Motion Estimation - Motion Correction in Thoracic Positron Emission Tomography - page 40

Coarse level
Intermediate level
Fine level

Transformation parameter

Fig. 2.6 Artificial example for the behavior of an objective function in a multi-level setting. For
each level, going from the coarse to the fine level, the value of the registration functional is plotted
against the (artificial) transformation parameter. With increasing level, the function is getting less
smooth due to the increased level of details in the images. Note the slight shift of the global
minimum between the different levels

2.1.5.1 Gauss-Newton

The minimization of the discretized registration functional can be solved by

means of the Gauss-Newton method. To this end, we can approximate the functional
in Eq. ( 2.1 ) by a Taylor expansion (leaving the regularization term aside)

J

1
2 ( ∇

T 2

J ( y + ∇ y ) ≈J ( y )+ ∇ J ( y ) ∇ y + y ) ( ∇ J ( x ))( ∇ y )

1
2 ( ∇

T H

= J + ∇ J ∇ y + y ) ( ∇ y ) , (2.50)

where H is the (approximated) Hessian matrix. Let us assume the functional J is

given by a residual r ( y ) and an outer function ψ ( · ) . In the case of SSD this is

1

2 x 2 . The Hessian matrix H is then approximated by

r ( y )= T◦ y −R and ψ ( x )=

2 T 2

H = ∇ J ( x )= ∇ r ( x ) ∇ ψ ∇ r ( x ) . (2.51)

According to [ 95 ] a minimizer of J is then given by

H ∇ y = − ∇ J . (2.52)

The last step in the Gauss-Newton method is the search for an adequate step size
for the motion update y . The updated transformation should reduce the objective

function's value compared to the previous guess. A simple way to do this is the
Armijo line search. The idea is to add the previously determined search direction
to the transformation with decreasing step size (it is halved in each step) until a
reduction of the objective function's value occurs.

∇