fMRI: Biological Basis and Experiment DesignLecture 20: Motion compensation
• Rotation matrices• Effects on data• Examples
1 light year = 5,913,000,000,000 miles?
Before
After
Rotation matrices
cos( ) sin ( )
-sin( ) cos( ) R =
r = (x,y)r' = R r r'
x
yTwo-dimensional rotation:
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:
An aside: matrix multiplication
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
y is [m x p]
x1,1 x1,2 x2,1 x2,2
x3,1 x3,2
. . .xn,1 xm,2
x =
n
jjjxA
11,,1
n
jjjxA
11,,2
n
jjjm xA
11,,
n
jjjxA
12,,1
n
jjjxA
12,,2
n
jjjm xA
12,,
x is [n x p]A is [m x n]
Rotation matrices
cos( ) sin ( )
-sin( ) cos( ) R =
r = (x,y)r' = R r r'
x
yTwo-dimensional rotation:
cos( ) sin ( )
-sin( ) cos( )
x
y=
x cos( ) + y sin ( )
-x sin( ) + y cos( ) r' =
Rotation example: 45 degree rotation of r = (1,1)
cos(45 ) sin (45 )
-sin(45 ) cos(45 ) R =
r = (1,1)
r' = R r
= 45x
yTwo-dimensional rotation:
1/2 1/2
- 1/2 1/2
1
1=
1/2 + 1/2
-1/2 + 1/2 r' = =x
2/2
0
r' = (2,0)
Strong activation affects center of mass calculation
Output of MoCo algorithms
Translation – one scan
Translation – all scans