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

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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, MI . Presentation Overview. Problem Statement Problem Relevance 1-D Blind Deconvolution of Even PSFs

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Andrew E. Yagle and Siddharth Shah

Dept. of EECS, The University of Michigan

Ann Arbor, MI

Presentation Overview
• Problem Statement
• Problem Relevance
• 1-D Blind Deconvolution of Even PSFs
• 2-D Blind Deconvolution of Even PSFs
• 3-D Blind Deconvolution of Even PSFs
• Conclusion
Problem Statement
• GIVEN: Observations y(x)=h(x)*u(x)+n(x)
• h(x)=unknown even PSF [h(x)=h(-x)]
• u(x)=unknown compact-support image
• n(x)=white Gaussian noise random field
• GOAL: Reconstruct u(x) from y(x)
Previous Work
• Iterative algorithms: alternating projections
• Sometimes don’t converge; usually stagnate
• NASRIF: requires small-support inverse PSF
• Often not true (e.g., Gaussian-like PSFs)
• Statistical methods: require stochastic image models; often insufficient for unique answer
Problem Relevance
• 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) which represents channel propagation effects
• Response at x to source at same x: u(x)
• GOAL: Reconstruct V(x) from u(x)
Problem Relevance

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

Assume: Born (single-scatter) approximation

Problem Ambiguities
• SCALE FACTOR: Solution {h(x),u(x)} implies solution {ch(x),u(x)/c} for any c.
• TRANSLATION: Solution {h(x),u(x)} implies solution {h(x+d),u(x-d)} for any d.
• EXCHANGE: Solution {h(x),u(x)} implies solution {u(x),h(x)} [but h(x)=h(-x) avoids]
• REDUCIBLE: Solution {h(x),u(x)} need irreducible z-transforms (almost surely).
1-D Blind Deconvolution
• Observe: y(n)=h(n)*u(n) [omit noise here]
• Even PSF: h(n)=h(-n) [symmetric]
• z-transforms: 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)
• Resultant: Equate coefficients gives Toeplitz
• Need: No U(z) zeros in conjugate reciprocal quadruples (in practice, none on unit circle)
1-D Blind Deconvolution: Example

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

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

Noisy Data Problem
• Goal: Compute maximum-likelihood (ML) estimator of image in white Gaussian noise
• Log-Likelihood: Need to find minimum perturbation of data {y(n)} such that:
• Overdetermined Toeplitz matrix has reduced rank, so null vector exists;
• Frobenius matrix norm ||Y|| minimized.
• How to solve this linear algebra problem?
Noisy Data Solution
• Two methods were investigated:
• Lift-and-Project(LAP):
• Lift to Toeplitz using “Toeplitzation”;
• Project to reduced-rank using SVD.
• Structured Total Least Squares (STLS):
• Perturb y(n) to satisfy constraints
2-D Blind Deconvolution
• Use Fourier transform to decouple the 2-D problem into 1-D problems:
• Analogous to 1-D, get 2-D equation
• Y(x,y)U(1/x,1/y)=Y(1/x,1/y)U(x,y)
• Set y=yk=exp{j2k/N} in this. Get:
• Y(x,yk)U*(1/x*,yk)=Y*(1/x*,yk)U(x,yk)
• Decoupled (in yk) 1-D problems as before
2-D Blind Deconvolution
• Scale factor between 1-D problems:
• Resolved by performing decoupling in both x and y; comparing solutions
• Additive WGN decouples into WGNs
• Even more interesting in 3-D problem:
• See papers for details and solutions
2-D Blind Deconvolution
• Unknowns are the pixel values u(i,j)
• No need to compute PSF and then deconvolve PSF from the noisy data
• Can incorporate irregular support of image explicitly (toss matrix columns)
• Can use edge-preserving regularization algorithms (linear system for u(i,j))
2-D Blind Deconvolution
• 452X452 image blurred with UNKNOWN
• 61X61 Gaussian PSF; noiseless example
2-D Blind Deconvolution
• 220X220 image blurred with UNKNOWN
• 37X37 Gaussian PSF; noiseless example
2-D Blind Deconvolution
• MSE vs. SNR for: TLS; LAP; STLN methods
• MSE vs. SNR for: Direct vs. Fourier methods
3-D Blind Deconvolution
• Use Fourier transform to decouple the 3-D problem into 1-D problems:
• Analogous to previous, get equation
• Y(x,yi,zj)U*(1/x*,yi,zj) = Y*(1/x*,yi,zj)U(x,yi,zj)
• where yi=exp{j2i/N} and zj similar.
• Decoupled (in yi and zj) 1-D problems.
3-D Blind Deconvolution
• 63X63X63 image blurred with UNKNOWN
• 9X9X9 3-D Gaussian PSF; noiseless example
3-D Blind Deconvolution

MSE vs. SNR for: STLN vs. TLS

Conclusion
• 2-D and 3-D blind deconvolution problem
• Require PSF to be an even function
• Application to scattering: channel effects
• Decouple 2-D and 3-D to 1-D problems
• Solve 1-D problems using resultant
• Use STLN or LAP for MLE in noisy data
Goals for Next Year
• Apply basis function inverse scattering to bases developed by Bownik
• Mine signature detection using transforms to detect hyperbolae (prestack) vs. lines
• Apply to channel identification for radar
• NEW: Do not require even PSF; can also handle non-compact image; Bezout lemma
Publications Supported:
• A.E. Yagle and S. Shah, “2-D Blind Deconvolution of Even Point-Spread Functions from Compact-Support Images,” submitted to IEEE Trans. Image Proc.
• A.E. Yagle and S. Shah, “3-D Blind Deconvolution of Even Point-Spread Functions from Compact-Support Images,” submitted to IEEE Trans. Image Proc.
• A.E. Yagle and S. Shah, “2-D Blind Deconvolution of Compact-Support Images using Bezout’s Lemma and a Spline-Based Image Model,” submitted to IEEE Trans. Image Proc.
Publications Supported:

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

• A.E. Yagle, “A Closed-Form Linear Algebraic Solution to 2-D Phase Retrieval,” submitted to IEEE Trans. Image Proc.
• A.E. Yagle, “Fast Spatially-Varying 2-D Blind Deconvolution of Binary Images,” submitted to IEEE Trans. Image Proc.
Publications Supported:

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

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

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