Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 1 (June 2016), 55-60

Automated Processing of Multiple-Brightness Peak Histogram Image
Using Curvature and Variance Estimation
Yusuke Kawakami 1, Tetsuo Hattori 2, Yoshiro Imai 2, Yo Horikawa 2, Kazuaki Ando 2, R. P. C. Janaka Rajapakse 3
1DynaxT Co., Ltd. 2217-6 Hayashi
Takamatsu City, Kagawa 761-0301, Japan
E-mail: riverjp2002@gmail.com
2Kagawa University, 2217-20 Hayashi
Takamatsu City, Kagawa 761-0396, Japan
E-mail: {hattori, imai, horikawa, ando}@eng.kagawa-u.ac.jp
3Tainan National University of the Arts, 66 Daci
Guantian District, Tainan 72045, Taiwan
E-mail: janakaraja@gmail.com

Abstract
Previously, we have illustrated that the Histogram Matching based on Gaussian Distribution (HMGD) is aneffective automated image processing method for obtaining a better feeling impression image. However, the simple
HMGD works only for the image whose histogram has just one peak. For the image whose histogram has multiplebrightness peak, it does not work as in the case of single peak histogram image. In this paper, we propose theimproved method for multiple-brightness peak (HMGD-MBP). This method can not only detect multiple peaks but
also estimate the variance of Gaussian distribution at each detected peak in the image histogram, using curvaturecomputation. This paper also presents the effectiveness of the proposed method by showing the experimental results.
Keywords: Image processing, Histogram, Curvature, Variance estimation, Histogram matching, HMGD

1. Introduction
Recently, automated image processing for theenhancement and arrangement of color images is getting
popular in our familiar electronic equipment andcomputer systems, for example, Digital Signage, Smart
Phone, etc.1-3
In the previous paper, we have presented that theHistogram Matching based on Gaussian Distribution(HMGD) processing is one of the effective automatedimage arrangement methods.4-5 And also, we haveillustrated that HMGD processing works to obtain better
feeling (or Kansei) impression than that of original

image, by the comparative experimental investigation.6
However, HMGD processing has just worked only forsingle brightness-peak histogram image.
In this paper, we present the improved HMGD for
multiple-brightness peak histogram image that we call
“HMGD-MBP”, using curvature computation andvariance estimation. We explain the principle, and also
give the procedure of peak detection and varianceestimation of Gaussian distribution of each peak in theimage histogram. Lastly, we show the effectiveness byillustrating the experimental results of the HMGDMBP.

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Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 1 (June 2016), 55-60

Yusuke Kawakami, Tetsuo Hattori, Yoshiro Imai, Yo Horikawa, Kazuaki Ando, R. P. C. Janaka Rajapakse

From the relationship between y=f(x) and y=γ(z) as
shown in Fig. 1, we have the following Eq. (6).

2. Principle∫x(1)zL

L f ( x )dx =In the section, we explain the principle of HMGD
processing.
Fig. 1 shows the conceptual image of HMGD. Let f(x)and h(y) be two probabilistic density functions (PDF) on
real variables x and y, respectively. The PDF iscorresponding to histogram of image brightness level
which is discretely defined.
In addition, let y=f(x) be a continuous and monotonicincrease function corresponding to cumulative
histogram of image brightness level between variables x
and y.7-9 And let f(x) be defined by Eq. (1).

y = f ( x ) = L ∫ f ( x )dx.

 (z − µ )2 
dz. (6)exp −2
22σ2pσ 0x2.1. Histogram Matching based on GaussianDistribution (HMGD)

0∫

And we can derive Eq. (7) from Eq. (6).
xz
 (z − µ )2   dz
dd  Ldz  .
L f ( x )dx = exp −2 dx 0dz  2pσ 2 0σ2  dx


∫∫

(7)

Then we have,f ( x )dx = Gauss(z )dz.

(8)

That is, we understand that f(x) is transformed into
Gaussian distribution Gauss(z).
Thus, we can say that HMGD processing is a kind of acumulative histogram transformation from the original
histogram into Gaussian histogram.80f (x )dx = h( y )dy = h ( y )f ' (x )dx= h ( y )Lf ( x )dx.

yyy=γ(z)h(y)

First, we have to expand the brightness level of original
image histogram and convert it into an uniform
distribution histogram, because we aim to match
Gaussian distribution. From Eq. (1) and Fig. 1, we canderive Eq. (2) and (3).

y=f(x)f(L)f(x0+dx)yf(x0)(2)

LL
x1h( y ) = .
Lz0(3)We understand the histogram of original image f(x)becomes uniform distribution h(y) by Eq. (3). It meansthat the distribution f(x) is expanded to h(y).
In the same way, let Gauss(z) and γ(z) be two functions
that are defined by Eq.(4) and Eq.(5), respectively.

Gauσσ ( z ) = ( z − µ )2 . (4)exp −22σ22pσ
1zy = γ (z ) = L Gauss (z )dz∫0= (z − µ )2 dz.
exp −222s2ps 0

Lz∫

(5)

z z0+dz

Gauss(z)

x0

z

x0+dx

1
Lxf(x)

Fig. 1. Conceptual image of HMGD.6,8-10

2.2. Peak Detection of Histogram in HMGD-MBP
The HMGD processing method for multiple-brightness
peak histogram image needs to detect peaks in image
histogram. In order to do so, we use the curvaturecomputation.
Let y be a function with respect to x, the definition
curvature R is given by Eq. (9).6-10

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Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 1 (June 2016), 55-60

Automated Processing of Multiple−32d 2 y   dy   2
R = 2 1 +    .
dx   dx  g (a ) =(9)(x0

 (u − a )2  (11)exp∫  − 2σ 2 du.
2pσ 2 0Kx

 ( x − a )2 dyK
. (12)()=g x =exp −2dx2σ2pσ 2

d 2 y dg ( x ) (a − x )g (x )==dx 2dx
σ2R=(a − x ) g (x )σ2(1 + g (x ) )32 2(a − x ) g (x ).
σ2σ 2π.

(15)()= g (a )e −1.

y0From Eq. (14), the curvature R varies the sign according
to x. That is, if x<a → R>0 (downward convex shape),and if x<a → R<0 (upward convex shape).9,10yIn this section, we propose how to optimize the shape ofthe reference histogram, which is used in the HMGDprocessing method for multiple-brightness peak image.
Fig. 2 shows the conceptual image of the original image
histogram which is variance σ2 and average a. And Fig.
3 shows its cumulative histogram. From Eq. (10), we
have the following Eq. (15) and Eq. (16).

L2σ(R a + 2σx)R(a)(R a − 2σ)
aa − 2σLa + 2σxFig. 3. Conceptual image of the original image cumulative
histogram.8

Let S = g(a)e-1. Then, from Eq.(14), the curvature R atx = a ± 2σ is represented as follows.

(R a ± 2σ

2.3. Variance Estimation

a+Fig. 2. Conceptual image of the original image histogram.8,10(13)(14)

a2σa−0≈Kg(a)Next, let y=f(x) be a function representing thecumulative histogram which is represented Eq. (11).
That is, dy/dx and d2y/dx2 be described as Eq. (12) andEq. (13), respectively. From Eq. (12) and Eq. (13), we
obtain the approximation of curvature R as Eq. (14).

f (x ) = ∫ g (u )du ==)K(10)2πσ2 a ± 2σ − a 2 Kg a ± 2σ =exp − (16)
2σ 2σ 2p

Let g(x) be a Gaussian distribution function as definedby the following equation, where K is a constant value.

 ( x − a )2 ,exp −g (x ) =2σ 2 
2pσ 2L
 (u − a )2 Kexp∫  − 2σ 2 du = 1.
2pσ 2 0K

2=  σ)(g (a )e ) =   2  S .

(1 + (g (a )e ) )  σ  (1 + S )−13−1 2 2(17)32 2Then we have Eq. (18).

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2
H .
R a ± 2σ =   σ (18)Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 1 (June 2016), 55-60

Yusuke Kawakami, Tetsuo Hattori, Yoshiro Imai, Yo Horikawa, Kazuaki Ando, R. P. C. Janaka Rajapakse

where

H=S

(1 + S )=32 2(g (a )e ) . (19)
(1 + (g (a )e ) )−13−1 2 2Let v = 2σ be the distance from the average a, wehave the following Eq. (20).

4R (a − v ) − R (a + v ) = H .
v(20)Then, we can obtain the variance σ2 from the curvature
R and peak value g (a ) , using the following procedure.
(i) Compute the H from the value g (a ) using Eq.(19).
(ii) Detect the v that satisfies the Eq.(20).
(iii) Obtain the σ 2 = v 2 / 2 .

Fig.8 shows another example of results and thecorresponding histogram for HMGD and HMGD-MBP,respectively.
In this case, we understand that HMGD-MBP
processed image has more quiet color than original
image. On the other hand, HMGD processed image isbrighter than the original image. But its color becomes
unnatural (located left side of image).

yp1
p2

3. HMGD Processing for Multiple-Brightness
Peak (HMGD-MBP)
In this section, we present the concrete procedure of
HMGD-MBP as follows.
(i) Detect the brightness-peak values xm (m=1,…,n) oforiginal image histogram (Fig. 4).
(ii) Perform variance estimation and generate thereference histogram for each brightness-peak valuex m.
(iii) Perform HMGD processing and make an image imwhich has histogram hm (Fig. 5).
(iv) Proceed to (v) if the original image has nobrightness-peak value. Otherwise, set thebrightness-peak value to xm+1 and return to (ii).
(v) Calculate the average of the HMGD processed(output) image Ia using Eq. (21) (Fig. 6).

Ia =1 n∑ imn m=1(21)

0x1x2

Lx

Fig. 4. Conceptual histogram of original image (multiplebrightness peak).9y

h1

p1
p2

0x1x2

Lx

Fig. 5. Conceptual histogram of HMGD processed image(brightness peak x1).9y4. Experimentation

h1p1

Fig.7 shows the example of results and thecorresponding histograms by HMGD and HMGD-MBP,respectively.
In this case of Fig.7, we understand that HMGD-MBP
processing is enhancing the contrast more naturally than
HMGD. And also in the HMGD-MBP processed image,the edges in the mountain (located left below of image)
becomes clear and detailed.

h2p20x1x2LFig. 6. Conceptual histogram of output image.9

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Automated Processing of Multiple(a) Original image and its histogram(a) Original image and its histogram(b) HMGD processed image and its histogram(b) HMGD processed image and its histogram(c) HMGD-MBP processed image and its histogram(c) HMGD-MBP processed image and its histogram

Fig. 7. Experimental results (I).

10

Fig. 8. Experimental results (II).

5. Conclusion
In this paper, aiming to obtain an improved automatedprocessing for multiple-brightness peak histogram
image, we have proposed “Histogram Matching basedon Gaussian Distribution for multiple-brightness peakhistogram image” that we call “HMGD-MBP”. As forthe concrete processing method, we have used curvaturecomputation for detecting brightness peaks and varianceestimation of Gaussian distribution at each detected
peak. After that, each of the peak shapes is transformed
into Gaussian shape.
From the experimental results, the HMGD-MBP
processing naturally enhances the contrast of image, andso the edges of image become clarified and detailed.

Also, we understand that the color of HMGD-MBP
processed image tends to become more moderate thanoriginal image.As a conclusion, we consider that theHMGD-MBP processing method is more useful thanHMGD.
For further study, we have to much more improve theproposed method so that it can arrange the colors andcontrast of input image easily and in detail. In order for
the image processing to have such arrangement
capability, we would like to introduce some related
processing methods of Signal Processing in Kansei
Engineering, Time Series Analysis, and Advection
Diffusion in Fluid Dynamics. 11-15

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Journal of Robotics, Networking and Artificial Life, Vol. 3, No. 1 (June 2016), 55-60

Yusuke Kawakami, Tetsuo Hattori, Yoshiro Imai, Yo Horikawa, Kazuaki Ando, R. P. C. Janaka Rajapakse

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