Motion Correction in Emission Tomography - Correction Techniques in Emission Tomography - page 175

consistency-based optical flow methods [16],[8] described before as it is based
on the continuity equation. This change in the basic model is necessary as
brightness consistency is not given in cardiac-gated PET data due to the
PVE.
The law of mass conservation says that the total mass in a closed system is
conserved. If we substitute activity by mass, the law must still hold in our case,
as the total activity in the system remains the same from systolic to diastolic
phases of the heart. It is blurred only in the diastolic phase. It should be
noted that our data is pre-corrected for the time-dependent radioactive decay
during the reconstruction process so that the decay itself plays no role in our
considerations.
The continuity equation for mass conservation is given as [11]

@I
@t + div(Iu) = 0

(8.15)

where I is the intensity value, u = (u;v;w) T is the velocity vector, i.e., the
optical flow. Deviations from this equation can be penalized by the following
functional:

Z

div(u)) 2 dxdydz: (rI · u + I t + I · (8.16)

The derivative in time I t can be calculated on discrete image volumes by
using the difference: I 2 I 1 , where I 2 is the floating and I 1 is the target
image volume. As with the intensity-based optical flow, this is again an under-
determined system of equations and therefore a smoothing term can be added
to solve it. The same smoothing term as given in the Horn{Schunck algorithm
above can be used here as well. The resulting optical flow functional is thus:

Z Z

D 2 dV +

f = argmin SdV (8.17)

V V

with

div(Iu) + I t ; S = jruj 2 + jrvj 2 + jrwj 2 :

D =

The minimization of the equation (8.17) can be achieved by using the
corresponding Euler{Lagrange equations. These are given by

0 = D x I + u

0 = D y I + v (8.18)

0 = D z I + w

where D x ;D y ;D z are the first derivatives of D in the corresponding directions.
Here, is a weighting parameter.

8.9.1 Correcting for motion
Once the optical flow is found (see Figure 8.7) the images have to be
transformed to get the motion-corrected data. Equation (8.15) can be used