Image Processing Reference
In-Depth Information
Most of the current image processing applications applied to MR image data
can be formulated as a parameter estimation problem. For example, in the case of
noise filtering the parameter to be estimated is given by the true signal component
underlying the noise-corrupted data, whereas in the construction of
maps
the parameters to be estimated are given by the relaxation time constants [1-9].
Nowadays, there exist several estimation procedures, each of which seems to be
slightly different from the other. So which one should we use? Which one is optimal
with respect to a specified error criterion? Throughout this chapter, we hope to give
the reader answers to these questions by analyzing commonly used signal and noise
estimation methods as well as the maximum likelihood (ML) method. Each method
is described in detail and evaluated in terms of precision and accuracy.
This chapter is organized as follows: Because optimal quantitative analysis
requires exploitation of the knowledge of the underlying data statistics, Section 4.2
describes various probability density functions (PDFs) that appear when dealing with
MR data. Section 4.3 reviews some results from statistical parameter estimation
theory, which are used in the remainder of the chapter. Different performance mea-
sures for estimators as well as the so-called Cramér-Rao lower bound (CRLB) and
the ML estimator are discussed. In Section 4.4 and Section 4.5 we explain how the
various PDFs can be exploited to estimate parameters from MR data. In particular,
in Section 4.4 we will focus on the estimation of (noiseless) signal components,
whereas in Section 4.5 we will consider the estimation of the image noise variance.
T
and
T
1
2
4.2
PDFs IN MRI
Whenever quantitative information needs to be extracted from MRI data, knowl-
edge of the PDF of the data is of vital importance. Indeed, if an incorrect PDF
is assumed
, systematic errors (bias) may be introduced when estimating
parameters from these data. Therefore, this section starts with an overview of the
various PDFs that would appear when dealing with (processed) MR data [10].
a priori
4.2.1
G
PDF
AUSSIAN
The raw MR data acquired in K-space during an MR acquisition scheme are
known to be complex valued. The complex data are composed of noiseless signal
components and noise contributions that are assumed to be additive and indepen-
dent and are characterized by a zero-mean Gaussian PDF [11-13]. An MR
reconstruction is then obtained by means of an inverse Fourier transform (FT).
Because of the linearity and orthogonality of the FT, the complex data resulting
from the transformation are still independent and Gaussian distributed* [14-16].
Hence, the PDF of a raw, complex data point
w
c
=,
(
w
)
is given by
r
i
2
2
1
(
ω
−
A
cos
ϕ
)
(
ω
−
A
sin
ϕ
)
r
i
−
−
p
c
(
ωω ϕσ
,
|
A
,
,
)
=
e
e
,
(4.1)
2
2
2
σ
2
σ
r
i
2
πσ
2
* It is assumed that the MR signals are sampled on a uniform grid in K-space. Furthermore, the
variance of the noise is assumed to be equal for each raw data point.
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