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Functional MRI brain activation directly from k-space

Daniel B. Rowea,b,*, Andrew D. Hahna, and Andrew S. Nenckaa

aDepartment of Biophysics, Medical College of Wisconsin, Milwaukee, WI 53226, USA

bDivision of Biostatistics, Medical College of Wisconsin, Milwaukee, WI 53226, USA

Abstract

In functional magnetic resonance imaging (fMRI), the process of determining statistically significant

brain activation is commonly performed in terms of voxel time series measurements after image

reconstruction and magnitude-only time series formation. The image reconstruction and statistical

activation processes are treated separately. In this manuscript, a framework is developed so that

statistical analysis is performed in terms of the original, pre-reconstruction, complex-valued k-space

measurements. First, the relationship between complex-valued (Fourier) encoded k-space

measurements and complex-valued image measurements from (Fourier) reconstructed images is

reviewed. Second, the voxel time-series measurements are written in terms of the original spatio-

temporal k-space measurements utilizing this k-space and image relationship. Finally, voxel-wise

fMRI activation can be determined in image space in terms of the original k-space measurements.

Additionally, the spatio-temporal covariance between reconstructed complex-valued voxel time

series can be written in terms of the spatio-temporal covariance between complex-valued k-space

measurements. 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.

Keywords

image reconstruction; k-space; fMRI; complex data; Rowe-Logan

1. Introduction

In functional magnetic resonance imaging (fMRI), an array of data for an individual image is

observed in an encoded form. The sampled data are generally Fourier encoded [2,4] and thus

are measured spatial frequencies. These spatial frequency (k-space) observations are then

reconstructed into an individual image array by the process of an inverse Fourier

transformation. A series of these arrays of encoded images are acquired and the reconstruction

process is applied to each array. For each voxel, temporally sequential voxel measurements

are collected into a time series for determination of statistically significant activation. The

originally sampled spatial frequencies are complex-valued and the inverse Fourier

transformation image reconstruction process may yield complex-valued data. Due to

*Corresponding author. Daniel B. Rowe, Department of Biophysics, Medical College of Wisconsin, 8701 Watertown Plank Road,

Milwaukee, WI 53226, Tel: +1 414 456 4027, fax: +1 414 456 6512, Email address: dbrowe@mcw.edu (D.B. Rowe).

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Magn Reson Imaging. Author manuscript; available in PMC 2010 December 1.

Published in final edited form as:

Magn Reson Imaging. 2009 December ; 27(10): 1370–1381. doi:10.1016/j.mri.2009.05.048.

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measurement error and imperfections in the Fourier encoding, voxel time series are generally

complex-valued.

The process of determining statistical activation in each voxel has, for the most part, been from

magnitude-only time series [1,12]. The process of converting a complex-valued time series

into a magnitude-only time series is to take the square root of the sum of the squares of the real

and imaginary parts of the complex-valued time series at each time point [11]. An activation

statistic from the magnitude-only time series for each voxel is determined by computing a

measure of association between the observed voxel time series and a preassigned ideal time

series based on the timing of the experiment and physiological considerations. This association

measure for each voxel is statistically compared to the association measure that would result

from a time series of random noise. A statistical threshold is chosen, a scale of color values for

the activation statistic is assigned, and each voxel above threshold is given the color

corresponding to its activation statistic.

The idea of computing an activation statistic from the complex-valued time series has been

previously discussed [5,8]. This idea of computing fMRI activation from complex-valued data

has recently been expanded upon [9-12]. Work has also been performed on computing fMRI

activation from phase-only time series [13]. However, the processes of image reconstruction

and statistical activation have been treated separately. Thus, activation is determined in terms

of complex-valued voxel measurements after reconstruction and not the original encoded

measurements.

In the current study, the relationship between the original encoded k-space measurements and

reconstructed voxel measurements for each image is summarized. For each image, a vector of

real-imaginary reconstructed voxel measurements is formed and written as a linear

combination of real-imaginary k-space measurements. A larger vector of reconstructed real-

imaginary voxel time series measurements is formed by stacking the individual vectors of real-

imaginary voxel measurements for each image in temporal order. This large vector is written

as a linear combination of a large vector of real-imaginary k-space time series measurements

that is ordered in a similar manor. A permutation matrix is utilized to reorder the voxel

measurements that are real then imaginary per image to be of real then imaginary per voxel.

Statistical functional brain activation can then be determined with the aforementioned recent

complex-valued activation models. A map of these activation statistics can then thresholded

to determine statistically significant activation while adjusting for multiple comparisons [6,

7].

Statistically significant voxel activation and correlation between voxels can thus be determined

in image space in terms of the originally acquired k-space measurements. This will allow the

modeling of the originally acquired measurements in their original state, as they are acquired,

and not in a transformed state. Implications of k-space preprocessing on voxel activation and

correlation can then be evaluated.

2. Background

Previous work has included the development of a real-valued representation of the standard

complex-valued Fourier transform [14]. In this section we review the representation and offer

a graphical example to illustrate the method. Magnetic resonance images are almost exclusively

Fourier encoded. That is, one ideally measures the Fourier transform of an image and

reconstruct the image via an inverse Fourier transform. The Fourier transform and inverse

Fourier transforms are complex-valued procedures that results in complex-valued arrays.

Standard, complex-valued Fourier matrices are defined as follows. If ΩC is a p×p Fourier

matrix, it is a matrix with (j,k)th element [ΩC]jk= κ (ωjk) where κ=1 and ω=exp[-i 2 π (j-1)

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(k-1)/p] for the forward transformation while κ=1/p and ω =exp[+i 2π (j-1)(k-1)/p] for the

inverse transformation, where j,k=1,…,p.

Consider the Fourier transform of an image that has dimensions py×px(py rows and px

columns). Often the image is square, although this is not necessary. More specifically, consider

an 8×8, ideal, noiseless, gray scale image as presented in Fig. 1. Since the Fourier transform

and inverse Fourier transform procedures operate on, and produce complex-valued arrays, the

real-valued image in Fig. 1 can be represented as a complex-valued image RC that has a real

part RR as in Fig. 1 an imaginary part RI that is the zero matrix so that RC=RR+iRI. The encoded

data, or Fourier transform of this image, can be found as in Eq. (1) by pre-multiplying the

py×px dimensional complex-valued matrix RC by a standard complex-valued forward Fourier

matrix Ω̄yC=Ω̄yR+iΩ̄yI, that is of dimensions py×py, and post-multiplying RC by the transpose

of another standard forward Fourier matrix

transposition, that is of dimensions px×px. The result of the pre- and post-multiplications is a

complex-valued array of spatial frequency (k-space) measurements, SC, with real part SR and

imaginary part SI as also shown in Eq. (1).

, where T denotes matrix

(1)

This mathematical procedure is graphically illustrated in Fig. 2 using the aforementioned 8×8

image. In Fig. 2 the 8×8 image, RC, is utilized to mimic an image from a magnetic resonance

echo planar imaging experiment. RC is displayed with real part, RR, in Fig. 2c and imaginary

part, RI, in Fig. 2d. The spatial frequency (k-space) values, SC=(SR+iSI), associated with this

complex-valued image, are found by pre-multiplying the complex-valued image by the

complex-valued forward Fourier matrix Ω̄yC (Fig. 2a and Fig 2b) and then post-multiplying

the result by the transpose of the symmetric forward Fourier matrix (Fig. 2e and Fig. 2f). The

spatial frequency (k-space) values, SC, for the complex-valued image RC are presented as an

image with real part, SR, in Fig. 2g and imaginary part, SI, in Fig. 2h. Note that, as mentioned

earlier, the image does not have to be square.

However, as previously described, in MRI encoded (k-space) measurements, SC, are made and

reconstructed (transformed) into an image. The inverse Fourier procedure is performed. This

reconstruction procedure, or inverse Fourier transform, of the spatial frequency (k-space)

measurements can be found as

(2)

by pre-multiplying the py×px dimensional complex-valued spatial frequency matrix, SC, by a

complex-valued inverse Fourier matrix, Ωy, that is of dimensions py×py, and post-multiplying

SC by the transpose of another Fourier matrix, Ω̄xT, that is of dimensions px×px, where T

denotes matrix transposition. The result of the pre- and post-multiplications is a complex-

valued array of image measurements RC, with real part RR and imaginary part RI as also shown

in Eq. (2).

The complex-valued image RC can be recovered as seen in Fig. 3. The process of recovering

the original complex-valued image RC is to pre-multiply the complex-valued spatial frequency

(k-space) values SC by the complex-valued inverse Fourier matrix ΩCy, (Fig. 3a and Fig. 3b)

then post-multiply the result by the transpose of the symmetric inverse Fourier matrix ΩCx,

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(Fig. 3e and Fig. 3f). The recovered complex-valued image, RC, is presented with real part,

RR, in Fig. 3g and imaginary part, RI, in Fig. 3h.

This complex-valued inverse Fourier transformation image reconstruction process can be

equivalently described as a linear transformation with a real-valued representation [14]. Such

a transformation is often called an isomorphism in mathematics. Define a real-valued vector,

s, to be a 2pxpy dimensional vector of complex-valued spatial frequencies from an image where

the first pxpy elements are the rows of the real part of the spatial frequency matrix, SR, shown

in Fig. 3c, and the second pxpy elements are the rows of the imaginary part of the spatial

frequency matrix, SI, shown in Fig. 3d. The real-valued vector of spatial frequencies is thus

formed as s=vec(SRT,SIT), where (SRT,SIT) is a px×2py matrix formed by joining the transpose

of the real and imaginary parts of SC as seen in Fig. 4a, and vec(·) denotes the vectorization

operator that stacks the columns, shown in Fig. 4b, of its matrix argument. This yields us a

real-valued vector representation of the matrix of spatial frequency (k-space) values that is

given in Fig. 5b.

Further define a matrix Ω that is another representation of the complex-valued inverse Fourier

transformation matrices as described in Eq. (3) where the matrix elements of Ω are

and ⊗ denotes the Kronecker product that multiplies every element of its first matrix argument

by its entire second matrix argument. Utilizing the complex-valued Fourier matrix ΩCy, with

real and imaginary parts ΩyR and ΩyI given in Fig. 3a and Fig. 3b, along with the complex-

valued Fourier matrix ΩCx, with real and imaginary parts ΩxR and ΩxI given in Fig. 3e and

Fig. 3f, the resulting Ω matrix is presented in Fig. 5a.

The real-valued vector representation s of the spatial frequency (k-space) values in Fig. 5b is

then pre-multiplied by the (inverse Fourier) reconstruction matrix Ω as in Eq. (3)

(3)

where the real-valued representation, r, of the complex-valued image has a dimension of

2pxpy×1, true mean and no measurement error.

This is pictorially represented in Fig. 5. Fig. 5b is the spatial frequency vector s and Fig. 5a is

the inverse Fourier transformation matrix Ω as described in Eq. (3). This matrix multiplication

produces a vector representation, r, of the image voxel measurements given in Fig. 5c as

described in Eq. (3). The vector of voxel measurements, r, is partitioned into column blocks

of length px. These blocks are then arranged as in Fig. 6a and formed into a single matrix image

as in Fig. 6b where the first (last) eight columns are the transpose of the real (imaginary) part

of the image. As can be seen, the same resultant complex-valued image is reconstructed with

the complex-valued inverse Fourier transformation procedure described in Eq. (2) and

presented in Fig. 3.

In the above described procedure, measurement noise was not considered. Redefine SC to be

the py×px dimensional complex-valued spatial frequency measurement of a slice with noise

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that consists of a py×px dimensional matrix of true underlying noiseless complex-valued spatial

frequencies, S0C, and a py×px dimensional matrix of complex-valued measurement error, EC.

This partitioning of the measured spatial frequencies in terms of true noiseless spatial

frequencies plus measurement error can be represented as

(4)

where i is the imaginary unit while S0R, S0I, ER, and EI are real and imaginary matrix valued

parts of the true spatial frequencies and measurement noise, respectively. Let ΩCx and ΩCy be

px×px and py×py complex-valued Fourier matrices as described above. Then, the py×px

complex-valued inverse Fourier transformation reconstructed image, RC, of SC can be written

as

(5)

where RC has a true mean R0C and measurement error NC. Note that the complex-valued

matrices for reconstruction, Ωx and Ωy in Eq. (5), need not be exactly Fourier matrices but may

be Fourier matrices that include adjustments for independently measured magnetic field

inhomogeneities or reconstruction matrices for other encoding procedures.

The real-valued inverse Fourier transformation method for image reconstruction can also be

directly applied to noisy measurements. We can represent the noisy complex-valued spatial

frequency matrix as s=s0+ε where this 2pxpy dimensional vectors includes the reals of the rows

stacked upon the imaginaries of the rows of the corresponding matrix. This implies that if the

mean and covariance of the spatial frequency measurement vector, s, that is of dimension

2pxpy×1, are s0 and Γ, then the mean and covariance of the reconstructed voxel measurements,

r, are Ωs0 and ΩΓΩT.

3. Theory

The previously described data for a single image is expanded upon to mimic an fMRI

experiment. In fMRI, a series of the previously described image slices are acquired. Denote

the py×px complex-valued spatial frequency matrix, corrupted by random noise, acquired at

time t as SCt=S0Ct+ECt and define

parts of SCt for time points t=1,…,n. Define the total number of voxels in the image, which is

the same as the number of complex-valued k-space measurements in fully sampled, Fourier

encoded, Cartesian acquisitions, to be p=pxpy. This sequence of measured spatial frequency

vectors can be collected into a 2p×n matrix S=(s1,…,sn) where the tth column contains the p

real k-space measurements stacked upon the p imaginary k-space measurements for time t.

Having done this, n reconstructed images can be formed by the 2p×n matrix R=ΩS where the

tth column of R contains the p real voxel measurements stacked upon the p imaginary voxel

measurements for time t, t=1,…,n.

, where SRt and SIt are the real and imaginary

The k-space measurements and the image voxel measurements can be stacked as s=vec(S) and

r=vec(R). Note that s and r and have been redefined from their initial definition. If the mean

and covariance of the 2np×1 vector of spatial frequency measurements, s, are s0 and Δ, then

the mean and covariance of the 2np×1 vector of reconstructed voxel measurements, r, are (In

⊗ Ω)s0 and (In ⊗ Ω)Δ(In ⊗ ΩT). For example, if the k-space measurements were taken to be

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