In fMRI, the signal equation s(k_{x},k_{y}t) in Equation 1 describes the signal that we measure for an image in terms of the physical parameters, proton spin density ρ(x,y), transverse relaxation T_{2}*(x,y), and differential from the main magnetic field ΔB(x,y)
Equation 1

s(k_{x},k_{y}t)=
_{∫∫}
ρ(x,y)
e^{t/T2*(x,y)}
e^{iγΔB(x,y)t}
e^{i2π(kxx+kyy)}
dxdy

where
k_{x}= ^{γ}⁄_{2π}
_{∫0t}
G_{x}(t')dt'

and
k_{y}= ^{γ}⁄_{2π}
_{∫0t}
G_{y}(t')dt'

are known spatial frequencies because we know the magnetic field gradients G_{x}(t') and G_{y}(t') that we appply over time t'. The process of changing the magnetic field gradients over time is called a pulse sequence. Note that fMRI activaton is in terms of a statistically significant change in regression beta coefficients and not any of the physical parameters ρ(x,y), T_{2}*(x,y), or ΔB(x,y).
The effects of the transverse relaxation T_{2}*(x,y) and differential from the main magnetic field ΔB(x,y) are generally ignored
ρ*(x,y)=ρ(x,y)
e^{t/T2*(x,y)}
e^{iγΔB(x,y)t}

and thus the usual Fourier relationship in Equation 2
Equation 2

s(k_{x},k_{y}t)=
_{∫∫}
ρ*(x,y)
e^{i2π(kxx+kyy)}
dxdy

is obtained.
A pulse sequence is presented in Figure 1 (left) where rf denotes the radio frequency pulse and ADC denotes the analog to digital converter (adapted from Haacke et al., 1999.).
Initially at time t=0, G_{x}=0 and G_{y}=0, indicating that k_{x}=0 and that G_{y}=0 and we are in the center of kspace in Figure 1 (center).
When G_{y} is turned on positive, we move up to the uppermost value of k_{y}space (k_{y}=3 in this example) and when G_{x} is turned on negative we move to the leftmost value for k_{x}space (k_{x}=3 in this example).
Then we turn on G_{x} positive which takes us from left to right on the uppermost line of k_{y}space (from k_{x}=3 to k_{x}=3 for k_{y}=3 in this example).
We then turn on k_{y} negative which takes us down one line in the k_{y}= direction (from k_{y}=3 to k_{y}=2 in this example) and also turn on k_{y} negative to take us fron right to left.
This process is repeated until all left right lines of kspace are covered all the while the ADCs are on sampling kspace at time intervals Δt. The kspace sampling locations are denoted by filled black circles in Figure 1 (center). At each sampled kspace location, the measurement is complexvalied (real and imaginary measurements).
The complexvalued (real and imaginary) sampled kspace points are recorded in a string of values (i.e. string out points along trajectory from top left to bottom right) as in Figure 1 (right). Each point in Figure 1 (right) denoted as a filled black circle consists of a real and imaginary measurement.
In order to generate an image, we first need to remove the turn around points (3 for each turn around in this example) in Figure 1 (center) that are not on the Cartesian array of interest. The next step is to reconstruct the complexvalued 2D kspace array using the complexvalued 2D inverse Fourier transform. The 2D forward and 2D inverse transform processes can be represented as complexvalued matrix multiplications. In Figure 2, it is seen that we can take a complexvalued image (as in the second column), premultiply it by a complexvalued forward Fourier matrix (as in the first column) and postmultiply it by the transpose of a complexvalued forward Fourier matrix (as in the third column), to obtain the complexvalued kspace (spatial frequency) representation of it (as seen in the fourth column).
Figure 2














+i

×

+i

×

+i

=

+i








In Figure 3 it is seen we can take the complexvalued kspace data (as in the second column), premultiply it by a complexvalued inverse Fourier matrix (as in the first column) and postmultiply it by the transpose of a complexvalued inverse Fourier matrix (as in the third column), to obtain the complexvalued image (as seen in the fourth column).
Figure 3














+i

×

+i

×

+i

=

+i








Ω_{y}

×

F

×

Ω_{x}'

=

Y

Instead of reconstructing an image from the kspace data by pre and postmultiplying by complexvalued inverse Fourier matrices, we can use an alternative but equivalent procedure involving a realvalued isomorphism (equivalent representation) for the complexvalued procedure. We can stack the rows of the real part of the kspace data (spatial frequencies) on top of the imaginary part of the kspace data (spatial frequencies) and form a single vector. Then the inverse Fourier transform image reconstruction procedure can be equivalently be represented as a matrix premultiplication of the vector of spatial frequencies by a larger inverse Fourier transformation matrix (Rowe et al., 2007). Later we will represent preprocessing operations as matrix premultiplications O which will allow us to keep track of the change in mean Of_{0} and covariance OO' (Nencka et al., 2009).
With the matrix of complexvalued kspace data represented as a single fector that is rows of reals upon rows of imaginaries as in Figure 4 (right), it can be premultiplied by this much larger inverse Fourier transform matrix as in Figure 4 (center), to obtain a vector of reconstructed image values as in Figure 4 (left) that are the rows of reals stacked upon the rows of the imaginaries.
The vector of reconstructed image values as in Figure 4 (left) that are the rows of reals stacked upon the rows of the imaginaries can be chopped up and reshaped into the complexvalued image in Figure 3 (fourth column).
Returning to Figure 1 (right), there are complexvalued data points (the turn around points) that need to be censored (removed). We can censor these points as in Figure 5 by premultiplying the complexvalued vector in Figure 1 (right) of real and imaginary values (170 in this example) by a censoring matrix. The censoring matrix is an identity matrix with rows removed to censor particular values.
We now need to premultiply by a rowreveral permutation matrix to flip alternating rows. A permutation matrix is a matrix of ones and zeros that results in a reordering.
Figure 6








=


×


×



=

R

×

C

×

f

We now have only the kspace data (real and imaginary values for each point) that we need and going in the proper direction. We now need to order the kspace data to be rows of reals stacked upon rows of imaginaries. We reorder values to be rows of reals stacked upon rows of imaginaries by using another permutation matrix as in Figure 7.
Figure 7










=


×


×


×


f

=

P_{C}

×

R

×

C

×

f

We now have the kspace vector from Figure 4 (right). The operations of censoring, reversing rows, and reordering using C, R, and P_{C} are simply book keeping. Image reconstruction can proceed as in Figure 4 with the realvalued isomorphism (Rowe et al., 2007).
In general, there are many preprocessing steps applied to the data. We can represent linear preprocessing in kspace and also in image space as matrix multiplications (Nencka et al., 2009).
The first preprocessing step is shifting odd and even lines of kspace to eliminate Nyquist ghosting. This can be described as in Equation 3 with an additional term in the signal equation s(k_{x},k_{y}t)
Equation 3

s(k_{x},k_{y}t)=
_{∫∫}
ρ(x,y)
e^{t/T2*(x,y)}
e^{iγΔB(x,y)t}
e^{i(1)l2π
Δkxx}
e^{i2π(kxx+kyy)}dxdy

where l represents the line number, Δk_{x} represents the line shift offset, and all other variables are as previously defined. Such shifts are often determined through the use
of navigator echoes (Jesmanowicz et al., 1993) or reference scans Bernstein et al. (2004).
An example of kspace line shifting Nyquist ghost correction (Nencka and Rowe, 2008) is presented in Figure 8.
The Nyquist ghost correction can be implemented when an estimate of Δk_{x} is available by permuting the kspace values to be real then imaginary for each row, then Fourier transforming each row, multiplying each row by a phase shift, then inverse fourier transforming each row, then repermuting to be the rows of the reals stacked upon the rows of the imaginaries.
Figure

9










P_{R}^{1}

×

Ω_{row}^{1}

×


×


×


×


P_{R}^{1}

×

Ω_{row}^{1}

×

Φ

×

Ω_{row}

×

P_{R}

×

f

If there is no line shifting, then Φ=I and P_{R}^{1}Ω_{row}^{1}ΦΩ_{row}P_{R} =I and does nothing. Recall that
f=P_{C}×R ×C×f
as in Figure 7.
Since it takes time to measure the complexvalued kspace points, often a portion is not measured to save time and the missed data interpolated. Conjugate symmetry ideally exists about the origin in kspace, as
the reconstructed image is expected to be realvalued. This symmetry allows half of kspace to be generated without being acquired. This is called partial or sometimes half kspace and a Homodyne interpolation is performed. Figure 11 shows Homodyne interpolation of kspace array acquired in Figure 10 (left) with 1 overscan line to "full" kspace array in Figure 10 (right).
Homodyne interpolation for one overscan line as in Figure 10 can be performed on a vector of kspace data via the H operator matrix in Figure 11.
Figure 11


Sometimes researchers artificially increase their image resolution by "zerofilling" the complexvalued kspace. Zerofilling is a process where zeros are placed around the measured kspace array then reconstructed as if they were measured. What you are saying when you do this is, I have measured these spatial frequencies and they are zero. When kspace is zerofilled, voxel values are sinc interpolated. Figure 12 shows zero filling of kspace from a 4×4 array is in Figure 12 (left) is zerofilled to an 8×8 array of kspace Figure 12 (right).
Zerofilling from a 4×4 kspace array to an 8×8 kspace array as in Figure 12 can be performed on a vector of kspace data via the F
operator in Figure 13.
Figure 13


As can be seen in Figure 13, the F operator is not square and appends zeros around the observed kspace measurements. After size of the kspace array size has been finalized, it is often smoothed (scanner manufactures may do this for you without your knowledge) using an Apodizer to reduce Gibbs ringing (ripples) in images. Apodization of kspace arrays can be carried out by multiplication by the apodization matrix A as presented in Figure 14.
Figure 14


Together these preprocessing operations in kspace are
O_{k}=AFH(P_{R}^{1}Ω_{row}^{1}ΦΩ_{row}P_{R})(P_{C}RC).
When Fourier inverse transformation image reconstruction was described above, the fact that
T_{2}*(x,y) is not infinity and ΔB(x,y) is not zero was ignored. These have an effect. We can adjust our reconstruction operator Ω to include and adjust for
T_{2}*(x,y) and ΔB(x,y). Subject specific T_{2}*(x,y) and ΔB(x,y) maps can be estimated and utilized in reconstruction. A version of personalized medicine. Our new reconstruction operator is Ω_{a} which is an inverse Fourier transform image reconstruction operator that incorporates estimates of T_{2}*(x,y) and ΔB(x,y).
Figure 15






Ω

Ω_{a}(T_{2}*)

Ω_{a}(ΔB)

Now after kspace preprocessing, we reconstruct with Ω_{a}. Some researchers like to perform image processing such as image smoothing (blurring). Image operations such as smoothing can be performed with a matrix operator O_{I}. An example of image smoothing through a matrix operation S_{m} upon a vector of image data is presented in Figure 16 for an 8×8 image.
Figure 16


Now the complete set of operations done on originally measured data are
O=O_{I}ΩO_{k}. With multivariate statistics we can see how the operations O changes our mean image vector and image vector covariance matrix. If the original kspace measurements have a mean of f_{0} and a covariance matrix of Γ=I, then the new mean is μ=Of_{0} and the new covariance is Σ=OO'. If the preprocessing operations matrices are not orthogonal, then they induce a correlation between voxels. This induced corrleation should be taken into account in fCMRI and fMRI.
Taking a closer look at the new covariance matrix,
Σ=(I_{2}⊗S_{m})
Ω_{a}
A
F
H
(P_{R}^{1}
Ω_{row}^{1}
Φ
Ω_{row}
P_{R}
)(P_{C}RC
)(C'R'P_{C}'
)(P_{R}'
Ω_{row}'
Φ'
Ω_{row}^{1}'
P_{R}^{1}')
H'
F'
A'
Ω_{a}'
(I_{2}⊗S_{m}').
It is well known that
CC'=I,
RR'=I, and
P_{C}P_{C}'=I, so the covariance matrix is
Σ
=
(I_{2}⊗S_{m})
Ω_{a}
A
F
H
(P_{R}^{1}
Ω_{row}^{1}
Φ
Ω_{row}
P_{R}
)(P_{R}'
Ω_{row}'
Φ'
Ω_{row}^{1}'
P_{R}^{1}')
H'
F'
A'
Ω_{a}'
(I_{2}⊗S_{m}').
The operations that censor turn around points, reverse alternating lines, and permute
from realimaginary for each kspace point
acquired to all reals then all imaginaries do not induce a correlation.
It is also well known that
P_{R}P_{R}'=I, and that
Ω_{row}Ω_{row}'=I, thus the first place that correlation could be induced is
ΦΦ'. However, it can be shown that ΦΦ'=I and thus the covariance matrix is
Σ
=
(I_{2}⊗S_{m})
Ω_{a}
A
F
H
H'
F'
A'
Ω_{a}'
(I_{2}⊗S_{m}').
It can be shown that H
H'≠I when there is a partial kspace acquisition and a portion of kspace is interpolated.
It can be shown that F
F'≠I when kspace is zero filled.
It can be shown that A
A'≠I when kspace is apodized.
It can be shown that Ω_{a}Ω_{a}'≠I when T_{2}* and/or ΔB are incorporated in image reconstruction.
It can be shown that S_{m}S_{m}'≠I when images are smoothed.
So, it has been shown that when preprocessing is done to MRI data in kspace, when physical parameters are incorporated in image reconstruction, or when images are processed, a correlation between voxels is induced!
Correlation matrices are very large and difficult to interpret. To illustrate induced correlations,
the correlation induced between a voxel (the center one) and all others are presented. Imagine that
we have a 5×5 image as in Figure 17 with voxels numbered 125 from top left to bottom right.
Figure 17


The 5×5 image yields a 25×25 image correlation matrix for example as in Figure 18 top left.
The row within the correlation matrix corresponding to the voxel of interest (the center one) can be extracted
as in Figure 18 top right and partitioned as in Figure 18 bottom left then assembled as in Figure 18 bottom
right to produce a correlation image for that voxel. Figure 18 bottom left is the correlation between
the voxel of interest (the center one) and all other voxels in the image.
Figure 18
To illustrate the effects of the above operators, exact theoretical calculations
(not Monte Carlo simulations) are coconducted. Using the signal equation in
Equation 1 above, true noiseless exact theoretical
kspace data is generated with BW=250 kHz
Δt=4 μs, EES=0.96 ms, TE=50.0 ms
and the the true parameter maps for
proton spin density ρ(x,y), transverse relaxation T_{2}*(x,y), and differential from the main magnetic field ΔB(x,y)
given in Figure 19 (Nencka, Hahn, Rowe, 2009).
Figure 19






ρ(x,y)

T_{2}*(x,y)

ΔB(x,y)




Δt

H

A

Examples of induced correlation from select kspace and image space preprocessing operations are presented in Figure 20. The correlation for reconstruction with:
no processing (ideal case) is in row 1 and column 1;
homodyne partial kspace reconstruction is in row 1 and column 2;
3 mm FWHM image space smoothing in row 1 and column 3;
an unaccounted for delta B field is in row 1 and column 4;
an unaccounted for T2* decay is in row 2 and column 1;
homodyne partial kspace reconstruction and 3 mm FWHM image space smoothing is in row 2 col 2;
homodyne partial kspace reconstruction and an unaccounted for T2* decay is in row 2 col 3;
an unaccounted for delta B field and a 3 mm FWHM image space smoothing in row 2 and column 4;
an unaccounted for T2* decay and 3 mm FWHM image space smoothing in row 3 and column 1;
an unaccounted for T2* decay and an unaccounted for delta B field in row 3 and column 2;
an unaccounted for T2* decay, a homodyne partial kspace reconstruction, and 3 mm FWHM image space smoothing in row 3 and column 3;
an unaccounted for T2* decay, an unaccounted for delta B field, and 3 mm FWHM image space smoothing in row 3 and column 4.
Figure 20
Our new reconstruction operator is Ω_{a} corrects
the effects of subject specific
T_{2}*(x,y)
and ΔB(x,y).
However, the new operator may potentially induce
correlations in the imagespace as a result of
changing the properties of the data. The examples of the induced magnitudesquared correlation
from the reconstruction operators,
Ω, Ω_{a}
(T_{2}*),
and
Ω_{a}
(T_{2}* and ΔB)
are illustrated with a magnitude underlay
in Figure 21.
Figure 21



Ω

Ω_{a}(T_{2}*)

Ω_{a}(T_{2}*
and ΔB)

The AMMUST framework can be expanded to describe the SENSE parallel image
reconstruction model and its statistical ramifications (Bruce et al., 2011, Bruce et al., 2012).
Consider the spatial
frequencies F_{j},
from j=1,...,N_{C}
receiver coils, subsampled by a reduction factor R,
where every R^{th} line of
kspace is acquired in the phaseencoding direction.
To apply the AMMUST framework, each of the N_{C}
spatial frequency matrices are reshaped into vectors, f_{j},
by stacking all rows of the real spatial frequencies upon all rows of the imaginary spatial frequencies.
These vectors are in turn stacked into a single vector, f
=(f_{1},...,f_{NC})',
with alternating real and imaginary components from the N_{C}
receiver coils. The N_{C} subsampled vectors of
spatial frequencies are inverse Fourier transformed into aliased images in image space with
N_{C} applications of the above Ω
operator using the Kronecker product (I_{NC}⊗Ω).
In order to apply the SENSE unfolding operation in its matrix form, it is first necessary to
permute the aliased images such that they are organized by voxel rather than by coil.
The result of such a permutation, P_{C},
is a vector with the N_{C} real aliased
voxel values stacked upon the N_{C}
imaginary aliased voxel values for all voxels in the reduced fieldofview images. For even reduction factors,
it was found that a second permutation performing a Fourier transform shift,
P_{S}, is necessary such that the edges of the aliased
images are appropriately aligned with the center of kspace
for a centered unfolding. Based on the SENSE model, the vector of
N_{C} complexvalued aliased voxel values,
a_{C}, are derived from a vector of complexvalued
unaliased voxel values, ν_{C}, premultiplied by
an N_{C}×R
complexvalued sensitivity matrix, S_{C},
with additive coil noise ε_{C} as
a_{C}
=
S_{C}×
ν_{C} +
ε_{C}.
Since ε_{C} is approximated by the complex
normal distribution, with a mean of 0 and an N_{C}
×N_{C} complexvalued covariance between coils of
Ψ_{C}, it can be shown through a transformation
of variables that the unaliased voxel values are derived using a complexvalued weighted least squares estimation
ν_{C} =
(
S_{C}^{H}
Ψ_{C}^{1}
S_{C})^{1}
S_{C}^{H}
Ψ_{C}^{1}
a_{C}
where H denotes the transpose complex conjugate (Hermitian).
In applying this to the AMMUST framework, the above equation is represented in terms of its realvalued
isomorphism by
ν =
(
S^{T}
Ψ^{1}
S)^{1}
S^{T}
Ψ^{1}
a
=
U_{SE}
a
where a realvalued representation of the SENSE unfolding operation has been used.
It should be noted that the SENSE model relies upon the complex normal distribution
and as such requires Ψ to be skewsymmetric.
The SENSE unfolding matrix U_{SE}
is applied in the AMMUST framework as a block diagonal operator, where each block unfolds
the 2N_{C} ×1 concatenated
vectors of real aliased voxel values stacked upon imaginary voxel values in
a
=
P_{S}
P_{C}
(I_{NC}⊗Ω)
f
into 2R×1 vectors of unaliased real voxel
values stacked upon imaginary voxel values corresponding to the
R folds. After unfolding
each of the aliased voxels, a final permutation,
P_{U}, is applied to organize the unaliased
voxel values from being organized by voxel to by row, and ultimately by fold.
Thus, a final vector, y, with all real unaliased voxel values stacked upon all
imaginary voxel values is derived from the N_{C} subsampled spatial frequency vectors
in a complete set of operations as
y=
P_{U}
U_{SE}
P_{S}
P_{C}
(
I_{NC}⊗Ω)
f .
Additional preprocessing operations, O_{k},
or postprocessing operations, O_{I},
could be performed in kspace or image space respectively as
y=
O_{I}
P_{U}
U_{SE}
P_{S}
P_{C}
(
I_{NC}⊗Ω)
O_{k}
f .
Assuming an identity covariance structure in the acquired data, the covariance induced by the SENSE reconstruction process is thus
Σ
= O_{SE} O_{SE}^{T}
where
O_{SE} =
O_{I}
P_{U}
U_{SE}
P_{S}
P_{C}
(I_{NC}⊗Ω)
O_{k}.
The SENSE operators, O_{SE},
for a 9×9 acquisition with a reduction factor R=3 from
N_{C}=4 coils are illustrated in
Figure 22 below where the shift permutation is treated as identity due to an odd choice in reduction factor:
Figure 22


P_{S}



P_{U}

U_{SE}

P_{S}

P_{C}

(I_{NC}⊗Ω)

In the event of smoothing applied in imagespace, using a Gaussian kernel with a FWHM of 3 voxels,
the correlations induced about the center voxel bythe SENSE reconstruction
operators O_{SE} are illustrated in Figure 23.
Figure 23




SENSE Real

SENSE imaginary

SENSE real/imaginary

SENSE magnitudesquared

The SENSE correlations in Figure 23 are illustrated with a magnitude underlay.
It can be seen that the SENSE model results in correlations between the folds that are accentuated by processing operations such as smoothing.
Current and future work involves the representation of the GRAPPA parallel image reconstruction model with an isomorphism representation and the examination of induced correlation. We intend to compare induced correlations of the SENSE model to the induced correlations of the GRAPPA model.
In the same way that spatial processes were examined and induced spatial correlation determined,
we can examine temporal processes (slice timing correction, time series filtering, dynamic B_{0} feld correction,
rigid body motion correction) and induced temporal correlation can be determined.
If we denote the kspace vector for each image as
f_{t}, t
=1,....,n (where f_{t}
could be made up from multi coil kspace vectors).
Then, the kspace vectors can be stacked to form a single larger
kspace f
=(f_{1},...,f_{n})'
and the kspace processing operators O_{kt},
reconstruction operators Ω_{at}
and image preprocessing operators O_{It}
of all image kspace vectors,
t=1,...,n
can be represented simultaneously as
ν_{1}
⋮
ν_{n}

=

O_{I}1
ΩO_{a1}
O_{k1}
0
⋱
0
O_{In}
ΩO_{an}
O_{kn}

f_{1}
⋮
f_{n}

or
ν=
IRK where
K has block diagonal elements
O_{kt},
R has block diagonal elements Ω
_{at},
and
I has block diagonal elements
O_{It}.
Now that each image has been processed, the observations can be reordered using a permutation matrix,
P,
and then time series processing implemented on each time series as
O_{Tj}
where
j=1,...,
p denotes voxel number as
y_{1}
⋮
y_{n}

=

O_{T}1
0
⋱
0
O_{Tn}

P

O_{I1}
ΩO_{a1}
O_{k1}
0
⋱
0
O_{In}
ΩO_{an}
O_{kn}

f_{1}
⋮
f_{n}

or
O=TPIRK where
T has block diagonal elements
O_{Tj}.
We can now in an exact mathematical framework precisely represent spatiotemporal covariance
Σ
induced in the data as
TPIRKΓK'R'I'P'T' where
Γ is the
kspace covariance matrix. Even if
Γ=I, unless all of
KK'=I,
RR'=I,
II'=I,
and
TT'=I, a spatiotemporal correlation is induced. (It is already the case that
PP'=I.)
The spatiotemporal correlation induced by preprocessing needs to be taken into account in fMRI and fCMRI.
Activation
After the inverse Fourier transform, we obtain a complex valued measured object that consists of a true complex valued object plus complex
valued noise due to random noise, phase imperfections, and possible biophysical processes that produce phase signal variation. Even though the
magnitude of a complexvalued observation at time t is Ricean distributed, it can be approximated by the normal distribution at high SNRs.
The complexvalued image at time t measured over time t can be described with a nonlinear multiple regression model that includes both a
temporally varying magnitude ρ_{t} and a phase imperfection θ given by
y_{t} =
[ρ_{t}
cosθ+
η_{R}_{t}]+
i[ρ_{t}
sinθ+
η_{I}_{t}]
where
ρ_{t} =
x_{t}^{'}β =
β_{0}+
β_{1}x_{1t}+...+
β_{q}x_{qt},
(η_{Rt},η_{It},)^{'} = N(0,Σ), and
Σ = σ^{2}I_{2}.
x_{t}^{'} is the t^{th} row of an
n×(q+1) design matrix X, and
β is the magnitude regression coefficient vector.
The phase imperfection
θ is assumed to be fixed and unknown, but may be estimated voxelbyvoxel (Rowe and Logan, 2004; Rowe and Logan, 2005b).
This model can also be written to describe the observations at all time points simultaneously as
y =
[X 0 ; 0 X][βcosθ ; βsinθ] +
η.
The observed vector of data
y = (y_{R}^{'}, y_{I}^{'})^{'}
is the vector of observed real values stacked on the vector of observed imaginary values and the vector of errors
η = (η_{R}^{'}, η_{I}^{'})^{'}~N(0,Σ⊗Φ ).
It is generally assumed that
Σ = σ^{2}I_{2} and
Φ = I_{n}.
The likelihood of the complexvalued data model can then be written as
p(yX,β,θ,σ^{2}) =
(2πσ^{2})^{n}×exp{(1/σ^{2})
[y[Xβcosθ ; Xβsinθ]]^{'}×[y[Xβcosθ ; Xβsinθ]]}.
Model parameters can then be estimated under appropriately constrained null and alternative hypotheses,
H_{0}:Cβ=0 vs.
H_{1}:Cβ≠0, then activation can be determined with a generalized likelihood ratio statistic. For example,
with a model with
β_{0} representing an intercept,
β_{1} representing
a linear drift over time, and
β_{2} representing a contrast effect of a stimulus, in order to test whether the coefficient for the reference
function or stimulus is 0, the contrast vector is set to be
C= (0,0,1), so that the hypothesis is
H_{0}:β_{2}=0.
Unrestricted maximum likelihood estimates (MLEs) of the parameters can be derived (Rowe and Logan, 2004) to be
θ^{^} = (1/2)tan^{1}

2β^{^}_{R}^{'}(X^{'}X)β^{^}_{I}^{'}

β^{^}_{R}^{'}(X^{'}X)β^{^}_{R}^{'}β^{^}_{I}^{'}(X^{'}X)β^{^}_{I}^{'}

β^{^} = β^{^}_{R}cosθ^{^}+β^{^}_{I}sinθ^{^}
σ^{^}^{2} = (1/2n)[y[Xβ^{^}cosθ^{^} ; Xβ^{^}sinθ^{^}]]^{'}×[y[Xβ^{^}cosθ^{^} ; Xβ^{^}sinθ^{^}]]
where β^{^}_{R} = (X^{'}X)^{1}X^{'}y_{R} and
β^{^}_{I} = (X^{'}X)^{1}X^{'}y_{I}.
The restricted MLEs can also be derived (Rowe and Logan, 2004) to be
θ^{~} = (1/2)tan^{1}

2β^{^}_{R}^{'}Ψ(X^{'}X)β^{^}_{I}^{'}

β^{^}_{R}^{'}Ψ(X^{'}X)β^{^}_{R}^{'}β^{^}_{I}^{'}Ψ(X^{'}X)β^{^}_{I}^{'}

β^{~} = Ψ[β^{^}_{R}cosθ^{~}+β^{^}_{I}sinθ^{~}]
σ^{~}^{2} = (1/2n)[y[Xβ^{~}cosθ^{~} ; Xβ^{~}sinθ^{~}]]^{'}×[y[Xβ^{~}cosθ^{~} ; Xβ^{~}sinθ^{~}]]
where Ψ is
Ψ = I_{q+1}(X^{'}X)^{1}C^{'}[C(X^{'}X)^{1}C^{'}]^{1}C.
Denoting the maximum likelihood estimators under the alternative hypothesis using hats, and those under the null hypothesis using tildes, the generalized
likelihood ratio statistics can be derived as,
2logλ=2nlog(σ^{~}^{2}/σ^{^}^{2}).
This statistic has an asymptotic χ_{r}^{2} distribution in large samples, where
r is the difference in the number of constraints between the alternative and
the null hypotheses or the full row rank of C. Note that, when r=1,
twosided testing can be performed using the signed likelihood ratio test given by
Z = sign(Cβ^{^,})sqrt(2logλ)
which has an approximate standard normal distribution under the null hypothesis (Rowe and Logan, 2004; Severini, 2001).
Thresholding
Something about thresholding of activation maps from complexvalued time series analysis here that incorporates
the induced/modified correlation from preprocessing.
Logan and Rowe 2004, Logan, Geliazkova, and Rowe, 2008; and complexvalued permutation resampling as mentioned in Rowe and Logan 2004.
To be continued..... Blah, Blah, blah.
Future Efforts
Stuff.
Bruce IP, Karaman MM, ROWE DB. A statistical examination of SENSE image reconstruction via an isomorphism representation. Magn. Reson. Imaging, 29(9):12671287, (2011). (Contribution: Extends the AMMUSTk framework to describe SENSE image reconstruction. Significance: Can precisely quantify the induced correlation among folds in subsampled kspace data including preprocessing.)
Bruce IP, Karaman MM, ROWE DB. The SENSEIsomorphism Theoretical Image Voxel Estimation (SENSEITIVE) Model for Reconstruction and Observing Statistical Properties of Reconstruction Operators (Status: Conditional Acceptance, Magn. Reson. Imaging), 2012. (Contribution: The SENSE model assumes a skewsymmetric covariance matrix. The SENSEITIVE model assumes a symmetric coil covariance matrix and introduces a voxel covariance matrix. Utilized a phase plane adjustment that eliminates ancillary correlation seen in Bruce (2011) due to gradient drift.)
Hahn AD, Nencka AS, Rowe DB. Dynamic compensation of B0 field inhomogeneities restores complex fMRI time series activation power. Proc. Intl. Soc. Magn. Reson. Med. 16:1251 (2008). (Contribution: Estimate and adjust for dynamic magnetic field changes. Significance: Removes nearly all temporally varying ancillary signals such as respiration to reduce the residual variance of fMRI activation models that use phase information. Allows statistical activation models that utilize phase information to be used for the extraction of more biological information and for increased detection power.)
Hahn AD, Nencka AS, Rowe DB. Improving robustness and reliability of phasesensitive fMRI analysis using temporal offresonance alignment of singleecho timeseries (TOAST). Neuroimage 44:742752 (2009a). (Contribution: Formal model for the estimation and adjustment for dynamic magnetic field changes both in simulated and real data. Significance: Removes nearly all temporally varying ancillary signals such as respiration to reduce the residual variance of fMRI activation models that use phase information. Allows complexvalued statistical activation models that utilize phase information to be used for the extraction of more biological information and for increased detection power.)
Hahn AD, Nencka AS, Rowe DB. Dynamic magnetic field corrections improve phaseonly fMRI activations. Proc. Intl. Soc. Magn. Reson. Med., 17:2789 (2009b). (Contribution: Applies dynamic field correction of Hahn, Nencka and Rowe 2009a to phase only functional activation analysis. Significance: Statistically significant detection of phase functional response, absent in raw data, can be found after removal of ancillary temporal signal such as respiration using dynamic field correction. Phase signal of interest can be separated intact for analysis from undesirable ancillary signals.)
Hahn AD, Nencka AS, Rowe DB: Enhancing the utility of complexvalued fMRI detection of neurobiological processes through postacquisition estimation and correction of dynamic B0 errors and motion. To appear, Hum. Brain. Mapp., (2011). (Contribution: Thorough examination of complexvalued fMRI activation with or without TOAST dynamic magnetic field correction for geometric distortion and motion regressors for rigid body displacement. Significance: Reliably demonstrates utility of Rowe's complexvalued activation with correction for dynamic magnetic field fluctuations in conjunction with estimated motion parameters.)
Hahn AD, Rowe DB. Methodology for robust motion correction of complexvalued fMRI time series. Proc. Intl. Soc. Magn. Reson. Med. 18:3051, Stockholm, Sweden, (2010). (Contribution:Framework for effective motion correction through image registration of complexvalued image time series. Significance: Eliminates temporally static and dynamic background phase and addresses the problem of interpolation of vector valued images when applying motion correction to complexvalued image time series through image registration. Allows improved motion correction for artifact removal prior to complexvalued analysis.)
HernandezGarcia L, Vazquez AL, Rowe DB. Complexvalued analysis of arterial spin labeling based FMRI signals. Magn. Reson. Med. 62(6):15971608, 2009. (Contribution: Applies the Rowe 2005b magnitude and phase activation . Significance: A simulation study indicated that the complexvalued activation model exhibits combined magnitude and phase detection power and thus maximizes sensitivity under ideal conditions. This suggests that, as arterial spin labeling imaging and image correction methods develop, the complexvalued detection model may become helpful in signal detection..)
Logan BR, Geliazkova MP, Rowe DB. An evaluation of spatial thresholding techniques in fMRI analysis. Hum. Brain Mapp. 29:13791389 (2008). (Contribution: Operating characteristics and properties of thresholding methods that utilize the activation status of neighboring voxels are examined. Significance: The Bayesian spatial mixture model performs optimally among the thresholding methods that were considered.)
Logan BR, Rowe DB. An evaluation of thresholding techniques in fMRI analysis. Neuroimage 22:95108 (2004). (Contribution: Evaluated the operating characteristics of multiple comparison voxel thresholding methods. Significance: In practice computationally intensive permutation resampling methods that account for spatial correlation do not need to be utilized.)
Nencka AS, Hahn AD, Rowe DB. A mathematical model for understanding the statistical (AMMUSTk) effects of kspace preprocessing on observed voxel measurements in fcMRI and fMRI. J. Neurosci. Meth. 181:268282 (2009). (Contribution: Extends and examines previous work (Nencka and Rowe, 2008) to correlation between magnitude time series to characterize voxel correlation induced by the multitude of preprocessing. Significance: The prereconstruction premagnitude time series formation preprocessing adjustments produce or modify the spatial correlation between voxels. The true correlation (connectivity) between voxels may be greater than or less than previously thought.)
Nencka AS, ROWE DB: The use of Three Navigator Echoes in Cartesian EPI Reconstruction Reduces Nyquist Ghosting. Proc. Intl. Soc. Magn. Reson. Med., 16:3032, 2008. (Contribution: Introduces a method of estimating the kspace line shift to eliminate Nyquist ghosting from three already acquired navigator echos. Significance:Line shift can be estimated without the acquisition of additional data.)
Nencka AS, Hahn AD, Rowe DB. Redundant spatial harmonic information in zeugmatography with linear encoding (RSHIZLE) theoretically encodes intraacquisition decay. Proc. Intl. Soc. Magn. Reson. Med. 16:3157 (2008). (Contribution: Introduces a method of estimating a T2* or T2 map within a single EPI acquisition, even under the circumstance of nonnegligible magnetic field inhomogeneity. Significance: Provides theoretical work for making quantitative measurements of T2* or T2 from a single image acquisition following a single excitation pulse by utilizing the expected symmetry of kspace observations.)
Nencka AS, Rowe DB. Apodization and smoothing alter voxel time series correlations. Proc. Intl. Soc. Magn. Reson. Med. 16:2457 (2008). (Contribution: Uses relationship between complexvalued kspace and voxel measurements to characterize the correlation between voxels induced by the consistent temporal application of apodization and smoothing of kspace data. Significance: This consistently added spatial correlation over a time series leads to correlation between voxel time series and thus can affect connectivity measurements. The effects of apodization on connectivity measurements are shown to be nonnegligible.)
Nencka AS, Rowe DB. Image space correlations induced by kspace processes. Proc. Org. Hum. Brain Mapp. S55:284 (2007b). (Contribution: Uses relationship between complexvalued kspace and complexvalued voxel measurements to characterize voxel correlation induced by preprocessing. Significance: These adjustments produce or modify the spatial correlation between voxel measurements. Possible kspace adjustments include the shifting of alternating lines to correct the signal (but also the noise) to eliminate ghosting, apodization of kspace measurements, and partial kspace acquisition since many spatial frequencies are identically the same numbers. The true correlation (connectivity) between voxels may be less than (or greater than) previously thought.)
Nencka AS, Rowe DB. Reducing the unwanted draining vein BOLD contribution in fMRI with statistical postprocessing methods. Neuroimage 37:177188 (2007a). (Contribution: Developed Monte Carlo simulations and examined Human echo planar imaging data with two activation methods. Significance: We found that the complexvalued model (Rowe and Logan, 2005) exhibits a strong bias against detecting magnitude signal changes in voxels that have task related phase changes (characteristic of unwanted signal from draining veins). Thus the complex model yields grey matter voxels that are a subset of those from the magnitudeonly model.)
Rowe DB. fMRI activation in image space from kspace data. Proc. Org. Hum. Brain Mapp. S114:377 (2007a). (Contribution: Computes brain activation in image space in terms of kspace measurements. Significance: 1) Activation is one step closer to the original data. 2) The relationship between kspace measurements and correlation can be incorporated instead of trying to model correlation between voxels that are more complicatedly related.)
Rowe DB. fMRI statistical brain activation from kspace data. Proc. Am. Stat. Assoc. (Biometrics Section) 12:107114 (2007b). (Contribution: Computes brain activation in image space in terms of kspace measurements. The correlation between voxel measurements can also be written in terms of correlation between kspace measurements. Significance: 1) Activation and association is one step closer to the original data. 2) The relationship between kspace measurements and correlation can be incorporated instead of trying to model correlation between voxels that are more complicatedly related.)
Rowe DB. Magnitude and phase signal detection in complexvalued fMRI data. Magn. Reson. Med. 62:1356–1357 2009.(Contribution: Examines the computation of magnitude and phase activation from complexvalued data. Significance: In special limited cases, magnitude and phase activation can be computed using a closed form solution but in useful cases it can not and the only mathematically correct model is that in Rowe 2005b.)
Rowe DB. Modeling both the magnitude and phase of complexvalued fMRI data. Neuroimage 25:13101324 (2005b). (Contribution: Developed a more general fMRI model that simultaneously describes both the magnitude and phase of complexvalued fMRI data, thus allowing the observed data to be fully utilized in answering important biological questions (Rowe, 2005b). Significance: 1) Can determine signal changes corresponding to true activation close to the activation site via blood oxygenation to the highlylocalized capillary bed. 2) These activation maps have drastically reduced contamination by unwanted draining veins carrying away the blood for long distances from the activation site that also exhibit task related phase changes (TRPCs).)
Rowe DB. Parameter estimation in the complex fMRI model. Neuroimage 25:11241132 (2005a). (Contribution: Two models were evaluated in terms of parameter estimation and brain activation statistics (Rowe, 2005a). Significance: 1) Showed that the unrestricted phase or magnitudeonly parameter estimates become increasingly biased as the SNR decreases whereas the complexvalued model is unbiased at all SNR levels. 2) The parameter estimates achieved their CramerRao variance lower bound for the complexvalued model regardless of SNR while the magnitudeonly model did not. 3) The complexvalued activation statistic was uniformly higher than the magnitudeonly model.)
Rowe DB, Bruce IP. Processing Induced Voxel Correlation in SENSE FMRI Via the AMMUST Framework. Proc. Second Biennial International Conference on Resting State Connectivity, Medical College of Wisconsin, Milwaukee, Wisconsin, F052 (2010). http://www.restingstate.com/. (Contribution: Previous work (Nencka et al., 2009) that theoretically describes induced correlation between image voxels from spatial preprocessing and reconstruction operations has been summarized and extended to include the SENSE multi coil image reconstruction method. Significance: This has null hypothesis fcMRI connectivity implications as the no connectivity scenario is not for no spatial correlation but is rather for the spatial correlation induced by preprocessing.)
Rowe DB, Hahn AD, Nencka AS. Functional magnetic resonance imaging brain activation directly from kspace. Magn. Reson. Imaging 27:13701381, 2009. (Contribution: In Rowe, Nencka, Hoffmann it was shown that complexvalued voxel measurements can be written as a the complexvalued kspace measurements that have been premultiplied by a reconstruction matrix. A framework is developed so that statistical analysis (brain activation) is computed in terms of the original, prereconstruction, complexvalued kspace measurements. Significance: This allows one to utilize the originally measured data in its more natural, acquired state rather than in a transformed state. The effects of modeling preprocessing in kspace on voxel activation and correlation can then be examined. .)
Rowe DB, Logan BR. A complex way to compute fMRI activation. Neuroimage 23:10781092 (2004). (Contribution: Determined magnitude fMRI activation in complexvalued voxel time series data while specifying the traditionally believed voxelwise temporally constant but spatially varying phase. Significance: 1) Uses the correct thermal noise statistical distribution of bivariate normality with phase coupled means and not the incorrect normal assumption for the magnitudes. 2) Possesses increased detection power at all SNRs by inclusion of all 2n realimaginary observations instead of n magnitude quantities. 3) Produces more highlyfocused activation regions that are better localized to grey matter which is exactly where firing neurons and functional activation should be localized.)
Rowe DB, Logan BR. Complex fMRI analysis with unrestricted phase is equivalent to a magnitudeonly model. Neuroimage 24:603606 (2005). (Contribution: Outlined a more general magnitude fMRI activation based on complexvalued data that assumed unrestricted or unique temporal phase values. Significance: 1) Derived the same regression coefficients and activation statistics as used for the usual magnitudeonly data model. 2) By deriving these statistics we now understand the complexvalued assumptions inherent in the commonly used magnitudeonly fMRI activation model.)
Rowe DB, Meller CP, Hoffmann RG. Characterizing phaseonly fMRI data with an angular regression model. J. Neurosci. Methods 161:331341 (2007). (Contribution: Phaseonly fMRI activation is examined using an angular regression model, linear independent variable (x) and angular dependent variable (y). Significance: 1) No longer need to unwrap phase time series. 2) Accurate regression coefficient and variance estimates.)
Rowe DB, Nencka AS, Hoffmann RG. Signal and noise of Fourier reconstructed fMRI data. J. Neurosci. Methods 159:361369 (2007). (Contribution: Related measured complexvalued kspace spatial frequencies and complexvalued images. Significance: 1) This allows the computation of fMRI brain activation directly from unreconstructed originally observed kspace measurements (Rowe, 2007). 2) This allows modeling of covariation between the originally acquired data in its more natural setting rather than a very nonintuitive complicated transformed version between voxels.)
Zhu H, Li Y, Ibrahim JG, Shi X, An H, Chen Y, Lin W, Rowe DB, Peterson BS. Rician regression models for magnetic resonance images. J. Am. Stat. Assoc. 104:623637 (2009). (Contribution:Introduce a more precise magnitudeonly Rician regression model to characterize background noise to properly estimate model parameters in various MRI techniques such as diffusion weighted images and functional MRI. Significance: Model parameters are able to be accurately estimated for lower SNRs.)
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