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1 Complex Single- and Multiple-Coil MR Signals

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



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





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)



36



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)



38



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)



40



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.



42



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)



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