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

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4.5.3.2 Nonlinear Reconstructions We investigated the use of the fixed NF method with an iterative static reconstruction algorithm. In this experiment we used fixed NF to calculate a single hyperparameter that was used for each iteration. Running the algorithm to convergence indicated that the λNF=1 was located in the minimum region of the resolution curve. It may be possible to use the fixed NF method to calculate a new hyperparameter for each step of the iterative algorithm, however, this was not pursued. 4.5.3.3 Inverse Crime The act of employing the same model to generate, as well as to invert, simulated data is known as an inverse crime [122]. In this work, data were simulated using a 1968 element mesh, so reconstructions using the same mesh constitute an inverse crime. Such reconstructions had noticeably better resolution than was achieved with other meshes and, as shown in figures 4.11(a) and 4.11(b), exhibited λBestRes and λLC corresponding to an uncharacteristically high NF. This suggests a method to detect inverse crimes: using the suspect FEM, and associated data, construct a resolution curve or L-curve with the simulated data and calculate the NF corresponding to λBestRes or λLC. If the NF >> 3 (for example, figure 4.11(a) had NF > 7) it is likely that the reconstruction algorithm is committing an inverse crime. The method was validated by the observation that reconstructions over the 1968 element mesh using tank data did not exhibit the large NF bias while the high NF phenomenon was observed every an “inverse crime” configuration was analyzed. One explanation for these results is that the optimal hyperparameter for the inverse crime case is significantly lower than that generally required, since the geometry matching between forward and inverse solutions is giving a regularizing effect. log10||Rˆx|| 10 −2 10 −3 10 −4 λ=1.0273e-006 NF=7.6022 λBestRes = λLC λNF=1 λ=13.9147 10 log10||Hˆx − z|| −3 (a) L-Curve, NF = 7.6022 at corner is excessive and indicates inverse crime Blur Radius 0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2 λBestRes = λLC λNF=1 NF=7.6022 10 −6 10 −4 10 −2 10 0 10 2 log10(λ) (b) Resolution curve, NF = 7.6022 is excessive and indicates an inverse crime. Figure 4.11: Either the L-curve or resolution curve can be used to detect an inverse crime. 4.6 Conclusion This paper proposes a new method of objective hyperparameter selection for use in onestep image reconstructions and compares it to some existing methods including heuristic selection. We present the following observations: 59
1. Heuristic selections of hyperparameter are inconsistent among experts and unrepeatable. This suggests that there is no single preferred value of λ, rather there is a preferred region of λ over which reconstructions are not subjectively distinguishable. Moreover, it was not possible for observers to differentiate reconstructions based on heuristic hyperparameter selections from those produced from the objective methods. 2. The GCV method is unreliable for the class of algorithms used in this work. 3. The L-Curve is, in general, shallow for EIT applications and is not reliable for all configurations (doesn’t indicate a hyperparameter). When the method does work it provides a lower hyperparameter value than the Fixed NF and BestRes methods. 4. With NF = 1 the Fixed NF Method calculates a hyperparameter that falls in the minimum region of the Resolution Curve. At low noise levels λNF=1 is very close to λBestRes. As AWGN is added to the simulated data λBestRes increases while λNF=1 remains constant. However at the noise level found in our EIT equipment a NF of 1 produces good reconstructions that are close to the optimal reconstructions achieved with BestRes method. 5. The Fixed NF Method provides a configuration independent method to select λ that is repeatable and is more consistent than expert selection. One could use the Fixed NF method with NF = 1 to calculate a minimum hyperparameter value for any configuration. This method is repeatable and in applications with realistic noise levels will produce consistent stable reconstructions that are as good as heuristic selection. 6. Hyperparameters taken from the minimum region of the Resolution Curve (BestRes method) always produce good solutions that are comparable to, but more consistent than expert selections. Moreover λBestRes is optimal in terms of our figure of merit. For the class of regularized reconstruction algorithms used in this work both the Fixed NF and BestRes methods provide objective methods to select a good value for λ. The values were indistinguishable from those selected by human experts. Both methods were developed using simulated data but shown to be applicable (validated) using Tank data. Although these methods do not completely solve the problem of obtaining an optimal hyperparameter value, they do provide a rationale and method for objectively and automatically selecting λ. Using the BestRes method with imaging equipment provides a sound engineering method for manufacturers or researchers to obtain a configuration-dependent hyperparameter that is optimal in terms of resolution. This allows end users to perform impedance imaging without the necessity of having to manually tweak parameters. 60
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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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Page 23 and 24: there are two primary types in EIT.
Page 25 and 26: impedance, σ(�x,t) + jω(�x,t)
Page 27 and 28: elying on a variational statement.
Page 29 and 30: with Y11 = −Y12 − Y13, Y22 =
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Page 33 and 34: with Ci being the following column
Page 35 and 36: Substitution of 2.24 into 2.21 yiel
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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.
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(a) L 2 solution with 2.5% AWGN, fi
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(a) i=1 (b) i=2 (c) i=3 (d) i=4 (e)
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Figure 7.15: Four layer tank used f
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can be solved with the algorithm. T
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Appendix: Variability in EIT Images
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where H is the Jacobian or sensitiv
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(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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