1 Complex Single- and Multiple-Coil MR Signals
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32
3 Statistical Noise Models for MRI
Under the assumption that the noise affects equally all the frequencies (i.e., to
all the samples in the k-space), it is both signal- and source-independent, and the
noise can be modeled as a complex Additive White Gaussian Noise (AWGN) process
with zero mean and variance σ 2Kl [42, 100]. The acquired signal in the lth coil in the
k-space can therefore be modeled as
sl (k) = al (k) + n l (k; 0, σ 2Kl (k)), l = 1, . . . , L ,
(3.2)
where al (k) is the noise-free signal at the lth coil (of a total of L coils) and sl (k) is
the received (noisy) signal at that coil. The term n l (k; 0, σ 2Kl (k)) is what we usually
refer to as noise, i.e., the complex AWGN process. If the noise in the RF signal is
considered to equally affect all the frequencies, it makes sense to consider n l itself
stationary, which implies that σ 2Kl (k) = σ 2Kl is a constant, and therefore the variance
of noise does not depend on the position. Under this assumption, we can write
n l (k; 0, σ 2Kl (k)) ≡ n l (k; 0, σ 2Kl ) = n lr (k; 0, σ 2Kl ) + j · n li (k; 0, σ 2Kl ).
(3.3)
This is the usual assumption for MRI data: the noise in each coil can be considered stationary in the k-space. (Note that this assumption could not hold if some
interpolation scheme is adopted in the k-space, such as the one needed after spiral
acquisition).
The complex image domain is obtained as the inverse discrete Fourier transform
(iDFT) of sl (k) for each slice and at each coil. Under the assumption that the data
is sampled on a Cartesian lattice and no interpolation, filtering, or apodization is
applied, and the noise in the complex image domain is again a stationary AWGN
random process for each receiving coil, the transformed signal can be obtained by
applying the iDFT (an orthogonal and linear operator):
Sl (x) = Al (x) + Nl (x; 0, σl2 ), l = 1, . . . , L ,
(3.4)
where Nl (x; 0, σl2 ) = Nlr (x; 0, σl2 ) + j · Nli (x; 0, σl2 ).
Note that, in this coil by coil analysis the correlations among coils have been left
aside. However, there may be an initial noise correlation between the receiver coils
due to electromagnetic coupling [95, 96, 195]. Even when the coils are decoupled,
correlations may exist since one of the main sources of noise is the object itself due
to thermal motion. As a consequence, the noise pattern in the complex image domain
may be seen as a complex multivariate (one variable per coil) AWGN process, with
zero mean and covariance matrix [41]
⎛
σ12 σ12
⎜ σ21 σ22
⎜
=⎜ . .
⎝ .. ..
σ L1 σ L2
···
···
..
.
⎞
σ1L
σ2L ⎟
⎟
.. ⎟ ,
. ⎠
· · · σ 2L
(3.5)
3.1 Complex Single- and Multiple-Coil MR Signals
33
with σi j = ρi j σi σ j the covariance between the ith and jth coils and ρi j the coefficient
of correlation between those coils. While ρi j depends only on the electromagnetic
coupling between corresponding coils, the variance of noise for each coil may be
easily calculated directly from the k-space [12, 100, 226]
σl2 =
1 2
σ .
|Ω| Kl
(3.6)
where |Ω| is the size of the Field of View (FOV), i.e., the number of points used in
the 2D iDFT.
3.2 Single-Coil MRI Data
For a single-coil acquisition, the complex model in Eq. (3.4) simplifies to
S(x) = A(x) + N (x; 0, σ 2 ),
with N (x; 0, σ 2 ) = Nr (x; 0, σ 2 ) + j · Ni (x; 0, σ 2 ) a complex AWGN and A(x) =
Ar (x) + Ai (x) the (complex) original non-noisy signal. The magnitude signal M(x)
is the Rician distributed envelope of the complex signal [89]:
M(x) = |S(x)| =
Ar (x) + Nr (x; 0, σ 2 )
2
2
+ Ai (x) + Ni (x; 0, σ 2 ) .
(3.7)
The probability density function (PDF) of the Rician distribution is defined as [75]
p M (M|A, σ) =
M
M 2 + A2
exp
−
σ2
2σ 2
I0
AM
σ2
u(M),
(3.8)
where I0 (.) is the 0-th order modified Bessel function of the first kind, u(.) Heaviside’s
step function, and A(x) = |Ar (x) + j · Ai (x)| = Ar2 (x) + Ai2 (x). In the image
background, where the signal to noise ratio (SNR) is zero due to the lack of waterproton density in the air, the Rician PDF simplifies to a Rayleigh distribution with
PDF, see Fig. 3.1:
M
M2
(3.9)
p M (M|σ) = 2 exp − 2 u(M).
σ
2σ
More information about these two distributions can be found in Appendix A. For the
sake of illustration, a pipeline with the distributions involved in single-coil acquisitions is depicted in Fig. 3.2.
When the SNR is high, i.e., the values of the signal A(x) are large with respect to
σ, the distribution is usually assumed to be Gaussian.
34
3 Statistical Noise Models for MRI
Fig. 3.1 In the background of a single-coil signal, the Rician distribution simplifies into a Rayleigh
Fig. 3.2 Single-coil acquisition process. The data in both the k-space and the image domain follow
a Gaussian distribution. The final signal after the magnitude is taken will follow a Rician distribution
Equation (3.7) may be approximated by a series expansion as
M(x) = A(x) + Nr (x; 0, σ 2 ) +
Ni (x; 0, σ 2 )2
+O
2 A(x)2
Ni (x; 0, σ 2 )
A(x)
3
,
and assuming high SNR it can be simplified to
M(x) ≈ A(x) + Nr (x; 0, σ 2 ),
which can be seen as a Gaussian noise with variance σ 2 .
(3.10)
3.3 Fully Sampled Multiple-Coil Acquisition
35
3.3 Fully Sampled Multiple-Coil Acquisition
The process of acquiring MR data from multiple receivers have been reviewed in
Sect. 2.4. Noise in each coil is assumed to be a stationary complex Gaussian. However,
the way the coils are combined and the assumptions over the covariance matrix will
produce different distributions for the final CMS. In this section, we will consider
three possible cases: (1) the reconstruction is done using SoS and the coils are not
correlated; (2) SoS is used, but there exist correlations between coils; and (3) the
reconstruction is done using a SMF.
3.3.1 Uncorrelated Multiple-Coil with SoS
For nonaccelerated acquisitions, one of the most direct approaches to fuse the coil
information into one single image is the so-called Sum of Squares (SoS) (see
Eq. (2.18) in Sect. 2.4). Although alternative reconstruction methods can be used,
the advantage of the SoS is that no extra parameters, such as the coil sensitivity, need
to be estimated.
In order to provide a feasible study of the noise distribution in the CMS, it is
necessary to consider an ideal scenario with the following assumptions:
1. The k-space is fully sampled (using a Cartesian lattice);
2. The variance of noise σl2 is the same for each coil.
σ2 =
1 2
σ .
|Ω| K
3. There are no correlations between coils.
4. The CMS is obtained using the SoS.
In this scenario, the covariance matrix
values
in Eq. (3.5) is diagonal with identical eigen= σ 2 · I,
where I is the L × L identity matrix. Under these assumptions, the CMS MT (x)
follows a noncentral χ (nc-χ) distribution [58, 214] with PDF
p MT (MT |A T , σ, L) =
A1−L
M 2 + A2
T
MTL exp − T 2 T
2
σ
2σ
I L−1
The signal A T (x) is the SoS of the non-noisy signal
L
A2T (x) =
|Al (x)|2 .
l=1
A T MT
σ2
u(MT ),
(3.11)
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3 Statistical Noise Models for MRI
For L = 1, the nc-χ reduces to the Rician distribution. In the background, this PDF
simplifies to a central χ (c-χ) distribution with PDF
p MT (MT |σ, L) =
21−L MT2L−1
MT2
exp
−
(L) σ 2L
2σ 2
u(MT ),
(3.12)
which also reduces to a Rayleigh distribution for L = 1. For the sake of illustration,
a pipeline with the distributions involved in multiple-coil acquisitions is depicted in
Fig. 3.3. Note the correlations between coils are the same in both the k-space and the
image domain.
Similar to the Rician distribution, when the SNR is high, the nc-χ can also be
approximated by a Gaussian distribution. The SoS of the signals Sl (x) can be simplified by truncating a series expansion to
MT (x) ≈ A T (x) + N T (x; 0, σ 2 ),
(3.13)
with N T (x; 0, σ 2 ) a real stationary Gaussian process with variance σ 2 .
Fig. 3.3 Multiple-coil acquisition process. The data in both the k-space and the image domain
follow a Gaussian distribution in each coil. The final composite magnitude signal will follow
different distributions depending on the method employed to aggregate coils and on the possible
correlations. If all the coils have the same variance, there is no correlation between them and the
CMS is calculated using SoS, the CMS will follow a noncentral χ distribution
3.3 Fully Sampled Multiple-Coil Acquisition
37
3.3.2 Correlated Multiple-Coil with SoS
The nc-χ distribution proposed in the previous section has been used to model the
noise in MRI when the signals at different receiving coils are combined with SoS [10,
58, 68, 117]. However, this CMS will only show nc-χ statistics if the variance of noise
is the same for all coils, and no correlation exists between them. Although it is well
known that in phased array systems noise correlations do exist [41, 96, 95, 195], this
effect is usually left aside due to their minimal effect and practical considerations,
as stated in [58]. However, for modern acquisition systems comprising up to 32 or
64 coils, the receivers usually show a certain coupling. This means that the noisy
samples at each k-space location are correlated from coil to coil. Assuming such
correlation is frequency-independent (i.e., the same for all k-space samples), the
linear iDFT operator will extend the correlation between coils in the complex image
domain, so that becomes a nondiagonal, symmetric, positive definite matrix, where
the off-diagonal elements stand for the correlations between each pair of coils.
In this case, the actual PDF is not strictly a nc-χ, though for small correlations it is expected that such model remains approximately valid [9]. Even
when the nc-χ assumption is feasible, correlations will affect the number of
Degrees of Freedom (DoF) of the distribution. If SoS is used, the PDF of the
CMS can indeed be accurately approximated with the traditional nc-χ model
in Eq. (3.11) with effective parameters (a reduced number of coils L and an
increased variance of noise σ 2 ) are used.
The effective values can be calculated as [9]
A2T (x) tr ( ) + (tr ( ))2
;
A H (x) A(x) + || ||2F
tr ( )
2
,
σeff
(x) =
L eff (x)
L eff (x) =
(3.14)
(3.15)
where ||.|| F is the Frobenius norm and
A(x) = [A1 (x), A2 (x), . . . , A L (x)]T .
These effective values can be calculated through the method of the moments.
We consider MT2 (x) instead of MT (x), since the former will follow a noncentral chi
square (nc-χ2 ) distribution, whose moments are more tractable than those of the
nc-χ. Note that the PDF of MT2 (x) cannot be theoretically derived if correlations are
considered, but its mean and variance are easily computed as
E{M L2 (x)} = A T (x)2 + 2 tr ( )
Var{M L2 (x)} = 4 A H (x) A(x) + 4 || ||2F
(3.16)
(3.17)
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3 Statistical Noise Models for MRI
The moments of the equivalent nc-χ2 are
2
L eff
E{M L2 } = A2T (x) + 2σeff
2
2 2
4
Var{M L } = 4 A T (x) σeff + 4L eff σeff
.
(3.18)
(3.19)
To deduce an effective nc-χ2 PDF, the method of moments can be used with
Eqs. (3.18) and (3.19) are respectively equated to Eqs. (3.16) and (3.17) to solve
for the effective parameters.
Note that, with this approximation, L eff (x) is not necessarily an integer number. As
an illustration, on Fig. 3.4 the effective number of coils for different configurations are
depicted as a function of the correlation between coils (assuming a similar value ρ for
all correlations). Note that as the correlation grows, all the values tend to one, i.e., for
very high correlation the system is equivalent to a single-coil. Those configurations
with a greater number of coils are the ones with the greater slope, i.e., they are much
more affected by correlations. For instance, a 32-coil configuration with ρ = 0.2
is equivalent to a 18-coil configuration, which gives an idea of the importance of
the correlations in the scanner performance. Little correlations will provide higher
effective number of coils, and therefore smaller effective noise powers.
One interesting feature of the effective values here defined is that
2
(x) = tr ( ) ,
L eff (x) · σeff
(3.20)
which only depends on . Note that both effective values are spatially dependent,
though their product is spatially independent. In the particular case in which the
variance of noise is equal in each coil it becomes
2
(x) = L · σ 2
L eff (x) · σeff
(3.21)
which is exactly equal to the uncorrelated case. This feature will be latter exploited
for noise estimation.
a
b
Fig. 3.4 Effective number of coils as a function of the coefficient of correlation, analytically
computed for synthetic data: a absolute value; b relative value (taken from [9])
3.3 Fully Sampled Multiple-Coil Acquisition
39
Simplified Scenarios
Consider an scenario in which the variance of noise σl2 is the same in every coil,
σl2 = σ02 . Under that assumption the covariance matrix becomes
⎛
1
⎜ ρ21
⎜
= σ02 · ⎜ .
⎝ ..
ρ L1
ρ12 · · ·
1 ···
.. . .
.
.
ρ L2 · · ·
⎞
ρ1L
ρ2L ⎟
⎟
.. ⎟
. ⎠
(3.22)
1
where ρi j ∈ [−1, 1] is the correlation coefficient between coils i and j. A further
simplification is to assume Ai (x) = A j (x) for all i, j. Although this premise might
not be completely realistic, its study provides a simple, yet powerful insight on the
behavior of the true statistics with regard to noise correlations. Under this assumption,
Eqs. (3.14) and (3.15) read
L eff
A2 ρ + L σ02 ρ2
= L 1 + (L − 1) T 2
A T + L σ02
2
σeff
= σ02 1 + (L − 1)
A2T ρ + L σ02 ρ2
A2T + L σ02
−1
(3.23)
,
(3.24)
with ρ the average of the values of ρi j and ρ2 the average of |ρi j |2 . Results in
Eqs. (3.23) and (3.24) make clear that the effective values of L and σ02 do depend
on the signal value A T (x). If A2T (x)ρ is comparable to L σ02 ρ2 , i.e., if the SNR is
low, the effective parameters will depend on the position x. Therefore the data is
no longer stationary and the noise power varying along with M L2 (x). Consider these
extreme cases
1. In the background, where no signal is present and hence SNR = 0, the effective
values are
L eff,B =
L
1+
ρ2 (L
− 1)
2
= σ02 (1 + ρ2 (L − 1)).
σeff,B
(3.25)
(3.26)
2. For high SNR areas, the effective values become
L
1 + ρ(L − 1)
= σ02 (1 + ρ(L − 1)).
L eff,S =
(3.27)
2
σeff,S
(3.28)
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3 Statistical Noise Models for MRI
In real cases, where the correlation between coils is positive, is positive,
0 ≤ ρi j ≤ 1,
the effective variance of noise in the signal areas will be greater or equal than
in the background, i.e.,
2
2
≥ σeff,B
.
σeff,S
Gaussian Approach for High SNR
For high SNR, the SoS of correlated signals can also be effectively approximated by
a Gaussian distribution:
MT (x) ≈ A T (x) + N T (x; 0, σT2 (x)),
(3.29)
with N T (x; 0, σ 2 ) a Gaussian process with spacial-dependent variance
σT2 (x) =
A H (x) A(x)
.
A2T (x)
(3.30)
Sometimes it could be even possible to carry out a further approximation to avoid
the spatial dependency with σT2 . If we assume that Ai (x) ≈ A j (x), Eq. (3.30) can be
simplified to
|| || F
.
(3.31)
σT2 =
L
This approximation is only valid in those areas where the SNR is high enough, and
never over the background of the image.
3.3.3 Multiple-Coil with SMF Reconstruction
The use of the SoS to merge the information of multiple-coil is the cause of the
appearance of the nc-χ distribution (actual or approximated) in multiple-coil data.
However, it is not the only method used to reconstruct the CMS. If the SMF in
Eq. (2.15) is used, the reconstructed data can be written as
L
SSMF (x) = W(x) · ST (x) =
Wl (x)Sl (x).
l=1
with W(x) a reconstruction vector for each point x
(3.32)
3.3 Fully Sampled Multiple-Coil Acquisition
W(x) = C H (x)C(x)
41
−1
C H (x).
(3.33)
The signal SSMF (x) is obtained as a linear combination of the samples in each coil,
Sl (x), where the noise is Gaussian distributed, multiplied by some weights. The
resulting signal is also Gaussian, with variance
σ2 = W W H .
(3.34)
Note that matrix W(x) is x dependent, so the variance of noise σ 2 is also spatial
dependent, σ 2 (x). The final magnitude signal is calculated by taking the module of
the complex signal, MT (x) = |SSMF (x)|. Let us consider two different scenarios:
Uncorrelated coils: Let us first assume that there are no correlations between coils
and that every coil has the same variance of noise σ02 . Under that assumption, the
covariance matrix becomes diagonal = σ02 I and Eq. (3.34) becomes
⎛
L
σ 2 = σ02 WW H = σ02
L
|Wl (x)|2 = σ02
l=1
l=1
⎞
⎜
⎜
⎜
⎝
|Cl (x)|2
L
|Cm (x)|
⎟
⎟
⎟ = σ02 .
⎠
2
m=1
This, in this case, is the final variance of noise does not depend on the position,
and therefore SSMF (x) follows a complex Gaussian distribution with variance σ02 .
When the magnitude is considered, the final image MT (x) will follow a Rician
distribution with σ = σ0 , totally equivalent to one coil systems.
Correlated coils: If we assume correlation between coils and different variances
of noise for each coil, the simplification of the previous case is no longer possible.
As a consequence, the variance of noise becomes:
σ 2 (x) = W(x) W H (x).
(3.35)
The final image will follow a complex Gaussian distribution, but the variance
of noise will become dependent on the position, i.e., it becomes a non-stationary
complex Gaussian distribution. When the magnitude is considered, the final image
MT (x) will also be Rician, but unlike the single-coil case, now the parameter
σ(x) becomes spatially variable. This distribution is exactly the same that follows
SENSE reconstructed data.
A survey with all the distributions generated from fully sampled multiple-coil
data is depicted on Fig. 3.5.
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3 Statistical Noise Models for MRI
Fig. 3.5 Distributions of noise in the CMS generated from multiple-coil data
3.4 Statistical Models for pMRI Acquisitions
3.4.1 General Noise Models in pMRI
In the previous section, we have reviewed the noise model for multiple-coil systems
when the k-space is fully sampled. When acceleration, subsampling, and pMRI are
considered, noise models in the resulting CMS can vary. Depending on the way
the information from each coil is combined, the statistics of the image will follow
different distributions. It is therefore necessary to study the behavior of the data for
a particular reconstruction method.
We will follow the notation and methods of Sect. 2.5. The acquired data in the lth
coil, slS (k), is a subsampled version of the original k-space signal, and therefore it
is also corrupted with AWGN with variance σ 2Kl . If the iDFT is directly applied to
the subsampled signal, we will have an AWGN process with variance (compare to
Eq. (3.6)):
r 2
σ ,
(3.36)
σl2 =
|Ω| Kl
with |Ω| the final number of pixels in the FOV. Note the final noise power is greater
than in the fully sampled case due to the reduced k-space averaging, as it will be the
case with SENSE. On the contrary, the iDFT may be computed after zero-padding
the missing (not sampled) k-space lines, and then the noise in each coil of the x-space
will be Gaussian with variance [12]:
σl2 =
1
σ2 .
|Ω| · r Kl
(3.37)