Source-linked AI summary
Deep Learning for Photoacoustic Tomography from Sparse Data
Stephan Antholzer, Markus Haltmeier, Johannes Schwab
TL;DR
Sparse spatial sampling in PAT produces artifact-prone reconstructions, while iterative methods can be computationally demanding and prior-dependent. The paper proposes a two-stage deep-learning reconstruction combining linear preprocessing with a trained CNN, and reports quality comparable to state-of-the-art iterative methods at direct-reconstruction computational effort.
Problem
Sparse-data PAT requires reconstruction from limited measurements, while iterative approaches can be time consuming and depend on object-prior assumptions.
Method
The method applies a linear reconstruction algorithm to sparse PAT data and then a trained CNN, implemented as FBP followed by a U-net.
Results
The approach reconstructs sparse-data PAT images with quality comparable to state-of-the-art iterative algorithms while requiring effort similar to direct FBP reconstruction.
Takeaways & Limitations
Deep learning is feasible and highly promising for efficient PAT image reconstruction from sparse data.
Takeaways & Limitations
The reported results assume an ideal impulse response of the acoustic measurement system, and future work may examine alternative CNNs and network design.
Abstract
from arXiv · showhide
The development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography. In this paper, we investigate this issue for the sparse data problem in photoacoustic tomography (PAT). We develop a direct and highly efficient reconstruction algorithm based on deep learning. In our approach image reconstruction is performed with a deep convolutional neural network (CNN), whose weights are adjusted prior to the actual image reconstruction based on a set of training data. The proposed reconstruction approach can be interpreted as a network that uses the PAT filtered backprojection algorithm for the first layer, followed by the U-net architecture for the remaining layers. Actual image reconstruction with deep learning consists in one evaluation of the trained CNN, which does not require time consuming solution of the forward and adjoint problems. At the same time, our numerical results demonstrate that the proposed deep learning approach reconstructs images with a quality comparable to state of the art iterative approaches for PAT from sparse data.
1 Introduction
The paper develops a deep-learning framework for sparse-data PAT, addressing artifact-prone and computationally costly reconstruction with a trained CNN-based approach. It combines a linear reconstruction step with learned artifact removal and targets efficient, high-quality reconstruction.
- PAT and the sparse sampling problem: PAT combines optical contrast with ultrasound resolution, but limited detector numbers create sparse spatial sampling and undersampling artifacts.Acoustic signals generated by short optical pulses are measured outside the object to recover its interior.
- PAT and the sparse sampling problem: Iterative methods can reduce sparse-data artifacts but repeatedly solve forward and adjoint problems and depend on object-prior assumptions.TV minimization, for example, assumes gradient sparsity, which may not hold strictly for real-world images.
- Proposed deep learning approach: CNN weights are adjusted using training data, creating an implicit data-driven model rather than requiring an explicit prior for the reconstructed object class.Training is performed before reconstruction, while CNN evaluation is direct and can be parallelized.
- Proposed deep learning approach: The proposed method first applies a linear PAT reconstruction to sparse data, then uses a CNN to map the artifact-containing intermediate image to an artifact-free output.The two stages can be viewed as a single network with the linear reconstruction as its first layer and the CNN in the remaining layers.
- Proposed deep learning approach: The framework uses FBP in the presented implementation, while other linear reconstruction methods and CNN architectures are possible alternatives.The paper interprets the implemented architecture as FBP followed by a U-net, and positions the work as an early application of deep learning to PAT.
2 Photoacoustic tomography
Photoacoustic tomography recovers an object's initial pressure distribution from measured acoustic waves, but sparse detector sampling produces severe undersampling artifacts. Iterative methods can reduce these artifacts but are time-consuming and depend on object priors that may not hold in practice.
- PAT generates acoustic waves with short optical pulses and measures them outside the sample to recover the interior initial pressure distribution.
- The paper focuses on two-dimensional PAT with a circular measurement geometry, where detectors surround the investigated object.
- Sparse data arise because acoustic pressure is sampled at finitely many detector locations, with each detector recording a time-dependent signal.
- Iterative reconstruction can reduce artifacts but repeatedly solves forward and adjoint problems and depends on assumptions such as gradient sparsity.
3 Deep learning for PAT image reconstruction
The proposed method learns a restoration map from sparse-data reconstructions to artifact-reduced images using a CNN built around FBP preprocessing and a U-net. Training adjusts the network weights before reconstruction, so inference is direct and non-iterative.
- Deep learning approach: The approach first applies a linear reconstruction to sparse measurements and then a CNN to map the intermediate image to an artifact-reduced reconstruction.
- Supervised reconstruction: The restoration function is trained on paired inputs containing undersampling artifacts and corresponding artifact-free target images.
- Training: CNN weights are adjusted during training before actual reconstruction by minimizing a training error with gradient-based optimization.
- CNN and U-net: The implemented CNN uses a U-net architecture with convolutional layers, ReLU activations, pooling, upsampling, and direct connections between layers.
- FBP and U-net reconstruction: The proposed PAT reconstruction applies FBP first and the U-net second; the two stages can also be combined into one network.
- Reconstruction outcome: The FBP-plus-U-net pipeline yields an image with significantly reduced undersampling artifacts after the two reconstruction steps.
4 Numerical realization
The numerical study evaluates the proposed CNN on simulated sparse-data PAT under clean, noisy, and mismatched-training conditions. Appropriate training substantially reduces artifacts, while performance degrades when test phantoms differ from the training class; reconstruction is also computationally efficient.
- Data generation and network training: The experiments use 128 × 128 images, 30 detector locations, and 300 pressure samples over the time interval [0, 2].The measurement curve has radius R = 1.
- Data generation and network training: Two ellipse-phantom training sets contain 1000 pairs each, using either noiseless or randomly noise-corrupted pressure data.Training outputs contain one to five randomly generated ellipses sampled on the imaging grid.
- Data generation and network training: The network is trained for 60 epochs with batch size one, learning rate η = 10^-3, momentum β = 0.99, mean absolute error, 32 initial channels, and 19 layers.Weights use a Glorot-style uniform initializer.
- Numerical results: CNN reconstructions remain almost artifact free for data with 2% Gaussian noise, using networks trained either with or without noise.The study compares both CNN variants with FBP and TV minimization, reporting relative ℓ2 errors in Table 1.
- Results for Shepp-Logan type phantom: When ellipse-trained CNNs process Shepp-Logan phantoms, they do not remove all artifacts, although many artifacts are still removed.Additional training on ellipse and Shepp-Logan classes is used to assess the effect of appropriate training data.
- Computational efforts: One CNN evaluation requires O(F^2Ld^2) FLOPS, whereas each PAT forward or adjoint evaluation requires O(d^3) FLOPS; TV minimization used 400 forward/adjoint evaluations.The authors therefore expect the deep-learning reconstruction to be faster than iterative TV-based approaches in these examples.
5 Conclusion
The proposed deep-learning approach reconstructs sparse PAT data with quality comparable to state-of-the-art iterative algorithms while using a non-iterative reconstruction procedure. Its effectiveness depends on similarity between the PAT images and training data, and the study assumes an ideal acoustic impulse response.
- The approach combines an initial linear reconstruction with a CNN trained on example data, producing sparse-data PAT reconstructions non-iteratively.
- Its reconstruction quality is comparable to state-of-the-art iterative algorithms for sparse-data PAT.
- The method requires PAT images to share similarities with the training data used to adjust the CNN weights.
- Future work includes testing realistic phantom classes, different geometries, larger discretizations, real-world data, and CNNs that may outperform the implemented U-net.
- The reported results assume an ideal impulse response; piezoelectric-sensor applications require modeling limited bandwidth and training the CNN for deconvolution.