Reconstruction Algorithms
ROBUSTNESS OF LINEAR AND NONLINEAR RECONSTRUCTIONS ALGORITHMS FOR BRAIN EITS Non-linear – is it worth the effort? Rebecca J. Yerworth1a, Lior Horesha, Richard H. Bayfordb, Andrew Tizzardb, David S. Holdera a
b
Department of Medical Physics and Bioengineering, UCL, London, UK; Middlesex University, Archway Campus, Furnival Building, Highgate, London, UK
ABSTRACT: When choosing a reconstruction algorithm for clinical use accuracy, speed and robustness, with respect to errors in the measured data, must be considered. Until recently the UCL brain EIT group have used a 3D, Truncated Singular Value Decomposition algorithm (TSVD), and published time-difference images of evoked responses using this. However, the group is extending its interests to stroke, where time difference imaging cannot be used and additional reconstruction algorithms must be implemented. This paper compares two approaches – TSVD with a computer model for the reference image, and Non-linear Conjugate Gradients. These are assessed using simulated and tank data with known sources of measurement error. The solutions are found to be very sensitive to discrepancies between the position of the electrodes used to obtain the boundary voltages, and those used for reconstruction. Position errors of 1% causing significant artefacts. The TSVD algorithm was also unable to return the true conductivity of a homogenous image if the reference voltage was out by more than 10%. The non-linear algorithm could compensate for the reference voltage being out by 30%, but is much more computationally expensive than TSVD, and would require speeding up before routine clinical use. Keywords: EIT; Multifrequency; Stroke; reconstruction algorithms.
1. INTRODUCTION When choosing a reconstruction algorithm for clinical use accuracy, speed and robustness, with respect to errors in the measured data, must be considered. Until recently the UCL brain EIT group have used a 3D, Truncated Singular Value Decomposition Algorithm (TSVD) (Bagshaw et al. 2003), to produce timedifference images of evoked responses. However, the group is extending its interests to stroke, with the aim of differentiating between ischemic and hemorrhagic causes, within two hours of onset. This would enable the decisions on the use of thrombolytic treatment to be swiftly made (Harraf et al. 2002).In this context time difference imaging cannot be used and so different reconstruction approaches must be considered. Linear algorithms, including TSVD, are fast and robust for time difference imaging but have no track record in other paradigms (e.g. frequency difference imaging), non-linear algorithms have theoretical advantages, but are computationally expensive and sensitive to errors in the forward model. This is the first of two papers to be presented at this conference and addresses the relative suitability of linear and non-linear approaches. The second paper (Horesh 2004) discusses the issue of what non-linear algorithms are suitable for large-scale problems.
2. METHODS The robustness of one linear algorithm: Truncated Singular Value Decomposition (TSVD) and one nonlinear algorithm: Non-linear Conjugate-Gradient (NLCG), have been compared. Homogeneous and layered spherical meshes, 100mm radius, have been used for data simulation and image reconstruction, eliminating confounding variables and allowing each noise factor to be studied independently. Conductivities for the spherical meshes were: scalp 0.35S/m, skull 0.02S/m, CFC 0.70S/m, brain 0.13S/m, (Gabriel et al. 1996). Finally tank data is shown from a simple, cylindrical geometry. All reconstructions were performed on an AMD Athlon XP 1800+ processor (1.54 GHz, 480MB RAM). For ease of comparison, the conductivity changes reconstructed using TSVD, are added to the reference conductivity image, before display. 1
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XII ICEBI – V EIT 2004 Gdańsk
2.1. Electrode position Accuracy Two homogeneous meshes were used to assess how accurately the position of electrodes must be known. A set of electrode coordinates was mapped onto the surface of each sphere and the nearest node identified. In addition the coordinates of the nodes identified on a 2,000-element mesh were mapped onto a 30,000element mesh, creating two position files for the 30,000 mesh, and 1 for the 2,000. The mean difference between the intended and actual electrode positions was calculated in each case. The 30,000 mesh was used, with each position file in turn, to simulate boundary voltages with a) no perturbation, and b) a 20mm radius perturbation, conductivity = CSF, centred at –60mm on the x axis. The 2,000 mesh was used to reconstruct these four sets of boundary voltages, using both the TSVD and NLCG algorithms. The conductivity of the reference was set to that of brain.
2,2, Conductivity estimate accuracy The boundary voltages, calculated in 0, were reconstructed with the conductivity of the reference set to 50%, 70% 90% and 110% of the brain conductivity.
2.3. Effect of layered structures Electrode positions were mapped on to the shelled mesh, and used to generate boundary voltages with a) no perturbation, and b) with a 20mm radius perturbation entirely within the brain region, centred at – 40mm on the x axis, with conductivity = CSF. The same mesh was used to reconstruct these boundary voltages, using both the TSVD and NLCG algorithms and with the conductivity of the reference set to 100% and 90% of that used to generate the boundary voltages.
2.4. Measurements on biological samples A cylinder of Swede (Brassica napobrassica) was cut to fill a cylindrical Perspex tank, diameter 88mm, height 45mm which had a ring of 16, equally spaced, 7mm diameter electrodes inserted in the wall at half the height. 0.1% w/v saline was used to ensure contact between the swede and the electrodes. A well, 20mm diameter, 2/3 of radius from centre, was cut in the swede for the insertion of perturbations. Data was collected using a UCLH mk2 EITS imaging device (Yerworth et al. 2003) with diametric current injection and adjacent measurement for the following cases: a)Perspex near electrode 12, b) Banana near electrode 8. c) Banana near electrode 8, additional, saline filled well near electrode 13. All measurements were performed at room temperature, using a protocol with 96 independent voltage measurements. After data collection rectangular samples were cut from the swede and banana and their impedance measured using an HP 4284A impedance analyser (Hewlett Packard, www.hewlettpackard.com).
3. RESULTS 3.1. Electrode position Accuracy The error, in electrode placement, between the two meshes was 7mm when analytical electrode positions were used. This was reduced to 1mm when positions chosen on the 2,000 element mesh were used. For both TSVD and NLCG quantitative image quality was lower with the larger position error (Figure 1).
3.2. Conductivity estimate accuracy Apart from the 50% reference conductivity, which did not converge, all the non-linear reconstructions, of the perturbation, showed an impedance increase of 0.24+/-2s/m at the 6 o’clock position (Figure 2). The TSVD reconstructions also had an impedance increase at the 6 o’clock position, but this varied from 2.2s/m (70% reference) to 3.7s/m (110% reference). No impedance increase was present in the 50% case.
3.3. Effect of layered structures For an ideal reference image, and perturbation at 6 o’clock, TSVD showed an impedance increase of 0.18s/m, in the brain region, at the 6 o’clock position (Figure 2). The NLCG reconstruction showed an increase in the same place of 0.15s/m. However with an inaccurate reference image the NLCG was still able to localize an impedance increase at 6 o’clock but TSVD shows an impedance decrease at 12 o’clock. NLCG required 3.5 hours to reconstruct each image, TSVD less than 1minute, including solving the normalised sensitivity matrix.
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Reconstruction Algorithms
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TSVD
