TimeReversal MUSIC Imaging of

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IEEE APS/URSI 2006
Time-Reversal MUSIC Imaging of Extended
Scatterers, and a Non-Iterative Exact
Inverse Scattering Formula
Edwin A. Marengo and Fred K. Gruber
Department of Electrical and Computer
Engineering and Center for Subsurface Sensing and
Imaging Systems Northeastern University Boston,
Massachusetts 02115, USA
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Problem Statement
Scattering or data matrix K
(inverse problem)
scatterers support

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Problem Statement
known
background Greens function
G
(unknown)
(unknown)
General remote sensing aperture
Scattering or data matrix K
(inverse problem)
scatterers support

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Visualization Approach
transmit and receive aperture
?
extended scatterer
image depicting scatterer
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Visualization Approach
known
background Greens function
G
?
(unknown)
(unknown)
General remote sensing aperture
Objective To obtain images of the scatterers
depicting their supports from knowledge of the
data matrix K.

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Literature Perspective
Previous work on shape reconstruction/inverse
support problem
Colton, Kirsch, Potthast, H. Zhao, and their
co-workers Sampling methods.
Chambers, Berryman, other, in forward aspects of
time-reversal.
- Near field, and limited view data. -
Assumes realistic noise, hence adopts
principal component analysis approach.
(e.g., Rayleigh fading environment, other
clutter.) - The regularized,
time-reversal MUSIC version. - New enclosure
form of the approach. - Unifying
handling of space-time information. -
Non-iterative nonlinear inverse scattering
(computational grids).
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Context and Generalizations

General propagators and scattering potentials.
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Random Media Perspective
Diffusion equation context for certain cluttered
random media.
Guillaume Bal Eric Miller

• Self-averaging imaging properties of
  time-reversal
• using broadband fields
• Papanicolaous group.
• Finks group.
• Greens function mismatch problem
• Carins group.

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Exact Forward Scattering
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Support Regions
transmitter
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Support Regions
receiver
transmitter
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Support Regions
dominant targets plus clutter
receiver
transmitter
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The Scattering or Data Matrix K
Output, received field
Input, transmitter signal
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Mappings and Hilbert Spaces
Usual inner products
Transmit space
Frequency domain context for simplicity of
exposition.
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Mappings and Hilbert Spaces (2)
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Mappings and Hilbert Spaces (2)
Transmit space
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Mappings and Hilbert Spaces (3)
Transmitter Propagator
Masked field space
Transmit space
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Mappings and Hilbert Spaces (4)
(Exact scattering theory)
Interaction
Masked field space
Transmit space
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Mappings and Hilbert Spaces (4)
(Exact scattering theory)
Interaction
induced source
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Mappings and Hilbert Spaces (4)
(Exact scattering theory)
Interaction
induced source
nonsingular
(Born series)
(exact scattering)
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Mappings and Hilbert Spaces (5)
Interaction
Induced source space
Masked field space
Transmit space
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Mappings and Hilbert Spaces (6)
Receiver Propagator
Interaction
Induced source space
Masked field space
Receive space
Transmit space
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Scattering or Data Matrix K
Known
Depend only on illumination region and G
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Scattering or Data Matrix K
unknown
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Scattering or Data Matrix K
Known structure, unknown details
Depend only on unknown scatterer support and G
Multiple scattering, nonsingular
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Forward Scattering Description in Essentially
Finite Dimensional Spaces
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Effective Transmitter Propagator
Transmit space
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Effective Transmitter Propagator
Singular Value Decomposition (SVD)
(for some index gt Mtot. singular values are
negligible (Pierri))
Effective transmitter space
Transmit space
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Effective Receiver Propagator
Singular Value Decomposition (SVD)
Effective number of receiver modes
Effective receiver space
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Nature of the Data Matrix
Probing signals in effect. transmitter space and
measured signals in effective receiver space.
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Nature of the Data Matrix
Probing signals in effect. transmitter space and
measured signals in effective receiver space.
Key intrinsic to the unknown support.
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Nature of the Data Matrix (2)
Approximation in essentially finite dimensional
spaces
(negligible singular functions taken out)
Regularization or numerical filtering
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Rank of the Data Matrix
effective rank
effective
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Rank of the Data Matrix (2)
Transmit space
Receive space
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Rank of the Data Matrix (2)
Transmit space
Receive space
If and only if
dmax(0,c-b)
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Rank of the Data Matrix (3)
Opposite of sufficient condition
At least one linear superposition of the transmit
signals such that
field0
Transmit space
Receive space
non-scattering or invisible object to that
linear superposition
Such a superposition represents no new
information, hence rank diminishes.
Let us ignore this possibility next, assuming
that the sufficient condition holds so that
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Support Inversion
Transmit space
Receive space
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Singular Value Decomposition
From the actual data
Controlled by transmit space
(negligible singular values being taken out)
Defines range and space of visibly
scattering signals
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Key Observations
Dimensionality Ntot.
having non-negligible singular values
Visible subspace
known from SVD
Dimensionality Mtot.
having non-negligible singular values
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Key Observations (2)
actual
assumed
Visible subspace
Null
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Pseudospectrum for Imaging
Transmit Mode
If the rank R is controlled by the transmit space
then
if
Visible subspace
known from data
(Null)
assumed
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Extended Versus Point Targets
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Computer Simulation Study
transmit and receive aperture
extended scatterer
image depicting scatterer
20 dB SNR
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Computer Simulation Study (2)
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Computer Simulation Study (3)
Using 1 null subspace singular value (smallest
singular value) only in the construction of the
null subspace.
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Computer Simulation Study (4)
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Computer Simulation Study (5)
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Computer Simulation Study (6)
Broadband version
100 300 MHz, 20 dB SNR
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Full Inverse Scattering - Non-iteratively
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Active Target Isolation
(Cepni, Stancil and their collaborators)
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Active Target Isolation
scattered field 0
Also minimally invasive focusing
scattered field 0
(Cepni, Stancil and their collaborators)
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Active Target Isolation (2)
receivers


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Active Target Isolation (3)
receivers


multiple scattering
Cannot isolate by conventionally a priori
focusing on that target.
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Forward Model
scattering strengths
Background Green function vectors
Total (background plus targets) Green function
vectors
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Foldy-Lax Multiple Scattering Model
Nonlinearity
scattering strengths
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Key Observations
Time-reversal MUSIC assumption
A.J. Devaney, Super-resolution processing of
multi-static data using time-reversal and
MUSIC, 2000.
Key Linear Independence Fact
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Active Target Isolation (4)
A post-interaction approach
receivers


?
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Active Target Isolation (5)
receivers


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Non-Iterative Nonlinear Inversion Formula
Using the linear independence fact,
and substituting above
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Computational Example
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Estimation Error Versus S/N Ratio
Location
Born
Iterative
Non-iterative
(JASA, 2005)
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Conclusion and Future Directions

• Time-reversal MUSIC approach for support
  inversion of
• extended targets.
• Recent focus Piecewise constant case.
• Broadband fields.
• Support inversion as a prior to full
  reconstruction of
• constitutive properties or scattering
  potential, which
• we wish to address non-iteratively for certain
  classes
• of scatterers, modeled computationally.

Support
AFOSR Grant FA9550-06-01-0013