Use of an optimized spatial prior in D-bar reconstructions of EIT tank data

Melody Alsaker; Jennifer L. Mueller

doi:10.3934/ipi.2018037

Article Contents

2018, Volume 12, Issue 4: 883-901. Doi: 10.3934/ipi.2018037

This issue Previous Article Convergence theorems for the Non-Local Means filter Next Article Recursive reconstruction of piecewise constant signals by minimization of an energy function

Use of an optimized spatial prior in D-bar reconstructions of EIT tank data

Melody Alsaker^1, , and
Jennifer L. Mueller^2,

1.
Department of Mathematics, Gonzaga University, MSC 2615, Spokane, WA 99258, USA
2.
Department of Mathematics and School of Biomedical Engineering, Colorado State University, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA

* Corresponding author: Melody Alsaker
* Corresponding author: Melody Alsaker

Received: February 28, 2017

Revised: November 30, 2017

Published: July 2018

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Abstract

The aim of this paper is to demonstrate the feasibility of using spatial a priori information in the 2-D D-bar method to improve the spatial resolution of EIT reconstructions of experimentally collected data. The prior consists of imperfectly known information about the spatial locations of inclusions and the assumption that the conductivity is a mollified piecewise constant function. The conductivity values for the prior are constructed using a novel method in which a nonlinear constrained optimization routine is used to select the values for the piecewise constant function that give the best fit to the scattering transform computed from the measured data in a disk. The prior is then included in the high-frequency components of the scattering transform and in the computation of the solution of the D-bar equation, with weights to control the influence of the prior. In addition, a new technique is described for selecting regularization parameters to truncate the measured scattering data, in which complex scattering frequencies for which the values of the scattering transform differ greatly from those in the scattering prior are omitted. The effectiveness of the method is demonstrated on EIT data collected on saline-filled tanks with agar heart and lungs with various added inhomogeneities.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 65N21, 94A08.

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Figure 1. The five experimental data sets considered in this paper

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Figure 2. Illustration of artifacts and distortions in reconstructions of experimental case (iv) when ${\bf{t}}_{\text{pr}}$ is a poor match for ${\bf{t}}$. These images are reconstructions using the a priori method described here, with the scattering prior ${\bf{t}}_{\text{pr}}$ computed from output ${{\bf{c}}}^j$ of six iterative steps of the optimization routine. The reconstruction with the initial guess ${\bf{c}}^0$ is depicted in (a) and the reconstruction using the final result of the optimization routine is depicted in (f). The value of the objective function $J({\bf{c}})$ is given below each reconstruction. Smaller $J({\bf{c}})$ indicates increased goodness of fit between ${\bf{t}}_{\text{pr}}$ and ${\bf{t}}$

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Figure 6. Data Collection 2, case (ⅳ) agar heart and lungs. The influence of the prior increases from left to right in each row

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Figure 3. Data Collection 1, case (ⅰ) agar heart and lungs. The influence of the prior increases from left to right in each row

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Figure 4. Data Collection 1, case (ⅱ) agar heart and lungs with a conductive copper pipe added to right lung. The influence of the prior increases from left to right in each row

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Figure 5. Data Collection 1, case (ⅲ) agar heart and lungs with a resistive PVC pipe added to right lung. The influence of the prior increases from left to right in each row

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Figure 7. Data Collection 2, case (ⅴ) agar heart and lungs with top half of left lung removed. The influence of the prior increases from left to right in each row

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Table 1. Results from experiments with the optimization routine, in which values in the initial guess vector ${{\bf{c}}}^0$ were varied. We include statistical summary values for the initial objective function $J({{\bf{c}}}^0)$, the number of iterations required (NumIter), the resulting output vector ${{\bf{c}}}$, the optimized value of the objective function $J({{\bf{c}}})$, and the quantity $D({{\bf{c}}}): = \| {\bf{t}}_{\text{pr}}^{\text{vec}}({{\bf{c}}}) - {\bf{t}}_{\text{pr}}^{\text{vec}}({{\bf{c}}}_*) \|_{\infty}$ where ${{\bf{c}}}_*$ corresponds to the "control experiment" in our tests, which is a measure of variation between test cases of the resulting scattering data. Statistical values presented include the mean, max, min, range, standard deviation, and coefficient of variation of each quantity

	Initial Guess Vector ${{\bf{c}}}^0$						Output Vector ${{\bf{c}}}$
	$c^0_b$	$c^0_h$	$c^0_r$	$c^0_l$	$J({{\bf{c}}}^0)$	NumIter	$c_b$	$c_h$	$c_r$	$c_l$	$J({{\bf{c}}})$	$D({{\bf{c}}})$
Mean	0.9978	1.1116	0.9312	0.9312	62.19	14.140	1.0475	1.1689	0.9083	0.9515	9.8517867	3.33E-06
Max	1.1769	1.1769	1.1769	1.1769	145.41	19	1.0547	1.1769	0.9145	0.9580	9.8517867	7.15E-06
Min	0.8658	0.8658	0.8658	0.8658	11.78	9	1.0288	1.1480	0.8921	0.9344	9.8517867	8.31E-07
Range	0.3112	0.3112	0.3112	0.3112	133.63	10	0.0259	0.0289	0.0225	0.0235	1.470E-09	6.32E-06
StdDev	0.0753	0.0904	0.0904	0.0904	46.17	2.406	0.0078	0.0087	0.0067	0.0071	3.636E-10	1.51E-06
CoeffVar	7.55%	8.13%	9.70%	9.70%	74.24%	17.02%	0.74%	0.74%	0.74%	0.74%	3.69E-09%	45.39%

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