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

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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

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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)

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Previous Work

• Iterative algorithms alternating projections
• Sometimes dont 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

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Problem Relevance

• GIVEN 1 monopole point source antenna 1
  frequency, moving platform (e.g., plane)
• Unknown scatterer V(x) compact support
• Unknown Greens 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)

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Inverse Scattering Formulation
G(x,x)
V(x)

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Problem Relevance
Reciprocity G(x-y)G(y-x) even PSF
Assume Born (single-scatter) approximation
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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(xd),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).

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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)

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1-D Blind Deconvolution Example
Solve 24,57,33h(0),h(0)u(0),u(1)
Solution u(0),u(1)8,11 to scale factor
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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?

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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

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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 yykexpj2?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

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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

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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))

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2-D Blind Deconvolution

• 452X452 image blurred with UNKNOWN
• 61X61 Gaussian PSF noiseless example

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2-D Blind Deconvolution

• 220X220 image blurred with UNKNOWN
• 37X37 Gaussian PSF noiseless example

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2-D Blind Deconvolution

• MSE vs. SNR for TLS LAP STLN methods
• MSE vs. SNR for Direct vs. Fourier methods

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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 yiexpj2?i/N and zj similar.
• Decoupled (in yi and zj) 1-D problems.

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3-D Blind Deconvolution

• 63X63X63 image blurred with UNKNOWN
• 9X9X9 3-D Gaussian PSF noiseless example

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3-D Blind Deconvolution

• MSE vs. SNR for STLN vs. TLS

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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

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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

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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 Bezouts Lemma
  and a Spline-Based Image Model, submitted to
  IEEE Trans. Image Proc.

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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.

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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