Marquette University Wisconsin - Dr Daniel Rowe
 
 
 

Professor of Computational Sciences
Head of Functional Magnetic Resonance Image Analysis Lab
coDirector of Bioinformatics Masters of Science
Department of Mathematics, Statistics, and Computer Science
Marquette University

313 Cudahy Hall,
1313 W. Wisconsin Ave.
Milwaukee, WI 53233

Phone: 414-288-5228
Fax:  414-288-5472
E-mail: daniel.rowe{at}marquette.edu

Applications are now being accepted for:
MU Computational Sciences PhD Program.
MU Bioinformatics MS Program.

Adjunct Professor of Biophysics
Faculty of the Center for Imaging Research
Department of Biophysics
Medical College of Wisconsin

2062 MAAC Fund Research Center
8701 Watertown Plank Road
Milwaukee, WI 53226-0509

Phone: 414-456-4027
Fax:  414-456-6512
E-mail: dbrowe{at}mcw.edu

Adjunct Professor in Electrical Engineering
Department of Electrical Engineering and Computer Science
Engineering and Mathematical Sciences Building
3200 North Cramer St
University of Wisconsin-Milwaukee
Milwaukee, WI 53201-0413
E-mail: drowe{at}uwm.edu

 

Functional Magnetic Resonance Image Analysis Lab


  Don't change the data to fit the model, change the model to fit the data. -- DB Rowe

Lab Members

Current
Daniel B. Rowe, PhD
Laboratory Head 
Associate Professor
Andrew S. Nencka, PhD
Assistant Professor, MCW 
Iain P. Bruce, PhD
Postdoc, Duke 
M. Muge Karaman, MS
Postdoc, UIC
Mustafa Farrah
Doctoral Candidate, UWM
Mary C. Kociuba, MS
Doctoral Student, MU
Emily M. Paulson, BS
Doctoral Student, MU
Former
Andrew D. Hahn, PhD
Postdoctoral Fellow, UWisc 

complex-valued time series, Rowe 2005

activation map, Hahn et al. 2009


Click on a heading to expand the text.
+ Research Description

Background

The long-term goal of the Rowe lab research efforts is to develop a unified mathematical model for functional magnetic resonance imaging (fMRI). This mathematical model is to include the fundamental physics of the nuclear magnetic resonance signal, the compensation for biological signals not of interest, the preprocessing of the MR images, the statistical modeling of complex-valued time series, and the statistically significant determination of focal brain activation. This mathematical model will extract the most information from the acquired data in the most efficient way possible. This unified model will contain important physiological information that may not be available by other means or is only available by more time-consuming elaborate means. This model will allow us to address important fundamental neuroscience questions with fMRI. The Rowe lab's primary research efforts are focused on working deeper in the data acquisition and processing stream while expanding the unified model at each step.

Accomplishments

The fMRI technical process can be categorized into five main research areas: hardware improvements, pulse sequence development, preprocessing and image reconstruction, time-series modeling to determine activation, and activation-map thresholding. The efforts of the Rowe lab have involved working deeper into the processing stream.

Thresholding

Initial and recent research efforts have examined the thresholding of statistical activation maps. Thresholding can be localized to utilize only a voxel's activation level (Logan & Rowe, 2004) or utilize neighboring voxel activation levels since activations generally occur in clusters (Logan, Geliazkova, & Rowe, 2008). From statistical modeling, each voxel will have a statistical measure of association between its observed time series and the ideal experimental design. Thresholding, while accounting for multiple comparisons, allows the objective statistical separation of voxels that contain signal and noise (active) and those that contain noise (not active). Thresholding is important for the neurological interpretation of fMRI results.

Activation

Further efforts have been aimed at a complete model of the true complex-valued (magnitude-phase or real-imaginary) voxel time series to produce statistical activation maps (Rowe & Logan, 2004; Rowe & Logan, 2005; Rowe, 2005a; Rowe, 2005b; Rowe, 2009) in contrast to the commonly used magnitude-only model. In fMRI models, the phase time series (half of numbers) are generally discarded, and only the magnitude half of the numbers used. Additionally, phase-only activation has been examined with an angular regression model (Rowe, Meller, & Hoffmann, 2007) as has more precise magnitude-only activation (Rowe, 2005a; Zhu et al., 2009). These models yield statistical activation maps that can be thresholded with the methods examined in the Rowe lab. The mathematics for Dr. Rowe’s activation models convincingly demonstrate that more biological information, such as vascularity within voxels (and potentially direct magnetic field change due to neuronal current), can be extracted with the use of the phase portion of the complex-valued time series (Nencka & Rowe, 2007a) and that increased statistical detection power can be achieved (Rowe, 2005a). The Rowe 2005b magnitude and-or phase activation model was applied to arterial spin labeling (ASL) experiments (Hernandez, Vazquez, Rowe 2009).

Reconstruction Isomorphism for Images

Continuing research efforts have connected complex-valued voxel measurements to the original pre-reconstruction (inverse Fourier transformation) complex-valued k-space (spatial frequency) measurements (Rowe, Nencka, & Hoffmann, 2007). The array of complex-valued k-space measurements can be written as a vector of reals of rows over imaginaries of rows then the inverse Fourier transform reconstruction process represented as a pre-multiplication of the k-space vector by an inverse Fourier transform matrix. With this representation, voxel measurements can be written as a linear combination of k-space measurements. This relationship has allowed the connection of Dr. Rowe’s complex-valued time series activation models to k-space and thus has led to a brain activation model that determines statistical activation in terms of the original complex-valued k-space (spatial frequency) measurements (Rowe, 2007a; Rowe, 2007b; Rowe, 2009).

Spatial Preprocessing of Images

With the above linear relationship between complex-valued k-space and voxel measurements, it has made possible the examination of modified voxel means and variances in addition to induced correlation between voxels due to pre-reconstruction preprocessing methods (Nencka & Rowe, 2007b; Nencka & Rowe, 2008; Nencka, Hahn, & Rowe, 2009). This will allow the examination of preprocessing steps and settings to optimize information extraction. It has been found that k-space preprocessing changes the underlying voxel properties (mean, variance, covariance, correlation), and this needs to be accounted for in any statistical model. Apodization (k-space smoothing) and image smoothing with a Gaussian kernel induces a local Gaussian corelation between voxels. Zero-filling of k-space induces a local sinc correlation between voxels. Other procesing has been examined including a series of processes.

Magnetic B-Field Correction of images

Challenges have existed experimentally with humans (not with phantoms) to robustly demonstrate an improvement with the activation models that Dr. Rowe have developed. However, these challenges have recently been overcome with the correction of magnetic field inhomogeneities through time to estimate and account for the physiological signal that is not of interest, while preserving signal variation related to stimulated functional response (Hahn, Nencka, & Rowe, 2008; Hahn, Nencka, & Rowe, 2009a; Hahn, Nencka, & Rowe, 2009b). In humans, ancillary temporally varying physiological processes such as respiration produce dynamic magnetic field changes that manifest as temporally varying signals that are not of interest (and geometric warping of images due to improper Fourier encoding). These physiologic signals that are not of interest manifest largely in the phase portion of the complex-valued time series and are very difficult to mathematically model. The unmodeled physiologic signals increase the models’ residual variance (while leaving the estimated signal of interest nearly unchanged) and decrease detection power. Preliminary results demonstrate that the estimation and adjustment for the dynamic magnetic field changes accounts for nearly all of the ancillary signals (and geometrically unwarps the images), leaving the signal variation unrelated to these processes, and of potential interest, unchanged. This allows Dr. Rowe’s statistical activation models to be utilized for the extraction of more biological information such as the vascularity within voxels (and potentially direct magnetic field change due to neuronal current) in addition to increased detection power. For fMRI to address fundamental neuroscience questions such as the diagnosis of human brain disease and the precise localization of brain function for clinical presurgical mapping, the signal needs to be considered in light of the advances in the previously described research.

Motion and Physiologic Correction of Complex-Valued Images

Bulk or rigid body subject motion is a common problem faced in fMRI experiments, and is generally accounted for with magnitude image registration. However, similar motion correction prior to analysis using Dr. Rowe’s statistical model requires complex-valued image registration. Recent work has shown that this is a more difficult process than typical magnitude registation, particularly due to background phase and challenges associated with the interpolation of vector valued images (Hahn & Rowe, 2010). A general framework has been proposed, utilizing the previously mentioned dynamic field correction technique to overcome the difficulties. It has been shown that motion parameter regression along with dynamic magnetic field compensation improves complex-valued activation (Hahn, Nencka, & Rowe, 2011). In addition to including motion regressors in the analysis, a complex-valued RETROICOR type analysis of regressing out signal from physiologic processes such as respiration from a bellows belt and cardiac signal from a pulse-ox monitor has shown an improvement in reducing ancillary signals (Hahn & Rowe, 2012). Effective minimization of motion and physiology related artifacts in complex-valued time series should further improve the utility of Dr. Rowe’s statistical models for biophysical study.

Multi Coil Reconstruction of Complex-Valued Images

In MRI, the parallel acquisition of sub-sampled spatial frequencies from an array of multiple receiver coils has become a common means of reducing data acquisition time. This is an important topic that recent efforts have been devoted to. Preliminary results demonstrate that there is an artificial correlation induced between reconstructed voxels from the different folds of aliased images (Rowe & Bruce, 2010; Bruce, Karaman, Rowe, 2011 and 2012).

Future Efforts

Current work involves a mathematical description of the effects of time-series preprocessing on voxel correlations, motion correction, and the examination of auxillary regressors in fMRI. The Rowe lab is also interested in statistical image reconstruction of multicoil parallel imaging data. Of interest is extending the brain activation model to include and estimate both the proton spin density and the transverse relaxation (Nencka, Hahn, & Rowe, 2008). Estimating proton spin density and transverse relaxation will allow the extraction of more biological information within voxels in terms of tangible physiologic parameters. This may allow the extraction of sub-voxel information such as vascularity or possibly direct detection of neuronal firing via local magnetic field changes. Future plans include the examination of parallel imaging methods and the linking of them to the aforementioned body of work. The Rowe lab’s research efforts involve the theoretical development of new mathematical methods, their statistical characterization by computer simulation, and validation by human experiments. This will enable better understanding of brain function and better understanding of how it is affected by neurodegenerative diseases, mental illnesses, and brain injuries.

+  Research Details

In fMRI, the signal equation s(kx,ky|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 T2*(x,y), and differential from the main magnetic field ΔB(x,y)

Equation 1
s(kx,ky|t)= ∫∫ ρ(x,y) e-t/T2*(x,y) e-ΔB(x,y)t e-i2π(kxx+kyy) dxdy

where

kx= γ 0t Gx(t')dt'

and

ky= γ 0t Gy(t')dt'

are known spatial frequencies because we know the magnetic field gradients Gx(t') and Gy(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), T2*(x,y), or ΔB(x,y).

The effects of the transverse relaxation T2*(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-ΔB(x,y)t

and thus the usual Fourier relationship in Equation 2

Equation 2
s(kx,ky|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, Gx=0 and Gy=0, indicating that kx=0 and that Gy=0 and we are in the center of k-space in Figure 1 (center). When Gy is turned on positive, we move up to the uppermost value of ky-space (ky=3 in this example) and when Gx is turned on negative we move to the leftmost value for kx-space (kx=-3 in this example). Then we turn on Gx positive which takes us from left to right on the uppermost line of ky-space (from kx=-3 to kx=3 for ky=3 in this example). We then turn on ky negative which takes us down one line in the ky= direction (from ky=3 to ky=2 in this example) and also turn on ky negative to take us fron right to left. This process is repeated until all left right lines of k-space are covered all the while the ADCs are on sampling k-space at time intervals Δt. The k-space sampling locations are denoted by filled black circles in Figure 1 (center). At each sampled k-space location, the measurement is complex-valied (real and imaginary measurements). The complex-valued (real and imaginary) sampled k-space 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.

Figure 1
pulse sequence k-space points k-space points strung out

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 complex-valued 2D k-space array using the complex-valued 2D inverse Fourier transform. The 2D forward and 2D inverse transform processes can be represented as complex-valued matrix multiplications. In Figure 2, it is seen that we can take a complex-valued image (as in the second column), pre-multiply it by a complex-valued forward Fourier matrix (as in the first column) and post-multiply it by the transpose of a complex-valued forward Fourier matrix (as in the third column), to obtain the complex-valued k-space (spatial frequency) representation of it (as seen in the fourth column).

Figure 2
omega bar real image real omega bar real k space real
                    +i ×                     +i ×                     +i =                     +i
omega bar imag image imag omega bar imag k-space imag

In Figure 3 it is seen we can take the complex-valued k-space data (as in the second column), pre-multiply it by a complex-valued inverse Fourier matrix (as in the first column) and post-multiply it by the transpose of a complex-valued inverse Fourier matrix (as in the third column), to obtain the complex-valued image (as seen in the fourth column).

Figure 3
omegay real k-space real omegax real image real
                    +i ×                     +i ×                     +i =                     +i
omegay imag k-space imag omegax imag image imag
                Ωy ×                 F ×                 Ωx' =                 Y

Instead of reconstructing an image from the k-space data by pre- and post-multiplying by complex-valued inverse Fourier matrices, we can use an alternative but equivalent procedure involving a real-valued isomorphism (equivalent representation) for the complex-valued procedure. We can stack the rows of the real part of the k-space data (spatial frequencies) on top of the imaginary part of the k-space data (spatial frequencies) and form a single vector. Then the inverse Fourier transform image reconstruction procedure can be equivalently be represented as a matrix pre-multiplication 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 pre-multiplications O which will allow us to keep track of the change in mean Of0 and covariance OO' (Nencka et al., 2009).

With the matrix of complex-valued k-space data represented as a single fector that is rows of reals upon rows of imaginaries as in Figure 4 (right), it can be pre-multiplied 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.

Figure 4
image vector = omega × k-space vector
        ρ =                                       Ω ×         f

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 complex-valued image in Figure 3 (fourth column).

Returning to Figure 1 (right), there are complex-valued data points (the turn around points) that need to be censored (removed). We can censor these points as in Figure 5 by pre-multiplying the complex-valued 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.

Figure 5
= censor mat × k-space vector strung
=                             C ×        f

We now need to pre-multiply by a row-reveral permutation matrix to flip alternating rows. A permutation matrix is a matrix of ones and zeros that results in a reordering.

Figure 6
= row rev mat × censor mat × k-space vector strung
=                           R ×                             C ×         f

We now have only the k-space data (real and imaginary values for each point) that we need and going in the proper direction. We now need to order the k-space 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
k-space vector = perm to RI × row rev mat × censor mat × k-space vector strung
         f =                       PC ×                           R ×                             C ×         f

We now have the k-space vector from Figure 4 (right). The operations of censoring, reversing rows, and reordering using C, R, and PC are simply book keeping. Image reconstruction can proceed as in Figure 4 with the real-valued isomorphism (Rowe et al., 2007). In general, there are many pre-processing steps applied to the data. We can represent linear preprocessing in k-space and also in image space as matrix multiplications (Nencka et al., 2009).

The first preprocessing step is shifting odd and even lines of k-space to eliminate Nyquist ghosting. This can be described as in Equation 3 with an additional term in the signal equation s(kx,ky|t)

Equation 3
s(kx,ky|t)= ∫∫ ρ(x,y) e-t/T2*(x,y) e-ΔB(x,y)t e-i(-1)l2π Δkxx e-i2π(kxx+kyy)dxdy

where l represents the line number, Δkx 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 k-space line shifting Nyquist ghost correction (Nencka and Rowe, 2008) is presented in Figure 8.

Figure 8
before k-space line shifting after

The Nyquist ghost correction can be implemented when an estimate of Δkx is available by permuting the k-space 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
  PR-1 × Ωrow-1 × shift matrix × FT rows × perm rows × k-space vector
  PR-1 × Ωrow-1 ×                       Φ ×                       Ωrow ×                       PR ×        f

If there is no line shifting, then Φ=I and PR-1Ωrow-1ΦΩrowPR =I and does nothing. Recall that f=PC×R ×C×f  as in Figure 7.

Since it takes time to measure the complex-valued k-space points, often a portion is not measured to save time and the missed data interpolated. Conjugate symmetry ideally exists about the origin in k-space, as the reconstructed image is expected to be real-valued. This symmetry allows half of k-space to be generated without being acquired. This is called partial or sometimes half k-space and a Homodyne interpolation is performed. Figure 11 shows Homodyne interpolation of k-space array acquired in Figure 10 (left) with 1 overscan line to "full" k-space array in Figure 10 (right).

Figure 10
partial k-space real
 
 
 
 
 
Homodyne real
                    +i                     +i
partial k-space real
 
 
 
 
 
Homodyne imag

Homodyne interpolation for one overscan line as in Figure 10 can be performed on a vector of k-space data via the H operator matrix in Figure 11.

Figure 11
Homodyne matrix

Sometimes researchers artificially increase their image resolution by "zero-filling" the complex-valued k-space. Zero-filling is a process where zeros are placed around the measured k-space 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 k-space is zero-filled, voxel values are sinc interpolated. Figure 12 shows zero filling of k-space from a 4×4 array is in Figure 12 (left) is zero-filled to an 8×8 array of k-space Figure 12 (right).

Figure 12
orig k-space real zero-filled real
                    +i                     +i
orig k-space imag zero-filled imag

Zero-filling from a 4×4 k-space array to an 8×8 k-space array as in Figure 12 can be performed on a vector of k-space data via the F operator in Figure 13.

Figure 13
zero-fill matrix

As can be seen in Figure 13, the F operator is not square and appends zeros around the observed k-space measurements. After size of the k-space 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 k-space arrays can be carried out by multiplication by the apodization matrix A as presented in Figure 14.

Figure 14
apodization matrix

Together these preprocessing operations in k-space are
Ok=AFH(PR-1Ωrow-1ΦΩrowPR)(PCRC).

When Fourier inverse transformation image reconstruction was described above, the fact that T2*(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 T2*(x,y) and ΔB(x,y). Subject specific T2*(x,y) and ΔB(x,y) maps can be estimated and utilized in reconstruction. A version of personalized medicine. Our new reconstruction operator is Ωa which is an inverse Fourier transform image reconstruction operator that incorporates estimates of T2*(x,y) and ΔB(x,y).

Figure 15
Omega matrix Omega matrix adjusted T2 Omega matrix adjusted B0
                             Ω                          Ωa(T2*)                          ΩaB)

Now after k-space preprocessing, we reconstruct with Ωa. Some researchers like to perform image processing such as image smoothing (blurring). Image operations such as smoothing can be performed with a matrix operator OI. An example of image smoothing through a matrix operation Sm upon a vector of image data is presented in Figure 16 for an 8×8 image.

Figure 16
image smoothing matrix

Now the complete set of operations done on originally measured data are O=OIΩOk. With multivariate statistics we can see how the operations O changes our mean image vector and image vector covariance matrix. If the original k-space measurements have a mean of f0 and a covariance matrix of Γ=I, then the new mean is μ=Of0 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=(I2Sm) Ωa A F H (PR-1 Ωrow-1 Φ Ωrow PR )(PCRC )(C'R'PC' )(PR' Ωrow' Φ' Ωrow-1' PR-1') H' F' A' Ωa' (I2Sm').
It is well known that CC'=I, RR'=I, and PCPC'=I, so the covariance matrix is
Σ = (I2Sm) Ωa A F H (PR-1 Ωrow-1 Φ Ωrow PR )(PR' Ωrow' Φ' Ωrow-1' PR-1') H' F' A' Ωa' (I2Sm').
The operations that censor turn around points, reverse alternating lines, and permute from real-imaginary for each k-space point acquired to all reals then all imaginaries do not induce a correlation.
It is also well known that
PRPR'=I, and that ΩrowΩrow'=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 = (I2Sm) Ωa A F H H' F' A' Ωa' (I2Sm').
It can be shown that H H'≠I when there is a partial k-space acquisition and a portion of k-space is interpolated. It can be shown that F F'≠I when k-space is zero filled. It can be shown that A A'≠I when k-space is apodized. It can be shown that ΩaΩa'≠I when T2* and/or ΔB are incorporated in image reconstruction. It can be shown that SmSm'≠I when images are smoothed. So, it has been shown that when preprocessing is done to MRI data in k-space, 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 1-25 from top left to bottom right.

Figure 17
5x5 image

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
correlation mat censor mat censor mat
censor mat censor mat

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 k-space 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 T2*(x,y), and differential from the main magnetic field ΔB(x,y) given in Figure 19 (Nencka, Hahn, Rowe, 2009).

Figure 19
proton spin density map transverse relaxation map static B0 map
                        ρ(x,y)                         T2*(x,y)                         ΔB(x,y)
k-space time map Homodyne reconstruction Apodization filter
                             Δt                          H                          A

Examples of induced correlation from select k-space 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 k-space 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 k-space reconstruction and 3 mm FWHM image space smoothing is in row 2 col 2; homodyne partial k-space 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 k-space 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
correlation mat censor mat censor mat censor mat censor mat
censor mat censor mat censor mat censor mat
censor mat censor mat censor mat censor mat

Our new reconstruction operator is Ωa corrects the effects of subject specific T2*(x,y) and ΔB(x,y). However, the new operator may potentially induce correlations in the image-space as a result of changing the properties of the data. The examples of the induced magnitude-squared correlation from the reconstruction operators, Ω, Ωa (T2*), and Ωa (T2* and ΔB) are illustrated with a magnitude underlay in Figure 21.

Figure 21
Mag2Ideal Mag2T2 Mag2T2andDeltaBE colorbar
Ω Ωa(T2*) Ωa(T2* 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 Fj, from j=1,...,NC receiver coils, sub-sampled by a reduction factor R, where every Rth line of k-space is acquired in the phase-encoding direction.

To apply the AMMUST framework, each of the NC spatial frequency matrices are reshaped into vectors, fj, 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 =(f1,...,fNC)', with alternating real and imaginary components from the NC receiver coils. The NC sub-sampled vectors of spatial frequencies are inverse Fourier transformed into aliased images in image space with NC applications of the above Ω operator using the Kronecker product (INC⊗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, PC, is a vector with the NC real aliased voxel values stacked upon the NC imaginary aliased voxel values for all voxels in the reduced field-of-view images. For even reduction factors, it was found that a second permutation performing a Fourier transform shift, PS, is necessary such that the edges of the aliased images are appropriately aligned with the center of k-space for a centered unfolding. Based on the SENSE model, the vector of NC complex-valued aliased voxel values, aC, are derived from a vector of complex-valued un-aliased voxel values, νC, pre-multiplied by an NC×R complex-valued sensitivity matrix, SC, with additive coil noise εC as

aC = SC×νC + εC.

Since εC is approximated by the complex normal distribution, with a mean of 0 and an NC ×NC complex-valued covariance between coils of ΨC, it can be shown through a transformation of variables that the un-aliased voxel values are derived using a complex-valued weighted least squares estimation

νC = (SCH ΨC-1 SC)-1 SCH ΨC-1 aC

where H denotes the transpose complex conjugate (Hermitian). In applying this to the AMMUST framework, the above equation is represented in terms of its real-valued isomorphism by

ν = (ST Ψ-1 S)-1 ST Ψ-1 a = USE a

where a real-valued 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 skew-symmetric. The SENSE unfolding matrix USE is applied in the AMMUST framework as a block diagonal operator, where each block unfolds the 2NC ×1 concatenated vectors of real aliased voxel values stacked upon imaginary voxel values in a = PS PC (INC⊗Ω) f into 2R×1 vectors of un-aliased real voxel values stacked upon imaginary voxel values corresponding to the R folds. After unfolding each of the aliased voxels, a final permutation, PU, is applied to organize the un-aliased voxel values from being organized by voxel to by row, and ultimately by fold. Thus, a final vector, y, with all real un-aliased voxel values stacked upon all imaginary voxel values is derived from the NC sub-sampled spatial frequency vectors in a complete set of operations as

y= PU USE PS PC (INC⊗Ω) f .

Additional pre-processing operations, Ok, or post-processing operations, OI, could be performed in k-space or image space respectively as

y= OI PU USE PS PC (INC⊗Ω) Ok f .

Assuming an identity covariance structure in the acquired data, the covariance induced by the SENSE reconstruction process is thus Σ = OSE OSET where OSE = OI PU USE PS PC (INC⊗Ω) Ok.

The SENSE operators, OSE, for a 9×9 acquisition with a reduction factor R=3 from NC=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
censor mat censor mat PS censor mat censor mat
                    PU                               USE PS                     PC                     (INC⊗Ω)

In the event of smoothing applied in image-space, using a Gaussian kernel with a FWHM of 3 voxels, the correlations induced about the center voxel bythe SENSE reconstruction operators OSE are illustrated in Figure 23.

Figure 23
realSE imagSE reimSE mgsqSE colorbar
SENSE Real SENSE imaginary SENSE real/imaginary SENSE magnitude-squared

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 B0 feld correction, rigid body motion correction) and induced temporal correlation can be determined. If we denote the k-space vector for each image as ft, t =1,....,n (where ft could be made up from multi coil k-space vectors). Then, the k-space vectors can be stacked to form a single larger k-space f =(f1,...,fn)' and the k-space processing operators Okt, reconstruction operators Ωat and image preprocessing operators OIt of all image k-space vectors, t=1,...,n can be represented simultaneously as

ν1

νn
= OI1 ΩOa1 Ok1                  0
                         ⋱
        0                       OIn ΩOan Okn
f1

fn
or ν=IRK where K has block diagonal elements Okt, R has block diagonal elements Ωat, and I has block diagonal elements OIt.

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 OTj where j=1,...,p denotes voxel number as
y1

yn
= OT1                 0
            ⋱
 0                 OTn
P OI1 ΩOa1 Ok1                  0
                         ⋱
        0                       OIn ΩOan Okn
f1

fn
or O=TPIRK where T has block diagonal elements OTj.
We can now in an exact mathematical framework precisely represent spatio-temporal covariance Σ induced in the data as TPIRKΓK'R'I'P'T' where Γ is the k-space covariance matrix. Even if Γ=I, unless all of KK'=I, RR'=I, II'=I, and TT'=I, a spatio-temporal correlation is induced. (It is already the case that PP'=I.)

The spatio-temporal 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 complex-valued observation at time t is Ricean distributed, it can be approximated by the normal distribution at high SNRs. The complex-valued image at time t measured over time t can be described with a nonlinear multiple regression model that includes both a temporally varying magnitude ρt and a phase imperfection θ given by

yt = [ρt cosθ+ ηRt]+ i[ρt sinθ+ ηIt]

where

ρt = xt'β = β0+ β1x1t+...+ βqxqt,

(ηRtIt,)' = N(0,Σ), and Σ = σ2I2.

xt' is the tth 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 voxel-by-voxel (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 = (yR', yI')' is the vector of observed real values stacked on the vector of observed imaginary values and the vector of errors η = (ηR', ηI')'~N(0,ΣΦ ). It is generally assumed that Σ = σ2I2 and Φ = In. The likelihood of the complex-valued data model can then be written as

p(y|X,β,θ,σ2) = (2πσ2)-n×exp{(-1/σ2) [y-[Xβcosθ ; Xβsinθ]]'×[y-[Xβcosθ ; Xβsinθ]]}.

Model parameters can then be estimated under appropriately constrained null and alternative hypotheses, H0:=0 vs. H1:≠0, then activation can be determined with a generalized likelihood ratio statistic. For example, with a model with β0 representing an intercept, β1 representing a linear drift over time, and β2 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 H0:β2=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^Rcosθ^^Isinθ^

σ^2 = (1/2n)[y-[Xβ^cosθ^ ; Xβ^sinθ^]]'×[y-[Xβ^cosθ^ ; Xβ^sinθ^]]

where β^R = (X'X)-1X'yR and β^I = (X'X)-1X'yI.

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^Rcosθ~^Isinθ~]

σ~2 = (1/2n)[y-[Xβ~cosθ~ ; Xβ~sinθ~]]'×[y-[Xβ~cosθ~ ; Xβ~sinθ~]]

where Ψ is

Ψ = Iq+1(X'X)-1C'[C(X'X)-1C']-1C. 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~2^2).

This statistic has an asymptotic χr2 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, two-sided testing can be performed using the signed likelihood ratio test given by

Z = sign(^,)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 complex-valued time series analysis here that incorporates the induced/modified correlation from preprocessing.

Logan and Rowe 2004, Logan, Geliazkova, and Rowe, 2008; and complex-valued permutation resampling as mentioned in Rowe and Logan 2004.

To be continued..... Blah, Blah, blah.

Future Efforts Stuff.
+  Cited Publications

Bruce IP, Karaman MM, ROWE DB. A statistical examination of SENSE image reconstruction via an isomorphism representation. Magn. Reson. Imaging, 29(9):1267-1287, (2011). (Contribution: Extends the AMMUST-k framework to describe SENSE image reconstruction. Significance: Can precisely quantify the induced correlation among folds in subsampled k-space data including preprocessing.)

Bruce IP, Karaman MM, ROWE DB. The SENSE-Isomorphism Theoretical Image Voxel Estimation (SENSE-ITIVE) Model for Reconstruction and Observing Statistical Properties of Reconstruction Operators (Status: Conditional Acceptance, Magn. Reson. Imaging), 2012. (Contribution: The SENSE model assumes a skew-symmetric covariance matrix. The SENSE-ITIVE 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 phase-sensitive fMRI analysis using temporal off-resonance alignment of single-echo timeseries (TOAST). Neuroimage 44:742-752 (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 complex-valued 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 phase-only 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 complex-valued fMRI detection of neurobiological processes through post-acquisition estimation and correction of dynamic B0 errors and motion. To appear, Hum. Brain. Mapp., (2011). (Contribution: Thorough examination of complex-valued 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 complex-valued activation with correction for dynamic magnetic field fluctuations in conjunction with estimated motion parameters.)

Hahn AD, Rowe DB. Methodology for robust motion correction of complex-valued fMRI time series. Proc. Intl. Soc. Magn. Reson. Med. 18:3051, Stockholm, Sweden, (2010). (Contribution:Framework for effective motion correction through image registration of complex-valued 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 complex-valued image time series through image registration. Allows improved motion correction for artifact removal prior to complex-valued analysis.)

Hernandez-Garcia L, Vazquez AL, Rowe DB. Complex-valued analysis of arterial spin labeling based FMRI signals. Magn. Reson. Med. 62(6):1597-1608, 2009. (Contribution: Applies the Rowe 2005b magnitude and phase activation . Significance: A simulation study indicated that the complex-valued 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 complex-valued 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. 29:1379-1389 (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 22:95-108 (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 (AMMUST-k) effects of k-space preprocessing on observed voxel measurements in fcMRI and fMRI. J. Neurosci. Meth. 181:268-282 (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 pre-reconstruction pre-magnitude time series formation pre-processing 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 k-space 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 (R-SHIZLE) theoretically encodes intra-acquisition 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 k-space 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 complex-valued k-space and voxel measurements to characterize the correlation between voxels induced by the consistent temporal application of apodization and smoothing of k-space 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 non-negligible.)

Nencka AS, Rowe DB. Image space correlations induced by k-space processes. Proc. Org. Hum. Brain Mapp. S55:284 (2007b). (Contribution: Uses relationship between complex-valued k-space and complex-valued voxel measurements to characterize voxel correlation induced by preprocessing. Significance: These adjustments produce or modify the spatial correlation between voxel measurements. Possible k-space adjustments include the shifting of alternating lines to correct the signal (but also the noise) to eliminate ghosting, apodization of k-space measurements, and partial k-space 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 post-processing methods. Neuroimage 37:177-188 (2007a). (Contribution: Developed Monte Carlo simulations and examined Human echo planar imaging data with two activation methods. Significance: We found that the complex-valued 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 magnitude-only model.)

Rowe DB. fMRI activation in image space from k-space data. Proc. Org. Hum. Brain Mapp. S114:377 (2007a). (Contribution: Computes brain activation in image space in terms of k-space measurements. Significance: 1) Activation is one step closer to the original data. 2) The relationship between k-space 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 k-space data. Proc. Am. Stat. Assoc. (Biometrics Section) 12:107-114 (2007b). (Contribution: Computes brain activation in image space in terms of k-space measurements. The correlation between voxel measurements can also be written in terms of correlation between k-space measurements. Significance: 1) Activation and association is one step closer to the original data. 2) The relationship between k-space 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 complex-valued fMRI data. Magn. Reson. Med. 62:1356–1357 2009.(Contribution: Examines the computation of magnitude and phase activation from complex-valued 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 complex-valued fMRI data. Neuroimage 25:1310-1324 (2005b). (Contribution: Developed a more general fMRI model that simultaneously describes both the magnitude and phase of complex-valued 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 highly-localized 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 25:1124-1132 (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 magnitude-only parameter estimates become increasingly biased as the SNR decreases whereas the complex-valued model is unbiased at all SNR levels. 2) The parameter estimates achieved their Cramer-Rao variance lower bound for the complex-valued model regardless of SNR while the magnitude-only model did not. 3) The complex-valued activation statistic was uniformly higher than the magnitude-only 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 k-space. Magn. Reson. Imaging 27:1370-1381, 2009. (Contribution: In Rowe, Nencka, Hoffmann it was shown that complex-valued voxel measurements can be written as a the complex-valued k-space 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, complex-valued k-space 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 k-space on voxel activation and correlation can then be examined. .)

Rowe DB, Logan BR. A complex way to compute fMRI activation. Neuroimage 23:1078-1092 (2004). (Contribution: Determined magnitude fMRI activation in complex-valued voxel time series data while specifying the traditionally believed voxel-wise 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 real-imaginary observations instead of n magnitude quantities. 3) Produces more highly-focused 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 magnitude-only model. Neuroimage 24:603-606 (2005). (Contribution: Outlined a more general magnitude fMRI activation based on complex-valued data that assumed unrestricted or unique temporal phase values. Significance: 1) Derived the same regression coefficients and activation statistics as used for the usual magnitude-only data model. 2) By deriving these statistics we now understand the complex-valued assumptions inherent in the commonly used magnitude-only fMRI activation model.)

Rowe DB, Meller CP, Hoffmann RG. Characterizing phase-only fMRI data with an angular regression model. J. Neurosci. Methods 161:331-341 (2007). (Contribution: Phase-only 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 159:361-369 (2007). (Contribution: Related measured complex-valued k-space spatial frequencies and complex-valued images. Significance: 1) This allows the computation of fMRI brain activation directly from unreconstructed originally observed k-space 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. 104:623-637 (2009). (Contribution:Introduce a more precise magnitude-only 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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