Title: TimeReversalBased Noniterative Exact Inverse Scattering of Multiply Scattering Point Targets
1Time-Reversal-Based Noniterative Exact Inverse
Scattering of Multiply Scattering Point Targets
- Edwin Marengo and Fred Gruber
- Department of Electrical and Computer
Engineering, Northeastern University, Boston,
Massachusetts 02115
2Problem Statement
Active array of point transmitters and receivers
Generally non-coincident
background medium
?
M
point targets
?
K
Data matrix
At given frequency ?
3Two-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.
4Two-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.
5Two-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.
6Forward Model
where the scattering potential
total medium
reflectivities
- Greens functions. - Diffusion.
Other PDEs.
7Forward Model
8Forward Model
Background Green function vectors
9Forward Model
Background Green function vectors
Total (background plus targets) Green function
vectors
10Foldy-Lax Multiple Scattering Model
11Foldy-Lax Multiple Scattering Model
Nonlinearity
12Previous 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.
13Previous 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.
14Non-Iterative Nonlinear Inversion Formula
Assumption
A.J. Devaney, Super-resolution processing of
multi-static data using time-reversal and
MUSIC, 2000.
Key Linear Independence Fact
15Active Target Isolation
receivers
16Active Target Isolation
receivers
multiple scattering
Cannot isolate by conventionally a priori
focusing on that target.
17Active Target Isolation
A post-interaction approach
receivers
?
(known background Green function)
18Active Target Isolation
A post-interaction approach
receivers
?
(known background Green function)
19Non-Iterative Nonlinear Inversion Formula
20Non-Iterative Nonlinear Inversion Formula
From the linear independence fact,
21Non-Iterative Nonlinear Inversion Formula
From the linear independence fact,
Using the Foldy-Lax model
22Computational Example 1
23(No Transcript)
24(No Transcript)
25Estimation 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.
26Estimation Error Versus S/N Ratio (Assuming
Correct Positions)
Iterative
gt 80 iterations
Non-iterative
27Estimation Error Versus S/N Ratio (Other Values)
(2)
(1)
Convergence question!
(3)
(4)
28High Dimensional Position Estimation
Born Approximated Case
vectorizing
M linearly independent columns
29Reflectivity estimation
30Equivalent 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
31Multiple Scattering Case
32Computational Example 2
33Born approximation
34Multiple scattering
35Conclusions
- 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.)