Image Processing Reference
In-Depth Information
Several wavelet-domain filtering methods to reduce the noise power from
images are reported in the literature [6-11,25-27].
In particular, Nowak [25] described a method that explicitly accounts for the
Rician nature of the data. He demonstrated that the filter weight
that minimizes
α
the mean squared error of each coefficient is given by
(
)
(
)
2
2
d
o
()
k
−
3
σ
o
()
k
j
j
+
(6.6)
α
=
(
)
2
d
j
o
()
k
where (
0, and is the
variance of the observed coefficients. It means that the filter sets small wavelet
coefficients with squared magnitude less than three times the estimated variance
to zero and leaves larger coefficients approximately unaltered.
In the presence of additive white Gaussian noise with variance
+
) means that (x)
=
x if x
≥
0 and (x)
=
0 if x
<
σ
o
()
k
+
+
σ
2
(high SNR),
the filter weight
given by Equation 6.6 can be evaluated with
σ
o
()
σ
α
k
=
n
(assuming that each wavelet coefficient has equal variance at any spatial
position
).
A simple procedure for estimating
k
is reported in Reference 25, and it
consists of evaluating one half of the mean of the squared pixel values in the
region outside the patient within the scanner.
In low-SNR situations, the mean of the magnitude image is not equal to the
noise-free image (as in the Gaussian approximation) and, hence, the magnitude
image is biased. Nowak demonstrated that when operating on the squared mag-
nitude image rather than the magnitude, the wavelet coefficients are approxi-
mately Gaussian distributed. In such situations, the filter weight
σ
n
can still be
used. He also suggested a method for estimation and compensation. The
advantage of such an algorithm is that it can be used in both high- and low-SNR
imaging situations [25-27]. An advantage of the algorithm in Equation 6.6 in
the high-SNR case is that its complexity is slightly lower than that in the low-
SNR case.
α
σ
o
()
k
6.4
ADAPTIVE TEMPLATE FILTERING
An interesting approach to MRI image filtering is the adaptive template filtering
technique proposed by C.B. Ahn et al. [3]. Ahn proposed a local shape-adaptive
template filtering for the enhancement of the SNR without resolution loss. Unlike
conventional filtering, in which the template shape and coefficients are predefined
and the filtered output is given as the weighted sum of the image gray levels
surrounding the current pixel, in adaptive template filtering, multiple templates
are defined. It prevents the edge blurring usually observed when using fixed-
coefficient
to obtain SNR enhancement in
almost constant regions. Using the proposed process, edge blurring is minimized
and SNR maximized by selecting the optimally matched template.
filters that employ spatial averaging
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