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

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using total variation as a regularization functional [2][118][45][32], where TV (σ) is approximated with �� Lkσ�2 + β, it has different implications in this context. Particularly, k it was shown in [12] that the centering condition is the complementary condition of the following pair of smooth optimization problems � n� � (Pβ) min (Dβ) min � i=1 (�zi� 2 + β 2 ) 1 2 : (y,z) ∈ Y c T x + β n� i=1 (1 − �xi� 2 ) 1 2 : x ∈ X � (7.21) The problem Pβ and Dβ are a primal dual pair. Specifically, Dβ has the solution (y(β),z(β)) and Pβ has the solution x(β), all satisfying equations 7.19a, 7.19b, and 7.20. Introducing the perturbation β in the complementary condition for the original pair of problems is therefore equivalent to smoothing the norms in (P) and introducing a cost into (D). Particularly the cost function n� i=1 (1 − �xi� 2 ) 1 2 can be understood to keep the dual solution away from its boundary (�xi� = 1), from which the name of centring condition for equation 7.20, and of interior point method for the algorithm. The concept of keeping iterates away from the boundary of feasible regions originates from interior point methods for linear programming (LP) [124]. In LP optimal points are known to lie on vertices of the feasible set; traditional algorithms, such as the simplex method, exploited this by working on the frontier of the feasible region and examining vertices to find the solution. This approach changed in the mid 80s with Karmarkar’s [80] introduction of interior point methods, which work by following a smoother path inside the feasible region called a central path (identified by a centering condition), and possibly making larger steps at each iteration. In MSN the central path is defined by the solutions (y(β),z(β),x(β)) of Pβ, Dβ for β > 0, β → 0. Using these results Andersen et al realised an efficient PD–IPM algorithm that works maintaining feasibility conditions, equations 7.19a, 7.19b and applies the centering condition, equation 7.20, with a centering parameter β which is reduced during iterations, following the central path to the optimal point. In the next Section we describe the application of the PD-IPM framework to TV regularized linear inverse problems. 7.2.5 Duality for Tikhonov Regularized Inverse Problems In inverse problems, with linear forward operators, the discretized TV regularized inverse problem, can be formulated as (P) min x 1 2 �Ax − b�2 + α � |Lkx| (7.22) k where L, as in 7.9, is a discretization of the gradient operator. We will label it as the primal problem (P). The dual problem, can be derived noting, as for the MSN problem, that |Lkx| = ||Lkx|| = max y:�y�≤1 y Lkx (7.23) By applying 7.23 to (P), the dual problem (D) is obtained as follows [23] (D) max y:�yi�≤1 min 1 x 2 �Ax − b�2 + αy T Lx (7.24) 101
The optimization problem min x 1 2 �Ax − b�2 + αy T Lx (7.25) has an optimal point defined by the first order conditions A T (Ax − b) + αL T x = 0 (7.26) the dual problem can be written therefore as (D) max y : �yi� ≤ 1 A T (Ax − b) + αL T x = 0 The primal–dual gap for (P) and (D) is therefore: 1 2 �Ax − b�2 + αy T Lx (7.27) 1 2 �Ax − b�2 + α � |Lkx| − k 1 2 �Ax − b�2 − αy T Lx = � � α |Lkx| − y T � (7.28) Lx k The complementary condition, which nulls the primal–dual gap, for 7.22 and 7.27 is therefore: � |Lkx| − y T Lx = 0 (7.29) k which with the dual feasibility �yi� ≤ 1 is equivalent to requiring that Lix − yi�Lix� = 0 i = 1,... ,n (7.30) The PD-IPM framework for the TV regularized inverse problem can thus be written as �yi� ≤ 1 i = 1,... ,n (7.31) A T (Ax − b) + αL T x = 0 (7.32) Lix − yi�Lix� = 0 i = 1,... ,n (7.33) It is not possible to apply the Newton method directly to equations 7.31,7.32,7.33 as equation 7.33 is not differentiable for Lix = 0. A centering condition has to be applied [13][23], obtaining a smooth pair of optimization problems (Pβ) and (Dβ) and a central path parameterised by β. This is done by replacing Lix by (�Lix�2 + β) 1 2 in 7.33. 7.2.6 PD-IPM for EIT The PD-IPM algorithm described in section 7.2.5 was developed by Chan et al [30] for inverse problems with linear forward operators. We next describe a numerical implementation of the PD-IPM algorithm, based on [23], to calculate the non-linear solution to σrec = arg min σ 1 2 �F(σ) − Vmeas� 2 + αTV (σ) (7.34) 102
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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.
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
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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
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Page 99 and 100: case normalized to the first and la
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Page 117 and 118: PD-IPM Algorithm ψ(σ) = 1 2 �F(
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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%
Page 137 and 138: Figure A.4: Reconstructions with ho
Page 139 and 140: Figure A.7: Reconstructions with no
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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