Image Processing Reference
In-Depth Information
In this chapter, we will review some of the recent methods to improve SNR
and CNR in MR images, also accounting for the statistical nature of the process.
Emphasis will be given to an application dealing with edge enhancement in
myocardium image segmentation.
6.2
THE MR IMAGE MODEL
Usually, noise in MR images is defined simply as a deviation from the true value
considered representative of a tissue category; noise is expressed by the standard
deviation, whereas the true value is evaluated by the mean value of a set of pixels
expected to belong to the same tissue category. It has been shown [23,37] that
due to nonlinearity of the magnitude reconstruction process introduced to obtain
real images, such assumptions about the image model are too weak. In fact, the
first step in reconstructing MR images is to compute the inverse discrete Fourier
transform of raw frequency-domain (k-space) measurements. Let y(m, n) denote
a complex image with additive noise as follows:
y(m, n)
=
r(m, n)
+
n (m, n)
+
j[i(m, n)
+
n (m, n)
]
(6.1)
r
i
where r(m, n) and i(m, n) are real and imaginary parts of the noiseless image,
with n
(m, n) representing the Gaussian noise in the real and imag-
inary images, respectively, with standard deviations
(m, n) and n
r
i
.
By considering the magnitude reconstruction process of MR images, the
magnitude of y(m, n) is given by:
σ
n
|
y m,n
(
) |
=
{ (
r m,n
)
+
n
(
m,n
)}
2
+
{ (
i m,n
)
+
n
(
m,n
)}
2
(6.2)
r
i
We recall that in MR images, the signal magnitude is simply the square root
of the sum of two independent Gaussian random variables, and the magnitude image
data are described by a Rician distribution.
If the image intensity is much larger than the noise standard deviation,
Equation 6.2 may be approximated as [3]:
|(
ym,n
)| (,)
=
xmn
+
n mn
z
(,)
(6.3)
where
xmn
(,)
=
[(, ] [(, ]
rmn
2
+
imn
2
rmn n
(,)
⋅
((,) (,) (,)
(,)
mn imn n mn
xmn
+
⋅
nmn
(,)
=
r
i
;
z
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