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

Figure 3.2: Typical Static Imaging System (from [3]).
1. Obtain an initial approximation for the conductivity distribution. The initial conductivity,
σ0, distribution of the model reflects an a priori assumption about the
conductivity distribution of the medium. However, it is often a crude estimate of the
equivalent homogenous conductivity of the medium based on the data [69].
2. Solve the forward problem to determine the simulated measurements, vsimulated.
3. Calculate the change in conductivity,
where
∆σ = (H T H + λ 2 R T R) −1 H T z (3.11)
z = vmeasured − vsimulated
4. Update the absolute conductivity,
(3.12)
σk+1 = σk + ∆σ (3.13)
where σ0 is a vector of length E and is the initial, a priori, conductivity.
In terms of the Jacobian used for static imaging, each element of the Jacobian is
Hij = ∂zi
�
�
and relates a small change in the ith error measurement, zi where z is as
� ∂xj σk
defined in 3.12, to a small change in the conductivity of jth element. H is a function
of current injection pattern and the kth conductivity estimate. Thus calculation of
the Jacobian is identical for both difference and static imaging however with static
imaging the signal is the error signal, 3.12, and the conductivity change is used in the
iterative build up of the absolute conductivity via 3.13.
5. Update the admittance matrix with the current estimation of the conductivity. In
other words form Y(σk+1).
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