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)

Rotation example: parabola (ICE10.m)

Three dimensional rotation

Excerpt from Wolfram MathWorld

Affine transformation

(Wikipedia)

Strong activation affects center of mass calculation

Output of MoCo algorithms

Translation – one scan

Translation – all scans

Effect of MoCo – increased activation size

Before After

Effect of MoCo – voxels maintain identityBefore

After

Oakes et al., Fig. 1 & 6

Interpolation in motion correction

uncorrected corrected