A Novel Evaluation Standard Combining Gini-index and Variation Coe�cient for Double Plateaus Histogram Equalization

—Image contrast enhancement or boosting is normally referred to as one of the most crucial tasks in image processing, and histogram equalization (HE) is one of the most pervasive methods applied to address this task. HE and its variants have been proven a simple and effective technique. However, no one consistent image quality evaluation standard has been built for them, not to say other relevant approaches. In other words, it is lack of enough attention to image quality evaluation for contrast enhancement algorithms. The authors proposed a novel evaluation standard combining Gini-index and variation coefficient. They verified the effectiveness of the proposed evaluation standard especially for double plateaus histogram equalization (DPHE) algorithm. Their experimental results showed that the proposed objective standard could provide an additional objective basis for the quality evaluation of DPHE, which may be extended to pervasive image enhancement algorithms.


I. INTRODUCTION
MAGE contrast enhancement or boosting is normally referred to one of the most crucial tasks in image processing [1], many algorithms have been proposed to address such task, and histogram equalization (HE) [2] is one of the most pervasive methods.HE also has a lot of variants such as plateaus histogram equalization(PHE) [3] and double plateaus histogram equalization (DPHE) [4], etc.Most of HE related algorithms focus much more on the global contrast.To make it easier and more reliable to value these contrast enhancement algorithms especially for DPHE, an effective evaluation approach is highly required.However, there is no standard evaluation index or criteria built and its conception has been kept a hard task, so it is lack of attention.In fact, most of image enhancement efficiency or effect evaluation studies are subjective as they are based only on visual and qualitative criteria, thus the provided results are biased, time consuming, and sometimes wrong.To overcome these problems, various functions and criteria have been developed or borrowed, for example, the probabilistic rand coefficient [5], the Rosenberger criterion [6], and the contrast of Zeboudj criterion [7].These functions and criteria are considered as reliable, trustworthy, and objective as they are based on a quantitative measurement and a segmented image.Meanwhile, these functions are not directly and specifically for image enhancement algorithms, and have not been verified intensively, then can not be applied or adopted broadly especially for DPHE.
Till now, DPHE is known a very effective method for image contrast, which used a upper plateau threshold value to limit the background and noise and a lower plateau threshold is added to further improve the details [3].In order to more accurately evaluate image quality for DPHE processing and facilitate enhancement of image contrast, a novel evaluation criterion concerning Gini-index [8] and variation coefficient [9] based on the equalized image is proposed.Though our criterion is proposed based on DPHE example, it can be extended to be applied to a broad range of the image contrast enhancement algorithms [10].

II. ALGORITHM FOR DOUBLE PLATEAUS HISTOGRAM EQUALIZATION
DPHE algorithm maps statistical histogram of an image by introducing a correction.Specifically, DPHE algorithm needs to define both an upper threshold Tup and a lower threshold Tdown (where Tup is greater than Tdown, which are the pixel number of corresponding upper and lower pixel values, respectively).Meanwhile, in order to compare the difference between the two thresholds of different images, statistical probability histogram is adopted and an upper probability threshold (TH) and lower probability threshold (TL) are introduced to correct the statistical probability histogram.Normally, both TH and TL are obtained by ADPHE algorithm [11], and both Tup and Tdown are obtained by traversal method [12].

A. Gini-index and Coefficient of Variation
The coefficient of variation (CV) is ratio of the standard deviation and the mean of a data set, which is a standard measurement of the dispersion of probability distributions [13].CV is calculated by I where σ is the standard deviation and μ indicates the data mean.
Gini-index has traditionally been an important analytical indicator to comprehensively examine the difference of income distribution among residents [14].Maryam Habba [15] introducted it to characterize the distribution difference of grayscale statistical histogram, which gave us enlightenment.In order to further characterize the distribution of gray values in rows and columns, we make certain modifications, that is, it is further modified to represent the pixel difference between rows or columns of an image, as defined in ,  denotes the cumulative size of pixels in row/column  as a proportion of the total cumulative value, and  denotes the number of pixels in row/column  as a proportion of the total number of pixels (for the image, row and column  are two fixed values).
(4) Ratio cvgn shows the difference in an image rows and columns.Within a certain range, gncv shows a certain extent of discrete characteristics of pixel value distribution of an image.For grayscale images such as infrared images, the change of image brightness and darkness can be characterized by both cvgn and gncv, usually higher is gncv value, the discrete difference of pixel value distribution of an image is greater, and cvgn after normalization is basically consistent with image contrast C [16].
As shown in Fig. 1, taking three 5*5 binary images [17] as an example, pixel statistical values of Fig. 1 (a), (b), and (c) are obviously exactly the same, so only from the statistical histogram analysis and comparison, the Gini-index for the statistical histogram cannot demonstrate the difference among these three images.G_1 represents the Gini-index calculated statistically for image pixels in columns and G_2 represents the Gini-index calculated statistically for image pixels in rows.If from column analysis comparison, the first and second columns of Fig. 1   In Experiment I, both cvgn and gncv were studied by keeping the upper threshold unchanged and the lower threshold changing from small to large within a certain range three urban architectural images Fig. 2 (a), (b), and (c).
In Experiment II, by replacing three images in different scenes where a tree in Fig. 2 (d), a road with vehicles in Fig. 2 (e), and a walking person in Fig. 2 (f), the applicability of cvgn and gncv in different scenarios are explored.

B. Quantitative evaluation
In order to clearly and intuitively validate the effectiveness of our proposed method, we assessed the variation of the proposed evaluation criteria with other quantitative criteria: 1. Image entropy (H) [18], H measures the richness of details in output image.H is specified by equations (5): .
where p(k) is to a function of probability distribution of grayscale statistical histograms.

Peak signal-to-noise ratio (PSNR),
It is one of the most commonly one of the performance measurement techniques to evaluate the quality of the output image [19].The following equations ( 6) and ( 7) illustrates how PSNR works, , and , (7) where g(x, y) represents a source image of P by Q pixels, and h(x, y) denotes a reconstructed image.The g(x, y) and h(x, y) pixel values take black (0) to white (255) values.MSE is the mean of the square of the error between the input image and enhanced image [19].RMSE is the root mean square error of MSE.

Image contrast (C),
C is used to calculate gray deviation.The following equations ( 8) and ( 9) illustrates how C works, , (8) and , (9) where P is the width of an image and Q is the height of an image, g(x, y) is the gray level of the pixel at (x,y).Ccontrast represents the deviation of gray levels [11].In order to facilitate the comparison of the magnitude of image contrast in different experiments, the logarithmic method was used to obtain C for the calculated value of Ccontrast .

C. Experimental Results and Comparison
As shown in Fig. 3, Fig. 4, and Fig. 5, max-th and min-tl represent the maximum and minimum values of the gray value statistical histogram of an image dealt with by DPHE, Gini_hist represents Gini-index based on statistical histogram, Gini_rc represents the average Gini-index based on rows and columns, and CV represents image variance coefficient.entropy GE equals to Gini_hist plus H.
In order to compare experimental results, the max-th and the min-tl of pixel statistics of an image by DPHE using different lower thresholds will be recorded in a histogram as shown in Fig. 3 (a), Fig. 4 (a), and Fig. 5 (a) ， where the vertical axis represents proportion of pixel number with the maximum or minimum gray value to the total pixel number in an image, and the abscissa represents TL increasing gradually from 0 to TH with a step size 0.0005; the normalized quantitative metrics of Fig. 2  From Fig. 3 (a), Fig. 4 (a), and Fig. 5 (a), we found that as TL approaches TH, the extreme values of the reconstructed statistical histogram gradually approach each other, the image entropy and Gini entropy gradually increase and then stabilize.At the same time, cvgn and C gradually decrease, which indicates that the image contrast gradually decreases, thus we can conclude that cvgn can reflect the change of image contrast to a certain extent.Futherly, gncv gradually increases, which indicates the pixel dispersion of an image.This is also shown by PSNR and image entropy.When TH is determined by ADPHE algorithm, and TL changes from 0 to 0.002 ~ 0.004, H and GE remain unchanged, and as shown in Fig. 4 (a) and Fig. 5 (a), when TL reach a certain level, the PSNR also stabilizes.Therefore, for images with essentially unchanged H or PSNR, H or PSNR is not suitable be applied to discriminate them.In contrast, as shown in Fig. 3 (a), Fig. 4 (a), and Fig. 5 (a),when TL changes from 0 to 0.004 and beyond, in the region where H and GE remain basically unchanged, both C and cvgn still exhibit the same trend of changes.Therefore, both cvgn and gncv can better describe different images than H.
Moreover, it can be seen from Fig. Therefore, when TH and TL approach a certain level, the discrete entropy and Gini coefficient based on statistical histogram calculation(as shown in ( 5)) cannot effectively describe the differences in small changes between these image states.Therefore, image entropy (including GE) is no longer applicable.In contrast, the row/column Gini coefficient describes the differences between image rows and columns.Clearly, the variation in the changes based on the differences between image rows and columns is greater and more sensitive compared to the differences based on statistical histogram, so cvgn and gncv, which describe the changes in the differences between rows and columns of the image, can better describe these differences.As shown in Fig. 6, Fig. 7, and Fig. 8, when TL changes from 0 to 0.002 ~ 0.004, the image entropy, PSNR, Gini_hsit, gncv, and Gini_rc gradually increase, and when TL continues to increase, image entropy and Gini_hsit tend to be stabilized, and when TL changes to 0.004 ~ 0.006 and beyond, the growth of the PSNR in Fig. 7 begins to slow down , the PSNR of Fig. 8 starts to stabilize, however, the gncv and Gini_rc based on row-column differences still gradually increase, and C and cvgn keep the same trend of decreasing.
It is worth noting that in Fig. 6, as TL approaches towards TH, PSNR maintains the same increasing trend with gncv and Gini_rc.This is because in this scenario, the value of TH obtained by the ADPHE algorithm is higher, at this time, there is still a large difference between TH and TL, and therefore PSNR increments in this range.In Fig. 8, the TL only changes from 0 to 0.01 because the TH value obtained by the ADPHE algorithm is small in this scenario.However, the trend of each quantitative index obtained from these three scenarios is similar to that of the urban building scenario.Finally, the experiment Ⅱ yielded experimental results similar to those obtained in experiment I, so we can still make the corresponding experimental conclusions.
In summary, when the maximum and minimum values of the pixel statistics of the histogram-corrected image(max-th and min-tl) gradually approached each other, until max-th and min-tl were equal, the image entropy and Gini entropy (including PSNR for some images) gradually increased to the maximum value and remained unchanged.Therefore, both cvgn and gncv can better describe different images than image entropy (Including Gini entropy and PSNR).Through the similar experimental results and experimental conclusions of Experiment I and Experiment II, thus, it can be concluded that gncv and cvgn are still applicable in scenarios such as trees, roads with vehicles, walking people, etc., and thus can be further generalized to other scenarios.

V. CONCLUSION
In this paper, a new evaluation method for DPHE based on Gini-index and variation coefficient is proposed.The proposed method is extended from the beginning Gini-index application in image processing field.Two experiments are designed and explored to validate the proposed criteria.The experimental results showed that the proposed objective standard could provide an additional objective basis for the quality evaluation of DPHE, which may be extended to pervasive image enhancement algorithms.
(a) have five 1s, and the other columns are zero; each column of Fig.1 (b) has two 1s.Obviously, it is necessary to use G_1 to distinguish Fig. 1 (a) and (b) , Fig. 1 (a) and (c).Furthermore, each column of Fig. 1 (c) has also two 1s, and a single column Gini-index is obviously still unable to distinguish Fig. 1 (b) and (c), so we calculate G_2, and their mean (G_1+G_2)/2 as whole Gini-index of the image.Interestingly, Fig. 1 (a) rotates 90 degrees in an anticlock wise manner will become Fig. 1 (b), thus the G_2 of Fig. 1 (a) is the same as the G_1 of Fig. 1 (b), and vise versa it keeps.In other words, the image Gini-index of Fig.1 (a) is equal to that of Fig. 1 (b), then Fig. 1 (a) and Fig. 1 (b) is regarded as the same.
Fig.5 (b) that the row/column Gini coefficient Gini_rc gradually increases as TL moves closer to TH, which intuitively indicates that Gini_rc is more sensitive to subtle changes in an image compared to Gini_hist which is based on statistical histogram.The change trend of Gini_hist is the same as that of H, while the change trend of Gini_rc is the same as that gncv.These change trends are also more stable and predictable, also demonstrate feasibility and accuracy of cvgn and gncv.In fact, this is because the max-th and min-tl have become very close enough , indicating a high level of uniformity.At this point, traversing TL within this range and performing dual-platform histogram equalization will result in minimal changes to the modified statistical histogram.However, the changes in other quantitative indicators (such as C) can still indicate that the image state (including the discrete state) is still changing.According to (8) and (9), since C is calculated based on how an image pixel points change, subtle changes in an image can be described.In contrast, in (5), H is calculated based on the change situation of the gray scale statistical histogram.Further, as shown in Fig.3 (b), Fig.4 (b), and Fig.5 (b), when TL changes from 0 to 0.004, Gini_hist remains largely stable, and Gini_rc continues to increase.This is further evidenced by the difference in the results of the same metric Gini-index in the Gini_hist calculated based on the gray scale statistical histogram and the Gini_rc calculated based on the row and column pixel values.Therefore, when TH and TL approach a certain level, the discrete entropy and Gini coefficient based on statistical histogram calculation(as shown in (5)) cannot effectively describe the differences in small changes between these image states.Therefore, image entropy (including GE) is no longer applicable.In contrast, the row/column Gini coefficient describes the differences between image rows and columns.Clearly, the variation in the changes based on the differences between image rows and columns is greater and more sensitive compared to the differences based on statistical histogram, so cvgn and gncv, which describe the changes in the differences between rows and columns of the image, can better describe these differences.

Fig. 7
Fig. 7 Normalized quantization index change trend of Fig.2 (e) after DPHE (where TH is determined and kept constant at TH=0.0157 by the ADPHE algorithm, and TL changes from 0 to 0.0120 by step 0.0005).

Fig. 8
Fig. 8 Normalized quantization index trend chart of Fig.2 (f) after DPHE (where TH is determined and kept constant at TH=0.0109 by the ADPHE algorithm, and TL changes from 0 to 0.0100 in intervals of 0.0005 In Experiment II, images of several scenarios were