Image Reconstruction for 3D Lung Imaging - Department of Systems ...

6. Evaluate a stopping rule. For example stopping after a single iteration [35], stopping
after some fixed number of iterations, or stopping after the difference between the two
sets of measurements drops below some threshold [114], i.e. ε ≤ �vmeasured − vsimulated�.
If the current solution satisfies the stopping rule then exit, otherwise continue to step
7.
7. Update the Jacobian based on the current estimate of the conductivity. Some researchers
update the Jacobian at each iteration, others do not.
8. go to step 2. Note that vsimulated calculated at step 2 is a function of the iteration
number, k.
Equation 3.11 is similar to the difference image equation 3.7 with ˆx = ∆σ and z defined
as the difference between the measured voltages and the set of simulated voltages, z =
vmeasured − vsimulated. Although the interpretation of x and z are different the Jacobian is
the same as those used for difference imaging. Often the regularization matrices are also
the same as those used in difference imaging.
3.3.1 MAP Regularized Inverse
The most clearly formulated reconstruction model for 2D difference imaging at the start of
this work was the Maximum a Posteriori (MAP) algorithm of Adler and Guardo [4]. The
MAP approach to image reconstruction defines the solution as the most likely estimate of
ˆx given the measured signal z and certain statistical information about the medium. This
approach allows an elegant interpretation of the image reconstruction algorithm in terms of
statistical properties of the experimental situation. It is explained in the following section.
In order to simplify the reconstruction algorithm the image statistical properties are
modeled by a Gaussian distribution of mean x∞ and covariance Rx
x∞ = E [x]
Rx = E [x − x∞] = E � x T x � − x T ∞ x∞
With these parameters the distribution function of the image, f(x), is modeled as
f(x) =
1
(2π) N/2 � |Rx| e−(1/2)(x−x∞)T R −1
x (x−x∞)
(3.14)
(3.15)
The a posteriori distribution function of z given a conductivity distribution x is derived
from the definition of the inverse problem, equation 3.1:
f(z|x) =
1
(2π) M/2 � |Rn| e−(1/2)(z−Hx)T R −1
n (z−Hx)
(3.16)
The difference (z − Hx) is due entirely to the noise n, which is assumed to be Gaussian,
white, zero mean with covariance Rn. Thus
⎡
⎤
Rn = E � n T n � ⎢
= ⎢
⎣
σ2 1 0 · · · 0
0 σ2 2 0
.
. ..
0 0 · · · σ 2 M
.
⎥
⎦
(3.17)
where σ in this and all subsequent equations in this section, represents the square root of
the variance and not the conductivity.
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