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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 )()(
Basis Function RepresentationBasis Function Representation
• 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]
Basis Function RepresentationBasis Function Representation
• 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 ::
Matrix Problem FormulationMatrix Problem Formulation
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
Matrix Problem FormulationMatrix Problem Formulation
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
Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem
RewriteRewrite: Previous (NM)X(NM) system as:: Previous (NM)X(NM) system as:
MN
N
NM vg
vg
AA
u
u
11
1
1
Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem
RewriteRewrite: Multiparam eigenvalue problem:: Multiparam eigenvalue problem:
MN
NN
NM vg
vg
AgAg
u
u
11
11
1
)...(
Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem
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:
Use of Left Null MatrixUse of Left Null Matrix
• 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!
Use of Left Null MatrixUse of Left Null Matrix
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)
1-D Illustrative Numerical Example1-D Illustrative Numerical Example
• 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)
1-D Illustrative Numerical Example1-D Illustrative Numerical Example
• 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
1-D Illustrative Numerical Example1-D Illustrative Numerical Example“Tiny” Linear System of Equations“Tiny” Linear System of Equations
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
1-D Illustrative Numerical Example1-D Illustrative Numerical Example“Tiny” Linear System of Equations“Tiny” Linear System of Equations
• 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.
Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem
RecallRecall: This form of large linear system:: This form of large linear system:
MN
NN
NM vg
vg
AgAg
u
u
11
11
1
)...(
Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem
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.
Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem
[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]}
Reformulation as Overdetermined Reformulation as Overdetermined Multiparameter Eigenvalue ProblemMultiparameter Eigenvalue Problem
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
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