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3.1 Simultaneous Iterative Reconstructive Technique (SIRT) 17
The idea about Cimmino’s reflection method is that the next iterate can be
described using an equal weighting of the reflections of x k on Hi. Reflections
on hyperplanes is the following:
Ri(z) = z + 2 bi − 〈ai , z〉
ai2 a
2
i .
The reflection method then uses the average of the reflections of x k onto the
hyperplanes Hi to determine the direction of the step to the new iteration.
Figure 3.1 illustrates the concept in R 2 for a consistent problem. The method
can then be written as follows:
x k+1 = x k + λk
m 1
wi Ri(x
m
k ) − x k ,
i=1
where the relaxation parameter λk determines how much of the step is taken
from x k to the new iterate x k+1 and wi > 0 are user-defined weights. Using the
definition of reflections we get the following:
x k+1 = x k + λk
m 2
m
i=1
wi
bi − 〈ai , xk 〉
ai2 ai for k = 0, 1, . . ..
2
Cimmino’s reflection method can be written using matrix notation on the form
wi
for i = 1, . . .,m and T = I.
(3.1), where M = 2
m diag
a i 2 2
We will now introduce Cimmino’s projection method. Using an equal weighting
of all the equations the next iterate in Cimmino’s projection method can be
described using orthogonal projections of x k on Hi. As shown in appendix A.1
an orthogonal projection of the vector z on the hyperplane Hi is the following:
Pi(z) = z + bi − a i , z
a i 2 2
a i . (3.3)
Cimmino’s projection method uses the average of the projections of x k onto the
hyperplanes Hi to determine the direction of the step to the new iterate. Figure
3.2 illustrates the concept in R 2 for a consistent problem.
The new iterate can then be described as the current iterate plus a contribution
of the average of the found step direction. We can therefore write Cimmino’s
projection method as the following:
x k+1 = x k + λk
1
m
m
i=1
wi Pi(x k ) − x k ,