2.1 Introduction
2.1.5 Frequency difference reconstruction algorithms
If EIT is to be used for imaging acute stroke, then TD imaging is not possible; baseline data cannot be acquired as patients present after the event. Consequently, absolute orFD
imaging must be used, and FDimaging, as a differential calculation, has the advantages of reducing instrumentation and geometric errors. In FDimaging, multi-frequency data are collected at a single time point, and the boundary voltage changes, ∆v, calculated as the difference in voltages between two frequencies.
A common approach in EIT, is to use a linear approximation to relate a map of conductivity changes, ∆σ, typically within a FEM mesh, to boundary voltage changes, ∆v:
∆v = A∆σ, (2.1)
where A is the sensitivity matrix, a matrix defined by mesh, electrode positions, and current injection and measurement protocol. The calculation of the pseudo-inverse of this, A†, termed the inverse solution, allows for the mapping of measured ∆v to ∆σ
∆σ = A†∆v, (2.2)
with ∆σ being rasterised, or otherwise displayed, as theEITimage. As A is underdeter- mined the solution is ill-posed, therefore the pseudo-inverse is performed with regulari- sation of A. The regularisation method employed in this study istSVD, which is centred upon a decomposition of A:
A = U ΣVT, (2.3)
where the singular values are the diagonal entries of the matrix Σ and are the length of the number of measurements. The singular values are ordered from largest to smallest and each can be thought of as describing the contribution to the solution that a given measurement provides. Essentially the singular values indicate the degrees of freedom,
and inclusion of all values will result in a fuller, but noisier solution, as ∆σ includes errors, while the inclusion of fewer singular values will result in a lower noise, spatially smoother solution. The need to remove smaller singular values can be understood by the fact that each singular value will be inverted so small values will become exceptionally large. Therefore those singular values/measurements determined to be below a given threshold can be suppressed in the inverse solution by replacing the entries with ≈ 0 entries, this is referred to as truncation, to result in Σ. The truncation point can bee chosen based upon the data noise, the decay of the singular values, or through other methods[179]. Following this A† can be calculated as:
A†= VΣUe T. (2.4)
For the same A† it is possible to alter the inverse solution by calculating ∆v in different ways: it is this alteration that is referred to in this study as different reconstruction algorithms.
The advantage of difference imaging is error suppression. The ill-posed nature of the inverse problem in EIT means that comparatively small errors in the measured data can potentially lead to arbitrarily large errors in the estimated internal conductivity. In difference imaging, time invariant noise and error, due to stray capacitance or electrode- electrolyte interaction, are better cancelled out than in absolute imaging. InTDimaging, this is achieved by calculating the voltage difference, ∆vT D, as the normalized difference between the voltages at a time with the perturbation, vtp, and the voltages at a time without the perturbation, vtb:
∆vT D= vtp− vtb. (2.5)
In FD, only voltages with the perturbation in place are used. The voltage difference, ∆vF Di, is calculated over frequency. This difference is the normalized difference between voltages at one frequency, vfi to a, typically lower, reference frequency vfref:
∆vF Di = vfi− vfref. (2.6)
FD imaging may not benefit completely from the advantages of difference imaging as different errors are present at different frequencies. An additional consideration is that the sensitivity matrix approach assumes a linear relationship between changes in a voxel in an object and the boundary voltages, whereas this relationship is governed by Poisson’s equation and is therefore non-linear. However, this approximation has been shown to be reasonable for resistance changes of less than 20 %[79]. In practice, this linear assumption produces acceptable images inEITproviding that either changes are less than about 20 %, or else that a perturbation of more than this is present against a uniform non
changing background. When the background also changes by more than 20 %, such as for
FD imaging with a frequency-dependant conductivity, such as carrot cubes in saline or the human brain, then images appear as widespread dome-shaped changes against which a smaller perturbation cannot be discerned[77;172]. In order to minimize errors from this assumption of linearity, imaging may be undertaken by normalization of two frequencies spaced close together so that resistivity contrasts lie in the relatively linear range of less than 20 %, which is done when using the frequency difference adjacent (FDA) algorithm. InFDA, the voltage difference, ∆vF DAi, is the normalized difference between voltages at
one frequency, vfi, to the voltages at the preceding frequency, vfi−1:
∆vF DAi = vfi− vfi−1. (2.7)
However, any perturbation may still be obscured against the background and the choice of closely spaced frequency pairs may also reduce theSNR, as the contrast of the pertur- bation will also be reduced.
Another approach is the weighted frequency difference (WFD) algorithm, in which a correction is made for the frequency-dependent variability in the conductivity of the background[180–182]. This has been shown to be successful in the presence of a background with frequency-dependent conductivity in numerical modelling, 2D cylindrical tank and hemispherical tank experiments. It may be implemented by either normalisation to a single reference frequency (WFD) or by comparing sequential frequency pairs (weighted frequency difference adjacent (WFDA)). In WFD, the voltage difference, ∆vW F Di, is cal- culated as the normalised difference between voltages at one frequency, vfi, to a reference
frequency, vfref, where the latter is weighted by a factor α:
∆vW F Di = vfi− αvfref, (2.8) where α = D vfi, vfref E D vfref, vfref E. (2.9)