4 Case Study: The LMMSE Signal Estimator
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5 Noise Filtering in MRI
where A(x) is the unknown intensity value in pixel x and M(x) the observation
vector. The covariance matrices can be defined as
T
C M 2 M 2 (x) = E{ M2 (x) − E{M2 (x)} M2 (x) − E{M2 (x)} }
2
= E{ M 2 (x) − E{M 2 (x)} } = E{M 4 (x)} − E{M 2 (x)}2
= C M 2 M 2 (x)
T
C A2 M 2 (x) = E{ A2 (x) − E{A2 (x)} M2 (x) − E{M2 (x)} }
= E{ A2 (x) − E{A2 (x)}
= C A2 M 2 (x)
M 2 (x) − E{M 2 (x)} }
and making use of the signal relation in Eq. (5.18):
C A2 M 2 (x) = E{A4 (x)} + 2E{A2 (x)}σ 2 − E{A2 (x)}E{M 2 (x)}
Finally, the LMMSE estimator becomes
E{A(x)4 } + 2E{A2 (x)}σ 2 − E{A2 (x)}E{M 2 (x)}
E{M 4 (x)} − E{M 2 (x)}2
2
2
× M (x) − E{M (x)}
(5.20)
A2 (x) = E{A2 (x)} +
Assuming local ergodicity, the expectation may be replaced by its sample estimator.
Using the relation from Eq. (5.18)
E{M 2 (x)} = E{A2 (x)} + 2σ 2
E{M 4 (x)} = E{A4 (x)} + 8σ 2 E{A2 (x)} + 8σ 4
the LMMSE estimator may finally be written as
A2 (x) = M 2 (x) x − 2σ 2 + K (x) M 2 (x) − M 2 (x)
with K (x)
K (x) = 1 −
(5.21)
x
4σ 2 M 2 (x) x − σ 2
.
M 4 (x) x − M 2 (x) 2x
(5.22)
The estimators so defined can be applied to stationary Rician data. It requires a
prior estimation of the parameter σ, which can be done using some of the methods
proposed in the following sections. The extension to non-stationary Rician MR data,
like the one produced by SENSE of GRAPPA+SMF, is straightforward: the single
value of σ must be replaced by a spacial noise map, σ(x):
A2 (x) = M 2 (x)
x
− 2σ 2 (x) + K (x) M 2 (x) − M 2 (x)
x
(5.23)
5.4 Case Study: The LMMSE Signal Estimator
with K (x)
K (x) = 1 −
113
4σ 2 (x) M 2 (x) x − σ 2 (x)
.
M 4 (x) x − M 2 (x) 2x
(5.24)
An illustration of this filter can be found in Figs. 5.9f, 5.10f and 5.11f.
5.4.2 Extension to Multiple Samples
The original formulation of the LMMSE presented in the previous section assumed
that only one single image is available for estimation. Thus, the covariance matrices
C A2 M 2 and C M 2 M 2 become a scalar for each position x. In [2], an extension for
multiple samples is made.
Let us assume that N measures of every pixel are available:
M(x) = [M1 (x) M2 (x) . . . M N (x)]T
where M(x) is the measure vector. For the sake of simplicity, in what follows we
will remove the dependency with x. M2 must be understood element-wise, i.e., M2 =
[M12 . . . M N2 ]T . C M 2 M 2 is the N × N covariance matrix of M2 , defined as
T
C M 2 M 2 = E{ M2 − E{M2 } M2 − E{M2 } }
After some algebra and replacing expectations by their sample estimator . , we can
finally write this matrix as
CM 2 M 2 =
M4
k
+ 4σ 4 − 4σ 2 M2
− 4σ 4 − 4σ 2 M2
k
k
− M2
2
k
1 N 1TN
IN
(5.25)
where 1 N is an all 1 vector of length N , and I N is the N × N identity matrix. The
sample estimators must be understood along the samples dimension:
Mn
k
=
1
N
N
Mkn
k=1
Matrix C A2 M 2 is
T
C A2 M 2 = E{ A2 − E{A2 } M2 − E{M2 } }
using sample operator becomes:
C A2 M 2 =
M4
k
+ 4σ 4 − 4σ 2 M2
k
− M2
2
k
1TN
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5 Noise Filtering in MRI
Finally, for each point in the image, the estimator is
2
2
A2 (x) = M2 (x) k − 2σ 2 + C A2 M 2 (x)C−1
M 2 M 2 (x) M (x) − M (x)
k
(5.26)
5.4.3 Recursive LMMSE Filter
Once the image is filtered with the LMMSE estimator, the output model may no
longer be considered Rician. However, since for high SNR the Rician and Gaussian
model converge, in [7], authors proposed a recursive method that experimentally
improved the original formulation of the LMMSE. The main idea behind the design
is that, if the noise is dynamically estimated in each iteration, the filter should reach
a steady state as the estimated noise gets smaller and smaller. The recursive LMMSE
filter (RLMMSE) is then defined as
2
(x) = Mt2 (x)
Mt+1
x
− 2 σt2 + K t (x) Mt2 (x) − Mt2 (x)
x
(5.27)
with
K t (x) = 1 −
4 σt2
Mt2 (x)
Mt4 (x)
x
x
− σt2
− Mt2 (x)
2
x
,
where Mt (x) is the magnitude image after t iterations of the filter. By definition
M0 (x) = M(x). This estimator is the starting point of the anisotropic diffusion extension of the LMMSE proposed in [120].
An illustration of this filter can be found in Figs. 5.9g, 5.10g and 5.11g.
5.4.4 Extension to nc-χ Data
Similar to what happens with many filtering schemes, the original formulation of
the LMMSE signal estimator was designed to cope with stationary Rician data. We
have already seen that the extension to non-stationary Rician data is straightforward.
However, in order to cope with nc-χ data, some algebra is needed. In [40], authors
presented a new formulation to cope with multicoil data. Once more, in order to
avoid the odd moments of the distribution, the square magnitude is considered. If the
CMS is reconstructed with SoS we can write:
L
MT2 (x) =
|Al (x) + Nrl (x; 0, σ 2 )) + j · Nil (x; 0, σ 2 )|2 ,
l=1
5.4 Case Study: The LMMSE Signal Estimator
115
and the LMMSE can be reformulated as:
A2T (x) = MT2 (x)
x
− 2Lσ 2 + K T (x) MT2 (x) − MT2 (x)
x
,
(5.28)
where K T (x) is defined as
K T (x) = 1 −
4σ 2
MT2 (x)
MT4 (x)
x
x
− Lσ 2
− MT2 (x)
2
x
.
(5.29)
The method derived this way depends on two parameters related with the nc-χ
distribution, σ and the number of coils, L. Note that this feature makes this filter unfit
to be used with correlated data, when noise becomes non-stationary and effective
parameters must be used. Thus, in order to cope with correlations and effective values,
a reformulation of the method must be done again [9]:
A2T (x) = Mt2 (x)
K T (x) = 1 −
x
− 2σ 2L (x) + K T (x) MT2 (x) − MT2 (x)
2
(x) MT2 (x) x − σ 2L
4σeff
MT4 (x) x − MT2 (x) 2x
x
,
.
(5.30)
(5.31)
2
Note that the product σ 2L (x) = σeff
(x) · L eff (x) is a constant for correlated multicoil
data with SoS, while for GRAPPA+SoS it becomes a spacial map depending on x.
For the sake of illustration, a T − 1 slice acquired in a multicoil system is simulated. Complex Gaussian noise is added to each coil, and an initial correlation
between coils of ρ = 0.15 is considered. The CMS is reconstructed using SoS. Note
that, according to the results in Sect. 3.3.2, the final CMS can be approximated by a ncχ if effective parameters are considered. To evaluate the effect of using an inaccurate
model for filtering, the noisy signal is filtered using a Rician LMMSE, the χ-LMMSE
without considering effective values and the adapted scheme in Eq. (5.30). Results
are shown in Fig. 5.12. According to the theoretical model, the Rician LMMSE uses
a overestimated level of noise, and hence oversmooths the image showing an excessive blurring of the edges. On the other hand, the χ-LMMSE (without considering
correlations between coils) implicitly uses an underestimated value of noise (note
that the effective value will always be greater). As a result, its filtering capability is
smaller than the cχ-LMMSE. Visual inspection of Fig. 5.12d, e shows that the signal
area of the image is indeed noisier for χ-LMMSE.
5.4.5 Extension for an Specific Application: DWI Filtering
The LMMSE schemes defined so far were designed for general MRI filtering. Thus,
on one hand, they are flexible and they can be applied to many different images from
different acquisition modalities. However, on the other hand, they do not take advantage of the properties of specific modalities. Note that, for instance, in modalities like
116
(a)
5 Noise Filtering in MRI
(b)
(c)
(d)
Fig. 5.12 Visual comparison of filtering of correlated multicoil data. A synthetic T1 slice is considered, L = 8 coils, σ = 15, ρ = 0.15
relaxometry, DWI or fMRI, multiple acquisitions of the same slice under different
condition are acquired. An adaptive filtering scheme must be capable of using the
redundant information of such data in order to improve the filtering of the data and
the estimation of the original signal.
In what follows we will show some extensions done to the LMMSE to better
cope with the inherent redundancy present in diffusion-weighted imaging (DWI).
The LMMSE was selected for (a) being a statistical method that ensures an unbiased
estimation; (b) its conservative nature for tensor estimation in non filtered data is
better than overfiltered data.
The signal estimator discussed so far made use of the local structure of the image
to estimate the underlaying signal A(x). However, as already said, there are some
MRI modalities in which, due to the specific acquisition process, some redundant
information is acquired, that can be exploited in order to improve the estimation.
That is the case of DWI. A methodology to use the LMMSE for this modality was
proposed in [227, 228].
The diffusion signal A = A(x) in a DWI acquisition may be modeled for the case
of one single fiber orientation inside a voxel as a Gaussian process [25]:
Ai = A0 exp −bgiT Dgi ,
(5.32)
where A0 is the baseline image, which is a conventional non-diffusion-weighted T2 MRI image; b is the weighting parameter given by the scanner; Ai is the amplitude
of the DWI image when a gradient in the direction gi is applied to the magnetic
field. Finally, D is the diffusion tensor, a symmetric, positive-definite rank 2 tensor. If
more than one fiber population with different orientations are present within the same
voxel, Eq. (5.32) is no longer valid. In these cases the positive-definite quadratic form
defined by bgiT Dgi may be replaced by a more general positive function D(b, gi ).
Note that both models yield an attenuation with respect to the baseline image A0 ,
which is greater for the directions of maximum diffusion. The attenuation becomes
more important for higher values of b. In practice the received signal Mi is corrupted
with noise, which, for this model, it is considered as stationary Rician, though other
models from the ones seen in Chap. 3 could be assumed.
5.4 Case Study: The LMMSE Signal Estimator
117
In order to estimate the original signal Ai from the noise image Mi , we make use
of the LMMSE scheme proposed in Eq. (5.21), which can be rewritten for multiple
DWIs as:
A2 (x) = A2 (x) + C A2 M 2 (x)C M 2 M 2 (x)−1 M2 (x) − M2 (x)
(5.33)
where X denotes the expected value of X is a column vector which contains the
squared measurements of all DWI baselines and gradients at x:
2
]T
M2 = [M02 , M12 , . . . , M L−1
and A2 (x) is a column vector that contains and the squared estimated values of all
DWI baselines and gradients at x
A2 = [A20 , A21 , . . . , A2L−1 ]T .
Here L = L b + L g , being L b the number of baselines and L g the number of gradient directions. We assume that A20 . . . A2L b −1 correspond to the baselines, and
A2L b . . . A2L−1 correspond to the gradient directions. C X Y denotes the covariance
matrix between X and Y. The estimation of C X Y would require the computation
of all crossed moments E{(X i − X i )(Y j − Y j )}, heavily increasing the computational load. In the approach proposed in [227] it is assumed that all expected values
Ai2 are completely correlated, since the expected value of each DWI may be predicted
as well in terms of the expected value of the baseline image and the tensor structure.
However, in [228] authors prove that the estimation of C X Y in fact does not depend
on the underlying tensor structure, and therefore:
C A2 M 2 = K A2 A2
T
T
C M 2 M 2 = K A2 A2 + 4σ 2 diag A2 + 4σ 4 I L
2
(5.34)
2
where K = (A4b − A2b )/A2b and b is any of the indices corresponding to baseline
images, where 0 ≤ b < L b . I L is the identity matrix. A4b is estimated as:
A4b = Mb4 − 8σ 2 A2b − 8σ 4 .
(5.35)
The inversion of C M 2 M 2 at each image location is a clear computational burden,
so an approximation is proposed instead:
C−1
M2 M2
ij
=
−1/4σ 2
4σ 2
K
+
2
l Al
+
1
4σ 2 Ai2
δi j ,
(5.36)
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5 Noise Filtering in MRI
where δi j is the Kronecker delta function. In practice, it is enough to use only one
term in the recursion. Note as well that the products with C−1
M 2 M 2 are very easy to
compute due to its simple structure.
The use of the baseline together with the gradient images in [227], although better than the conventional LMMSE, it may show ringing artifacts and overblurring
under some circumstances. Those effects are no longer an issue in the formulation
in [228]. In order to improve the performance of the previous filtering scheme, and
to avoid any kind of undesired blur of edges an structures, a new anisotropic formulation were proposed in [230]. The sample moments are calculated in an anisotropic
neighborhood that depends on the anatomical content.
An example to illustrate the efficiency of an adaptive methodology for noise filtering is now carried out over a DWI data set with 51 gradient directions, 8 baselines
and b = 700 s/mm2 . Results for the original LMMSE formulation, the joint LMMSE
in [227] and the version in [228] for 15 and 51 gradient directions are depicted in
Fig. 5.13. For the sake of reference, the UNLM has also be considered. Note that
the original LMMSE is not able to properly remove the noise, and the structural
information is mostly lost; with LMMSE-N the noise is removed and not only the
structures are preserved but they are even enhanced (see for example the left-bottom
part of the slice, red circle). Comparing LMMSE-15 and LMMSE-51, no noticeable differences may be found, except for a slightly higher blurring of the latter.
Fig. 5.13 Results for the first gradient direction (central slice) of V2. a Original noisy volume; b
Original LMMSE; c Joint LMMSE; d LMMSE-15; e LMMSE-51; f UNLM (Example taken from
[228].)
5.4
Case Study: The LMMSE Signal Estimator
119
But comparing to joint LMMSE, the details with LMMSE-N are much better preserved, the overblurring is avoided, and moreover the ringing artifacts near the ventricles and in the outer contours of the brain mostly disappear.
5.5 Some Final Remarks
Different filtering methods have been reviewed in this chapter. Many more can be
found in the literature, and many more will arise along the years to come. It is not
our purpose to make a complete survey of all possible methodologies to reduce the
influence of noise in MRI, since that would require a whole book, but to give an
outlook of the many possibilities and opportunities that noise filtering methods offer
to MRI processing. As stated at the beginning of the chapter, noise affects not only to
the visual quality, but it also makes harder many numerical procedures can make less
precise certain automatic measurements. Thus, in many occasions, the use of a proper
filter will help in the processing. However, the selection of the specific filter must be
totally tuned to the purpose of the noise filtering. In medical imaging, denoising is
not a cosmetic operation, but a method to enhance the quality of the data.
Knowing the underlying noise model helps in the design of more accurate filtering
methods. Besides, as we showed in the LMMSE example, the filtering method must
be adapted to the features of the data in such a way that important information can be
incorporated to the estimation problem. The main drawback of model based schemes
is that they rely on the parameters of certain probability distribution and those parameters have to be estimated. That is, precisely the purpose of the following chapters
is to define estimators for the noise parameters following the different configurations
found in MRI data.
Part II
Noise Analysis in Nonaccelerated
Acquisitions
Chapter 6
Noise Estimation in the Complex Domain
Noise estimation in MR is usually done over the composite magnitude signal (CMS),
since it is the usual output of the scanning process and, therefore, it is usually the
only data available. However, there are situations in which the raw data is available,
either in the k- or x-space. In those cases, the analysis and noise estimation can be
simplified and more accurate results can be achieved.
Any kind of noise analysis and processing done over the complex Gaussian data
presents many advantages when compared to CMS. First, the Gaussian noise model
is commonly assumed in image processing and many methods and techniques have
been developed along the years. Thus, there is a great library of well-tested methods
available to be used. In addition, the Gaussian distribution has some major advantages
when compared to the Rician and nc-χ: the moments have a closed form, and the noise
and signal can be easily identified as parameters of the distribution, where the mean
is precisely the original signal. Gaussian-based methods, either for noise estimation
or for noise filtering, are usually simpler than Rician or nc-χ. Additionally, the
acquisition before the CMS may take advantage of the higher number of samples,
since the detected signal is a complex number, with equally Gaussian-distributed
noise in both real and imaginary parts.
Despite the clear advantages of using the complex Gaussian data, the analysis
is usually moved to the CMS due to the unavailability of access to the raw data.
However, we must recall the importance of implementing these techniques inside
the scanner, so most of them are to be carried out before the magnitude is taken.
In this chapter, we will focus on noise estimation assuming a complex Gaussian
model for single and multiple-coil. The methods here defined are the base for more
complex methods that will be reviewed in following chapters. Finally, note that most
noise models in MRI can be simplified to Gaussian for high SNR. So, under certain
conditions, the methods here described could also be used over the CMS.
© Springer International Publishing Switzerland 2016
S. Aja-Fern´andez and G. Vegas-S´anchez-Ferrero,
Statistical Analysis of Noise in MRI, DOI 10.1007/978-3-319-39934-8_6
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6 Noise Estimation in the Complex Domain
6.1 Single-Coil Estimation
As previously studied in Chap. 3, noise in both the k- and x-spaces follows a Gaussian
distribution with zero mean and variance σ 2 in the real and imaginary parts. Noise
is assumed to equally affect all the frequencies in the k-space, independent, and
identically distributed for both the real and imaginary components. In the k-space
we can write the complex signal as
s(k) = ar (k) + n r (k; 0, σ 2K ) + j · ai (k) + n i (k; 0, σ 2K ) ,
(6.1)
and its equivalence on the x-space as
S(x) = Ar (x) + Nr (x; 0, σ 2 ) + j · Ai (x) + Ni (x; 0, σ 2 ) .
(6.2)
Noise estimation can be carried out assuming that the noise is spatially uncorrelated and with identical variance in each pixel. The advantage of working in the
complex space is that the problem reduces to the estimation of the variance of noise
assuming a Gaussian distribution, a problem very well known by the image processing community, with many different solutions and a great range of validated procedures.
The estimation can be done equally over the data in the k-space or over the x-space.
In what follows we will focus on the x-space, but results can be easily extrapolated
to k-space. In addition, note that both the real an imaginary parts of the acquired
signals are corrupted with noise with the same variance, so the estimation can be
done just over one of the component or over both of them.
Although the estimation benefits from the existence of a uniform background,
it is not necessary to have one. The only requirement for the image is not to be a
texture [14], which is the case for MR. So, most of the estimators proposed in the
literature are valid and accurate in the MR case.
Estimators for Gaussian Noise
Many different methods have been proposed to estimate the variance of noise out
of Gaussian data. Early works propose the estimation using multiple images [98],
where statistics can be easily applied as a multiple sample problem. However, in
most applications, only one single image is available, and new algorithms based on
very different approaches were proposed: wavelet decomposition [73, 219]; singularvalue-decomposition [119]; fuzzy logic [199] or block-wise operations [134, 206].
The number of methods currently defined to estimate Gaussian noise is huge and a
comprehensive review falls out of the scope of this chapter. We will review some of
the methods that can easily be adapted to MRI data.
In what follows we will assume that the estimation is only done over the real
part of the complex x-space signal, Sr (x). However, it could also be applied to the
imaginary part, or even over both together in order to duplicate the number of samples
available for estimation.