Biomedical Engineering Reference
In-Depth Information
Z
g
l
=
f(x)dx
L
l
Z
X
=
k
k
(x)dx
L
l
k
Z
X
=
k
(x)dx
k
L
l
k
X
=:
a
lk
k
k
or
g = A
where A = (a
lk
) is an M by N matrix, = (
l
) and g = (g
k
) are vectors. In
the case of pixels, a
lk
is just the length of the intersection of Ll
l
with pixel k.
All we have to do is compute the system matrix A once and for all for
a given system and solve the linear equation for each dataset g. Since A is
extremely sparse, it is not economical to compute the (pseudo) inverse of A
directly, and iterative methods are the appropriate tool for inversion.
Notice the big flexibility of these algorithms: valid for 2D and 3D, can deal
with any scanner geometry, can easily incorporate additional measurements
(cf. time of flight, TOF), and can incorporate information about f by choosing
the
k
appropriately. The major disadvantage is the usually slow runtime
when compared to analytical algorithms.
3.3.1 ART|Algebraic reconstruction technique
ART is used for CT rather than PET, but we will gain some insight also
for the EM algorithm.
In each iterative algorithm, we start with an initial guess and update
that value in each step, hopefully producing a sequence that converges toward
a solution f. The main idea in the ART or Kaczmarz algorithm is to use only
a small portion of the equations in each step. If the number of equations is
small, the corresponding linear system can easily be inverted and used for the
update.
The idea of ART or Kaczmarz can then be summarized as:
1. Start with an initial guess for , typically, = (0).
2. Choose one single measurement line L with measurement value g
L
.
Change the values of such that the integral over L in the image be-
comes g
L
.
3. Go back to 2.
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