2-D and 3-D Blind Deconvolution of Even Point-Spread Functions Andrew E. Yagle and Siddharth Shah Dept. of EECS, The University of Michigan Ann Arbor,

2-D and 3-D Blind Deconvolution 2-D and 3-D Blind Deconvolution of Even Point-Spread Functionsof Even Point-Spread Functions

Andrew E. Yagle and Siddharth ShahAndrew E. Yagle and Siddharth Shah

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

• Problem Relevance Problem Relevance

• 1-D Blind Deconvolution of Even PSFs1-D Blind Deconvolution of Even PSFs

• 2-D Blind Deconvolution of Even PSFs2-D Blind Deconvolution of Even PSFs

• 3-D Blind Deconvolution of Even PSFs3-D Blind Deconvolution of Even PSFs

• ConclusionConclusion

Problem StatementProblem Statement

• GIVENGIVEN: Observations : Observations y(x)=h(x)*u(x)+n(x)y(x)=h(x)*u(x)+n(x)

• h(x)=unknown even PSF [h(x)=unknown even PSF [h(x)=h(-x)h(x)=h(-x)]]

• u(x)=unknown compact-support imageu(x)=unknown compact-support image

• n(x)=white Gaussian noise random fieldn(x)=white Gaussian noise random field

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

Previous WorkPrevious Work

• Iterative algorithmsIterative algorithms: alternating projections: alternating projections

• Sometimes don’t converge; usually stagnateSometimes don’t converge; usually stagnate

• NASRIF:NASRIF: requires small-support inverse PSF requires small-support inverse PSF

• Often not true (e.g., Gaussian-like PSFs)Often not true (e.g., Gaussian-like PSFs)

• Statistical methodsStatistical methods: require stochastic image : require stochastic image models; often insufficient for unique answermodels; often insufficient for unique answer

Problem RelevanceProblem Relevance

• 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) which Unknown Green’s function G(x-y) which represents channel propagation effectsrepresents channel propagation effects

• 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

Problem RelevanceProblem Relevance

dyxyGyVyxGxu )()()()(

Reciprocity:Reciprocity: G(x-y)=G(y-x) [even PSF] G(x-y)=G(y-x) [even PSF]

dyyVyxGxu )()()( 2

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

Problem AmbiguitiesProblem Ambiguities

• SCALE FACTORSCALE FACTOR: Solution {h(x),u(x)} : Solution {h(x),u(x)} implies solution {ch(x),u(x)/c} for any c.implies solution {ch(x),u(x)/c} for any c.

• TRANSLATIONTRANSLATION: Solution {h(x),u(x)} : Solution {h(x),u(x)} implies solution {h(x+d),u(x-d)} for any implies solution {h(x+d),u(x-d)} for any dd..

• EXCHANGEEXCHANGE: Solution {h(x),u(x)} implies : Solution {h(x),u(x)} implies solution {u(x),h(x)} [but solution {u(x),h(x)} [but h(x)=h(-x)h(x)=h(-x) avoids] avoids]

• REDUCIBLEREDUCIBLE: Solution {h(x),u(x)} need : Solution {h(x),u(x)} need irreducible z-transforms (almost surely).irreducible z-transforms (almost surely).

1-D Blind Deconvolution1-D Blind Deconvolution

• ObserveObserve: : y(n)=h(n)*u(n)y(n)=h(n)*u(n) [omit noise here] [omit noise here]

• Even PSFEven PSF: : h(n)=h(-n)h(n)=h(-n) [symmetric] [symmetric]

• z-transformsz-transforms: : Y(z)=H(z)U(z)=H(1/z)U(z).Y(z)=H(z)U(z)=H(1/z)U(z).

• Y(z)U(1/z)=H(z)U(z)U(1/z)=Y(1/z)U(z)Y(z)U(1/z)=H(z)U(z)U(1/z)=Y(1/z)U(z)

• ResultantResultant: Equate coefficients gives Toeplitz: Equate coefficients gives Toeplitz

• NeedNeed: No U(z) zeros in conjugate reciprocal : No U(z) zeros in conjugate reciprocal quadruples (in practice, none on unit circle)quadruples (in practice, none on unit circle)

1-D Blind Deconvolution: Example1-D Blind Deconvolution: Example

SolveSolve: : {24,57,33}={h(0),h(0)}*{u(0),u(1)}{24,57,33}={h(0),h(0)}*{u(0),u(1)}

0

0

0

0

)0(

)1(

)1(

)0(

330240

57335724

24573357

024033

u

u

u

u

SolutionSolution: : {u(0),u(1)}={8,11} {u(0),u(1)}={8,11} [to scale factor][to scale factor]

Noisy Data ProblemNoisy Data Problem

• GoalGoal: Compute maximum-likelihood (ML) : Compute maximum-likelihood (ML) estimator of image in white Gaussian noiseestimator of image in white Gaussian noise

• Log-LikelihoodLog-Likelihood: Need to find minimum : Need to find minimum perturbation of data {y(n)} such that:perturbation of data {y(n)} such that:

• Overdetermined Toeplitz matrix has Overdetermined Toeplitz matrix has reduced rank, so null vector exists;reduced rank, so null vector exists;

• Frobenius matrix norm ||Frobenius matrix norm ||Y|| minimized.Y|| minimized.• How to solve this linear algebra problem?How to solve this linear algebra problem?

Noisy Data SolutionNoisy Data Solution

• Two methods were investigated:Two methods were investigated:

• Lift-and-ProjectLift-and-Project (LAP):(LAP):

• Lift to Toeplitz using “Toeplitzation”;Lift to Toeplitz using “Toeplitzation”;

• Project to reduced-rank using SVD.Project to reduced-rank using SVD.

• Structured Total Least SquaresStructured Total Least Squares (STLS): (STLS):

• Perturb y(n) to satisfy constraintsPerturb y(n) to satisfy constraints

2-D Blind Deconvolution2-D Blind Deconvolution

• Use Use Fourier transformFourier transform to decouple the to decouple the 2-D problem into 1-D problems:2-D problem into 1-D problems:

• Analogous to 1-D, get 2-D equationAnalogous to 1-D, get 2-D equation

• Y(x,y)U(1/x,1/y)=Y(1/x,1/y)U(x,y)Y(x,y)U(1/x,1/y)=Y(1/x,1/y)U(x,y)

• Set Set y=yk=exp{j2y=yk=exp{j2k/N}k/N} in this. Get: in this. Get:• Y(x,yk)U*(1/x*,yk)=Y*(1/x*,yk)U(x,yk)Y(x,yk)U*(1/x*,yk)=Y*(1/x*,yk)U(x,yk)

• Decoupled (in yk) 1-D problems as beforeDecoupled (in yk) 1-D problems as before

2-D Blind Deconvolution2-D Blind Deconvolution

• Scale factor between 1-D problems:Scale factor between 1-D problems:

• Resolved by performing decoupling in Resolved by performing decoupling in both x and y; comparing solutionsboth x and y; comparing solutions

• Additive WGN decouples into WGNsAdditive WGN decouples into WGNs

• Even more interesting in Even more interesting in 3-D3-D problem: problem:

• See papers for details and solutionsSee papers for details and solutions

2-D Blind Deconvolution2-D Blind Deconvolution

• Unknowns Unknowns are the pixel values u(i,j)are the pixel values u(i,j)

• No need to compute PSF and then No need to compute PSF and then deconvolve PSF from the noisy datadeconvolve PSF from the noisy data

• Can incorporate Can incorporate irregular supportirregular support of of image image explicitlyexplicitly (toss matrix columns) (toss matrix columns)

• Can use Can use edge-preserving regularizationedge-preserving regularization algorithms (linear system for u(i,j))algorithms (linear system for u(i,j))

2-D Blind Deconvolution2-D Blind Deconvolution

• 452X452 image blurred with UNKNOWN452X452 image blurred with UNKNOWN

• 61X61 Gaussian PSF; noiseless example61X61 Gaussian PSF; noiseless example

2-D Blind Deconvolution2-D Blind Deconvolution

• 220X220 image blurred with UNKNOWN220X220 image blurred with UNKNOWN

• 37X37 Gaussian PSF; noiseless example37X37 Gaussian PSF; noiseless example

2-D Blind Deconvolution2-D Blind Deconvolution

• MSE vs. SNR for: TLS; LAP; STLN methodsMSE vs. SNR for: TLS; LAP; STLN methods

• MSE vs. SNR for: Direct vs. Fourier methodsMSE vs. SNR for: Direct vs. Fourier methods

3-D Blind Deconvolution3-D Blind Deconvolution

• Use Use Fourier transformFourier transform to decouple to decouple the 3-D problem into 1-D problems:the 3-D problem into 1-D problems:

• Analogous to previous, get equationAnalogous to previous, get equation

• Y(x,yi,zj)U*(1/x*,yi,zj) = Y(x,yi,zj)U*(1/x*,yi,zj) = Y*(1/x*,yi,zj)U(x,yi,zj)Y*(1/x*,yi,zj)U(x,yi,zj)

• where where yi=exp{j2yi=exp{j2i/N}i/N} and zj similar. and zj similar.• Decoupled (in yi and zj) 1-D problems.Decoupled (in yi and zj) 1-D problems.

3-D Blind Deconvolution3-D Blind Deconvolution

• 63X63X63 image blurred with UNKNOWN63X63X63 image blurred with UNKNOWN

• 9X9X9 3-D Gaussian PSF; noiseless example9X9X9 3-D Gaussian PSF; noiseless example

3-D Blind Deconvolution3-D Blind Deconvolution

MSE vs. SNR for: STLN vs. TLSMSE vs. SNR for: STLN vs. TLS

ConclusionConclusion

• 2-D and 3-D blind deconvolution problem2-D and 3-D blind deconvolution problem

• Require PSF to be an even functionRequire PSF to be an even function

• Application to scattering: channel effectsApplication to scattering: channel effects

• Decouple 2-D and 3-D to 1-D problemsDecouple 2-D and 3-D to 1-D problems

• Solve 1-D problems using resultantSolve 1-D problems using resultant

• Use STLN or LAP for MLE in noisy dataUse STLN or LAP for MLE in noisy data

Goals for Next YearGoals for Next Year

• Apply basis function inverse scattering to Apply basis function inverse scattering to bases developed by Bownikbases developed by Bownik

• Mine signature detection using transforms Mine signature detection using transforms to detect hyperbolae (prestack) vs. linesto detect hyperbolae (prestack) vs. lines

• Apply to channel identification for radarApply to channel identification for radar

• NEW: Do not require even PSF; can also NEW: Do not require even PSF; can also handle non-compact image; Bezout lemmahandle non-compact image; Bezout lemma

Publications Supported:Publications Supported:

1.1. A.E. Yagle and S. Shah, “2-D Blind Deconvolution of A.E. Yagle and S. Shah, “2-D Blind Deconvolution of Even Point-Spread Functions from Compact-Support Even Point-Spread Functions from Compact-Support Images,” submitted to IEEE Trans. Image Proc.Images,” submitted to IEEE Trans. Image Proc.

2.2. A.E. Yagle and S. Shah, “3-D Blind Deconvolution of A.E. Yagle and S. Shah, “3-D Blind Deconvolution of Even Point-Spread Functions from Compact-Support Even Point-Spread Functions from Compact-Support Images,” submitted to IEEE Trans. Image Proc.Images,” submitted to IEEE Trans. Image Proc.

3.3. A.E. Yagle and S. Shah, “2-D Blind Deconvolution of A.E. Yagle and S. Shah, “2-D Blind Deconvolution of Compact-Support Images using Bezout’s Lemma and Compact-Support Images using Bezout’s Lemma and a Spline-Based Image Model,” submitted to IEEE a Spline-Based Image Model,” submitted to IEEE Trans. Image Proc.Trans. Image Proc.

Publications Supported:Publications Supported:

4. A.E. Yagle, “A Simple Closed-Form Linear 4. A.E. Yagle, “A Simple Closed-Form Linear Algebraic Solution to the Single-Blur 2-D Blind Algebraic Solution to the Single-Blur 2-D Blind Deconvolution Problem,” submitted to LAADeconvolution Problem,” submitted to LAA

5.5. A.E. Yagle, “A Closed-Form Linear Algebraic A.E. Yagle, “A Closed-Form Linear Algebraic Solution to 2-D Phase Retrieval,” submitted to Solution to 2-D Phase Retrieval,” submitted to IEEE Trans. Image Proc.IEEE Trans. Image Proc.

6.6. A.E. Yagle, “Fast Spatially-Varying 2-D Blind A.E. Yagle, “Fast Spatially-Varying 2-D Blind Deconvolution of Binary Images,” submitted to Deconvolution of Binary Images,” submitted to IEEE Trans. Image Proc.IEEE Trans. Image Proc.

Publications Supported:Publications Supported:

7. A.E. Yagle and F. Al-Salem, Fast Non-Iterative 7. A.E. Yagle and F. Al-Salem, Fast Non-Iterative Single-Blur 2-D Blind Deconvolution of Single-Blur 2-D Blind Deconvolution of Separable and Low-Rank PSFs from Compact-Separable and Low-Rank PSFs from Compact-Support Images,” Proc. SPIE, San Diego, 2003Support Images,” Proc. SPIE, San Diego, 2003

8. A.E. Yagle, “Blind Superresolution from 8. A.E. Yagle, “Blind Superresolution from Undersampled Blurred Measurements,” Proc. Undersampled Blurred Measurements,” Proc. SPIE, San Diego, August 2003SPIE, San Diego, August 2003

9. J. Marble, “A Method for Determining Size and 9. J. Marble, “A Method for Determining Size and Burial Depth of Landmines using Ground-Burial Depth of Landmines using Ground-Penetrating Radar” Tech. Report, May 2003Penetrating Radar” Tech. Report, May 2003