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Bayesian Despeckling of Structured Sources

Ali Zafari, Shirin Jalali

TL;DR

This work introduces BD-QMAP, a Bayesian despeckling method for structured sources under multiplicative noise, by extending Quantized MAP (QMAP) to the multiplicative noise setting and formulating a two-term objective that combines a negative log-likelihood with a learned, quantization-based regularizer. For classic structured sources, including memoryless and piecewise-constant 1-Markov processes, BD-QMAP simplifies to symbol-by-symbol or segment-averaging rules and can be optimized efficiently via the Viterbi algorithm on the quantized space. The authors derive a theoretical lower bound on the mean-squared error for piecewise-constant first-order Markov sources and demonstrate that BD-QMAP approaches this bound in practice, achieving state-of-the-art despeckling performance on structured signals. Empirical results validate the approach against traditional despecklers and illustrate the impact of quantization level and regularization on reconstruction quality, with a clear path toward extending to image despeckling and broader source models.

Abstract

Speckle noise is a fundamental challenge in coherent imaging systems, significantly degrading image quality. Over the past decades, numerous despeckling algorithms have been developed for applications such as Synthetic Aperture Radar (SAR) and digital holography. In this paper, we aim to establish a theoretically grounded approach to despeckling. We propose a method applicable to general structured stationary stochastic sources. We demonstrate the effectiveness of the proposed method on piecewise constant sources. Additionally, we theoretically derive a lower bound on the despeckling performance for such sources. The proposed depseckler applied to the 1-Markov structured sources achieves better reconstruction performance with no strong simplification of the ground truth signal model or speckle noise.

Bayesian Despeckling of Structured Sources

TL;DR

This work introduces BD-QMAP, a Bayesian despeckling method for structured sources under multiplicative noise, by extending Quantized MAP (QMAP) to the multiplicative noise setting and formulating a two-term objective that combines a negative log-likelihood with a learned, quantization-based regularizer. For classic structured sources, including memoryless and piecewise-constant 1-Markov processes, BD-QMAP simplifies to symbol-by-symbol or segment-averaging rules and can be optimized efficiently via the Viterbi algorithm on the quantized space. The authors derive a theoretical lower bound on the mean-squared error for piecewise-constant first-order Markov sources and demonstrate that BD-QMAP approaches this bound in practice, achieving state-of-the-art despeckling performance on structured signals. Empirical results validate the approach against traditional despecklers and illustrate the impact of quantization level and regularization on reconstruction quality, with a clear path toward extending to image despeckling and broader source models.

Abstract

Speckle noise is a fundamental challenge in coherent imaging systems, significantly degrading image quality. Over the past decades, numerous despeckling algorithms have been developed for applications such as Synthetic Aperture Radar (SAR) and digital holography. In this paper, we aim to establish a theoretically grounded approach to despeckling. We propose a method applicable to general structured stationary stochastic sources. We demonstrate the effectiveness of the proposed method on piecewise constant sources. Additionally, we theoretically derive a lower bound on the despeckling performance for such sources. The proposed depseckler applied to the 1-Markov structured sources achieves better reconstruction performance with no strong simplification of the ground truth signal model or speckle noise.
Paper Structure (22 sections, 4 theorems, 54 equations, 3 figures, 1 table)

This paper contains 22 sections, 4 theorems, 54 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Related work
  4. Our contributions
  5. Notations and definitions
  6. Paper organization
  7. Bayesian Despeckling via QMAP
  8. Analysis of BD-QMAP for classic structured sources
  9. Structured memoryless source
  10. Piecewise constant source
  11. Proofs
  12. Proof of Lemma \ref{['lemma:Markov-QMAP-opt']}
  13. Proof of Theorem \ref{['thm:MSE-lower-bound']}
  14. Proof of \ref{['cor:mse-bound-convergence']}
  15. Numerical Experiments
  16. ...and 7 more sections

Key Result

Lemma 1

Solving the optimization in eq:QMAP-Markov-main is equivalent to solving the following optimization where for any $k\in\{0,\ldots,n-1\}$, the inner optimization is over $(n_1,\ldots,n_{k+1})\in(\mathbb{N}^+)^{k+1}$, such that $\sum_{i=1}^{k+1}n_i=n$. Moreover, for $i\in \mathcal{I}_j({\bf n})$,

Figures (3)

  • Figure 1: Piecewise constant source ($X$) with parameter $q_0=0.001$ sampled under speckle noise ($Y$), enhanced Lee and BD-QMAP$_b$ despeckled reconstructions, as discussed in \ref{['sec:experiments']}.
  • Figure 2: MSE lower bound of the genie-aided ML despeckler as derived in \ref{['thm:MSE-lower-bound']} and the proposed BD-QMAP despecker for piecewise constant source with parameter $q_0$. ($\operatorname{PSNR}=10\log_{10}(\operatorname{MAX}^2/\operatorname{MSE})$ where $\operatorname{MAX}$ is the maximum value of signal's support set)
  • Figure 3: Effect of hyperparameter $\lambda$ in BD-QMAP$_b$ and BD-QMAP optimization for different choices of bits ($b$) and source structure ($q_0$).

Theorems & Definitions (5)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Lemma 2: Concentration of Geometric janson2018tail