Reduced-Dimensionality Inverse Scattering Using Basis Functions

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Reduced-Dimensionality Inverse Scattering Using
Basis Functions

• Andrew E. Yagle
• Dept. of EECS, The University of Michigan
• Ann Arbor, MI

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Presentation Overview

• Problem Statement
• Basis Function Representation
• Matrix Problem Formulation
• Reformulation as Overdetermined Multiparameter
  Eigenvalue Problem
• Use of Left Null Matrix
• 1-D Illustrative Numerical Example

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Inverse Problem Statement

• GIVEN 1 monopole point source antenna 1
  frequency, moving platform (e.g., plane)
• Unknown scatterer V(x) compact support
• Unknown Greens function G(x,y)
• Response at x to source at same x u(x)
• GOAL Reconstruct V(x) from u(x)

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Inverse Scattering Formulation
G(x,x)
V(x)

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Inverse Problem Statement
Reciprocity G(x,y)G(y,x) x and y in ?3
Assume Born (single-scatter) approximation
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Basis Function Representation
Assume Unknown linear combinations of known
basis functions, as follows
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Basis Function Representation

• Should not be separable in receiver x and source
  y locations (precludes deconvolution)
• Dont confuse this with separable in (x,y,z)
• Need not be orthonormal, complete, or
  biorthogonal to each other
• Sample observations spatially uku(x-xk)
  special case impulse basis functions

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Basis Function Representation

• Selections of all of these basis functions are
  problem-dependent
• Multilayered media Greens functionsum of
  several terms with unknown reflections
• Multipole, wavelet, Fourier representations
• Need (NM) independent observations u(x) either
  samples or coefficient dimensionality

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Matrix Problem Formulation
Method-of-Moments (MoM) linear system
Insert expansions into integral equation
Where
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Matrix Problem Formulation
Rewrite as huge (NM)X(NM) linear system
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Matrix Problem Formulation
In principle Could solve this, and then
BUT Far too large to be practical!
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Reformulation as Overdetermined Multiparameter
Eigenvalue Problem
Define N matrices Ai, each (NM)XN, as
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Reformulation as Overdetermined Multiparameter
Eigenvalue Problem
Rewrite Previous (NM)X(NM) system as
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Reformulation as Overdetermined Multiparameter
Eigenvalue Problem
Rewrite Multiparam eigenvalue problem
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Reformulation as Overdetermined Multiparameter
Eigenvalue Problem
1. Heavily overdetermined (NMgtgtNM) 2. Actually
(NM) simultaneous polynomial equations in
(NM) unknowns gi and vj 3. But solution not easy
(see below) 4. Make use of (NM) data points as
follows
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Use of Left Null Matrix

• Apply recent procedure for multichannel blind
  deconvolution (both 1-D and 2-D)
• Tall matrices (rowsgtcolumns) have left
  nullspaces basis can be computed
• null vectorsXmatrix of unknowns0
• This becomes linear system in unknowns
• Now adapt this to the present problem

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Use of Left Null Matrix

• There is a reclining matrix B so that
• BA1A2AN0 00
• Ai is (NM)XM as defined previously
• A1A2AN is thus (NM)X(N-1)M
• B is MX(NM) where MNM-(N-1)M
• B can be PRECOMPUTED from Ai!

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Use of Left Null Matrix
Premult Huge linear system by known B
BUT MXM linear system, not (NM)X(NM)!
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Use of Left Null Matrix

• Instead of the huge (NM)X(NM) linear system, have
  small MXM linear system!
• Precompute the left null vector B from known
  basis-function-derived A matrix Off-line
  computation do for many bases
• Solve system directly for vi coefficients Can
  incorporate a priori information
• Sufficient statistic M-point YBu

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Use of Left Null Matrix Stochastic Formulation

• Usually have a priori pdfs for coefficients
• Compute MAP (Maximum A posteriori Probability)
  estimator instead of the ML (Maximum Likelihood)
  estimator
• If noise and a priori information pdfs are
  Gaussian, get least-squares solution
• Otherwise, use iterative algorithm (EM)

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1-D Illustrative Numerical Example

• 1-D problem entirely discrete space-time
• u(i)response at i to impulsive source at i
• G(i,j)response at i to impulse at j
• u(i)? G(i,j)V(j)G(j,i)? G2(i,j)V(j)
• GOAL Reconstruct V(j) from u(i)

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1-D Illustrative Numerical Example

• BASIS FUNCTION EXPANSIONS
• G2(i,j)g1/(i-j)2g2/(ij)2 N2
• Toeplitz-plus-Hankel structure (not exploited
  here, but not uncommon)
• Symmetric G(i,j)G(j,i) (reciprocity)
• V(j)v1?(j-1)v2?(j-2) M2
• 2-point support for scatterer
• u(i)? G(i,j)V(j)G(j,i)? G2(i,j)V(j)
• GOAL Reconstruct V(j) from u(i)

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1-D Illustrative Numerical Example

• BASIS FUNCTIONS Greens function
• ?1(i,,j)1/(i-j)2 ?2(i,j)1/(ij)2
• ?j(n)?(n-j) (scatterer support 1,2)
• ??k(n)?(n2-j) (sampled observations)
• OBSERVATIONS

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1-D Illustrative Numerical ExampleHuge Linear
System of Equations
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1-D Illustrative Numerical ExampleHuge Linear
System of Equations
Solving this and arranging into matrix
SOLUTION V(j)3?(j-1)4?(j-2) to an unknowable
scale factor
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1-D Illustrative Numerical ExampleTiny Linear
System of Equations
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1-D Illustrative Numerical ExampleTiny Linear
System of Equations
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1-D Illustrative Numerical ExampleTiny Linear
System of Equations

• POINT Solving tiny 2X2 linear system
  instead of solving huge 4X4 linear system
• Sufficient statistic YBu 4-vector to 2-vector
• Null matrix B precomputed from basis functions
  ahead of time, off-line.

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Reformulation as Overdetermined Multiparameter
Eigenvalue Problem
Recall This form of large linear system
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Reformulation as Overdetermined Multiparameter
Eigenvalue Problem

Left-multiply by matrix C where
Cu0 0g1A1gnAnv so that we
have g1A1gnAn is rank-deficient
and Vecg1A1gnAn is linear
combination vecA1vecAnknown basis set.
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Reformulation as Overdetermined Multiparameter
Eigenvalue Problem

• g1A1gnAn can be computed
• iteratively using Lift-and-Project method
• Project g1A1gnAn rank-deficient
• using SVD and setting smallest SV to 0
• 2. Project vecg1A1gnAn onto
• spanvecA1vecAn

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Reformulation as Overdetermined Multiparameter
Eigenvalue Problem

Both of these are (Frobenius matrix)
norm-reducing operations. By Composite Mapping
Theorem, this is guaranteed to converge (maybe to
0!) Problem Takes long time to converge.
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CONCLUSION

• Solve inverse scattering problem in Born
  approximation with coincident point source and
  receiver on moving platform
• Using precomputed null vectors, reduce (NM)X(NM)
  system to MXM system Mcoefficients
  representing scatterer Ncoefficients
  representing Greens
• Sufficient statistic reduce data dimension

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FUTURE WORK

• Should need much less data NMltltNM
• Apply the algorithms we are presently developing
  to solve non-overdetermined multiparameter
  eigenvalue problem
• Sample data for well-conditioned problem
  adaptively choose the vehicle trajectory