Reduced-Dimensionality Inverse
Scattering Using Basis Functions
Andrew E. Yagle
Dept. of EECS, The University of Michigan
Ann Arbor, MI
Presentation Overview
•
•
•
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Problem Statement
Basis Function Representation
Matrix Problem Formulation
Reformulation as Overdetermined
Multiparameter Eigenvalue Problem
• Use of Left Null Matrix
• 1-D Illustrative Numerical Example
Inverse Problem Statement
• GIVEN: 1 monopole point source antenna
1 frequency, moving platform (e.g., plane)
• Unknown scatterer V(x); compact support
• Unknown Green’s function G(x,y)
• Response at x to source at same x: u(x)
• GOAL: Reconstruct V(x) from u(x)
Inverse Scattering Formulation
G(x,x’)
V(x’)
Inverse Problem Statement
u ( x)   G( x, y )V ( y )G( y, x)dy
Reciprocity: G(x,y)=G(y,x); x and y in 3
u ( x)   G ( x, y )V ( y )dy
2
Assume: Born (single-scatter) approximation
Basis Function Representation
Assume: Unknown linear combinations
of known basis functions, as follows:
N
G ( x, y )   g i  i ( x, y )
2
i 1
M
V ( y)   v j  j ( y)
j 1
uk   u ( x) k ( x)dx
Basis Function Representation
• Should not be separable in receiver x and
source y locations (precludes deconvolution)
• [Don’t confuse this with separable in (x,y,z)]
• Need not be orthonormal, complete, or
biorthogonal to each other
• Sample observations spatially: uk=u(x-xk)
[special case: impulse basis functions]
Basis Function Representation
• Selections of all of these basis functions are
problem-dependent
• Multilayered media: Green’s function=sum of
several terms with unknown reflections
• Multipole, wavelet, Fourier representations
• Need (NM) independent observations u(x)
[either samples or coefficient dimensionality]
Matrix Problem Formulation
Method-of-Moments (MoM) linear system:
Insert expansions into integral equation:
N
M
uk   g i  v j Ai , j ,k
i 1
j 1
Where :
Ai , j ,k     i ( x, y ) j ( y ) k ( x)dx dy
Matrix Problem Formulation
Rewrite as huge (NM)X(NM) linear system
 u1   A111  ANM 1   g1v1 

   






  

u NM   A11( NM )  ANM ( NM )   g N vM 
Matrix Problem Formulation
In principle: Could solve this, and then:
 g1v1  g1vM   g1 
  



   v1  vM 

  
 g N v1  g N vM   g N 
BUT: Far too large to be practical!
Reformulation as Overdetermined
Multiparameter Eigenvalue Problem
Define: N matrices Ai, each (NM)XN, as:
 Ai11

Ai 

 Ai1( NM )
AiM 1 


AiM ( NM ) 
Reformulation as Overdetermined
Multiparameter Eigenvalue Problem
Rewrite: Previous (NM)X(NM) system as:
 u1 
 g1v1 
     A  A   
1
N
 


u NM 
 g N vM 
Reformulation as Overdetermined
Multiparameter Eigenvalue Problem
Rewrite: Multiparam eigenvalue problem:
 u1 
 g1v1 
    ( g A  ...  g A )   
1 1
N N
 


u NM 
 g N vM 
Reformulation as Overdetermined
Multiparameter Eigenvalue Problem
1. Heavily overdetermined (NM>>N+M)
2. Actually (NM) simultaneous polynomial
equations in (N+M) unknowns gi and vj
3. But solution not easy (see below)
4. Make use of (NM) data points as follows:
Use of Left Null Matrix
• Apply recent procedure for multichannel
blind deconvolution (both 1-D and 2-D):
• “Tall” matrices (#rows>#columns) have
left nullspaces; basis can be computed
• [null vectors]X[matrix of unknowns]=[0]
• This becomes linear system in unknowns
• Now adapt this to the present problem:
Use of Left Null Matrix
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•
•
•
•
•
There is a “reclining” matrix [B] so that:
[B][A1|A2|…|AN]=[0 0…0]
Ai is (NM)XM as defined previously
[A1|A2|…|AN] is thus (NM)X(N-1)M
B is MX(NM) where M=NM-(N-1)M
B can be PRECOMPUTED from Ai!
Use of Left Null Matrix
Premult: Huge linear system by known B:
 u1 
 v1 




B   Y  g N BAN 
 
 
vM 
u NM 
BUT: MXM linear system, not (NM)X(NM)!
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 Y=B[u]
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)
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)= G^2(i,j)V(j)
• GOAL: Reconstruct V(j) from u(i)
1-D Illustrative Numerical Example
• BASIS FUNCTION EXPANSIONS:
• G^2(i,j)=g1/(i-j)^2+g2/(i+j)^2 [N=2]
• 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) [M=2]
• 2-point support for scatterer
1-D Illustrative Numerical Example
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•
•
•
•
BASIS FUNCTIONS: Green’s function:
1(i,,j)=1/(i-j)^2; 2(i,j)=1/(i+j)^2
j(n)=(n-j) (scatterer support: [1,2])
k(n)=(n+2-j) (sampled observations)
OBSERVATIONS:
I
I=3 I=4
I=5 I=6
U(I) 5.445 1.796 0.962 0.617
1-D Illustrative Numerical Example
“Huge” Linear System of Equations
1
22
5.445   1
1.796   2

  3
0.962   1
2
0.617   4


1
 52
1
2
1
1
2
2
1
2
3
1
2
4
1
2
4
1
2
5
1
2
6
1
2
7
1
2 
5
g
v

1
1 
1


2  g v
6  1 2 
1   g 2 v1 
2
7   g v 
2 2
1
2 
8 
1-D Illustrative Numerical Example
“Huge” Linear System of Equations
Solving this and arranging into matrix:
 g1v1
g v
 2 1
g1v2  3 4 1 


3
4







g 2 v2   6 8   2 
SOLUTION: V(j)=3(j-1)+4(j-2)
[to an unknowable scale factor]
1-D Illustrative Numerical Example
“Tiny” Linear System of Equations
1 / 2
1 / 32
BA1  B 
2
1 / 4
1 / 52
2
1/1 
2
0
1/ 2
2   
1 / 3  0
2
1 / 4 
2
0

0
.070  .645 .501 .573
B

.000 .119  .677 .726
1-D Illustrative Numerical Example
“Tiny” Linear System of Equations
2
u
(
3
)


1 / 4
u ( 4) 
1 / 5 2
  Y  g2 B
B
2
u (5) 
1 / 6
2
u (6) 

1 / 7


1/ 5 
2 
1 / 6  v1 
2 
1 / 7   v2 

2
1/ 8 

2
57.38 4.166 4.048  v1 

11.54  0.792 0.849 v 

 
 2 
1-D Illustrative Numerical Example
“Tiny” Linear System of Equations
• POINT: Solving “tiny” 2X2 linear system
instead of solving “huge” 4X4 linear system
• Sufficient statistic Y=B[u]: 4-vector to 2-vector
• Null matrix B precomputed from basis functions
ahead of time, off-line.
Reformulation as Overdetermined
Multiparameter Eigenvalue Problem
Recall: This form of large linear system:
 u1 
 g1v1 
    ( g A  ...  g A )   
1 1
N N
 


u NM 
 g N vM 
Reformulation as Overdetermined
Multiparameter Eigenvalue Problem
Left-multiply by matrix C where C[u]=[0]:
[0]=[g1A1+…+gnAn][v] so that we have:
g1A1+…+gnAn is rank-deficient; and:
Vec[g1A1+…+gnAn] is linear combination
{vec[A1]…vec[An]}=known basis set.
Reformulation as Overdetermined
Multiparameter Eigenvalue Problem
[g1A1+…+gnAn] can be computed
iteratively using Lift-and-Project method:
1. Project [g1A1+…+gnAn] rank-deficient
using SVD and setting smallest SV to 0;
2. Project vec[g1A1+…+gnAn] onto
span{vec[A1]…vec[An]}
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.
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;
M=#coefficients representing scatterer;
N=#coefficients representing Green’s
• Sufficient statistic reduce data dimension
FUTURE WORK
• Should need much less data: N+M<<NM
• 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