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

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(a) i=1 (b) i=2 (c) i=3 (d) i=4 (e) i=5 (f) i=7 Figure 7.7: TV reconstructions of Phantom A at increasing iterations. Vertical axis is absolute conductivity. Normalized to 0. No Noise added. (a) TV solution at 7 th iteration. (b) L 2 solution at 7 th iteration. Figure 7.8: Reconstructions of Phantom A with 0.6% AWGN. example, the use of BestRes in chapter 5. The PD-IPM algorithm did not show to be strongly sensitive to the value of λ0. We varied the value of λ0 three orders of magnitude above and one order of magnitude below λBR without appreciably changing the TV solution at convergence or the rate of convergence. The initial hyperparameter, λ0, was always selected using Best Res, however, several numerical experiments were performed to determine the effect of the iterative hyperparameter, λi, on algorithm performance. Although λi could be changed at each iteration, in the reconstructions shown in this manuscript λi was maintained constant, thus λi = λi+1. Figure 7.14 shows the results of running the algorithm to convergence six times with a different value λi for each run. It is obvious from the figure that the algorithm is sensitive to the value of λi; too small a value of λi prevents a “blocky” solution, too large a value of λi will allow blocky reconstructions but suppress the amplitude. The BestRes method was originally used to calculate λi however the method was unable to find a good value for λi. Best results were obtained by the ad hoc visual inspection of figures such as figure 7.14 for various values of λi. Further work is required to develop an objective method to select λi. The original PD-IPM methods includes updating the Jacobian matrix at each iteration. 109
(a) L 2 solution with 2.5% AWGN, first step. Noisy but useful reconstruction. (b) TV solution with 1.5% AWGN, first step. Noise dominated solution. Figure 7.9: Reconstructions of Phantom A. Iteration 8 L1 L2 phantom Figure 7.10: Phantom B profiles. In our work numerical experiments this did not result always in a significant improvement in reconstructed images. We adopted therefore a the arrangement of not updating the Jacobian at each single iteration. This provides a reduction in the reconstruction computational time. As an additional numerical experiment, we evaluated the use of the same regularization matrix L as for TV regularization, (equation 7.9), with the quadratic algorithm (7.3). Although reconstructions from the first step were identical to TV reconstructions, the quadratic solutions rapidly degraded, producing noisy reconstructions that were dominated by noise artefacts after the 10 th iteration. The TV prior is not recommended for use with the quadratic algorithm. 7.4.6 Preliminary testing in 3D The generality of the PD-IPM scheme allows its use for the 3D EIT reconstructions. The method was expected to work equally well in three dimensions, and to be easily extended to this case. To validate this a single experiment with the simulated tank of figure 7.15 was performed. The tank has 315 nodes, 1104 elements, 32 electrodes and is constructed of 4 identical layers of tetrahedrons and was used for both simulating data and reconstruction. A single object in the shape of a crescent was used to generate simulated data. Reconstructions were made on the same mesh. The convergence of the PD-IPM algorithm is shown in figure 7.16. Convergence occurred rapidly with a reasonable image appearing in the first iteration and convergence being achieved by the 8 th iteration - there was no appreciable improvement in the image or change in the error norm after the 8 th iteration. Figure 7.17 shows slices taken at the five layer boundaries (including top and bottom tank surfaces) of the simulated tank. Figure 7.17 shows reconstructed conductivities after the first iteration, Figure 7.18 shows reconstructed conductivities after 8 iterations. The results were not as good as 110
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Enhancements in Electrical Impedanc
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Acknowledgements I would like to de
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3.5 3D Considerations . . . . . . .
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Bibliography 127 VITA 135 vii
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4.8 GCV curves for different priors
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7.15 Four layer tank used for 3D re
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Discrete Variables • I is the mat
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as opposed to anatomical imaging. W
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to know how various alternative con
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to noise however both algorithms pr
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Chapter 2 Forward Problem 2.1 Descr
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there are two primary types in EIT.
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impedance, σ(�x,t) + jω(�x,t)
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elying on a variational statement.
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with Y11 = −Y12 − Y13, Y22 =
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2.3.1.3 Derivation of Linear Interp
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with Ci being the following column
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Substitution of 2.24 into 2.21 yiel
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where the superscript identifies th
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2.3.3.4 Numerical Implementation of
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is selected, measurements are taken
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measurements of which only 104 are
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Chapter 3 Reconstruction The proces
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where the Jacobian is � H = T −
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The condition number of a matrix is
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2. As λ goes to zero, the un-regul
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6. Evaluate a stopping rule. For ex
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Finally Adler and Guardo define the
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3.5 3D Considerations In EIT it is
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Chapter 4 Objective Selection of Hy
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initial conductivity x = ∆σ/σ0.
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λ=0.0008 λ=0.0302 λ=0.0616 λ=6.
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4.3.3 Generalized Cross-Validation
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1. simulated data, generated using
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GCV 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0
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Page 75 and 76: human lung data. Keywords: regulari
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Page 81 and 82: 5.2.5 Nodal Gaussian Filter The Gau
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Page 91 and 92: 6.1 Introduction EIT attempts to ca
Page 93 and 94: 6.2.1 Image Reconstruction In order
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Page 113 and 114: etween primal and dual optimal poin
Page 115 and 116: The optimization problem min x 1 2
Page 117 and 118: PD-IPM Algorithm ψ(σ) = 1 2 �F(
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Page 131 and 132: Appendix: Variability in EIT Images
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Page 141 and 142: [15] Barber DC, Brown BH, Applied p
Page 143 and 144: [49] Frerichs I, Hahn G, Hellige G,
Page 145 and 146: [83] Kunst PWA, Vonk Noordegraaf A,
Page 147 and 148: [116] Vauhkonen M, Lionheart WRB, H
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