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

Reduced-Dimensionality Inverse Reduced-Dimensionality Inverse Scattering Using Basis FunctionsScattering Using Basis Functions

Andrew E. YagleAndrew E. Yagle

Dept. of EECS, The University of MichiganDept. of EECS, The University of Michigan

Ann Arbor, MI Ann Arbor, MI

Presentation OverviewPresentation Overview

• Problem StatementProblem Statement

• Basis Function RepresentationBasis Function Representation

• Matrix Problem FormulationMatrix Problem Formulation

• Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem

• Use of Left Null MatrixUse of Left Null Matrix

• 1-D Illustrative Numerical Example1-D Illustrative Numerical Example

Inverse Problem StatementInverse Problem Statement

• GIVENGIVEN: 1 monopole point source antenna : 1 monopole point source antenna 1 frequency, moving platform (e.g., plane)1 frequency, moving platform (e.g., plane)

• Unknown scatterer V(x); compact supportUnknown scatterer V(x); compact support

• Unknown Green’s function G(x,y)Unknown Green’s function G(x,y)

• Response at Response at xx to source at same to source at same xx: u(x): u(x)

• GOALGOAL: Reconstruct V(x) from u(x): Reconstruct V(x) from u(x)

G(x,x’)

V(x’)

Inverse Scattering Formulation

Inverse Problem StatementInverse Problem Statement

dyxyGyVyxGxu ),()(),()(

Reciprocity:Reciprocity: G(x,y)=G(y,x); x and y in G(x,y)=G(y,x); x and y in 33

dyyVyxGxu )(),()( 2

Assume:Assume: Born (single-scatter) approximation Born (single-scatter) approximation

Basis Function RepresentationBasis Function Representation

),(),(1

2 yxgyxG i

N

ii

Assume:Assume: Unknown linear combinations Unknown linear combinationsof known basis functions, as follows:of known basis functions, as follows:

)()(1

yvyV j

M

jj

dxxxuu kk )()(


• Should not be separable in receiver Should not be separable in receiver xx and and source source yy locations (precludes deconvolution) locations (precludes deconvolution)

• [Don’t confuse this with separable in (x,y,z)][Don’t confuse this with separable in (x,y,z)]

• Need not be orthonormal, complete, or Need not be orthonormal, complete, or biorthogonal to each otherbiorthogonal to each other

• Sample observations spatially: uk=u(x-xk) Sample observations spatially: uk=u(x-xk) [special case: impulse basis functions][special case: impulse basis functions]


• Selections of all of these basis functions are Selections of all of these basis functions are problem-dependent problem-dependent

• Multilayered media: Green’s function=sum of Multilayered media: Green’s function=sum of several terms with unknown reflectionsseveral terms with unknown reflections

• Multipole, wavelet, Fourier representationsMultipole, wavelet, Fourier representations

• Need (NM) independent observations u(x) Need (NM) independent observations u(x) [either samples or coefficient dimensionality][either samples or coefficient dimensionality]

Matrix Problem FormulationMatrix Problem Formulation

kji

M

jj

N

iik Avgu ,,

11

Method-of-Moments (MoM) linear system:Method-of-Moments (MoM) linear system:

dydxxyyxA kjikji )()(),(,,

Insert expansions into integral equation:Insert expansions into integral equation:

Where Where ::


Rewrite as huge (NM)X(NM) linear systemRewrite as huge (NM)X(NM) linear system

MNNMNMNM

NM

NM vg

vg

AA

AA

u

u

11

)()(11

11111


In principle: In principle: Could Could solvesolve this, and then: this, and then:

BUTBUT: Far too large to be practical!: Far too large to be practical!

M

NMNN

M

vv

g

g

vgvg

vgvg

1

1

1

111

Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem

DefineDefine: N matrices Ai, each (NM)XN, as:: N matrices Ai, each (NM)XN, as:

)()(1

111

NMiMNMi

iMi

i

AA

AA

A


RewriteRewrite: Previous (NM)X(NM) system as:: Previous (NM)X(NM) system as:

MN

N

NM vg

vg

AA

u

u

11

1

1


RewriteRewrite: Multiparam eigenvalue problem:: Multiparam eigenvalue problem:

MN

NN

NM vg

vg

AgAg

u

u

11

11

1

)...(


1. Heavily overdetermined (NM>>N+M)1. Heavily overdetermined (NM>>N+M)2. Actually (NM) simultaneous polynomial2. Actually (NM) simultaneous polynomial equations in (N+M) unknowns gi and vjequations in (N+M) unknowns gi and vj3. But solution not easy (see below)3. But solution not easy (see below)4. Make use of (NM) data points as follows:4. Make use of (NM) data points as follows:

Use of Left Null MatrixUse of Left Null Matrix

• Apply recent procedure for multichannel Apply recent procedure for multichannel blind deconvolution (both 1-D and 2-D):blind deconvolution (both 1-D and 2-D):

• ““Tall” matrices (#rows>#columns) have Tall” matrices (#rows>#columns) have left nullspaces; basis can be computedleft nullspaces; basis can be computed

• [null vectors]X[matrix of unknowns]=[0][null vectors]X[matrix of unknowns]=[0]

• This becomes linear system in unknownsThis becomes linear system in unknowns

• Now adapt this to the present problem:Now adapt this to the present problem:


• There is a “reclining” matrix [B] so that:There is a “reclining” matrix [B] so that:

• [B][A1|A2|…|AN]=[0 0…0][B][A1|A2|…|AN]=[0 0…0]

• Ai is (NM)XM as defined previouslyAi is (NM)XM as defined previously

• [A1|A2|…|AN] is thus (NM)X(N-1)M[A1|A2|…|AN] is thus (NM)X(N-1)M

• B is MX(NM) where M=NM-(N-1)MB is MX(NM) where M=NM-(N-1)M

• B can be PRECOMPUTED from Ai!B can be PRECOMPUTED from Ai!


BUTBUT: : MXMMXM linear system, not linear system, not (NM)X(NM)!(NM)X(NM)!

M

NN

NM v

v

BAgY

u

u

B 11

PremultPremult: Huge linear system by known B:: Huge linear system by known B:

Use of Left Null MatrixUse of Left Null Matrix• Instead of the Instead of the hugehuge (NM)X(NM) linear (NM)X(NM) linear

system, have system, have smallsmall MXM linear system! MXM linear system!

• PrecomputePrecompute the left null vector B from the left null vector B from known basis-function-derived A matrix: known basis-function-derived A matrix: Off-lineOff-line computation; do for many bases computation; do for many bases

• Solve system Solve system directlydirectly for vi coefficients: for vi coefficients: Can incorporate Can incorporate a prioria priori information information

• Sufficient statisticSufficient statistic: M-point : M-point Y=B[u]Y=B[u]

Use of Left Null Matrix: Use of Left Null Matrix: Stochastic FormulationStochastic Formulation

• Usually have Usually have a prioria priori pdfs for coefficients pdfs for coefficients

• Compute MAP (Maximum A posteriori Compute MAP (Maximum A posteriori Probability) estimator instead of the ML Probability) estimator instead of the ML (Maximum Likelihood) estimator(Maximum Likelihood) estimator

• If noise and If noise and a prioria priori information pdfs are information pdfs are Gaussian, get least-squares solutionGaussian, get least-squares solution

• Otherwise, use iterative algorithm (EM)Otherwise, use iterative algorithm (EM)

1-D Illustrative Numerical Example1-D Illustrative Numerical Example

• 1-D problem; entirely discrete space-time1-D problem; entirely discrete space-time

• u(i)=response at u(i)=response at ii to impulsive source at to impulsive source at i i

• G(i,j)=response at i to impulse at jG(i,j)=response at i to impulse at j

• u(i)=u(i)= G(i,j)V(j)G(j,i)= G(i,j)V(j)G(j,i)= G^2(i,j)V(j) G^2(i,j)V(j)

• GOAL:GOAL: Reconstruct V(j) from u(i) Reconstruct V(j) from u(i)


• BASIS FUNCTION EXPANSIONSBASIS FUNCTION EXPANSIONS::

• G^2(i,j)=g1/(i-j)^2+g2/(i+j)^2G^2(i,j)=g1/(i-j)^2+g2/(i+j)^2 [N=2] [N=2]

• Toeplitz-plus-Hankel structure (not exploited here, but not Toeplitz-plus-Hankel structure (not exploited here, but not uncommon)uncommon)

• Symmetric: G(i,j)=G(j,i) (reciprocity)Symmetric: G(i,j)=G(j,i) (reciprocity)

• V(j)=v1V(j)=v1(j-1)+v2(j-1)+v2(j-2)(j-2) [M=2] [M=2]

• 2-point support for scatterer2-point support for scatterer

• u(i)=u(i)= G(i,j)V(j)G(j,i)= G(i,j)V(j)G(j,i)= G^2(i,j)V(j) G^2(i,j)V(j)

• GOAL:GOAL: Reconstruct V(j) from u(i) Reconstruct V(j) from u(i)


• BASIS FUNCTIONSBASIS FUNCTIONS: Green’s function:: Green’s function:1(i,,j)=1/(i-j)^2; 1(i,,j)=1/(i-j)^2; 2(i,j)=1/(i+j)^22(i,j)=1/(i+j)^2j(n)=j(n)=(n-j) (scatterer support: [1,2])(n-j) (scatterer support: [1,2])k(n)=k(n)=(n+2-j) (sampled observations)(n+2-j) (sampled observations)

• OBSERVATIONSOBSERVATIONS::

I I=3 I=4 I=5 I=6U(I) 5.445 1.796 0.962 0.617

1-D Illustrative Numerical Example1-D Illustrative Numerical Example“Huge” Linear System of Equations“Huge” Linear System of Equations

22

12

21

11

2222

2222

2222

2222

81

71

41

51

71

61

31

41

61

51

21

31

51

41

11

21

617.0

962.0

796.1

445.5

vg

vg

vg

vg

1-D Illustrative Numerical Example1-D Illustrative Numerical Example“Huge” Linear System of Equations“Huge” Linear System of Equations

432

1

86

43

2212

2111

vgvg

vgvg

SOLUTIONSOLUTION: : V(j)=3V(j)=3(j-1)+4(j-1)+4(j-2)(j-2)[to an unknowable scale factor][to an unknowable scale factor]

Solving this and arranging into matrix:Solving this and arranging into matrix:

1-D Illustrative Numerical Example1-D Illustrative Numerical Example“Tiny” Linear System of Equations“Tiny” Linear System of Equations

00

00

4/15/1

3/14/12/13/1

1/12/1

22

22

22

22

1 BBA

726.677.

573.501.

119.000.

645.070.B


2

1

849.0792.0

048.4166.4

54.11

38.57

v

v

2

1

22

22

22

22

2

8/17/1

7/16/16/15/1

5/14/1

)6(

)5(

)4(

)3(

v

vBgY

u

u

u

u

B


• POINTPOINT: Solving “tiny” 2X2 linear system : Solving “tiny” 2X2 linear system instead of solving “huge” 4X4 linear systeminstead of solving “huge” 4X4 linear system

• Sufficient statisticSufficient statistic Y=B[u]: 4-vector to 2-vector Y=B[u]: 4-vector to 2-vector

• Null matrix B Null matrix B precomputedprecomputed from basis functions from basis functions ahead of time, off-line.ahead of time, off-line.


RecallRecall: This form of large linear system:: This form of large linear system:

MN

NN

NM vg

vg

AgAg

u

u

11

11

1

)...(


Left-multiply by matrix C where Left-multiply by matrix C where C[u]=[0]:C[u]=[0]:[0]=[g1A1+…+gnAn][v][0]=[g1A1+…+gnAn][v] so that we have: so that we have:

g1A1+…+gnAng1A1+…+gnAn is rank-deficient; and: is rank-deficient; and:Vec[g1A1+…+gnAn]Vec[g1A1+…+gnAn] is linear combination is linear combination{vec[A1]…vec[An]}={vec[A1]…vec[An]}=known basis set.known basis set.


[g1A1+…+gnAn][g1A1+…+gnAn] can be computed can be computediteratively using Lift-and-Project method:iteratively using Lift-and-Project method:

1.1. Project Project [g1A1+…+gnAn][g1A1+…+gnAn] rank-deficient rank-deficient using SVD and setting smallest SV to 0;using SVD and setting smallest SV to 0;2. Project 2. Project vec[g1A1+…+gnAn]vec[g1A1+…+gnAn] onto onto span{vec[A1]…vec[An]}span{vec[A1]…vec[An]}


Both of these are (Frobenius matrix) Both of these are (Frobenius matrix) norm-reducing operations.norm-reducing operations.

By Composite Mapping Theorem, this isBy Composite Mapping Theorem, this isguaranteed to converge (maybe to 0!)guaranteed to converge (maybe to 0!)

Problem: Takes long time to converge.Problem: Takes long time to converge.

CONCLUSIONCONCLUSION

• Solve inverse scattering problem in Born Solve inverse scattering problem in Born approximation with coincident point approximation with coincident point source and receiver on moving platformsource and receiver on moving platform

• Using precomputed null vectors, reduce Using precomputed null vectors, reduce (NM)X(NM) system to MXM system; (NM)X(NM) system to MXM system; M=#coefficients representing scatterer; M=#coefficients representing scatterer; N=#coefficients representing Green’sN=#coefficients representing Green’s

• Sufficient statistic reduce data dimensionSufficient statistic reduce data dimension

FUTURE WORKFUTURE WORK

• Should need much less data: N+M<<NMShould need much less data: N+M<<NM

• Apply the algorithms we are presently Apply the algorithms we are presently developing to solve developing to solve nonnon-overdetermined -overdetermined multiparameter eigenvalue problemmultiparameter eigenvalue problem

• Sample data for well-conditioned problem: Sample data for well-conditioned problem: adaptively choose the vehicle trajectory adaptively choose the vehicle trajectory

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

Documents

ux slide

mi slide

multiparam eigenvalue

receiver x

basis functions andrew

source y locations

huge nmxnm linear system

yunknown greens function