\) is the probability of the intensity \({{i}_{l}}\). \(P\) indicates the total number of intensities.

2.3 Measure of Enhancement (EME)

The improvement in images after enhancement is difficult to measure. Based on the human visual system weber law states that the detection of objects depends on the ratio of intensity values in the local region. Measure of enhancement (EME) is developed based on the weber’s law [8]. It is defined as

$$EME = \frac{1}{{{{K}_{1}}{{K}_{2}}}}\mathop \sum \limits_{k = 1}^{{{k}_{1}}} \,\mathop \sum \limits_{l = 1}^{{{k}_{2}}} \,20\ln \left( {\frac{{{{I}_{{\max ,k,l}}}}}{{{{I}_{{\min ,k,l}}}}}} \right),$$
(3)

where \({{K}_{1}}{{K}_{2}}\) represents the number of non- overlapping blocks with each block size \({{k}_{1}}{{k}_{2}}\). \({{I}_{{{\text{max}},k,l}}}\) and \({{I}_{{{\text{min}},k,l}}}\) denote the maximum and minimum intensities in the block respectively. Here \({{k}_{1}}{{k}_{2}}\) values are considered as \(8 \times 8\). EME value increases with the increment in contrast.

2.4 Universal Quality Index (UQI)

Universal quality index quantifies the loss of correlation, mean luminance preservation between two images, and similarity of contrast between the input and the enhanced image [9]. It is represented as:

$$Q = \frac{{{{\sigma }_{{I,J}}}}}{{{{\sigma }_{I}}{{\sigma }_{J}}}} \cdot \frac{{2{{\mu }_{I}}{{\mu }_{J}}}}{{(\mu _{I}^{2} + \mu _{J}^{2})}} \cdot \frac{{2{{\sigma }_{I}}{{\sigma }_{J}}}}{{\sigma _{I}^{2} + \sigma _{J}^{2}}},$$
(4)

where \(\frac{{{{\sigma }_{{I,J}}}}}{{{{\sigma }_{I}}{{\sigma }_{J}}}}\) indicates loss of correlation, \(\frac{{2{{\mu }_{I}}{{\mu }_{J}}}}{{(\mu _{I}^{2} + \mu _{J}^{2})}}\) denotes the closeness of mean luminance and \(\frac{{2{{\sigma }_{I}}{{\sigma }_{J}}}}{{\sigma _{I}^{2} + \sigma _{J}^{2}}}\) represents the similarity of contrast between the enhanced and the high contrast original images.

$$Q = \frac{{(2{{\mu }_{I}}{{\mu }_{J}})(2{{\sigma }_{{I,J}}})}}{{(\mu _{I}^{2} + \mu _{J}^{2})(\sigma _{I}^{2} + \sigma _{J}^{2})}}.$$
(5)

It varies from 0 to 1, where ‘1’ indicates there is no distortion in the processed image, ‘0’ indicates a higher degree of distortion.

2.5 Absolute Mean Brightness Error

To prevent the enhanced image from annoying artifacts and deterioration of visual quality, preservation of the mean brightness of the image is important during enhancement [10]. AMBE is used as an objective measure to examine the preservation of mean brightness in the enhanced image [11], [12]. It is defined as:

$$AMBE = \left| {{{\mu }_{I}} - {{\mu }_{J}}} \right|.$$
(6)

Lower values of AMBE indicates that the mean brightness is well preserved.

2.6 Contrast Improvement Index (CII)

CII uses a local window of dimension 3 × 3 to compute the contrast improvement in a spatial location of the enhanced image over the low contrast image [13], [14]. It is given as:

$$CII = \frac{{E\left( {{{C}_{{loc}}}\left( J \right)} \right)}}{{E\left( {{{C}_{{loc}}}\left( I \right)} \right)}},$$
(7)

where

$${{C}_{{loc}}} = \frac{{max\; - min}}{{max\; + min}},$$
(8)

where max and min represent the maximum and minimum intensity values in a 3 × 3 window respectively. Greater CII represents a better locally enhanced image.

2.7 Structural Similarity Index (SSIM)

SSIM is a perceptual quality measurement of a processed image to the reference image [15]. It varies from 0 to 1, where ‘1’ indicates the structural information of the image is prevented and ‘0’ indicates structural information is lost during enhancement. It is calculated from the statistical parameters of input and enhanced images. It is defined as,

$$SSIM\left( {I,J} \right) = \frac{{\left( {2{{\mu }_{I}}{{\mu }_{J}} + {{c}_{1}}} \right)\left( {2{{\sigma }_{{I,J}}} + {{c}_{2}}} \right)}}{{(\mu _{I}^{2} + \mu _{J}^{2} + {{c}_{1}})(\sigma _{I}^{2} + \sigma _{J}^{2} + {{c}_{2}})}},$$
(9)

where \({{c}_{1}}\) and \({{c}_{2}}\) are constants. They are defined as:

$${{c}_{1}} = {{\left( {{{k}_{1}}L} \right)}^{2}},\quad {{c}_{2}} = {{\left( {{{k}_{2}}L} \right)}^{2}},$$
(10)

\({{k}_{1}} \ll 1\), \({{k}_{2}} \ll 1\) and \(L\) is the maximum intensity of the image. The typical value of \(L\), \({{k}_{1}}\) and \({{k}_{2}}\) are considered to be 255, 0.01, and 0.04, respectively in this study. \({{\sigma }_{{I,J}}}\) is the covariance of images \(I\) and \(J\).

2.8 Contrast

It measures the deviation of pixel intensity values from the mean intensity of the image through this, contrast improvement of the output image is calculated [16]. It is defined as:

$$\begin{gathered} C = \frac{1}{{M \times N}}\mathop \sum \limits_{x = 1}^M \,\mathop \sum \limits_{y = 1}^N \,{{J}^{2}}\left( {x,y} \right) \\ - \;{{\left| {\frac{1}{{M \times N}}\mathop \sum \limits_{x = 1}^M \,\mathop \sum \limits_{y = 1}^N \,J\left( {x,y} \right)} \right|}^{2}}. \\ \end{gathered} $$
(11)

A higher value of \(C\) indicates the more utilization of the entire dynamic range and it is expressed in decibels as follows:

$${{C}_{{indB}}} = 10{{\log }_{{10}}}C.$$
(12)

2.9 Quality Index Based on Local Variance (QILV)

QILV is used to measure the structural information of the image. It is similar to SSIM, where the global statistic parameters are considered, but in QILV, local statistic parameters are considered to make an accurate assessment. In SSIM the degradation caused by blur is considered the minimum, whereas lower values of QILV indicate that higher the blur in the processed image [17].

$${\text{QILV}}\left( {I,J} \right) = \frac{{2{{\mu }_{{{{V}_{I}}}}}{{\mu }_{{{{V}_{J}}}}} \times 2{{\sigma }_{{{{V}_{I}}}}}{{\sigma }_{{{{V}_{J}}}}} \times {{\sigma }_{{{{V}_{I}}{{V}_{J}}}}}}}{{(\mu _{{{{V}_{I}}}}^{2} + \mu _{{{{V}_{J}}}}^{2}) \times (\sigma _{{{{V}_{I}}}}^{2} + \sigma _{{{{V}_{J}}}}^{2}) \times {{\sigma }_{{{{V}_{I}}}}}{{\sigma }_{{{{V}_{J}}}}}}},$$
(13)

where \({{\mu }_{{{{V}_{I}}}}}\) and \({{\mu }_{{{{V}_{J}}}}}\) represents the mean of the local variance of the input and processed images \(I\) and \(J\), respectively. \({{\sigma }_{{{{V}_{I}}}}}\) and \({{\sigma }_{{{{V}_{J}}}}}\) denotes the standard deviation of the input and processed images, respectively.

$${{\mu }_{{{{V}_{I}}}}} = \left( {\frac{1}{{MN}}} \right)\mathop \sum \limits_{i = 1}^M \,\mathop \sum \limits_{j = 1}^N \,Var\left( {{{I}_{{i,j}}}} \right),$$
(14)

and

$${{\sigma }_{{{{V}_{I}}}}} = {{(E\{ {{(Var\left( {{{I}_{{i,j}}}} \right) - {{\mu }_{{{{V}_{I}}}}})}^{2}}\} )}^{{1/2}}},$$
(15)

where \(Var({{I}_{{i,j}}}) = E({{({{I}_{{i,j}}} - \overline {{{I}_{{i,j}}}} )}^{2}})\) and \({{I}_{{i,j}}} = E\left( {{{I}_{{i,j}}}} \right)\).

2.10 Standard Deviation (σ)

To measure contrast globally, the standard deviation is used [18]. It is defined as

$$\sigma = \sqrt {\mathop \sum \limits_{k = 0}^{L - 1} {{{\left( {{{J}_{k}} - {{\mu }_{J}}} \right)}}^{2}} \times pdf\left( {{{J}_{k}}} \right)} ,$$
(16)

where \({{J}_{k}}\) indicates kth the intensity of the enhanced image and \(pdf\left( {{{J}_{k}}} \right)\) represents the probability density of kth intensity. A higher value of the standard deviation indicates better enhancement.

2.11 Michelson-Law Measure of Enhancement (AME)

Michelson law measures the contrast in terms of the ratio of the difference and sum of maximum and minimum intensities. AME is developed based on the Michelson law to measure the contrast in the enhanced image [19]. It is defined as

$$AME = - \frac{1}{{{{K}_{1}}{{K}_{2}}}}\mathop \sum \limits_{k = 1}^{{{k}_{1}}} \,\mathop \sum \limits_{l = 1}^{{{k}_{2}}} \,20\ln \left( {\frac{{{{I}_{{\max ,k,l}}} - {{I}_{{\min ,k,l}}}}}{{{{I}_{{\max ,k,l}}} + {{I}_{{\min ,k,l}}}}}} \right),$$
(17)

where \({{K}_{1}}{{K}_{2}}\) represents the number of non- overlapping blocks with each block size \({{k}_{1}}{{k}_{2}}\). \({{I}_{{{\text{max}},k,l}}}\) and \({{I}_{{{\text{min}},k,l}}}\) denote the maximum and minimum intensities in the block respectively. Here \({{k}_{1}}{{k}_{2}}\) values are considered as \(8 \times 8\). AME value decrease with the increment in contrast.

2.12 Michelson-Law Measure of Enhancement by Entropy (AMEE)

To measure the contrast by employing the entropy, AMEE is developed to evaluate the enhancement [19]. It is defined as

$$\begin{gathered} AMEE = \frac{1}{{{{K}_{1}}{{K}_{2}}}}\mathop \sum \limits_{k = 1}^{{{k}_{1}}} \,\mathop \sum \limits_{l = 1}^{{{k}_{2}}} \,\alpha {{\left( {\frac{{{{I}_{{\max ,k,l}}} - {{I}_{{\min ,k,l}}}}}{{{{I}_{{\max ,k,l}}} + {{I}_{{\min ,k,l}}}}}} \right)}^{\alpha }} \\ \times \;\ln \left( {\frac{{{{I}_{{\max ,k,l}}} - {{I}_{{\min ,k,l}}}}}{{{{I}_{{\max ,k,l}}} + {{I}_{{\min ,k,l}}}}}} \right), \\ \end{gathered} $$
(18)

where \({{K}_{1}}{{K}_{2}}\) represents the number of non- overlapping blocks with each block size \({{k}_{1}}{{k}_{2}}\). \({{I}_{{{\text{max}},k,l}}}\) and \({{I}_{{{\text{min}},k,l}}}\) denote the maximum and minimum intensities in the block respectively. Here \({{k}_{1}}{{k}_{2}}\) values are considered as \(8 \times 8\). AMEE value increases with the increment in contrast. \(\alpha \) is constant. In this paper, \(\alpha \) value is considered as 1. AMEE is having a positive correlation with subjective ranking [20].

2.13 Contrast per Pixel (CPP)

It measures the average difference between the intensity value of a pixel and its neighbour [21]. CPP is defined as,

$$CPP = \frac{{\sum\limits_{i = 1}^M {\sum\limits_{j = 1}^N {\left( {\sum\limits_{\left( {m,n} \right) \in R_{3}^{{\left( {i,j} \right)}}} {\left| {\gamma \left( {i,j} \right) - \gamma \left( {m,n} \right)} \right|} } \right)} } }}{{M \times N \times 8}}.$$
(19)

It is a measure of contrast. where \(\gamma \left( {i,j} \right)\) represents the center pixel and \(\gamma \left( {m,n} \right)\) represents the neighbourhood pixel in a local window of size 3 × 3. \(M \times N\) represents the size of the image.

2.14 Measure of Enhancement by Entropy (EMEE)

Entropy measures the randomness of the image. To incorporate the entropy for measuring the contrast of an image, EMEE is developed by modifying EME which is based on the weber contrast measure [22]. It is represented as:

$$EMEE = \frac{1}{{{{K}_{1}}{{K}_{2}}}}\mathop \sum \limits_{k = 1}^{{{k}_{1}}} \,\mathop \sum \limits_{l = 1}^{{{k}_{2}}} \,\alpha {{\left( {\frac{{{{I}_{{\max ,k,l}}}}}{{{{I}_{{\min ,k,l}}}}}} \right)}^{\alpha }}\ln \left( {\frac{{{{I}_{{\max ,k,l}}}}}{{{{I}_{{\min ,k,l}}}}}} \right),$$
(20)

where \({{K}_{1}}{{K}_{2}}\) represents the number of non- overlapping blocks with each block size \({{k}_{1}}{{k}_{2}}\). \({{I}_{{{\text{max}},k,l}}}\) and \({{I}_{{{\text{min}},k,l}}}\) denote the maximum and minimum intensities in the block respectively. Here \({{k}_{1}}{{k}_{2}}\) values are considered as \(8 \times 8\). EMEE value increases with the increment in contrast. \(\alpha \) is constant. In this paper, \(\alpha \) value is considered as 1.

2.15 Gradient Magnitude Similarity Deviation (GMSD)

GMSD is a perceptual image quality index with high prediction accuracy. It foresees the quality of the image by merging pixel-wise gradient magnitude similarity (GMS) with the standard deviation of the GMS map [23]. The horizontal and vertical gradient magnitude of images \(I\) and \(J\) are defined as:

$${{h}_{r}}\left( i \right) = \sqrt {{{{\left( {I \otimes {{h}_{x}}} \right)}}^{2}}\left( i \right) + {{{\left( {I \otimes {{h}_{y}}} \right)}}^{2}}\left( i \right)} ,$$
(21)
$${{{v}}_{d}}\left( i \right) = \sqrt {{{{\left( {J \otimes {{h}_{x}}} \right)}}^{2}}\left( i \right) + {{{\left( {J \otimes {{h}_{y}}} \right)}}^{2}}\left( i \right)} ,$$
(22)

where a symbol \( \otimes \) represents convolution. The \({{h}_{x}}\) and \({{h}_{y}}\) are Prewitt filters in the horizontal direction (x) and vertical direction (y) respectively. The gradient magnitude similarity (GMS) map is represented as

$$G\left( i \right) = \frac{{2{{h}_{r}}\left( i \right){{{v}}_{d}}\left( i \right) + \delta }}{{h_{r}^{2}\left( i \right) + {v}_{d}^{2}\left( i \right) + \delta }},$$
(23)

where \(\delta \) is a positive constant for stability. Then mean value is calculated by

$$GM = \frac{1}{{M \times N}}\mathop \sum \limits_{i = 1}^{M \times N} G\left( i \right),$$
(24)

where \(M \times N\) is the total number of pixels in an image. GMSD is defined by

$$GMSD = \sqrt {\frac{1}{{M \times N}}\mathop \sum \limits_{i = 1}^{M \times N} {{{\left( {G\left( i \right) - GM} \right)}}^{2}}} .$$
(25)

Lower GMSD score denotes better image perceptual quality.

2.16 Mutual Information-based Contrast Measure (MICM)

It is the no-reference metric used to measure the contrast based on the spatial correlation [24]. To calculate the MICM the gray level co-occurrence matrix is used. It is represented as

$$MICM = \mathop \sum \limits_{i = 1}^M \,\mathop \sum \limits_{j = 1}^N \,p\left( {i,j} \right){{\log }_{2}}\left( {\frac{{p\left( {i,j} \right)}}{{p\left( i \right)p\left( j \right)}}} \right),$$
(26)

\(p\left( {i,j} \right)\) represents the joint probability of intensity values which is obtained from the gray level co-occurrence matrix. \(p\left( i \right)\) and \(p\left( j \right)\) denote the marginal probabilities in row and column of co-occurrence matrix respectively. \(M\) and \(N\) represent the size of the co-occurrence matrix respectively.

Table 1 shows the complete taxonomy of the objective measures used for contrast enhancement. In Table 1 the second column gives the various performance measures, the third column shows the mathematical representation for the measures, the fourth column highlights the importance of the measures and the fifth column gives the details about the limitations of each performance measure.

Table 1. Performance Measures
Full size table

2.17 Suggested Measures

2.17.1. New contrast measure (New_cont). Based on the above discussions by considering the strengths of AMBE, contrast, and Standard Deviation, a new single performance measure is proposed for measuring the mean brightness shifting and improvement in contrast. There is no bound for the AMBE, contrast, and Standard Deviation values mentioned in the literature. But the proposed measure has an ideal range [0,1]. The value ‘0’ indicates that the mean value is shifted and there is no significant improvement in the contrast. The value ‘1’ indicates that the mean value is preserved and there is an improvement in the contrast. It is defined as:

$$New\_cont = \frac{{\sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {{{{\left( {J\left( {x,y} \right) - {{\mu }_{J}}} \right)}}^{2}}} } }}{{\sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {{{{\left( {J\left( {x,y} \right)} \right)}}^{2}}} } }},$$
(27)

where \(J\left( {x,y} \right)\) represents the enhanced image.2.17.2 Difference in Entropy (Diff_E).

In contrast enhancement, the entropy value of the processed image is the same as that of a low contrast input image, indicates entropy preservation or there is no loss of information in the processed image. Entropy preservation also quantifies that, there is no loss of gray levels in the processed image [25]. To measure the entropy preservation a new measure is proposed. The low value of the measure indicates the good preservation of entropy. It is defined as:

$$Diff\_{\kern 1pt} E = \frac{{\sum\limits_{x = 1}^M {\sum\limits_{y = 1}^N {\left( {\sum\limits_{i,j \in R_{3}^{{\left( {x,y} \right)}}} {\left| {E\left( j \right) - E\left( i \right)} \right|} } \right)} } }}{{\left| {E\left( J \right) - E\left( I \right)} \right|}},$$
(28)

where \(E\left( j \right)\) and \(E\left( i \right)\) represent the entropy values of the local neighbourhood of \(3 \times 3\) of enhanced and low contrast images.

3 RESULTS AND DISCUSSIONS

To analyze the performance measures and their execution time, low contrast images from CSIQ [26] database are considered. These images are enhanced by the residual spatial entropy-based contrast enhancement (RESE) method. Then the enhanced images, high contrast reference images, and low contrast images are used to obtain the performance metrics.

3.1 Residual Spatial Entropy-based Image Contrast Enhancement (RESE)

To utilize contextual information around each pixel and to overcome the drawbacks of using a one-dimensional histogram, methods based on two-dimensional histograms are proposed [27]. The two-dimensional histogram and the spatial entropy are obtained by using the following equations:

$${{h}_{i}} = {{h}_{i}}\left( {m,n} \right),\quad 1 \leqslant m \leqslant M,\quad 1 \leqslant n \leqslant N,$$
(29)

where \({{h}_{i}}\left( {m,n} \right)\) represents the two-dimensional histogram of the ith intensity value and \(m,n\) represents the spatial grid values on the image. \(M\) and \(N\) represent the total number of grids in the input image. From the two- dimensional histogram, spatial entropy is obtained to calculate the mapping function.

$${{S}_{i}} = - \mathop \sum \limits_{m = 1}^M \,\mathop \sum \limits_{n = 1}^N \,{{h}_{i}}\left( {m,n} \right){{\log }_{2}}\left( {{{h}_{i}}\left( {m,n} \right)} \right),$$
(30)

where \({{S}_{i}}\) represents the spatial entropy, and a discrete function \({{f}_{i}}\) is derived from \({{S}_{i}}\), which is defined as

$${{f}_{i}} = \frac{{{{S}_{i}}}}{{\sum\limits_{l = 1,l \ne i}^k {{{S}_{l}}} }}.$$
(31)

From the two-dimensional histogram, a joint spatial histogram is calculated by

$${{h}_{{i,l}}}\left( {m,n} \right) = \max \left( {{{h}_{i}}\left( {m,n} \right){{h}_{l}}\left( {m,n} \right)} \right).$$
(32)

From the joint spatial histogram, joint spatial entropy is obtained by

$${{S}_{{i,l}}} = - \mathop \sum \limits_{m = 1}^M \,\mathop \sum \limits_{n = 1}^N \,{{h}_{{i,l}}}\left( {m,n} \right){{\log }_{2}}{{h}_{{i,l}}}\left( {m,n} \right).$$
(33)

With the help of spatial and joint entropies, residual entropy is calculated by using the following formula

$${{R}_{i}} = \frac{{{{w}_{i}}{{s}_{i}} - {{w}_{{i,r}}}\sum\limits_{l = 1,l \ne k}^K {{{s}_{{i,l}}}} }}{{{{w}_{i}} + {{w}_{{i,r}}}}},$$
(34)

where \({{R}_{i}}\) represents the residual entropy. \({{w}_{i}}\) and \({{w}_{{i,r}}}\) are weighting functions obtained from two-dimensional histograms and joint histograms.

$${{w}_{i}} = \mathop \sum \limits_{m = 1}^M \,\mathop \sum \limits_{n = 1}^N \,{{h}_{k}}\left( {m,n} \right),$$
(35)
$${{w}_{{i,r}}} = \mathop \sum \limits_{l = 1,l \ne k}^K \,\mathop \sum \limits_{m = 1}^M \,\mathop \sum \limits_{n = 1}^N \,{{h}_{{i,l}}}\left( {m,n} \right).$$
(36)

Residual function \({{R}_{i}}\) is used to compute the discrete function \({{f}_{i}}\) and a cumulative distribution function \({{F}_{i}}\)

$${{f}_{i}} = \frac{{{{R}_{i}}}}{{\sum\limits_{l = 1,l \ne i}^k {{{R}_{l}}} }},$$
(37)

\({{f}_{i}}\) is further normalized to obtain cumulative distribution function \({{F}_{i}}\). From \({{F}_{i}}\) a mapping function is derived to obtain an enhanced image.

$${{y}_{i}} = \left\lfloor {{{F}_{i}} \times \left( {{{y}_{u}} - {{y}_{d}}} \right) + {{y}_{d}}} \right\rfloor ,$$
(38)

where \({{y}_{i}}\) represents the mapping function for the input intensity \({{x}_{i}}\), \({{y}_{u}}\) and \({{y}_{d}}\) represents the minimum and maximum intensities of the dynamic scale, respectively.

3.2 Qualitative and Quantitative Analysis

The above-mentioned method is applied to the low contrast images and the obtained high contrast image is evaluated by various performance measures. Figure 1 shows the low contrast images, obtained high contrast images, and reference high contrast images in the first, second, and third columns respectively. The high contrast reference images are used for calculating full reference metrics such as PSNR, UQI, and SSIM. For subjective analysis, the enhanced images are similar to high contrast images with the least variations. From the observation, it is interpreted that the enhancement algorithm works well to produce a high contrast image.

Fig. 1.
figure 1

Contrast-enhanced images by RESE method applied on CSIQ database; first column: low contrast images; second column: obtained high contrast image by RESE; third column: high contrast reference images.

Full size image

Due to various complications, the subjective analysis is not sufficient for the assessment of image quality. An automated image quality assessment is needed for machine vision applications. It can be achieved with the help of a suitable mathematical model of quality assessment. These mathematical models are referred to as quantitative or objective metrics. Various quantitative metrics are calculated and listed in Table 2. In Table 2, the second column gives the different performance measures, the third and fourth column provides the average value of performance measures for the low contrast and enhanced images respectively, the fifth column accentuates the range of values for each measure and the sixth column gives the average execution time for each objective measures.

Table 2. Comparison Table
Full size table

From Table 2 it is observed that the processing time is less for calculating the AMBE metric among all the measures. Most of the researchers used this metric to quantify their enhancement algorithm. This metric is related to the characteristics of display devices. Lower AMBE value indicates that the enhanced image will be displayed with lower artifacts. Entropy is used to measure the information content which is visible for an observer in an image. From the table, it is found that the low contrast image has low entropy value that indicates the viewer could not extract information from the image, due to the low visibility. After enhancement, the processed image has high entropy value and that ensures, the information in the image is observed.

Contrast, Standard Deviation, CII, and CPP metrics quantify the overall contrast of the image. A higher value of these measures indicates a significant improvement in contrast. EME, EMEE, and AMEE metrics are also used to measure the contrast. These measures may support the subjective analysis of contrast enhancement in the processed image. Structural information loss is quantified by the measures UQI, SSIM, and QILV. From Fig. 1 it is observed that structural information is preserved in the processed image. The tabulated values of SSIM, UQI, and QILV support the observation. Less GMSD value indicates that the enhanced image is not distorted due to the enhancement process. The suggested measure New_cont may quantify the contrast improvement and the amount of mean brightness shifting is less. Diff_E value shows that information loss may be very less in the processed image. From the above discussion, it is inferred that objective metrics are quantifying the subjective analysis and they may help to evaluate the performance of the algorithms.

4 CONCLUSIONS

In this paper, the taxonomy of performance measures have been discussed and analyzed thoroughly. A table was prepared to discuss the strength and weaknesses of various contrast enhancement objective measures. These objective measures are applied to the high contrast images which are obtained by the RESE method on the CSIQ database. Based on the objective measure values, information is extracted such as mean brightness preservation, contrast improvement, information loss, and structural information loss, etc. These values are considered along with subjective measures. The mean objective measure values along with mean processing time for the measures are tabulated in Table 2 for the reader inference. From these values, AMBE has low computation time (0.56 sec) and it can be used to evaluate the contrast enhancement algorithm. From the analysis, a new measure called New_cont is suggested to measure both mean brightness preservation and contrast improvement by considering the benefits of AMBE, Contrast, and Standard Deviation. To quantify the loss of information another measure based on entropy called Diff_E is proposed. New cont and Diff_E have the average execution time of 0.073 and 77.51 seconds, respectively. Both measures have a range of [0,1]. The values also coincide with subjective measures. This may lead to the development of new performance measures based on the human visual system and this will help the reader to identify the better algorithm for specific applications.