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Multilook Coherent Imaging: Theoretical Guarantees and Algorithms

Xi Chen, Soham Jana, Christopher A. Metzler, Arian Maleki, Shirin Jalali

TL;DR

This work addresses the problem of recovering high-resolution images in multilook coherent imaging under speckle noise when the measurements are undersampled (m<n). It develops a likelihood-based recovery framework constrained by a deep image prior (DIP) and proves a sharp high-probability bound on the reconstruction error that scales with the DIP parameter count k, the ambient dimension n, the number of looks L, and the undersampling m. Algorithmically, it introduces Bagged-DIP PGD, which combines a bagging strategy over patch-based DIPs with Newton-Schulz approximations to efficiently invert large matrices, achieving state-of-the-art performance in simulations. The results bridge theory and practice by linking MSE bounds to problem dimensions and demonstrating practical gains from the proposed numerically efficient, DIP-driven approach in undersampled, speckle-corrupted multilook imaging.

Abstract

Multilook coherent imaging is a widely used technique in applications such as digital holography, ultrasound imaging, and synthetic aperture radar. A central challenge in these systems is the presence of multiplicative noise, commonly known as speckle, which degrades image quality. Despite the widespread use of coherent imaging systems, their theoretical foundations remain relatively underexplored. In this paper, we study both the theoretical and algorithmic aspects of likelihood-based approaches for multilook coherent imaging, providing a rigorous framework for analysis and method development. Our theoretical contributions include establishing the first theoretical upper bound on the Mean Squared Error (MSE) of the maximum likelihood estimator under the deep image prior hypothesis. Our results capture the dependence of MSE on the number of parameters in the deep image prior, the number of looks, the signal dimension, and the number of measurements per look. On the algorithmic side, we employ projected gradient descent (PGD) as an efficient method for computing the maximum likelihood solution. Furthermore, we introduce two key ideas to enhance the practical performance of PGD. First, we incorporate the Newton-Schulz algorithm to compute matrix inverses within the PGD iterations, significantly reducing computational complexity. Second, we develop a bagging strategy to mitigate projection errors introduced during PGD updates. We demonstrate that combining these techniques with PGD yields state-of-the-art performance. Our code is available at https://github.com/Computational-Imaging-RU/Bagged-DIP-Speckle.

Multilook Coherent Imaging: Theoretical Guarantees and Algorithms

TL;DR

This work addresses the problem of recovering high-resolution images in multilook coherent imaging under speckle noise when the measurements are undersampled (m<n). It develops a likelihood-based recovery framework constrained by a deep image prior (DIP) and proves a sharp high-probability bound on the reconstruction error that scales with the DIP parameter count k, the ambient dimension n, the number of looks L, and the undersampling m. Algorithmically, it introduces Bagged-DIP PGD, which combines a bagging strategy over patch-based DIPs with Newton-Schulz approximations to efficiently invert large matrices, achieving state-of-the-art performance in simulations. The results bridge theory and practice by linking MSE bounds to problem dimensions and demonstrating practical gains from the proposed numerically efficient, DIP-driven approach in undersampled, speckle-corrupted multilook imaging.

Abstract

Multilook coherent imaging is a widely used technique in applications such as digital holography, ultrasound imaging, and synthetic aperture radar. A central challenge in these systems is the presence of multiplicative noise, commonly known as speckle, which degrades image quality. Despite the widespread use of coherent imaging systems, their theoretical foundations remain relatively underexplored. In this paper, we study both the theoretical and algorithmic aspects of likelihood-based approaches for multilook coherent imaging, providing a rigorous framework for analysis and method development. Our theoretical contributions include establishing the first theoretical upper bound on the Mean Squared Error (MSE) of the maximum likelihood estimator under the deep image prior hypothesis. Our results capture the dependence of MSE on the number of parameters in the deep image prior, the number of looks, the signal dimension, and the number of measurements per look. On the algorithmic side, we employ projected gradient descent (PGD) as an efficient method for computing the maximum likelihood solution. Furthermore, we introduce two key ideas to enhance the practical performance of PGD. First, we incorporate the Newton-Schulz algorithm to compute matrix inverses within the PGD iterations, significantly reducing computational complexity. Second, we develop a bagging strategy to mitigate projection errors introduced during PGD updates. We demonstrate that combining these techniques with PGD yields state-of-the-art performance. Our code is available at https://github.com/Computational-Imaging-RU/Bagged-DIP-Speckle.

Paper Structure

This paper contains 27 sections, 12 theorems, 79 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Main Theoretical Result
  4. Assumptions and their justifications
  5. Main theorem and its implications
  6. Key steps in the proof of Theorem \ref{['thm:maintheorem']}
  7. Details of the proof of thm:maintheorem
  8. Main Algorithmic Contributions
  9. Summary of Projected Gradient Descent and DIP
  10. Challenges of DIP-based PGD
  11. Challenge 1: Right choice of DIP
  12. Solution to Challenge 1: Bagged-DIP
  13. Challenge 2: Matrix inversion
  14. Solution to Challenge 2
  15. Simulation Results
  16. ...and 12 more sections

Key Result

Theorem 3.1

Let the elements of the measurement matrix $A_{ij}$ be iid $\mathcal{N}(0,1)$. Suppose that $m<n$ and that the function $g_{\boldsymbol{\theta}} ({\boldsymbol u})$, as a function of $\boldsymbol{\theta} \in B_k \left(\mathbf{0}, x_{\max}\sqrt{\frac{n}{k}}\right)$, is Lipschitz with Lipschitz constan with probability $1- C_2\left( e^{-\frac{m}{2}} + e^{-\frac{Ln}{8}} + e^{-C_3k \log n}+ e^{k \log

Figures (4)

  • Figure 1: PSNR (averaged over 8 images) versus iteration count is depicted for four DIP models fitted to both clean (left panel) and noisy images with only additive noise, noise level $\sigma=25$ (right panel). The 4-layer networks used in DIP are specified in the legend.
  • Figure 2: Newton-Schulz approximation compared with computing exact inverse for all interations, the rest of the curves correspond to stopping the update of the inverse after the first $5$, $10$, and $20$ iterations respectively. The number of looks is $L=32$, sampling rate is $m/n=0.5$. The test image is "Cameraman".
  • Figure 3: (Left) We compare a Bagged-DIP with three sophisticated DIP estimates, where $L=32$, $m/n=0.5$. (Right) We compare PGD with simple and Bagged-DIP across different looks $L=16,32,64$. The test image is "Cameraman".
  • Figure 4: The structure of DIP and Output Blocks.

Theorems & Definitions (21)

  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Lemma 6.1
  • proof
  • Lemma 6.2
  • Lemma 6.3: Concentration of $\chi^2$ jalali2014minimum
  • Theorem 6.4: Hanson-Wright inequality
  • ...and 11 more