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Deep learning versus $\ell^1$-minimization for compressed sensing photoacoustic tomography

Stephan Antholzer, Johannes Schwab, Markus Haltmeier

TL;DR

This paper addresses the challenge of reconstructing high-resolution PAT images from under-sampled measurements by comparing a joint $\ell^1$-minimization approach with two deep learning methods (a residual network and an approximate nullspace network). It introduces a joint reconstruction framework that recovers both ${\boldsymbol{f}}$ and ${\boldsymbol{h}} = c^2 \mathcal{L}_{\mathbf r}{\boldsymbol{f}}$ to exploit sparsity, and develops data-consistent deep learning architectures to enhance reconstructions from limited data. Numerical results show that the nullspace network generally outperforms the residual network, and performance depends on the sampling scheme: $\ell^1$-minimization excels with Bernoulli measurements, while deep learning methods outperform traditional FBP under both measurement types. The work provides practical guidance for selecting reconstruction strategies in CS PAT and demonstrates the potential of data-consistent DL approaches for real-time imaging.

Abstract

We investigate compressed sensing (CS) techniques for reducing the number of measurements in photoacoustic tomography (PAT). High resolution imaging from CS data requires particular image reconstruction algorithms. The most established reconstruction techniques for that purpose use sparsity and $\ell^1$-minimization. Recently, deep learning appeared as a new paradigm for CS and other inverse problems. In this paper, we compare a recently invented joint $\ell^1$-minimization algorithm with two deep learning methods, namely a residual network and an approximate nullspace network. We present numerical results showing that all developed techniques perform well for deterministic sparse measurements as well as for random Bernoulli measurements. For the deterministic sampling, deep learning shows more accurate results, whereas for Bernoulli measurements the $\ell^1$-minimization algorithm performs best. Comparing the implemented deep learning approaches, we show that the nullspace network uniformly outperforms the residual network in terms of the mean squared error (MSE).

Deep learning versus $\ell^1$-minimization for compressed sensing photoacoustic tomography

TL;DR

This paper addresses the challenge of reconstructing high-resolution PAT images from under-sampled measurements by comparing a joint -minimization approach with two deep learning methods (a residual network and an approximate nullspace network). It introduces a joint reconstruction framework that recovers both and to exploit sparsity, and develops data-consistent deep learning architectures to enhance reconstructions from limited data. Numerical results show that the nullspace network generally outperforms the residual network, and performance depends on the sampling scheme: -minimization excels with Bernoulli measurements, while deep learning methods outperform traditional FBP under both measurement types. The work provides practical guidance for selecting reconstruction strategies in CS PAT and demonstrates the potential of data-consistent DL approaches for real-time imaging.

Abstract

We investigate compressed sensing (CS) techniques for reducing the number of measurements in photoacoustic tomography (PAT). High resolution imaging from CS data requires particular image reconstruction algorithms. The most established reconstruction techniques for that purpose use sparsity and -minimization. Recently, deep learning appeared as a new paradigm for CS and other inverse problems. In this paper, we compare a recently invented joint -minimization algorithm with two deep learning methods, namely a residual network and an approximate nullspace network. We present numerical results showing that all developed techniques perform well for deterministic sparse measurements as well as for random Bernoulli measurements. For the deterministic sampling, deep learning shows more accurate results, whereas for Bernoulli measurements the -minimization algorithm performs best. Comparing the implemented deep learning approaches, we show that the nullspace network uniformly outperforms the residual network in terms of the mean squared error (MSE).

Paper Structure

This paper contains 22 sections, 4 theorems, 29 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. CS PAT recovery algorithms
  3. Outline
  4. Compressed photoacoustic tomography
  5. Photoacoustic tomography
  6. Discretization
  7. Compressive measurements in PAT
  8. Joint $\ell^1$-minimization for CS PAT
  9. Background from $\ell^1$-minimization
  10. Sparsification strategy
  11. Joint $\ell^1$-minimization framework
  12. Numerical minimization
  13. Deep learning for CS PAT
  14. Image reconstruction by deep learning
  15. Residual network
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let $\mathcal{A} \in \mathbb R^{mQ \times n}$ and ${\boldsymbol{h}} \in \mathbb R^{n}$ Assume where $\mathrm{sign}({\boldsymbol{h}})$ is the set valued signum function and $\operatorname{supp} ({\boldsymbol{h}})$ the set of all nonzero entries of ${\boldsymbol{h}}$, and that the restriction of $\mathcal{A}$ to the subspace spanned by $e_i$ for $i \in \operatorname{supp} ({\boldsymbol{h}})$ i sati

Figures (5)

  • Figure 1.1: (a) An object is illuminated with a short optical pulse; (b) the absorbed light distribution causes an acoustic pressure; (c) the acoustic pressure is measured outside the object and used to reconstruct an image of the interior.
  • Figure 4.1: Architecture of the residual network $\operatorname{Id} + \mathcal{U}_\theta$. The number written above each layer denotes the number of convolution kernels (channels). The numbers written on the left are the image sizes. The long arrows indicate direct connections with subsequent concatenation or summation.
  • Figure 4.2: Test phantoms for results presented below. Top: vessel phantom (left) and head phantom (right). Bottom: FBP reconstruction from full data of vessel phantom (left) and head phantom (right).
  • Figure 5.1: Reconstructions of blood vessel image from sparse measurements (left) and Bernoulli measurements (right). Top row: FBP reconstruction. Second row: Joint $\ell^1$-minimization. Third row: Residual network. Bottom row: Nullspace network.
  • Figure 5.2: Reconstructions of Shepp-Logan type phantom from sparse measurements (left) and Bernoulli measurements (right). Top row: FBP reconstruction. Second row: Joint $\ell^1$-minimization. Third row: Residual network. Bottom row: Nullspace network.

Theorems & Definitions (8)

  • Theorem 3.1: $\ell^1$-minimization
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof