Histogram Matching Based on Gaussian Distribution Using Variance Estimation - Comparing between Curvature Computation and Regression Analysis -

This paper describes variance estimation method comparing between regression analysis and curvature computation which is used in Histogram Matching based on Gaussian Distribution (HMGD). In the previous paper, we have described and illustrated that the variance estimation method have been considered of value for HMGD processing results. Though we have considered that histogram of original image is not always ideal. So, in this paper we propose improvement variance estimation method using regression analysis. First of all, we describe the principle of variance estimation methods using curvature computation, and regression analysis. Then, through some HMGD processing experiment, we compare between curvature computation results and regression analysis


Introduction
These days, automated image processing for enhancement of color images has been more familiar to us, for example, Digital Signage, Smart Phone, etc [1][2][3] .In the previous paper, we have presented that the Histogram Matching based on Gaussian Distribution (HMGD) processing is one of the automated image arrangement method using Elastic Transformation [4][5] based on the brightness axis.And through the comparative investigation, we have illustrated that HMGD processing could improve feeling (or Kansei) impression better than original image 6 .And we have aimed to improve HMGD processing, we have proposed that how to estimate the variance of reference histogram, which is used in HMGD processing based on curvature computation, and also illustrated these results.However, we have considered that the histogram of original image is not always ideal shape (i.e., Gaussian distribution, etc).That is, the variance estimation method based on curvature computation might not have high reliability.

P -579
In this paper, we describe principle of estimate brightness peak of the original image, and also describe two principle of variance method; (1) curvature computation based, (2) regression analysis based.Then, we illustrate some HMGD processing experiment and compare between curvature computation results and regression analysis results.

Brightness Peak Detection of Original Image
In the section, we describe the principle of brightness peak detection of original image.The Histogram Matching based on the Gaussian Distribution 4-9 processing need to calculate transforms function for brightness peak of histogram.And the solution to detect it is curvature computation of the histogram.Let y be a function with respect to x, the definition curvature R(x) is given by Eq. (1) 6-9 . .
Let g(x) and K be Gaussian distribution function and a coefficient which is defined by following equation, respectively.
From Eq. ( 6), we understand that the curvature R(x) varies the sign according to the value of x 9 .That is, if x<a → R>0 (downward convex shape), and if x<a → R<0 (upward convex shape).

Variance Estimation
In this section, we describe how to optimize the shape of the reference histogram, which is used in the HMGD processing 9 .First, we explain the conventional method which is based on the curvature computation, and second, explain the proposed method which is based on the regression analysis.

Variance Estimation based on the Curvature Computation
Fig. 1 shows the conceptual image of the original image histogram which is variance 2 and average a.And Fig. 2 shows its cumulative histogram.From Eq. (2), we have the following Eq.( 7) and Eq. ( 8). .
Then, the value of curvature R at 2 a is represented Eq. ( 9).Let S=g(a)e -1 , we have Eq.(10).That is, we understand that we can obtain reference histogram variance 2 .For example, let 2 v be the distance from average a,

Variance Estimation based on the Regression Analysis
In the previous subsection, we show the Fig. 1 as the conceptual image of the original image histogram which is variance 2 and average a.From Eq. ( 6), we can describe R(x) following Eq.( 12). .
x CH x R (13) Now, we can calculate a constant C by using leastsquare regression analysis 10 following Eq.( 14).

Experimentation
Fig. 3 shows the example of results and the corresponding histogram for original image and for HMGD image which applied variance estimation, respectively.In this case, we understand that HMGD which applied curvature computation based variance estimation image is softer and more natural contrast and than original image.However, HMGD which applied regression analysis based variance estimation image is reducing contrast than original image, and the color tones become unnatural.

Conclusion
In this paper we described the principle of brightness peak detection, estimate the variance of reference histogram, which is used in HMGD processing by using curvature computation, and we propose variance estimation based on the regression analysis.Through the comparing experimentation, HMGD processing image which applied variance estimation based on the curvature computation image is becomes soften and natural contrast from original.In contrast, HMGD processing image which applied the variance estimation based on the regression analysis image is reducing contrast than the others, and its color tones become unnatural.That is, we consider that we have to improve algorithm of variance estimation based on the regression analysis.