Motion Correction in Emission Tomography - Correction Techniques in Emission Tomography - page 168

Klein [35] was probably one of the first to apply optical flow to cardiac
PET studies for motion correction. Vemuri et al. used optical flow to register
medical images [65]. Dawood et al. have used optical flow to correct PET
images for the effects of motion [17],[16],[18].
Among other medical imaging modalities, Behar et al. used optical flow in
echocardiography [5].
We now turn to three optical flow algorithms based on the image gradients.

8.4 Lucas{Kanade optical ow
Lucas and Kanade use a non-iterative method which that a locally constant
flow [40]. Evaluations and comparisons by Barron [3], Bruhn [9] and Galvin
[30] have shown that the Lucas{Kanade algorithm is one of the best methods
for calculating optical flow fields especially in presence of noise.
Assuming that the ow (u;v;w) is constant in a small window of size
mmm with m > 1, which is centered at voxel x;y;z and numbering the
pixels as 1:::n we get a set of equations:

I x 1 u + I y 1 v + I z 1 w = I t 1
I x 2 u + I y 2 v + I z 2 w = I t 2
. . .
I x n u + I y n v + I z n w = I t n :
With this we get an over-determined system:
0
@

1
A

0
@

1
A

I x 1 I y 1 I z 1
I x 2 I y 2 I z 2
. . .
I x n I y n I z n

I t 1
I t 2
.
I t n

0 1

u
v
w

@ A =

or

AV = b:
To solve the this system of equations the least squares method can be used:

A T AV = A T (b)

or

V = (A T A) 1 A T (b)

(8.4)

or

0 1 0 1

1

I x I x I y I x I z
I x I y I y I y I z
I x I z I y I z I z

u
v
w

A T I t :

@ A = @ A