Design and Implementation - Segmentation of Hand Bone for Bone Age Assessment - page 65

The c ( q ) represents the diffusion coefficient function of SRAD where the q
denotes the instantaneous coefficient of variation represented by the ratio of stand-
ard deviation as shown in Eq. ( 3.36 ), the q 0 ( t ) denotes the coefficient of variation
at the time, t .
Diffusion coefficient function of speckle reducing anisotropic diffusion
(SRAD), c ( q ) is first incorporated by [ 13 ] in anisotropic diffusion SRAD to
remove the multiplicative speckle noises in ultrasound image. The problem of
this diffusion coefficient is the definition of standard coefficient of variance, q 0 .
Therefore, the decision to adopt the theory from [ 12 ] to modify c ( q ) so that it can
be automated by selecting the standard coefficient of variance, q 0 adaptively to dif-
ferent input image.
According to [ 12 ], the variance of noise is computed according to the expected
value of local variance distribution, S 2 in the light of the variance sampling coef-
ficient. To illustrate this idea, first looked at the relation of Chi square distribu-
tion to the definition of sampling variance distribution, S 2 at Eq. ( 3.35 ) with N − 1
degree of freedom [ 14 ]. Then, the definition of Chi square as a special case of
Gamma distribution as shown in Eq. ( 3.36 ). After that, the inspection is shifted
closer at the definition of Gamma distribution with α and β denoteing the shape
parameter and amplitude parameter respectively at Eq. ( 3.37 ). The expected mean
of Chi square is denoted in Eq. ( 3.38 ). Then, by computing the expected value of
S 2 in Eq. ( 3.35 ), we obtained Eq. ( 3.39 ). Finally, we obtain the expected value of
sample variance by substituting Eqs. ( 3.38 ) into ( 3.39 ). The conclusion is that the
expected sample variance amounts to the global variance.

2

(σ 2 )
N − 1

(3.35)

S 2 ∼ ( N − 1 )

χ 2 ( N − 1 ) ∼ γ( x , α = N − 1
2

(3.36)

, β = 2 )

1
Γ (α) x α− 1

1
β

e − β H ( x )

(3.37)

γ ( x , α , β) =

where H(x) denotes the Heaviside step function

(3.38)

χ 2 ( N − 1 ) E = E (γ( x , α , β)) = αβ = N − 1

(σ 2 )
N − 1 E

(3.39)

S 2 χ 2 ( N − 1 ) E =

σ 2

N − 1 ( N − 1 ) = σ 2

(3.40)

S 2 E =