Signal SubspaceBased NonIterative Exact Inverse Scattering of Point Targets

1
Signal Subspace-Based Non-Iterative Exact
Inverse Scattering of Point Targets

• Edwin Marengo and Fred Gruber
• Department of Electrical and Computer Engineering
  Northeastern University, Boston Massachusetts
  02115

2
Problem Statement
Scattering or data matrix K
Scattering potential describing
constitutive properties of the scatterers
V(r)
Inverse problem
3
Problem Statement

• Nonlinear Iterative?
• Non-iterative (direct) formula?
• (Mueller, Siltanen and Isaacson)
• For certain canonical classes of
• scatterers, it can be solved via
• a direct analytical formula.
• Computational non-iterative inverse
• scattering (real-time processing).

4
Problem Statement
Active array of point transmitters and receivers
Generally non-coincident


background medium
?
M
point targets
?
K
Data matrix
At given frequency ?
5
Two-Step Procedure

• Target location step via a) time-reversal MUSIC
  or
• b) a high-dimensional signal subspace method.

A.J. Devaney, E.A. Marengo and F.K. Gruber,
Time-reversal-based imaging and inverse
scattering of multiply scattering point targets,
J. Acoust. Soc. Am., Vol. 118, p. 3129-3138,
2005.
6
Two-Step Procedure

• Target location step via a) time-reversal MUSIC
  or
• b) a high-dimensional signal subspace method.
• Scattering strength estimation step via a new
  non-iterative
• approach which holds despite the nonlinearity
  of the
• mapping from the scattering strength to the
  data matrix.

A.J. Devaney, E.A. Marengo and F.K. Gruber,
Time-reversal-based imaging and inverse
scattering of multiply scattering point targets,
J. Acoust. Soc. Am., Vol. 118, p. 3129-3138,
2005.
7
Two-Step Procedure

• Target location step via a) time-reversal MUSIC
  or
• b) a high-dimensional signal subspace method.
• Scattering strength estimation step via a new
  non-iterative
• approach which holds despite the nonlinearity
  of the
• mapping from the scattering strength to the
  data matrix.

A.J. Devaney, E.A. Marengo and F.K. Gruber,
Time-reversal-based imaging and inverse
scattering of multiply scattering point targets,
J. Acoust. Soc. Am., Vol. 118, p. 3129-3138,
2005.
8
Two-Step Procedure

• Target location step via a) time-reversal MUSIC
  or
• b) a high-dimensional signal subspace method.
• Scattering strength estimation step via a new
  non-iterative
• approach which holds despite the nonlinearity
  of the
• mapping from the scattering strength to the
  data matrix.

CRB
A.J. Devaney, E.A. Marengo and F.K. Gruber,
Time-reversal-based imaging and inverse
scattering of multiply scattering point targets,
J. Acoust. Soc. Am., Vol. 118, p. 3129-3138,
2005.
E.A. Marengo and F.K. Gruber, to appear (2006).
9
Two-Step Procedure

• Target location step via a) time-reversal MUSIC
  or
• b) a high-dimensional signal subspace method.
• Scattering strength estimation step via a new
  non-iterative
• approach which holds despite the nonlinearity
  of the
• mapping from the scattering strength to the
  data matrix.

A.J. Devaney, E.A. Marengo and F.K. Gruber,
Time-reversal-based imaging and inverse
scattering of multiply scattering point targets,
J. Acoust. Soc. Am., Vol. 118, p. 3129-3138,
2005.
E.A. Marengo and F.K. Gruber, to appear (2006).
10
Two-Step Procedure

• Target location step via a) time-reversal MUSIC
  or
• b) a high-dimensional signal subspace method.
• Scattering strength estimation step via a new
  non-iterative
• approach which holds despite the nonlinearity
  of the
• mapping from the scattering strength to the
  data matrix.

A.J. Devaney, E.A. Marengo and F.K. Gruber,
Time-reversal-based imaging and inverse
scattering of multiply scattering point targets,
J. Acoust. Soc. Am., Vol. 118, p. 3129-3138,
2005.
E.A. Marengo and F.K. Gruber, to appear (2006).
11
Forward Model
where the scattering potential
total medium
reflectivities
- Greens functions. - Diffusion.
Other PDEs.
12
Forward Model
13
Forward Model
Background Green function vectors
14
Forward Model
Background Green function vectors
Total (background plus targets) Green function
vectors
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Foldy-Lax Multiple Scattering Model
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Foldy-Lax Multiple Scattering Model
17
Foldy-Lax Multiple Scattering Model
Nonlinearity
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Foldy-Lax Multiple Scattering Model
Born approximation
0,r
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Foldy-Lax Multiple Scattering Model
Nonlinearity
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Previous Work and Overview
(Reflectivity inversion)

• The calculation of the target reflectivities has
  been a topic of previous investigations1,2.
• While trivial for weak scatterers3, in the
  presence of multiple scattering the problem
  becomes nonlinear, and was tackled in previous
  research by means of an iterative technique3.
• Here we propose a new non-iterative technique
  with a performance comparable to the iterative
  one.

• M. Cheney, The linear sampling method and the
  MUSIC algorithm,' Inv. Probl., Vol. 17, pp.
  591-595, 2001.
• D. Colton and R. Kress, Eigenvalues of the far
  field operator for the Helmholtz equation in an
  absorbing medium'', SIAM J. Appl. Math., Vol. 55,
  pp. 1724-1735, 1995.
• A.J. Devaney, E.A. Marengo and F.K. Gruber,
  Time-reversal-based imaging and inverse
  scattering of multiply scattering point
  targets,'' J. Acoust. Soc. Am., Vol. 118, pp.
  3129-3138, 2005.

21
Previous Work and Overview
(Target localization)

• Early accounts of the high-dimensional signal
  subspace method can be found in a number of
  conference proceedings authored by the present
  authors4,5 and in work by Shi and Nehorai.6
• The present treatment differs in that
• addresses the question of number of localizable
  targets directly, demonstrating how the
  high-dimensional signal subspace method can
  significantly enhance the number of localizable
  targets if they are weakly interacting and
• comparatively studies the performance of the
  method relative to time-reversal MUSIC and the
  pertinent CRB.

• E.A. Marengo, Coherent multiple signal
  classification for target location using antenna
  arrays'', Proc. Natl. Radio Sci. Meeting,
  Boulder, Colorado, pp.169, January 2005.
• E.A. Marengo and F.K. Gruber, Single snapshot
  signal subspace method for target location'',
  Proc. IEEE Antennas and Propagat. Int. Symp. and
  USNC/URSI Natl. Radio Sci. Meeting, Washington,
  D.C., Vol. 2A, p.660-663 July 2005.
• G. Shi and A. Nehorai, Maximum likelihood
  estimation of point scatterers for computational
  time-reversal imaging, Commun. in Info. and
  Syst. Vol. 5, pp. 227-256, 2005.

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Non-Iterative Nonlinear Inversion Formula
Assumption
targets
transmitters receivers
A.J. Devaney, Super-resolution processing of
multi-static data using time-reversal and
MUSIC, 2000.
Key Linear Independence Fact

except rare, pathological configurations
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Non-Iterative Nonlinear Inversion Formula
Note If the MUSIC conditions hold, then
the conditions below also hold.
Assumption
targets
transmitters receivers
A.J. Devaney, Super-resolution processing of
multi-static data using time-reversal and
MUSIC, 2000.
Key Linear Independence Fact

except rare, pathological configurations
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Assume Targets Have Been Located
receivers


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Active Target Isolation
receivers


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Active Target Isolation
receivers


multiple scattering
Cannot isolate by conventionally a priori
focusing on that target.
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Active Target Isolation
A post-interaction approach
receivers


?
(known background Green function)
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Active Target Isolation
A post-interaction approach
receivers


?
(known background Green function)
Signal of minimum L2 norm giving rise to this
field
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Non-Iterative Nonlinear Inversion Formula
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Non-Iterative Nonlinear Inversion Formula
From the linear independence fact,
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Non-Iterative Nonlinear Inversion Formula
From the linear independence fact,
Using the Foldy-Lax model
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Non-Iterative Nonlinear Inversion Formula
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Computational Example 1
- coincident, N7 transceivers
1
1
Goal - MUSIC -- non-iterative
inversion
1
1
can use another method
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Estimation Error Versus S/N Ratio
Location
Born
Iterative
Non-iterative
A.J. Devaney, E.A. Marengo and F.K. Gruber,
Time-reversal-based imaging and inverse
scattering of multiply scattering point targets,
J. Acoust. Soc. Am., Vol. 118, p. 3129-3138, 2005.
37
Estimation Error Versus S/N Ratio (Assuming
Correct Positions)
Iterative
gt 80 iterations
Non-iterative
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Computational Example 1 (cont.)
- coincident, N7 transceivers
Another set of values for the reflectivities
1
2
Goal - MUSIC -- non-iterative
inversion
3
4
can use another method
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Estimation Error Versus S/N Ratio (Other Values)
(2)
(1)
Convergence question!
(initial guess sensitive)
(3)
(4)
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Key Observations The Non-Iterative Approach

• Under no noise, and under the time-reversal
  conditions, it is an exact direct
• analytical solver for the inverse scattering
  problem for multiply scattering
• point targets.
• In the presence of realistic noise, there are
  errors due to the localization
• step (e.g., via MUSIC, ML or other).
• Even if locations are known, in the presence of
  data perturbations there are
• errors.
• Variance of the estimates is comparable to that
  of the iterative approach
• for the tested examples.
• Iterative approach sometimes has convergence
  problems.
• For regimes of SNR non-iterative approach
  outperforms. So we conclude
• the method works comparably to the iterative
  approach at significant
• computational advantage.

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High Dimensional Position Estimation
Born Approximated Case
vectorizing
M linearly independent columns
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Reflectivity estimation
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Equivalent to ML
G. Shi and A. Nehorai, Maximum likelihood
estimation of point scatterers for computational
time-reversal imaging, Commun. in Info. and
Syst., Vol. 5, pp. 227-256, 2005.
Drawbacks

• If the space is 3D then the search has to be done
  in 3M dimensions
• We need to know M.

Benefits

• Less variance in the estimates
• Can detect up to

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Multiple Scattering Case
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Computational Example 2
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Born approximation
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Multiple scattering
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Conclusions

• Novel non-iterative computational approach for
  nonlinear inverse scattering of multiply
  scattering point targets. (Small targets, or
  points in a computational grid representing an
  extended scatterer whose scattering potential
  function one wishes to compute without
  iterations).
• It can be applied whenever time-reversal MUSIC
• can be applied.
• A high dimensional approach with an estimate
  variance lower than the time reversal MUSIC
  approach at the expense of much higher
  computational cost.
• In the Born approximated case the high
  dimensional approach is capable of detecting many
  more targets than the time reversal approach. (ML
  est.)

Support
AFOSR Grant FA9550-06-01-0013