Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 45

2.2.1 Lucas-Kanade Algorithm

As mentioned above, the flow vector u T =( u , v , w , 1 ) with three unknowns has to

be estimated at each spatial position x T

. To obtain an equation system so
that the problem can be solved, Lucas and Kanade [ 86 ] made the assumption that
the motion field is constant in a small area around x . With Eq. ( 2.56 )wehave

=( x , y , z )

I x u + I y v + I z w + I t = 0 (2.57)

or

u T ∇ I = 0 . (2.58)

Assuming a constant flow u in a small window of size m × m × m , m > 1, centered

at the voxel position x and denoting the neighbors as ( x i , y i , z i ) , i = 1 ,..., n , we get

a set of equations

I x 1 u + I y 1 v + I z 1 w + I t 1 = 0

I + I + I + I =

x 2 u y 2 v z 2 w 0 t 2

(2.59)

. . .

I x n u + I y n v + I z n w + I t n = 0

where I x i , I y i , I z i , and I t i , i = 1 ,..., n , represent the shortterm for I x ( x i , y i , z i , t ) ,

I y ( , respectively. Usually the central

voxel is given more weight to suppress noise and thus a Gaussian weighting function
is applied to obtain a modified data term [ 135 ]

x i , y i , z i , t ) , I z ( x i , y i , z i , t ) , and I t ( x i , y i , z i , t )

2 :

u T

T

D LK ( I, u ) = ( G ∗ ( ∇ I ∇ I )) u , (2.60)

where G is a Gaussian smoothing kernel which is convolved component-wisely with
the image gradients

T of the neighboring voxels. This minimization problem
is solved in a standard least-squares manner.
The optical flow estimated with the Lucas-Kanade algorithm fades out quickly
with increasing distance from the motion boundaries. The method is comparatively
robust in presence of noise [ 9 , 19 ].

∇ I ∇ I

Algorithm 2.2 Lucas-Kanade optical flow method
Input: Images I ( x , t ) and I ( x , t + 1 )
Output: Motion estimate u (for all voxels)
estimate ∇ I of each voxel for I ( x , t )
for each voxel x do
compute u ( x ) by least-squares based minimization of Eq. ( 2.60 )
end for