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A comprehensive framework for time series realignment
NeuroImage
11, Number
5, 2000, Part 2 of 2 Parts 10
E b
METHODS
1”
- ACQUISITION
A comprehensiveframework for time series realignment Jesper Andersson, John Ashburner,
Karl Friston
Wellcome Dept. of Cogn. Neurology, London, U.K. Subject movement during scanning is an important source of unwanted variance in functional neuroimaging studies with PET and fMRI. This is often alleviated by rigid body realignment of the images to some arbitrary reference image within the series. This is sufficient in many cases but not all, e.g. in fMR1 where there is considerable residual movement-related variance after realignment (1) or dynamic PET tracer studies where changes unrelated to movement confound the estimation of movement parameters (2). We present here a more comprehensive framework for realignment in which e.g. models for deformations due to field inhomogeneities or changes due to tracer uptake may be incorporated. The model One commonly used method for realignment (3) proceeds by linearising the problem such that the difference between two images f, and f2 is approximated by f,-f,EDp where D is the nx6 matrix of partial derivatives off, w.r.t. the rigid motion parameters. For each iteration this ib solved m a least squares sense with p=(D’D)-‘D’(fz-f,). Now consider extending this to m images, yielding the model
where p, is the vector of motion parameters mapping f, onto f,. This extension would yield the same results as a sequential pair wise registration and its only consequence would be to complicate the implementation. Its usefulness becomes evident when attempting to simultaneously model other sources of variance, which will be exemplified here by a stationary deformation field: One possible source of residual variance after realignment of EPI images is the presence of field inhomogeneities leading to a displacement of signal, particularly in the phase encoding direction (4). In the presence of subject movement this means that we image an object at different positions in a deformation field yielding slightly different deformations for each image. The effect is equivalent to looking oneself in a curved mirror at an amusement park. As one moves the shape of ones features appear to change, and there is no rigid body transformation that maps the face sampled at one position onto that sampled at another.This may be corrected through direct measurement of the field inhomogeneities (4.5). The framework suggested here may offer a way to estimate the differential deformations that are caused by the interaction between field inhomogeneity and subject movement. Consider a deformation field (in the phase encoding direction, x2) parameterised as a linear combination of basis warps such that [y,. yl, Y~]~=[xI, xz+0,~,(x)+13,B,(x)+. .&B,(x), x,IT where x=[x,, xZr xsJT represent the original coordinates, y the transformed, 8, to Br the k basis warps and 8, to Br the corresponding weights. For a fixed deformation field the differential effects at different positions are accommodated by sampling the basis functions at different positions such that if B(x)6 is the effect seen by f,, then B(x(p,))B, where x(p) is a vector function returning position, is the effect seen by f,. The full realignment model then becomes
@WP,))- B(x)) r,,; @W.P4 -B(x))
t
\
““1,
‘:’
(SQL))-B(X))
.F. 1
where o denotes the Kronecker tensor product and x denotes the direct (Hadamard) product. One possible choice of basis-warps is the discrete cosine set previously used for spatial normalisation (Ashburner & F&on, 1999). Simulations To examine the feasibility of estimating an unknown deformation field from multiple images of an unknown object at different positions a number of vimulation~ were performed in 2D. ‘The results indicate that it is feasible, that a reasonable estimate of the deformation field is obtained and that residual variance after realignment is reduced.
We have formulated a framework for modelling subject mwements m time series. This formulation allows incorporation of second order effects of movemenl potentially reducing residual variance and/or effects unrelated to movement potentially improving the estimate of movement itself. The model needs an efficient implementation for 3D to properly assess its usefulness. least squares and EM type algorithms. References 1. 2. 3. 4. 5. 6.
F&on K. Dagher A. F&on K. Jezzard P. Hutton C. Ashbumer
et al. (1996) Magn Reson Med, 35, 346. et al. (1997) NemoImage, 5, B75. et al. (1995) Hum Brain Map, 2, 165. & Glare S. (1999) Hum Brain Map, 8, 80. et al., Submitted to this meeting. I. & Friston K. (1999) Hum Brain Map, 7, 254
S488
We are presently
investigating
iterative linear
11, Number
5, 2000, Part 2 of 2 Parts 10
E b
METHODS
1”
- ACQUISITION
A comprehensiveframework for time series realignment Jesper Andersson, John Ashburner,
Karl Friston
Wellcome Dept. of Cogn. Neurology, London, U.K. Subject movement during scanning is an important source of unwanted variance in functional neuroimaging studies with PET and fMRI. This is often alleviated by rigid body realignment of the images to some arbitrary reference image within the series. This is sufficient in many cases but not all, e.g. in fMR1 where there is considerable residual movement-related variance after realignment (1) or dynamic PET tracer studies where changes unrelated to movement confound the estimation of movement parameters (2). We present here a more comprehensive framework for realignment in which e.g. models for deformations due to field inhomogeneities or changes due to tracer uptake may be incorporated. The model One commonly used method for realignment (3) proceeds by linearising the problem such that the difference between two images f, and f2 is approximated by f,-f,EDp where D is the nx6 matrix of partial derivatives off, w.r.t. the rigid motion parameters. For each iteration this ib solved m a least squares sense with p=(D’D)-‘D’(fz-f,). Now consider extending this to m images, yielding the model
where p, is the vector of motion parameters mapping f, onto f,. This extension would yield the same results as a sequential pair wise registration and its only consequence would be to complicate the implementation. Its usefulness becomes evident when attempting to simultaneously model other sources of variance, which will be exemplified here by a stationary deformation field: One possible source of residual variance after realignment of EPI images is the presence of field inhomogeneities leading to a displacement of signal, particularly in the phase encoding direction (4). In the presence of subject movement this means that we image an object at different positions in a deformation field yielding slightly different deformations for each image. The effect is equivalent to looking oneself in a curved mirror at an amusement park. As one moves the shape of ones features appear to change, and there is no rigid body transformation that maps the face sampled at one position onto that sampled at another.This may be corrected through direct measurement of the field inhomogeneities (4.5). The framework suggested here may offer a way to estimate the differential deformations that are caused by the interaction between field inhomogeneity and subject movement. Consider a deformation field (in the phase encoding direction, x2) parameterised as a linear combination of basis warps such that [y,. yl, Y~]~=[xI, xz+0,~,(x)+13,B,(x)+. .&B,(x), x,IT where x=[x,, xZr xsJT represent the original coordinates, y the transformed, 8, to Br the k basis warps and 8, to Br the corresponding weights. For a fixed deformation field the differential effects at different positions are accommodated by sampling the basis functions at different positions such that if B(x)6 is the effect seen by f,, then B(x(p,))B, where x(p) is a vector function returning position, is the effect seen by f,. The full realignment model then becomes
@WP,))- B(x)) r,,; @W.P4 -B(x))
t
\
““1,
‘:’
(SQL))-B(X))
.F. 1
where o denotes the Kronecker tensor product and x denotes the direct (Hadamard) product. One possible choice of basis-warps is the discrete cosine set previously used for spatial normalisation (Ashburner & F&on, 1999). Simulations To examine the feasibility of estimating an unknown deformation field from multiple images of an unknown object at different positions a number of vimulation~ were performed in 2D. ‘The results indicate that it is feasible, that a reasonable estimate of the deformation field is obtained and that residual variance after realignment is reduced.
We have formulated a framework for modelling subject mwements m time series. This formulation allows incorporation of second order effects of movemenl potentially reducing residual variance and/or effects unrelated to movement potentially improving the estimate of movement itself. The model needs an efficient implementation for 3D to properly assess its usefulness. least squares and EM type algorithms. References 1. 2. 3. 4. 5. 6.
F&on K. Dagher A. F&on K. Jezzard P. Hutton C. Ashbumer
et al. (1996) Magn Reson Med, 35, 346. et al. (1997) NemoImage, 5, B75. et al. (1995) Hum Brain Map, 2, 165. & Glare S. (1999) Hum Brain Map, 8, 80. et al., Submitted to this meeting. I. & Friston K. (1999) Hum Brain Map, 7, 254
S488
We are presently
investigating
iterative linear