Biomedical Engineering Reference
In-Depth Information
Obviously, in step 2, we have an infinite number of choices for the change.
We select the vector for that makes the smallest (minimum-norm) change.
Under mild conditions, this simple algorithm can be shown to converge.
Put in a more mathematical fashion, and using f
k
for the image in the kth
step of the algorithm:
1. Start off with an initial guess f
0
, typically, f
0
= 0, and set j = 0.
2. Choose M
0
of the available equations. Extract the corresponding lines
from A;g to A
j
;g
j
, such that A
j
f = g
j
.
3. Choose f
j+1
as the orthogonal projection of f
j
onto the subspace of all
f with A
j
f = g
j
, that is:
f
j+1
= f
j
+ !A
j
(A
j
A
j
)
1
(Af
j
g)
where A
j
is the adjoint of A
j
.
4. j = j + 1 and iterate from 2.
where we have introduced an iteration parameter !. In order to circumvent
the matrix inversion, (AA
)
1
can be replaced by any positive definite matrix
C
j
. We thus arrive at the final update step
f
j+1
= f
j
+ !A
j
C
j
(A
j
f
j
g
j
):
(3.1)
The iteration can be shown to converge towards the minimum norm solu-
tion of Af g, provided ! is small enough, and all equations are used for the
same number of times.
We remark that the convergence speed depends heavily on the order of
selected equations. If in the first two steps of the algorithm, perpendicular
matrix rows a
k
are used, the algorithm gives the correct answer after only two
iterations. However, if for any k, a
k+1
is almost parallel to a
k
, the improvement
per step will be very small, and the algorithm will converge very slowly on
the order of 1=k. This remark is also true for N large.
So, the implicit rule should be: choose the equations in such a way that
the corresponding matrix rows are as orthogonal as possible. While algorithms
for achieving such an optimal order exist, it turns out that the main rule is
to avoid the worst case of almost parallel lines. Choosing the equations at
random usually shows not much difference in speed to optimal orderings.
3.3.2 EM
To fix ideas, we restrict our discretization model to 3D and voxels in this
case.
Up to now, we completely neglected the fact that the data we have are
measurements of a random rather than continuous process. In fact, the number
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