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Complex-Valued Signal Recovery using the Bayesian LASSO

Dylan Green, Jonathan Lindbloom, Anne Gelb

Abstract

Recovering complex-valued image recovery from noisy indirect data is important in applications such as ultrasound imaging and synthetic aperture radar. While there are many effective algorithms to recover point estimates of the magnitude, fewer are designed to recover the phase. Quantifying uncertainty in the estimate can also provide valuable information for real-time decision making. This investigation therefore proposes a new Bayesian inference method that recovers point estimates while also quantifying the uncertainty for complex-valued signals or images given noisy and indirect observation data. Our method is motivated by the Bayesian LASSO approach for real-valued sparse signals, and here we demonstrate that the Bayesian LASSO can be effectively adapted to recover complex-valued images whose magnitude is sparse in some (e.g.~the gradient) domain. Numerical examples demonstrate our algorithm's robustness to noise as well as its computational efficiency.

Complex-Valued Signal Recovery using the Bayesian LASSO

Abstract

Recovering complex-valued image recovery from noisy indirect data is important in applications such as ultrasound imaging and synthetic aperture radar. While there are many effective algorithms to recover point estimates of the magnitude, fewer are designed to recover the phase. Quantifying uncertainty in the estimate can also provide valuable information for real-time decision making. This investigation therefore proposes a new Bayesian inference method that recovers point estimates while also quantifying the uncertainty for complex-valued signals or images given noisy and indirect observation data. Our method is motivated by the Bayesian LASSO approach for real-valued sparse signals, and here we demonstrate that the Bayesian LASSO can be effectively adapted to recover complex-valued images whose magnitude is sparse in some (e.g.~the gradient) domain. Numerical examples demonstrate our algorithm's robustness to noise as well as its computational efficiency.
Paper Structure (20 sections, 8 theorems, 86 equations, 59 figures, 5 algorithms)

This paper contains 20 sections, 8 theorems, 86 equations, 59 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Bayesian Formulation
  3. The likelihood
  4. The prior density function
  5. Sparse magnitude
  6. Sparse transform of magnitude
  7. The hyperprior
  8. The sparsifying transform operator
  9. Bayesian LASSO for a real-valued signal (RVBL)
  10. Complex-valued Bayesian LASSO (CVBL)
  11. CVBL for the Sparse Magnitude Case
  12. Sparsity in a Transform Domain of $|\mathcal{Z}|$
  13. Numerical Results
  14. Numerical Efficiency
  15. Sparsity in Magnitude
  16. ...and 5 more sections

Key Result

Theorem 4.1

\newlabelthm:magpost0 Let $F_1 = FD(e^{i\phi})$, $\tilde{F}_1=[\mathrm{Re}(F_1)^T \ \mathrm{Im}(F_1)^T]^T$, and $\tilde{\bm y}=[\mathrm{Re}(\bm y)^T \ \mathrm{Im}(\bm y)^T]^T$. Assume that $L$ has rank $n$. The function $\tilde{f}_{\mathcal{G}|\mathcal{Y},\Phi,\mathcal{T}^2}(\bm g|\bm y,\bm \phi,\

Figures (59)

  • Figure 1: Deblurring problem data
  • Figure 1: $F_F$ in \ref{['eq:Fouriertransform']}
  • Figure 2: Estimate of $\bm x$
  • Figure 3: Estimate of $\bm \tau^2$
  • Figure 4: $F_F$ in \ref{['eq:Fouriertransform']}
  • ...and 54 more figures

Theorems & Definitions (20)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 4.1
  • Corollary 4.2
  • Remark 6
  • Remark 7
  • Theorem 4.3
  • ...and 10 more