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

3.7 Summary
3.7.1 Reconstruction Summary
Here we summarize the state of the art in EIT difference imaging for clinical applications
such as pulmonary imaging. The framework is the non-linear optimization problem, equation
3.7, which is reproduced below:
ˆx = arg min
x
�
�Hx − z� 2 + λ 2 �Rx� 2�
(3.24)
This is solved using the MAP regularized framework of equation 3.22 again repeated below:
ˆx = (H T WH + λ 2 R T R) −1 H T Wz = B(λ)z (3.25)
where z = v2 − v1. The framework has several explicit parameters that must be selected
by the user:
1. The regularization hyperparameter, λ, is the subject of chapter 4.
2. The norm of the prior, �Rx� 2 , has historically been the ℓ 2 norm. The ℓ 1 norm has
been used for “blocky” reconstructions. An algorithm for solving the ℓ 1 norm is
evaluated in chapter 7.
3. The prior matrix, R, has many possibilities as discussed in this chapter.
4. The data weighting matrix W has the ability to consider noise and erroneous electrode
data. However with equal noise variance on each measurement channel and with good
electrodes (no accounting for erroneous electrodes), W becomes a scaled version of
the identity matrix.
In addition to these explicit parameters there are several implied parameters that conform
to some assumptions:
1. The initial conductivity, σ0, is typically assumed to be homogenous.
2. The conductivity used to calculate the Jacobian, σ ∗ , is typically assumed to be homogenous.
3. FEM modeling issues including degree of the shape functions (linear, quadratic),
isotropy of element conductivity, and mesh parameters such as number and degree
(triangle, quadrilateral) of elements, geometry, shape of the reconstructed mesh dimension
(2 or 3).
4. Electrode types, locations and size.
5. Current injection and measurement patterns.
None of these parameters appear explicitly in equations 3.24 or 3.25 but are important
parts of the problem. There is much work describing variations of the framework in terms
of explicit and implied parameters, however, there is little quantitative information on how
they compare and how important any one of them is.
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