Design and Implementation - Segmentation of Hand Bone for Bone Age Assessment - page 55

difference between the input image and the histogram equalized image. The func-
tion is designed such that the Relative Brightness Difference (RBD) are the func-
tion domain, is the parameter, and the resultant values are bound between the
range of 0 and 1 to enable comparisons. The step-by-step construction details of
BPS function are discussed as follows:

µ y − µ x

(3.1)

RBD =

µ x + c

The RBD denotes the relative brightness difference between the input radio-
graph and histogram equalized radiograph, denotes input image, denotes output
image, denotes mean of input image gray level intensity, denotes mean of output
image gray level intensity. It is too 'raw' and not yet suitable to be used solely
as component in the multiple objective regularization function as it lacks of two
desired important features: The first feature is the standard bound function range
value which is critical in the final regularization function to establish an integrated
performance from all the components; it is obvious to notice that the output value
can be as large as infinity in extreme cases when the brightness mean of input radi-
ograph is 0.

1
β ( m , n ) RBD ( m − 1 ) ( 1 − RBD ) ( n − 1 )

(3.2)

BPS ( RBD ; m , n ) =

BPS ( RBD ; m , n )
argmax ( BPS ( RBD ; m , n ))

(3.3)

NBPS ( RBD ; m , n ) =

The Eq. ( 3.2 ) denotes the Brightness Preserving Score (BPS) function in the
context of histogram equalization which bears a resemblance to Beta probability
distribution function where RBD satisfies with two real positive parameters a and b
manipulating the relative significance of initial brightness difference between out-
put and input image. Quantity denotes Beta function in terms of gamma function:

β ( m , n ) = Γ ( m ) Γ( n )
Γ( m + n )

(3.4)

where Γ ( a ) = ∫
0

e − t t ( a − 1 ) dt

(3.5)

Equation ( 3.3 ) denotes the Normalized Brightness Preserving Score (NBPS).
The effect of both parameters m and n are illustrated in Figs. 3.3 and 3.4 .
What is the difference between NBPS and BPS ? What BPS indicates? The
difference between NBPS and BPS is that the NBPS assures that the metric is
bound between 0 and 1. As in Fig. 3.5 , each BPS function has different maximum
value and the maximum value is not bound. However, as illustrated in Fig. 3.6 , the
maximum values of all the NBPS functions with different parameters are identi-
cal, which is 1. The BPS provides us an index to measure quantitatively the abil-
ity of brightness preservation of the histogram equalized algorithm. The NBPS is