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Learning end-to-end inversion of circular Radon transforms in the partial radial setup

This paper proposes a deep learning-based algorithm using a ResBlock U-Net to effectively invert circular Radon transforms in partial radial setups for photoacoustic tomography, overcoming the severe artifacts and limitations of traditional truncated singular value decomposition methods.

Original authors: Deep Ray, Souvik Roy

Published 2026-07-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Deep Ray, Souvik Roy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to take a perfect photo of a secret object hidden inside a foggy, round room. In the world of medical imaging, this "room" is a human body, and the "fog" is made of sound waves. This technique is called Photoacoustic Tomography (PAT). Here's how it works: you zap the body with a quick flash of laser light. The light gets absorbed, the body warms up just a tiny bit, and it expands, creating a ripple of sound—like a tiny, invisible thunderclap. Sensors placed around the edge of the body catch these ripples.

The big puzzle is: Can we look at those ripples and figure out exactly what the hidden object looks like inside?

The Old Way: A Broken Puzzle

For a long time, scientists tried to solve this puzzle using a traditional math method called "Truncated Singular Value Decomposition" (TSVD). Think of this like trying to solve a jigsaw puzzle where half the pieces are missing and the ones you have are covered in static.

The authors of this paper tried using this old method on a specific, tricky version of the problem called the "partial radial setup." This is like trying to solve the puzzle when the sensors listen for a shorter time than usual, meaning they only capture sound waves that haven't traveled the full distance across the room yet. The result? A disaster. The old math method produced images so full of weird streaks and ghostly artifacts that the picture was completely unusable. It was like trying to see your face in a mirror that had been shattered and glued back together wrong.

The New Way: A Super-Intelligent Detective

To fix this, the researchers built a new kind of "detective" using deep learning, specifically a neural network architecture called a U-Net. You can think of this U-Net as a super-smart student who has never seen the actual object but has studied millions of practice puzzles.

Here's the cool part: Instead of trying to solve the math equations step-by-step like a calculator, the U-Net looks at the messy sound data (the "sinogram") and directly guesses what the picture inside should look like. It's like showing the detective a blurry, noisy sketch and having them instantly draw the perfect, clear portrait.

The Training Camp

To teach this detective, the researchers didn't use real patients (since they didn't have data for this specific setup yet). Instead, they created thousands of fake "phantoms"—digital images made of squiggly ellipses that look like the famous Shepp-Logan phantom.

They took these fake images, ran them through a computer simulation to generate the sound data, and then fed that data to the U-Net. They taught the network in two different scenarios:

Full View: Sensors all around the room, listening for the full duration.
Limited View: Sensors all around the room, but listening for only half the angular range (effectively capturing data from half the directions), and the data is noisier.

Crucially, they didn't just teach the network with perfect, clean data. They also taught it with data that had 15% noise added to it. Why? Because in the real world, sensors aren't perfect; they get noisy. By training the network to handle the mess, it learned to ignore the static and focus on the real signal.

The Results: Clear vs. Chaos

When they tested their new detective, the results were a huge win for the U-Net and a total loss for the old math method.

The Old Method (TSVD): When the data was noisy or the view was limited, the images were ruined. The errors were so bad that for 15% noise, the reconstruction could be off by up to 400%. It was basically a mess of streaks.
The New Method (U-Net): The U-Net produced sharp, clear images that looked almost exactly like the original fake phantoms. Even when the input data was noisy or the view was limited, the network didn't get confused.

The researchers measured this using standard computer vision scores called PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity Index). In these simulations, the U-Net consistently scored much higher than the traditional method, proving it could reconstruct the image with high fidelity.

What's Next?

The paper shows that this deep learning approach works incredibly well in these computer simulations. However, the authors are careful to note that this is still a simulation. They haven't tested it on real human data yet. They suggest that future work will need to tackle more complex situations, like when sound travels at different speeds inside the body, and they want to figure out how to measure how "sure" the network is about its guesses.

But for now, in the world of these digital puzzles, the old math method has been left in the dust, and a new, noise-tolerant, deep-learning detective has taken the lead.

Technical Summary: Learning End-to-End Inversion of Circular Radon Transforms in the Partial Radial Setup

Problem Statement
This work addresses the inverse problem of reconstructing the initial acoustic wave pressure distribution, $f(r, \theta)$ , from circular Radon transform data, $g(\rho, \phi)$ , within the context of Photoacoustic Tomography (PAT). Specifically, the paper focuses on the partial radial setup, a scenario where data is collected only for radii $\rho \in (0, R)$ rather than the full range $(0, 2R)$ . This setup is motivated by practical cost constraints in data acquisition but presents significant mathematical challenges.

The primary difficulty lies in the fact that the only existing traditional numerical algorithm for this specific setup is the Truncated Singular Value Decomposition (TSVD) method, coupled with a product trapezoidal rule. The authors demonstrate that TSVD suffers from severe limitations:

It produces reconstructions contaminated by significant artifacts, rendering the fields unusable in many cases.
It requires a problem- and noise-dependent truncation parameter, which is difficult to select optimally.
In limited view scenarios (where angular data is restricted to a subset of $[0, 2\pi)$ due to inaccessible transducer zones), TSVD reconstructions degrade further, failing to recover the underlying phantom structure.

Methodology
To overcome the computational bottlenecks and artifact issues of TSVD, the authors propose an end-to-end deep learning algorithm based on a ResBlock-based U-Net architecture. The approach falls under the "direct approach" class of deep learning for inverse problems, where a neural network learns to map raw measurements directly to the reconstructed image without intermediate traditional reconstruction steps.

Forward Model: The problem is framed as $Y = F(X) + \eta$ , where $X$ is the ground truth pressure field (on a $128 \times 128$ grid) and $Y$ is the circular Radon data (sinogram) on a $\rho-\phi$ grid.
Network Architecture: A U-Net is employed, consisting of a contracting branch to extract multi-scale features and an expanding branch to recover spatial resolution. Skip connections combine features from both branches. The architecture is adapted for two specific input resolutions:
- Full View: Input size $128 \times 128$ ( $\rho \in [0, 1], \phi \in [0, 2\pi)$ ).
- Limited View: Input size $64 \times 64$ ( $\rho \in [0, 1], \phi \in [0, \pi)$ ), requiring an additional interpolation layer to match the output resolution.
Dataset Generation: Since experimental data for this specific partial radial setup is unavailable, the authors generate synthetic datasets using perturbed Shepp-Logan phantoms. These phantoms are modified by randomizing ellipse centers, axes, angles, and intensities, followed by rotation and translation.
Training Strategy: The networks are trained using a supervised learning strategy with a Mean Squared Error (MSE) loss function. Crucially, the authors investigate the impact of noise on training. They compare networks trained on clean data against those trained on a mix of clean and noisy data (specifically 15% noise). The goal is to ensure the network generalizes well to noisy test conditions.
Optimization: The networks are trained using the Adam optimizer for 300 epochs with $L2$ regularization.

Key Contributions

Demonstration of TSVD Failure: The paper provides numerical evidence that the standard TSVD method yields severe artifacts in the partial radial setup, particularly under limited view and noisy conditions, often resulting in errors up to 400% for high noise levels.
End-to-End Learning Framework: The authors introduce a U-Net architecture that directly learns the inverse physical mapping from the sinogram to the pressure field, bypassing the need for hand-crafted regularization or intermediate linear reconstructions.
Robustness to Noise: A significant finding is that training the network on a mixture of clean and noisy data (specifically up to 15% noise) significantly improves generalization. Networks trained solely on clean data fail to reconstruct accurately when tested on noisy inputs, whereas the noise-augmented training produces stable, artifact-free results across the noise spectrum.
Adaptation to Limited View: The study successfully extends the deep learning approach to the limited view setup (half angular coverage and coarser radial sampling), demonstrating that the network can handle reduced data dimensions effectively.

Results
The proposed method was evaluated using quantitative metrics: Peak Signal-to-Noise Ratio (PSNR) and Structural Similarity Index (SSIM).

Full View Performance: The U-Net trained with noisy data (UNet128cn15) significantly outperformed TSVD. While TSVD reconstructions degraded rapidly as noise increased from 0% to 15%, the U-Net maintained high PSNR and SSIM values, remaining stable across all noise levels.
Limited View Performance: Similarly, for the limited view setup (UNet64cn15), the deep learning approach produced accurate, artifact-free reconstructions. In contrast, TSVD reconstructions were so corrupted by artifacts that the underlying phantom structure was nearly indiscernible.
Visual Quality: Qualitative results confirmed that the U-Net reconstructions preserved the structural details of the Shepp-Logan phantoms without the streak artifacts characteristic of the TSVD method.

Significance and Claims
The paper claims that the proposed deep learning algorithm successfully overcomes the computational bottleneck and severe artifacting associated with the TSVD method for the partial radial setup in PAT. The authors emphasize that their approach is particularly significant because it operates directly on raw measurements, learning the underlying inverse physics without relying on a traditional linear reconstruction as a precursor.

The work highlights the necessity of training with noisy data to achieve robust performance in real-world scenarios where measurement noise is inevitable. The authors conclude that while the current work is limited to synthetic phantoms and constant sound speed assumptions, the proposed framework offers a viable path toward solving PAT inverse problems where traditional methods fail. Future work is identified as generalizing the algorithm to non-constant sound speeds and incorporating uncertainty quantification via Bayesian setups, but these are presented as future directions rather than current achievements.