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

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positions of the impulse phantom. The leftmost column is the RTik prior, the centre column is the Rdiag, the right most is the RLap prior. We do not show the RHPF solutions as they were similar to the RTik results. From a qualitative point of view the three priors provide similar reconstructions in that none of them appears superior to the others in terms of a qualitative assessment of figure 5.7. Analysis of the various plots of figure 5.8 show that the RTik is inferior to the others in terms image energy while the Rdiag prior is slightly superior in terms of resolution. (a) Target Height 23cm (b) Target Height 19cm (c) Target Height 15cm Figure 5.7: Quarter section reconstructions of contrasts located at radial offset of r/2. Left column is RTik prior, centre column is Rdiag prior, right column is RLap prior. Two electrodes per layer are shown Figure 5.8(a) shows the resolution for all three priors for the two sets of simulated data, r=0 and r/2. The resolution varies by 20% as a function of height. The best resolution for each prior occurs near the electrode planes with the worse resolution occurring in the 73
plane located halfway between the electrode planes. This is expected as resolution or its counterpart, sensitivity, decreases as position moves from current injecting or measuring electrodes. Thus resolution will be worse half way between the electrode planes. Radial position error as shown in figure 5.8(b) is lowest for contrasts at the centre of the tank and increases as contrasts move radially outward. In general however, the radial position error is small. true target height reconstructed target height Image Energy 25 20 25 20 15 10 15 10 5 0 0.38 0.4 0.42 0.44 0.46 0.48 Resolution (BR) 0.5 0.52 0.54 5 (a) Resolution Tik - centre diag - centre laplacian - centre Tik - half diag - half laplacian - half 0 0 5 10 15 true target height 20 25 30 13 12 11 10 9 8 7 6 5 4 (c) Height Error 3 0 5 10 15 20 25 true target height (e) Image Energy target height target height SNR 25 20 15 10 5 0 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 radial error (%) 30 25 20 15 10 5 (b) Radial Position Error Electrode Planes 0 -20 0 20 40 60 80 100 Position error (%) 0.3 0.28 0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 (d) Position Error Electrode Planes 0.1 0 5 10 15 20 25 target height (f) SNR Figure 5.8: Performance Measures for 3D Reconstructions of Two Simulated Data Sets. Legend in figure (c) applies to all figures. Electrode Planes are centered at heights of 8.5 and 19.5cm as indicated in 5.8(d) and 5.8(f) Height error as shown in figure 5.8(c) is common for all priors. There is a general tendency for contrasts to be reconstructed closer to the electrode planes than they actually are. Position error is shown in figure 5.8(d) is a combination of the radial and vertical position errors and mainly indicates an asymmetry in the vertical axis. Figure 5.8(e) shows the 74
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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
Page 35 and 36: Substitution of 2.24 into 2.21 yiel
Page 37 and 38: where the superscript identifies th
Page 39 and 40: 2.3.3.4 Numerical Implementation of
Page 41 and 42: is selected, measurements are taken
Page 43 and 44: measurements of which only 104 are
Page 45 and 46: Chapter 3 Reconstruction The proces
Page 47 and 48: where the Jacobian is � H = T −
Page 49 and 50: The condition number of a matrix is
Page 51 and 52: 2. As λ goes to zero, the un-regul
Page 53 and 54: 6. Evaluate a stopping rule. For ex
Page 55 and 56: Finally Adler and Guardo define the
Page 57 and 58: 3.5 3D Considerations In EIT it is
Page 59 and 60: Chapter 4 Objective Selection of Hy
Page 61 and 62: initial conductivity x = ∆σ/σ0.
Page 63 and 64: λ=0.0008 λ=0.0302 λ=0.0616 λ=6.
Page 65 and 66: 4.3.3 Generalized Cross-Validation
Page 67 and 68: 1. simulated data, generated using
Page 69 and 70: GCV 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0
Page 71 and 72: a hyperparameter selection method,
Page 73 and 74: 1. Heuristic selections of hyperpar
Page 75 and 76: human lung data. Keywords: regulari
Page 77 and 78: fields when reconstructing in 2D [1
Page 79 and 80: i and integrating across element j
Page 81 and 82: 5.2.5 Nodal Gaussian Filter The Gau
Page 83 and 84: Quantitative figures of merit are r
Page 85: (a) Rdiag BR=.236, SNR=.332 (b) Fil
Page 89 and 90: models that are difficult or imposs
Page 91 and 92: 6.1 Introduction EIT attempts to ca
Page 93 and 94: 6.2.1 Image Reconstruction In order
Page 95 and 96: Aligned, fig 6.2(a) Planar Zigzag Z
Page 97 and 98: of reconstructions were calculated
Page 99 and 100: case normalized to the first and la
Page 101 and 102: Resolution (BR) Image Radial Positi
Page 103 and 104: configuration gave the worse overal
Page 105 and 106: Planar and Planar-offset configurat
Page 107 and 108: Chapter 7 Total Variation Regulariz
Page 109 and 110: on the conductivity vector, σ, whi
Page 111 and 112: a regularization penalty term a muc
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(
Page 119 and 120: (a) First step, TV solution. BRTV =
Page 121 and 122: Iteration 1 (a) Iteration 4 (d) Ite
Page 123 and 124: (a) L 2 solution with 2.5% AWGN, fi
Page 125 and 126: (a) i=1 (b) i=2 (c) i=3 (d) i=4 (e)
Page 127 and 128: Figure 7.15: Four layer tank used f
Page 129 and 130: can be solved with the algorithm. T
Page 131 and 132: Appendix: Variability in EIT Images
Page 133 and 134: where H is the Jacobian or sensitiv
Page 135 and 136: (a) AσˆL = 33% (b) Aσ ˆL = 51%
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Figure A.4: Reconstructions with ho
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Figure A.7: Reconstructions with no
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[15] Barber DC, Brown BH, Applied p
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[49] Frerichs I, Hahn G, Hellige G,
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[83] Kunst PWA, Vonk Noordegraaf A,
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[116] Vauhkonen M, Lionheart WRB, H
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