Image Processing Reference
In-Depth Information
A
In addition, the dependence of both the bias and the variance of on the
number of data points has been investigated, yielding the following results:
ML
A
•
The bias of for complex data with identical phases as well as for
magnitude data with known and unknown noise variance generally
decreases with the number of data points used for the estimation. On
the other hand, it turns out that for complex data with different phases,
the bias of
ML
A
does not decrease with the number of data points.
ML
•
The variance, as may be expected, turns out to be inversely proportional
to the number of data points for all estimators.
4.4.5
S
IGNAL
A
MPLITUDE
E
STIMATION
FROM
PCMR D
ATA
From Subsection 4.2.2.4
we learned that PCMR data are derived from the square
root of the sum of the squares of a number of Gaussian-distributed variables,
which is again a nonlinear transformation. It has been shown that results are
biased when PCMR data are being used in quantitative analysis as an estimate
of the underlying flow-related signal component magnitude [22]. The bias is due
to the contributions from inherent random noise, which is not Gaussian distrib-
uted. Because the bias is not merely an additive component, it cannot be simply
subtracted out. To remove the bias, knowledge of the actual shape of the data
PDF becomes essential. In this section, the full knowledge of the PDF of the
PCMR data is exploited for optimal estimation of the underlying signal.
Although this section focuses on complex difference processed images, the
estimation techniques derived in this section can, under certain conditions, also
be applied to images obtained by phase difference processing. This is because for
both methods one has to estimate the underlying signal component from magnitude
images for which the pixel variable can be described by Equation 4.26. The
only difference is that for phase difference processing, the dimension, or number
of degrees of freedom
K
, directly equals the number of orthogonal Cartesian
directions in which flow is encoded, whereas for complex difference processing,
the dimension
K
is twice this physical dimension [22].
m
4.4.5.1
Region of Constant Amplitude and Known
Noise Variance
In the following discussion, it is assumed that an unknown deterministic signal
component
A
is to be estimated from
N
PCMR pixel values of a region
where
the signal component is assumed to be constant. Thereby, the noise variance is
assumed to be known.
Ω
4.4.5.1.1 CRLB
The Fisher information matrix of the data with respect to the parameter
A
is given by
∂
ln
p
A
2
m
I
=−
,
(4.121)
E
∂
2
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