Reduced-Dimensionality Inverse Scattering Using Basis Functions

1. Reduced-Dimensionality Inverse Scattering Using Basis Functions Andrew E. Yagle Dept. of EECS, The University of Michigan Ann Arbor, MI

2. 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

3. 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)

4. Inverse Scattering Formulation G(x,x’) V(x’)

5. Inverse Problem Statement Reciprocity: G(x,y)=G(y,x); x and y in 3 Assume: Born (single-scatter) approximation

6. Basis Function Representation Assume: Unknown linear combinations of known basis functions, as follows:

7. 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]

8. 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]

9. Matrix Problem Formulation Method-of-Moments (MoM) linear system: Insert expansions into integral equation: Where :

10. Matrix Problem Formulation Rewrite as huge (NM)X(NM) linear system

11. Matrix Problem Formulation In principle: Could solve this, and then: BUT: Far too large to be practical!

12. Reformulation as Overdetermined Multiparameter Eigenvalue Problem Define: N matrices Ai, each (NM)XN, as:

13. Reformulation as Overdetermined Multiparameter Eigenvalue Problem Rewrite: Previous (NM)X(NM) system as:

14. Reformulation as Overdetermined Multiparameter Eigenvalue Problem Rewrite: Multiparam eigenvalue problem:

15. 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:

16. 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:

17. Use of Left Null Matrix • 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!

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

19. 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]

20. 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)

21. 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)

22. 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 • u(i)= G(i,j)V(j)G(j,i)= G^2(i,j)V(j) • GOAL: Reconstruct V(j) from u(i)

23. 1-D Illustrative Numerical Example • 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:

24. 1-D Illustrative Numerical Example“Huge” Linear System of Equations

25. 1-D Illustrative Numerical Example“Huge” Linear System of Equations Solving this and arranging into matrix: SOLUTION: V(j)=3(j-1)+4(j-2) [to an unknowable scale factor]

26. 1-D Illustrative Numerical Example“Tiny” Linear System of Equations

27. 1-D Illustrative Numerical Example“Tiny” Linear System of Equations

28. 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.

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

30. 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.

31. Reformulation as Overdetermined Multiparameter Eigenvalue Problem • [g1A1+…+gnAn] can be computed • iteratively using Lift-and-Project method: • Project [g1A1+…+gnAn] rank-deficient • using SVD and setting smallest SV to 0; • 2. Project vec[g1A1+…+gnAn] onto • span{vec[A1]…vec[An]}

32. 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.

33. 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

34. 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