Benjamin Fuchs

Université de Rennes 1 - IETR, France

email: benjamin.fuchs@univ-rennes1.fr

webpage: http://perso.univ-rennes1.fr/benjamin.fuchs/

On the use of Regularizers

for Microwave Inverse Problems

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Given the effects (EM field), determine the causes (EM currents, scattering map,tissues characteristics, etc.)

Microwave Inverse Problems - Examples

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Given the effects (EM field), determine the causes (EM currents, scattering map,tissues characteristics, etc.)

Many applications

biomedical imaging, antenna diagnostic, humanitarian demining, archeologicprospection, security screening, etc…

Microwave Inverse Problems - Examples

DIATOOL, TICRA - DKM. Popovic, McGill University - CA

[2-4] GHz

Duke University – USA

[17.5-26.5] GHz

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From (possibly noisy) observations y, retrieve the signal x

Inverse Problems – Formulation

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From (possibly noisy) observations y, retrieve the signal x

Inverse problems are (very) often ill-posed

non uniqueness

ill conditioning (different causes lead to similar effects)

Inverse Problems – Formulation

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Ill posedness => Use of regularizer & prior knowledge

Inverse Problems – Regularizer

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Ill posedness => Use of regularizer & prior knowledge

Inverse Problems – Regularizer

data fitting regularity

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Ill posedness => Use of regularizer & prior knowledge

l > 0 regularization parameter

p=2 - Tikhonov – smooth solutions

p=1 - LASSO – sparse solutions

Inverse Problems – Regularizer

data fitting regularity

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Antenna diagnostic

with L. Le Coq and M.D. Migliore (U. of Cassino - IT)

Inverse problem: identify antenna defaults from its radiated field

Motivations : radiating structures are more and more complex (active elements)

non invasive method, well suited to on site diagnostic

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Antenna diagnostic

with L. Le Coq and M.D. Migliore (U. of Cassino - IT)

Inverse problem: identify antenna defaults from its radiated field

Motivations : radiating structures are more and more complex (active elements)

non invasive method, well suited to on site diagnostic

Our approach: exploit of priori knowledge to reduce the number of measurements

Differential scenario => sparsity prior

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Antenna diagnostic

Inverse problem – choice of the regularizer (which p-norm?)

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Antenna diagnostic

Inverse problem – choice of the regularizer (which p-norm?)

L1 minimization (point-wise sparsity)

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Antenna diagnostic

Inverse problem – choice of the regularizer (which p-norm?)

L1 minimization (point-wise sparsity)

L1/L2 minimization (group-wise sparsity)

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Antenna diagnostic

Inverse problem – choice of the regularizer (which p-norm?)

L1 minimization (point-wise sparsity)

L1/L2 minimization (group-wise sparsity)

TV-norm minimization (gradient sparsity)

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Antenna diagnostic

Application – reflectarray measurements @ 12GHz – 193 cells

Thalès Alenia Space – Project R3MEMS

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Antenna diagnostic

Back propagation 1024 points

Matrix inversion900 points

Application – reflectarray measurements @ 12GHz – 193 cells

Thalès Alenia Space – Project R3MEMS

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Antenna diagnostic

Back propagation 1024 points

Matrix inversion900 points

Application – reflectarray measurements @ 12GHz – 193 cells

Thalès Alenia Space – Project R3MEMS

Compressive sensing64 points

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Antenna diagnostic

Application – reflectarray measurements @ 12GHz – 193 cells

Compressive sensing approaches

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Computational imaging system

with M. Davy and C. Yoya Tondo

Goal: design of a fast & low cost microwave imaging system with high resolution

Low cost: Use of chaotic cavity: spatial diversity => frequential diversityTransfer the effort from physical layer to softwareFaster than SAR & cheaper than phased arrays !

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Computational imaging system

with M. Davy and C. Yoya Tondo

Goal: design of a fast & low cost microwave imaging system with high resolution

Low cost: Use of chaotic cavity: spatial diversity => frequential diversityTransfer the effort from physical layer to softwareFaster than SAR & cheaper than phased arrays !

High resolution => Regularized inverse problem!

y(w) = H(w,r) s(w)

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Computational imaging system

with M. Davy and C. Yoya Tondo

First resultsX bandcavity of 0.5 x 0.5 x 0.3 m3

Time reversal Least squares Norm l1 min

Active imaging

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Computational imaging system

with M. Davy and C. Yoya Tondo

First resultsX bandcavity of 0.5 x 0.5 x 0.3 m3

Time reversal Least squares Norm l1 min

Active imaging

Synthetic “target”

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Fast antenna characterization – NEAR FIELD

with L. Le Coq, B. Uguen, M.D. Migliore (U. Cassino – IT) & S. Rondineau (UnB - BR)

Goal: interpolation to speed up planar near field measurements

Inverse problem: given a small number of measurements, find the missing field

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Fast antenna characterization – NEAR FIELD

with L. Le Coq, B. Uguen, M.D. Migliore (U. Cassino – IT) & S. Rondineau (UnB - BR)

Goal: interpolation to speed up planar near field measurements

Inverse problem: given a small number of measurements, find the missing field

Main idea: only a priori knowledge, EM field => low complexity

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Fast antenna characterization – NEAR FIELD

Low complexity => minimum rank => minimum nuclear norm

Regularized inverse problem!

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Fast antenna characterization – NEAR FIELD

Sampling step of 2l=> 4 times less meas. points !

Low complexity => minimum rank => minimum nuclear norm

Regularized inverse problem!

SGH @60GHz

GBH @60GHz

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Goal: speed up spherical measurements (NF & FF)

Main idea: from coarse field sampling => projection on spherical harmonics with sparsity prior => Regularized inverse problem!

Fast antenna characterization – FAR FIELD

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Goal: speed up spherical measurements (NF & FF)

Main idea: from coarse field sampling => projection on spherical harmonics with sparsity prior => Regularized inverse problem!

To identify N modes, N/3 measurement points only!=> 70% saving !

=> More efficient use of FF measurement facilities

Coarse sampling(RA @12GHz)

Fast antenna characterization – FAR FIELD

Projection on SH

QUESTIONS ?

Collaborators: M. Bjelogrlic (EPFL - CH) and M. Mattes (DTU - DK)

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Differential microwave imaging

Chalmers

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Microwave imaging active technique to reconstruct a scene from emission/receptionof microwave

Advantages non invasive, non ionizing, low power, low cost

Targeted applications detection of tumor or brain stroke

differential scenario to follow the pathological evolution

Direct problem (EPFL-LEMA)

Very heterogeneous medium (high contrast and losses)

Large scale problem (100x100x100 voxels)

=> Formulation VIE

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Differential microwave imaging

Direct problem

Direct problem (EPFL-LEMA)

Very heterogeneous medium (high contrast and losses)

Large scale problem (100x100x100 voxels)

=> Formulation VIE

Inverse problem

Non linear => quantification of the non linearity degree (Born)

Ill posed => regularizer to stabilize the inversion

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Differential microwave imaging

Inverse problem

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Differential microwave imaging

Numerical results – simple model

Incident fields = 32 plane waves @ 1GHz

SNR=70dB

Reconstruction with regularizers

Differential scenario

Truncated SVD

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Differential microwave imaging

Numerical results – Duke model

Incident fields = 32 plane waves @ 1GHz

SNR=20dB

Next step: 3D problem

The Virtual Family-development of surface-based anatomical models of two adults and two children for dosimetric

simulations, Physics in Medicine and Biology, 2010.

Virtual Family Duke (IT’IS)

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Antenna diagnostic

Numerical Application – array of 10x10 open ended waveguides @10GHz

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Antenna diagnostic

Numerical Application – array of 10x10 open ended waveguides @10GHz

Back propagation 1024 points

Matrix inversion225 points

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Antenna diagnostic

Numerical Application – array of 10x10 open ended waveguides @10GHz

Compressive sensing approaches