Motion Correction in Emission Tomography - Correction Techniques in Emission Tomography - page 166

(x;y;z;t) in the corresponding directions. We shall write I x ,I y ,I z and I t for
the derivatives in the following.
Thus

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

or

V = I t : (8.3)
This is a single equation with three unknowns and therefore the system is
under-determined. To find the optical flow additional equations are required.

rI ·

8.3.2 Optical flow methods
Many optical flow methods have been proposed to solve the image con-
straint equation. Three main categories can be used to classify these methods:

• Block matching

• Frequency domain correlation

• Gradient based

In the block matching{based optical ow algorithms, small blocks of one
image are moved so that they match with a block on the other image. The
motion vector gives the transformation for this block only. All blocks in an
image are moved until vectors are found that minimize some error function.
It was shown by Davis/Freeman that block matching optical flow algorithms
are equivalent to gradient-based algorithms if the displacements are sub-voxel
and the deformations are rigid body deformations [14]. Zhang/Lu have even
combined both approaches to a hybrid block matching algorithm which utilizes
gradient information [66]. Behar et al. combined block matching with gradient-
based optical flow in a small window [5]. Correlation-based optical flow as
presented by Duan et al. [23] is also a form of block matching.
One way of detecting motion on the images is to analyze their frequency
transforms such as the Fourier transform. The diculty with such an approach
is that the Fourier transform is global and thus scenes including many objects
undergoing motion render it not suitable for this task. Reed [57] uses a Gabor
lter{based local frequency approach to overcome this problem and detect
motion in the frequency domain of the images. A similar approach was al-
ready used by Fleet/Japson [26]. They use a decomposition of the images into
band-pass channels and use a phase constraint equation in each channel to de-
tect motion. Prince et al. use a combination of band-pass and gradient based
optical flow [54]. Phase correlation-based optical flow [61] also falls under this
category.
There have been many proposals for image gradient{based methods of esti-
mating optical ow. Lucas{Kanade proposed a solution that assumed the ow
to be constant in a local neighborhood around the central voxel and used the
least means of square approach [40]. Bab-Hadiashar improved this method by