fMRI: Biological Basis and Experiment Design Lecture 20: Motion compensation

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fMRI Biological Basis and Experiment
DesignLecture 20 Motion compensation
Before

• Rotation matrices
• Effects on data
• Examples

After
1 light year 5,913,000,000,000 miles?
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Rotation matrices
Two-dimensional rotation
y
cos(? ) sin (? ) -sin(? ) cos(? )
R
x
?
r (x,y)
r' R r
r'
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An aside matrix multiplication
y Ax
A1,1 A1,2 A1,3 ... A1,n A2,1 A2,2 A2,3
... A2,n A3,1 A3,2 A3,3 ... A3,n . .
. Am,1 Am,2 Am,3 ... Am,n
A is an m x n matrix
x1,1 x1,2 x2,1 x2,2 x3,1 x3,2 . .
. xn,1 xm,2
x is an n x p matrix
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An aside matrix multiplication
y is m x p
x is n x p
A is m x n
A1,1 A1,2 A1,3 ... A1,n A2,1 A2,2 A2,3
... A2,n A3,1 A3,2 A3,3 ... A3,n . .
. Am,1 Am,2 Am,3 ... Am,n
x1,1 x1,2 x2,1 x2,2 x3,1 x3,2 . .
. xn,1 xm,2
x

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Rotation matrices
Two-dimensional rotation
y
cos(? ) sin (? ) -sin(? ) cos(? )
R
x
?
r (x,y)
r' R r
r'
cos(? ) sin (? ) -sin(? ) cos(? )
x y
x cos(? ) y sin (? ) -x sin(? ) y cos(? )

r'
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Rotation example 45 degree rotation of r (1,1)
Two-dimensional rotation
y
r (1,1)
cos(45? ) sin (45? ) -sin(45? ) cos(45?
)
R
? 45?
x
r' (?2,0)
r' R r
1/?2 1/?2 - 1/?2 1/?2
1 1
1/?2 1/?2 -1/?2 1/?2
2/?2 0

r'

x
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Rotation example parabola (ICE10.m)
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Three dimensional rotation
Excerpt from Wolfram MathWorld
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Affine transformation
(Wikipedia)
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Output of MoCo algorithms
Strong activation affects center of mass
calculation
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Effect of MoCo increased activation size
Before
After
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Effect of MoCo voxels maintain identity
Before
After
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Oakes et al., Fig. 1 6
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Interpolation in motion correction
uncorrected
corrected