By: Alireza Aghasi, Eric L. Miller, Andrew Ramsburg and Linda Abriola
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Transcript of By: Alireza Aghasi, Eric L. Miller, Andrew Ramsburg and Linda Abriola
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Parametric Shape-based Inversion in Electrical Impedance Tomography for the Characterization of Subsurface
Contaminant Distribution
By: Alireza Aghasi, Eric L. Miller,
Andrew Ramsburg and Linda Abriola
Tufts University, Medford, MA
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Introduction Motivation
Down-gradient mass flux and concentration signals have been linked to source zone distribution of DNAPL
Knowledge of distribution (e.g., location, configuration) can aid in remediation planning
Goal of this work Develop new inverse methods for
identification of DNAPL configuration Evaluate techniques in the context of
electrical impedance tomography
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Inverse Problems
Geophysics Medical Imaging
Astronomy and Remote Sensing Ocean Tomography
Physics
Object Measurement
MeasurementObject
Descriptor
Mathematically Complicated and hard to analyze
Ill-Posed (Obtaining a certain and unique answer is not possible)
Computationally very expensive
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Electrical Impedance Tomography Electrical Impedance Tomography (EIT) is a technique of imaging subsurface structures
through some measurements made on the surface or through boreholes. Poisson’s Equation:
:)(
:)(
:)(
:)(
rs
r
r
rv
Electric Potential
Electrical Conductivity
Electrical Permittivity
Current Source Distribution
)()(),( rsrvr
svK )(
Forward Problem:
Knowing electrical conductivity and permittivity of the media and the current s(x,y,z), we then find the electric potential.
Inverse Problem:Knowing the Electric Potential at some points (nodes) and the current s(x,y,z), we then estimate the conductivity and permittivity distribution throughout the media.
),,(
),(),(),(
zyxr
rirr
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Example of Pixel Based Reconstruction
40x40 Grid: 10m x 8m
30 sensors 40 experiments SNR=60 dB
Only Tikhonov Regularization
Poor Reconstruction Near sensor distortions Relative Error:191%
More intense near sensor regularization
Imposing positivity Better reconstruction Relative Error:87%
Simulations made for zero frequency (DC current) Parameter of interest is the resistivity: Simulation details:
Saturation distribution from UTChem Resistivity values from Newmark et al., JEEG., 3, 7–13.
(x, y) 1(x, y)
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Shape-Based Reconstruction
Problems with pixel based reconstruction:
• Too many unknowns• Low resolution• Very ill-posed
Level-Set Idea:
A low order representation of resistivity distribution
The footprint or basically the shape, an important quantity of interest
Level-set Curve
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Reconstruction Using Basis Expansions Parametric Shape Based Reconstruction (Level-Sets approach)
Assuming the resistivity to be piecewise constant in the background ( ) and foreground ( ), we estimate source zone boundaries, background resistivity and smooth (low order, constant in this talk) foreground resistivity.
In some sense, an adaptive thresholding approach aimed at recovering “average” resistivity of source zone along with better delineation of boundary.
Unknown
)()( xUxF bfb )(F
Nc
iiibfb rcUr
1
)()()(
bf
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What Are the Appropriate Basis Functions Representing every basis function as a monotonic radial basis function For every radial basis function (e.g. a Gaussian function), the center,
width and the height can vary and are the unknowns in the inversion process.
The unknown shape can be represented as some level-set of this basis expansion.
Center
Width
Height
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Choice of Compactly Supported Radial Basis Functions (CSRBF) They have the bell-shaped property of Gaussian
functions, and become strictly zero after a certain radius and causes a huge degree of sparsity in matrices.
Unlike other non-orthogonal basis functions, representation of any continuous function in terms of the summation of CSRBFs (collocation) is proved to be always stable.
Gaussian
CSRBF
Nc
iiiibfb rracUr
1
)()()(
icupdate
irupdate
iaupdate
fbupdate
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Shape Based Approach
SNR=60 dB
6.38b 4.194f 40x40 Grid: 10m x 8m 30 sensors 40 experiments SNR=60 dB
Initial level-set curve Final level-set curve
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Evolution Process (Movie)
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Performing the Parametric Reconstruction (3D)
30x30x30 Grid: 10m x 10m x 10m 100 sensors 120 experiments SNR=60 dB
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Shape Based Approach (3D) SNR=60 dB
Original Shape Initial Level-set Final Reconstruction
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Evolution Process(Movie)
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Using different frequencies SNR=60 dB The foreground and background conductivity and permittivity assumed to be constant and
a priori known 4 different frequencies used to generate more data: (0, 10, 1K, 10k) Hz
Original Reconstructed
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Conclusion A new representation of electrical conductivity-permittivity is
discussed, which is closer to the nature of unknowns. The use of adaptive CSRBFs shows to be
Better in reconstruction Faster in convergence More Reliable Bypassing the regularizations
This project is funded and supported by US National Science Foundation (NSF) under Grant EAR 0838313, and we thank
them for their support.
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Thank You!