Overcoming Activation-Induced Registration Errors in fMRI

Abstract

It has been shown that the presence of a blood oxygen level dependent (BOLD) signal in high-field (3T and higher) fMRI datasets can cause stimulus-correlated registration errors, especially when using a least-squares registration method. These errors can result in systematic inaccuracies in activation detection. The authors have recently proposed a new method to solve both the registration and activation detection least-squares problems simultaneously. This paper gives an outline of the new method, and demonstrates its robustness on simulated fMRI datasets containing various combinations of motion and activation. In addition to a discussion of the merits of the method and details on how it can be efficiently implemented, it is shown that, compared to the standard approach, the new method consistently reduces false-positive activations by two thirds and reduces false-negative activations by one third.

Full text

Overcoming Activation-Induced Registration Errors in fMRI
Jeff J. Orchard∗a, Chen Greifb, Gene H. Golubc, Bruce Bjornsondand M. Stella Atkinsa
aComputing Science, Simon Fraser University, Burnaby, BC, Canada V5A 1S6
bDept. of Computer Science, Univ. of British Columbia, Vancouver, BC, Canada V6T 1Z4
cSCCM Program, Stanford University, Stanford, CA, USA 94305-9025
dDept. of Pediatrics (Neurology), BC’s Children’s Hosp., Vancouver, BC, Canada V6H 3V4
ABSTRACT
It has been shown that the presence of a blood oxygen level dependent (BOLD) signal in high-field (3T and
higher) fMRI datasets can cause stimulus-correlated registration errors, especially when using a least-squares
registration method. These errors can result in systematic inaccuracies in activation detection. The authors
have recently proposed a new method to solve both the registration and activation detection least-squares
problems simultaneously. This paper gives an outline of the new method, and demonstrates its robustness on
simulated fMRI datasets containing various combinations of motion and activation. In addition to a discussion
of the merits of the method and details on how it can be efficiently implemented, it is shown that, compared
to the standard approach, the new method consistently reduces false-positive activations by two thirds and
reduces false-negative activations by one third.
Keywords: fMRI, least-squares, registration, BOLD, GLM, activation
1. INTRODUCTION
Patient motion can have a profound, negative impact on the accuracy of fMRI results. The misalignment of
volumes can artificially increase the variation of a voxel’s time-course, and prevent the detection of the blood
oxygen level dependent (BOLD) signal.1,2 The resulting activation maps are often subject to false-negatives,
the mis-classification of activated brain regions as inactive.
Patient motion has an opposite, and particularly insidious, effect when it is synchronized with the stimulus.
This type of motion causes voxel intensity fluctuations that mimic those of activated voxels. As a result,
unactivated voxels can take on the appearance of being activated.1A comprehensive study of the effects
of stimulus-correlated motion on fMRI activation maps was conducted by Hajnal et al.3Seven of the eight
in vivo datasets they acquired contained significant stimulus-correlated motion. They also showed that this
motion, by itself, could produce the observed activation even without the BOLD signal. That study illustrates
that motion cannot be assumed to be independent of the stimulation paradigm, and that stimulus-correlated
motion has an overwhelming potential to produce false-positive activations.
A common approach to overcome these issues is to apply motion compensation algorithms to the dataset.
The patient motion is estimated using an automatic motion detection program. These estimates can then
be used to resample the dataset and align its volumes. The resulting activation maps are more accurate.2
One of the most widely-used motion detection methods is least-squares registration. This method attempts to
minimize the sum of the squared residuals between the two volumes being compared. It is computationally
efficient, and the sum of squares cost function has been shown to be optimal when the two volumes differ only
by additive Gaussian noise.4
However, it has recently been shown5that the BOLD signal present in fMRI datasets acquired on high field
scanners (3 tesla and higher) may violate the assumptions underlying the least-squares registration method.
When two volumes are being compared, the presence of BOLD contrast in one image, but not the other, causes
∗jjo@cs.sfu.ca; phone 1 604 291 5509; www.cs.sfu.ca/∼jjo
Medical Imaging 2003: Image Processing, Milan Sonka, J. Michael Fitzpatrick, Editors,
Proceedings of SPIE Vol. 5032 (2003) © 2003 SPIE · 1605-7422/03/$15.00
130

the corresponding regions to be mismatched. The impact of this phenomenon was published by Freire et al.5
In his paper, he refers to these mismatched regions as “outliers” in the statistical fit. It is well-known that the
sum of squares cost function is particularly sensitive to outliers. The resulting inaccurate motion estimates are
then used in the motion “correction” process, removing some of the actual motion, but introducing erroneous
motion. Since the false motion is caused by the brain activation, it is correlated to the stimulus. Thus, the
resulting dataset has stimulus-correlated motion artifacts and is subject to the same problems as those with
genuine uncorrected stimulus-correlated patient motion: the false detection of activation near high-contrast
borders. Freire reported that, on datasets with voxel dimensions 3.75 ×3.75 ×4 mm, motion errors as small as
0.05◦and 0.05 mm resulted in the detection of false-positives. Freire’s findings have also been supported by a
theoretical study6in which the effect of the activation on motion estimates can be approximated linearly. The
theoretical results match well with the observed registration errors from simulated fMRI datasets.
Some methods have been proposed to deal with residual motion artifacts after a first pass of registration.1, 7, 8
In general, their approach is to remove the motion-related component from each voxel time-course. However,
when patient motion is authentically stimulus-correlated, these methods have difficulty distinguishing the true
BOLD signal from motion artifacts, and run the risk of discounting genuine activation.
With all the interdependencies between registration and activation, it seems that these two problems must
be considered coupled, and neither should be solved before the other. Hence, a simultaneous solution is required.
Such a solution method has been proposed by the authors,9and is called the Simultaneous Registration and
Activation (SRA) method.
In this paper, we compare the SRA method to the standard least-squares approach in which least-squares
registration is done first, followed by least-squares activation detection. Both methods are tested on a variety of
different simulated datasets involving mixtures of motion and activation. We motivate and provide algorithmic
details of our approach, and demonstrate its impressive performance on a comprehensive set of experiments.
2. THEORY
The registration and activation detection problems in fMRI can be combined into a single problem in which
motion parameters and activation are solved for simultaneously. In this section, we outline the models for
subject motion and activation separately, before combining them. This combined solution is achieved by
formulating the models using matrix notation, involving all voxels and all time steps. A dataset with m
volumes, each having nvoxels, can be stored in a single m×nmatrix, F, as shown in Fig. 1. Each of the m
volumes in the time series is stored in a column of F, and each row holds the time series for a single voxel.
2.1. Registration
A 3D rigid-body transformation can be specified using six parameters: three rotations (about each of the three
principal axes), and three translations (along each of the three principal axes). The problem of registration
amounts to finding, for each volume in F, the six motion parameters that align it with a reference volume.
If we let Gbe an m×nmatrix holding ncopies of the reference volume, we seek to minimize the difference
between Fand transformed versions of the reference volume, T(G,X), where Xis a 6×nmatrix, each column
of which holds the 6 motion parameters for one of the nvolumes:
X=









x(1) ··· x(n)
y(1) ··· y(n)
z(1) ··· z(n)
θ(1)
x··· θ(n)
x
θ(1)
y··· θ(n)
y
θ(1)
z··· θ(n)
z









.(1)
Proc. of SPIE Vol. 5032 131

volume 3
volume 2
volume 1
volume n
. . .
F=
space ( samples)
m
time ( samples)n
Figure 1.Storage of the time-series dataset in one matrix. Each of the nvolumes is stored in a column of length m.
The rigid-body transformation acts non-linearly on the volume. That is, it cannot be represented as a linear
operation on the intensities of a volume (not to be confused with the fact that it is a linear transformation on
the coordinates). By taking only the linear terms of the Taylor expansion of Twith respect to X, we get
T(G,X)=T(G,0) + ∇T(G,0)X
=G+∇T(G,0)X.(2)
Thus, registration is the task of finding Xso that
F=G+∇T(G,0)X
=G+AX ,(3)
where Aand ∇T(G,0) have six columns, each column holding the partial derivatives of all voxels in Gwith
respect to one of the motion parameters. In the presence of noise (and other sources of variation), we do
not expect to be able to solve Eqn. (3) exactly. Instead, we look to minimize some measure of the difference
between the two sides of the equal sign. For least-squares registration, we solve
min
XF−(G+AX),(4)
where · is the Frobenius norm (sum of the squares of the elements). The solution to Eqn. (4) can be
represented analytically as
X=A†(G−F),(5)
where A†is defined as ATA−1AT. Recall that Xholds the motion parameter estimates that bring Gcloser
to alignment with F. The volumes in Fare resampled using the inverse of these motion parameters. Since we
are approximating the non-linear rigid-body transformation with a linear operator (known as “linearization”),
this process needs to be iterated until the system converges or it reaches a maximum number of iterations.
This technique is known as fixed point iteration, and it is possible to show convergence under certain mild
conditions.
2.2. Activation
The presence of BOLD activation can be represented using the general linear model (GLM). Let Bbe a p×n
matrix with each of its prows holding a time-series regressor (one or more of which is the stimulus function).
132 Proc. of SPIE Vol. 5032

The m×pmatrix Yholds the corresponding spatial maps for the regressors. Each column of Yholds a
spatial distribution specifying the region of influence for a particular regressor, while each row of Yholds the
contribution of each regressor to a particular voxel. An unactivated volume Gis related to an activated volume
Fby the formula
F=G+YB .(6)
Equation (6) is only valid in the absence of motion, and can also be solved in the least-squares sense yielding
Y=(F−G)B†,(7)
where B†is equal to BTBBT−1.
2.3. Combined model
Equation (3) does not incorporate temporal fluctuations such as BOLD contrast, while Eqn. (6) assumes that
the volumes are perfectly aligned. In the standard implementation of fMRI processing, registration is performed
first, followed by activation detection. Because of this ordering, the registration algorithm has no information
about any expected intensity changes due to activation. This fact may violate the conditions under which the
least-squares cost function is effective. That is, the residuals resulting from the active voxels may not fit the
model of Gaussian noise, and act as outliers in the least-squares fit.
Combining Eqns. (3) and (6) gives a single model that includes both motion and activation,
F=G+AX +YB ⇔AX +YB −C=0,(8)
where Cis equal to (F−G). Solving Eqn. (8) means finding the Xand Ythat minimize
AX +YB −C.(9)
2.4. Simultaneous solution
A simultaneous solution of Eqn. (9) yields the (X,Y) pair that minimizes the error due to mis-registration and
activation at the same time. The solution can be found by using QR decomposition.10 The QR decomposition
of BTis
BT=QR
0,(10)
where Qis an n×northogonal matrix, and the second matrix is a 2 ×1 block matrix in which Ris p×pand
upper-triangular, and the remaining (n−p) rows are all zeros. Making the substitution for Bin Eqn. (9), and
right-multiplying by Q(which does not change the norm, since Qis unitary) gives us
A¯
X+YRT0−¯
C,(11)
where ¯
Xand ¯
Care equal to XQ and CQ, respectively. The cardinal feature that this algebraic manipulation
elicits is the fact that the elements of Yare only present in the first pcolumns of the matrix expression in
Eqn. (11). We will take advantage of this detail later on.
At this point, it is important to note that this problem does not have a unique solution. To demonstrate
this, assume that (X,Y) is an exact solution to Eqn. (8). Now, consider a perturbation of that solution,
(X+δX,Y+δY). Plugging the perturbation into Eqn. (8) yields
A(X+δX)+(Y+δY)B−C=0
⇒AX +YB −C+AδX+δYB=0
⇒AδX+δYB=0.(12)
Proc. of SPIE Vol. 5032 133

Hence, the perturbation is also a solution as long as (δX,δ
Y) satisfies Eqn. (12). The solution of Eqn. (12)
is δX=αBand δY=−Aα, where αis any 6 ×pmatrix. Thus, the general solution of Eqn. (9) is the space
(X+αB,Y−Aα)|∀α∈R6×p.(13)
To find a unique solution to our problem, we need to add a constraint. In our case, since we know the form
of the general solution, we can tackle the problem with a two-step approach. First, we find any particular
solution, (X,Y), and use it as the anchor for our general solution. Then, we use an additional constraint to
choose a unique solution from our solution space.
Since any solution will do for the first step, it is convenient to set the 6pelements in the first pcolumns
of ¯
Xto zero. This choice merely forces our corresponding solution for Xto be orthogonal to the rows in B.
We can do this without loss of generality because our general solution gives us the opportunity to add these
components back in later. Now the solution is easy to compute because ¯
Xand Yare completely decoupled:
min
(¯
X,Y)

A0¯
Xp+1:n+YRT0−¯
C1:p¯
Cp+1:n


= min
¯
X

A¯
Xp+1:n−¯
Cp+1:n

+ min
Y

YRT−¯
C1:p

.(14)
The notation Cp+1:nrepresents the sub-matrix of Ccontaining only columns (p+ 1) through n, and likewise
for ¯
X. As before, the least-squares solution can be expressed analytically as
¯
Xp+1:n=A†¯
Cp+1:n
Y=¯
C1:p(RT)−1.(15)
Finally, we calculate Xusing the formula X=¯
XQT. As with the registration procedure outlined in section 2.1,
the solution obtained from Eqn. (15) constitutes a single iteration in a fixed-point iterative method. The dataset
Fis resampled based on the inverse of the motion estimates, and the process is repeated until convergence.
2.5. Constrained simultaneous solution
As mentioned above, the solution to the combined least-squares problem is not unique. We know that if (X,Y)
is a solution, then every element of the set (X+αB,Y−Aα)|∀α∈R6×pis also a solution. However, we
can narrow down the set of plausible solutions and extract the one that is appropriate for our problem context
by adding a constraint. In the case of fMRI, we expect relatively small clusters of activation, suggesting that
most elements of Yshould be zero. However, the particular solution we obtain from the method outlined in
section 2.4 is likely to contain a significant Acomponent. Recall that the matrix Aholds the partial derivatives
of the reference volume with respect to the motion parameters (see Eqn. (3)). These volumes highlight edges
of the reference volume, and contain extended non-zero regions. We wish to remove these Acomponents so
that we are left with an activation volume, Y, that is very close to zero everywhere except at small clusters of
activation.
One way to enforce this preference is to minimize the 1norm of (Y−Aα). The 1norm has been shown
to favour images with large black regions.11 Other formulations that yield the same desired effect are also
commonly used. In our implementation, we use the constraint
min
αarctanc|Y−Aα|,(16)
for suitably-chosen constant c. Equation (16) is similar in essence to the Geman-McClure robust estimator.12
However, we point out that we are using this constraint to negotiate through our solution space for the least-
squares problem. Hence, any single solution we arrive at based on Eqn. (16) is still a least-squares solution of
Eqn. (8).
Once the desired, unique (X,Y) solution is found, a single iteration of the solution process is complete.
The dataset Fis then resampled, and the process continues until the element in Xwith the largest magnitude
is smaller than a user-given threshold, typically 0.001.
134 Proc. of SPIE Vol. 5032

3. METHODOLOGY
Both the standard least-squares algorithm and the SRA algorithm were implemented in C++ for testing
purposes. For the SRA method, we used the Nelder/Mead simplex method13 to minimize Eqn. (16). Both
methods used Fourier interpolation for dataset resampling.14
We compared the performance of the two algorithms on simulated fMRI datasets. Simulated datasets are
an effective tool for assessing registration accuracy because the motion and activation are known. Datasets of
40 frames were created by taking an original 64 ×64 ×30 EPI volume (voxel dimensions 3.75 ×3.75 ×4 mm),
applying a 3 ×3×3 median filter, and then duplicating the volume 40 times with varying amounts of BOLD
activation and motion. The motion was applied by the AFNI15 package, using the Fourier interpolation option.
The activation mask was hand-drawn over portions of the occipital and parietal regions, covering a total of
13% of the brain volume. During two stimulus epochs spanning frames 5 to 15 and frames 25 to 35, a 5%
signal increase was added to activated voxels. Then, Gaussian noise with a standard deviation of 2.5% was
added to all voxels, followed by smoothing using a 5 mm full-width at half-maximum Gaussian kernel.
To test the algorithms’ behaviour on datasets involving varying amounts of BOLD signal and patient
motion, 10 datasets were created for each of the following four scenarios.
Scenario 1: The dataset contains activation and random motion.
Scenario 2: The dataset contains activation and true stimulus-correlated random motion.
Motion profiles were generated using a random mixture of the stimulus function
and random motion.
Scenario 3: The dataset contains no activation, but contains stimulus-correlated random
motion.
Scenario 4: The dataset contains activation, but no motion.
The two registration methods were each used to generate motion estimates for each simulated dataset.
These estimates were then used to resample the datasets for motion “correction”.
The corrected datasets were used to generate binary activation masks, classifying each voxel as either “ac-
tive” or “inactive”. Classification was based on thresholding of each voxel’s linear fit coefficient and correlation
coefficient. A voxel’s linear fit coefficient is the value of its corresponding element in the matrix Y, and reflects
the size of the stimulus-correlated component present in the voxel’s time-course. The correlation coefficient
between a voxel time series {xi}and the stimulus function {bi}is calculated using the formula
CC(x, b)= i(xi−¯x)bi−¯
b
(i(xi−¯x))2ibi−¯
b2,(17)
where ¯xand ¯
bare the average values for the time series. If a voxel’s correlation coefficient was greater (in
magnitude) than 0.505 (corresponding to P<0.001), and its linear fit coefficient was greater than a given
threshold (about 5% of the maximum coefficient), then the voxel was classified as active and included in a
binary activation mask.
A true activation map was created by a similar means, but from a dataset that was neither motion-corrupted
nor motion corrected. In the true activation map, a voxel was considered active if its correlation coefficient
was greater (in magnitude) than 0.505. The true activation map contained 3454 active voxels in total. Based
on these activation maps, false-positive and false-negative activation counts were tabulated for each of the 40
trials.
Proc. of SPIE Vol. 5032 135

-0.02
-0.01
0
0.01
0.02
0 5 10 15 20 25 30 35 40
Deg.
Volume Number
Roll
-0.08
-0.06
-0.04
-0.02
0
0.02
0 5 10 15 20 25 30 35 40
Deg.
Pitch
-0.02
-0.01
0
0.01
0.02
0 5 10 15 20 25 30 35 40
Deg.
Yaw
-0.02
-0.01
0
0.01
0 5 10 15 20 25 30 35 40
mm
Volume Number
Superior
Std.
SRA
-0.01
0
0.01
0.02
0 5 10 15 20 25 30 35 40
mm
Left
-0.02
0
0.02
0.04
0.06
0.08
0 5 10 15 20 25 30 35 40
mm
Posterior
Figure 2. Typical motion detection error plots for the standard least-squares algorithm (Std) and the SRA algorithm.
Notice that some of the motion parameters for the standard registration method are highly correlated to the stimulus.
4. RESULTS
Figure 2 shows a typical set of motion errors resulting from the different methods. Notice that the standard
least-squares method often exhibits large excursions from zero during the two stimulus epochs (frames 5 to 15,
and frames 25 to 35). The motion errors for the SRA method exhibit no visible stimulus-correlation.
The false-positive and false-negative activation rates for the two methods are graphed in Fig. 3. For the
datasets in which activation was present (scenarios 1, 2 and 4), the SRA algorithm reduced the number of
false-positives by approximately two-thirds, and reduced the number of false-negatives by approximately one-
third compared to the standard method. The two methods performed equally well on scenario 3 datasets, each
averaging about 40 false-positives per trial.
Examples of typical activation masks produced in this study are shown in Fig. 4. The activation mask
produced by the SRA method is very similar to the actual activation mask, and does not show any additional
large clusters of activation. The mask resulting from the standard least-squares registration method shows
large clusters of false-positive activation. As expected, the clusters form near contrast borders.
5. DISCUSSION AND CONCLUSIONS
By combining the registration and activation detection steps into a single optimization problem, the SRA
algorithm properly decomposes the dataset into the baseline image, the BOLD signal, and motion. This
algorithm avoids stimulus-correlated registration errors while still allowing for true stimulus-correlated motion
to be corrected.
The SRA method consistently outperformed the standard least-squares algorithm on all simulated datasets
containing activation. Use of the SRA method instead of least-squares registration resulted in a drastic reduc-
136 Proc. of SPIE Vol. 5032

0
100
200
300
400
500
600
700
800
False−Positive Voxels
Scenario 1:
Activation,
Random
Motion
Scenario 2:
Activation,
Stim−Corr
Motion
Scenario 4:
Activation,
No Motion
Scenario 3:
No Activation,
Stim−Corr
Motion
SRA
Std.
0
100
200
300
400
500
600
700
800
False−Negative Voxels
Scenario 1:
Activation,
Random
Motion
Scenario 2:
Activation,
Stim−Corr
Motion
Scenario 4:
Activation,
No Motion
Scenario 3:
No Activation,
Stim−Corr
Motion
SRA
Std.
Figure 3. False-positive and false-negative activation rates for the standard least-squares algorithm and the SRA
algorithm. Scenario 3 has no false-negatives because there is no activation added to the dataset and the true activation
mask is blank.
a) b) c)
Figure 4. Activation masks for the actual (a), standard least-squares (b), and the SRA method (c) for a central slice
of a scenario 1 dataset, overlaid on the corresponding EPI slice for anatomic reference.
Proc. of SPIE Vol. 5032 137

tion in false-positive and false-negative activation rates, with a drop of 67% and 33% respectively. Both the
standard and SRA methods handle the activation-free trials appropriately.
Some methods attempt to remove residual motion artifacts as a part of activation detection, but run the
risk of ignoring activation if it is synchronized with motion. These methods are particularly problematic
when the patient moves in step with the stimulus, a phenomenon that is fairly common.3The SRA method
effectively handles these situations, performing equally well on datasets that contain either random or genuine
stimulus-correlated motion.
At the core of the simultaneous method outlined in this paper is the well-justified assumption that regions
of activation form localized clusters, while the rest of the activation map is close to zero. This principle is
incorporated into the model by applying an additional constraint to the optimization problem. Although
excellent results were attained with the aforementioned constraints (see Eqn. (16)), other forms may yield even
better results. Study of these forms remains a topic for future investigation.
Solving the combined least-squares problem is feasible on a modest computer system. This is made possible
in part by a careful consideration of the numerical aspects of our implementation. All of the analysis done for
this study was executed on a desktop computer with a 1.2 GHz AMD Athlon CPU and 512 MB of memory.
In conclusion, the SRA method is able to overcome the pitfalls encountered when performing registration
and activation detection sequentially. It is well motivated, has a solid theoretical basis, and is shown to be
robust in all tested combinations of motion and activation. Further testing of the SRA algorithm is ongoing,
including demonstration on in vivo data, as well as comparison to other registration methods such as mutual
information, robust estimators, and a customized two-stage method developed by Freire et al.16
ACKNOWLEGMENTS
The work of Jeff J. Orchard was supported in part by the Natural Sciences and Engineering Research Council of
Canada (NSERC), scholarship number PGSB-222602-1999. The work of Gene H. Golub was in part supported
by NSF grant NSF CCR-9505393.
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