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

a hyperparameter selection method, BestRes does not consider problems associated with
equipment such as uncertain electrode positioning, electrode and cable problems, and electromagnetic
interference (EMI) from other sources in the clinical environment. The problem
of uncertain electrode positioning is mitigated primarily by using difference measurements
and should not be addressed through hyperparameter adjustments. Electrode and cable
errors are best addressed by detecting the problem and alerting an operator to fix it, or by
compensating for the problem a posteriori through use of an algorithm such as that of [7].
Excessive EMI in a clinical environment [89] will swamp the signal such that extraction of
information is impossible regardless of hyperparameter. The BestRes method is a calibration
technique to choose a hyperparameter based on a noise level that is fixed, regardless
of whether the noise is representative of clinical data or added to simulated data.
4.5.3 Effect of radial position on λBestRes
Figure 4.10 is a plot of Resolution and λBestRes as a function of radial position of the
generating contrast. These curves were generated by reconstructing a set of simulated data
generated by an impulse contrast located at increasing radial positions. The best resolution
is achieved for contrasts located at 75% of radial distance from the centre. Although λBestRes
continues to increase as the radial position increases past 90%, the corresponding resolution
starts to decrease due to blurring caused by proximity to the edge.
4.5.3.1 Normalization
Although primarily concerned with proportional (normalized) difference imaging we also
investigated the performance of the Fixed NF method using simple (non-normalized) difference
algorithms. The simple difference problem is solved using equation 4.2 with x defined
as x = σ2 − σ1 and z defined as z = v2 − v1. (H is also modified in that its columns
are not divided by vref as in section 4.2). Similar to the proportional difference algorithm
the Fixed NF method was able to consistently calculate a hyperparameter located in the
minimum region of the associated resolution curve.
Comment on Normalization
Normalization has some advantages. By normalizing the data, one does not need to know
the amplitude current injected. Thus, the forward problem can be solved using a value of
injected current which is convenient for the algorithm (usually ±1). The authors of [90]
justify normalization with the argument that a normalized sensitivity matrix is less sensitive
to the boundary shape of the object and the position of the electrodes, which is a substantial
problem in clinical applications. Also, some EIT systems change the gain settings for each
electrode depending on the amplitude of signal expected. Normalization can compensate
for these cases as well as instances where the calibrated gain settings are not exact.
Normalization of the sensitivity matrix does not change the spectrum of singular values
of H T H. Although the condition number of H T H may improve by an order of magnitude
or more the relative change is not significant. For the meshes used in this work the condition
number of H T H improved by a factor of about 10. In other work involving non-circular
meshes we have seen changes from as low as 1.5 to as high as 1000. In some cases (nonhomogenous
background conductivity) the condition number increased, so it is not possible
to state that, as a rule, normalization always improves the condition number of H T H.
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