🤖 machine learning

Accurate 2D Reconstruction for PET Scanners based on the Analytical White Image Model

This paper proposes an accurate 2D PET reconstruction method that integrates a closed-form analytical white image model, derived from a precise crystal-to-crystal response, into a modified ray-driven MLEM algorithm to significantly outperform non-compensated approaches on both synthetic and real-world data.

Original authors: Tomislav Matulić, Damir Seršić

Published 2026-02-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tomislav Matulić, Damir Seršić

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 photograph of a glowing object inside a dark room, but you can't use a camera lens. Instead, you have a ring of thousands of tiny, super-sensitive eyes (detectors) surrounding the object. When the object emits a tiny flash of light (a positron annihilation), two of these eyes see it at the exact same time. They draw an invisible line between themselves, telling you the flash happened somewhere along that line.

This is how PET scanners (Positron Emission Tomography) work. They build an image by collecting millions of these invisible lines.

However, there's a problem. In the real world, these "eyes" aren't perfect. They are physical blocks of crystal with specific sizes and shapes. Because of this, the scanner doesn't see every flash equally.

Flashes happening right in the center of the ring are seen by many more pairs of eyes than flashes happening near the edge.
The edges of the crystals create "shadows" and blind spots.

If you just let a computer draw the picture based on raw data, the center of the image will look super bright and blurry, while the edges will look dark and distorted. It's like trying to paint a portrait where the paint is naturally thicker in the middle of the canvas.

The "White Image" Solution

The authors of this paper came up with a clever way to fix this distortion without needing a massive, slow computer to calculate every single possibility. They call their solution the "White Image."

Think of the "White Image" as a map of the scanner's own biases.
Imagine you have a perfectly white, glowing ball that fills the entire room. If you took a picture of this ball with your imperfect scanner, the resulting image wouldn't be a perfect white circle. It would be a weird shape: bright in the middle, dim at the edges, maybe with some rings or streaks.

That weird shape is the White Image. It tells the computer: "Hey, if I see a flash here, it's probably because my scanner is naturally better at seeing things in this spot, not because there's actually more light there."

How They Did It (The "Math Magic")

Usually, figuring out this "White Image" map is incredibly hard.

The Slow Way: You could simulate millions of flashes on a supercomputer (Monte Carlo simulation), but that takes forever.
The Messy Way: You could measure a real radioactive ball, but that's noisy and expensive.

The authors used math to solve it instantly.

The Crystal Dance: They realized that as the scanner rotates (or as the object rotates inside it), the relationship between any two "eyes" (crystals) follows a predictable pattern.
The Approximation: The exact math for how two crystals "see" each other is like a complex, tangled knot of equations. It's too messy to use in real-time.
The Shortcut: The authors found a way to untie that knot. They proved that you can approximate the complex shape of the crystal's vision with a simple triangle shape (like a tent) or even a rectangle.
The Result: By rotating this simple "tent" shape around the center and adding it all up, they got a perfect, closed-form formula. It's like finding a simple recipe that tastes exactly like a complex, 50-ingredient dish.

Putting It All Together: The New Recipe

They took this "White Image" map and plugged it into a standard image reconstruction algorithm (called MLEM).

Think of the standard algorithm as a chef trying to bake a cake but using a recipe that doesn't account for the fact that their oven has a hot spot in the middle. The cake comes out burnt in the center.

The authors' new method is like giving the chef a correction chart.

The chef bakes the cake (reconstructs the image).
The chef looks at the chart (the White Image) which says, "The oven is 20% hotter in the center."
The chef adjusts the recipe on the fly, dividing the center by 1.2 and the edges by 0.9.
Result: A perfectly even cake.

Why This Matters

Speed: Because their math is a simple formula, the computer doesn't need to store a massive library of data. It can calculate the correction instantly.
Accuracy: They tested this on real machines (the Raytest ClearPET) and fake data. The results were much sharper and had fewer "artifacts" (weird rings or blurs) than previous methods.
Simplicity: They proved that you don't need a supercomputer to understand how a PET scanner sees the world; you just need the right geometric insight.

In a nutshell: They figured out the "fingerprint" of the scanner's imperfections using simple geometry, turned it into a quick mathematical formula, and used it to clean up PET images, making them clearer and faster to produce.

1. Problem Statement

Positron Emission Tomography (PET) image reconstruction faces significant challenges due to the physical limitations of the scanner hardware, such as crystal geometry, detector efficiency, and the finite size of scintillating crystals.

System Matrix Complexity: Traditional iterative reconstruction methods (like MLEM) rely on a "system matrix" that maps the image space to the projection space. In 3D PET, this matrix is massive ( $>10^{12}$ elements), making storage and computation prohibitive.
Inaccurate Models: Analytical methods often simplify the physical model (e.g., ignoring scatter or using simple geometric intersections), leading to distorted images. Conversely, Monte Carlo (MC) simulations are accurate but computationally slow and noisy.
Artifacts: Without precise compensation for the system's response, reconstructed images suffer from central artifacts (due to high probability of detection at the center of rotation) and ring-like artifacts (due to quantization effects of finite crystal lengths).

The authors aim to develop a precise, fast, and memory-efficient 2D reconstruction method that overcomes these limitations without requiring a pre-computed, massive system matrix.

2. Methodology

The core of the proposed methodology is the derivation of an Analytical White Image Model and its integration into a modified Maximum Likelihood Expectation Maximization (MLEM) algorithm.

A. Crystal-to-Crystal Response Modeling

The authors derive a precise mathematical model for the probability density function (PDF) of an annihilation event occurring at a point $T(x,y)$ given a detection by a specific pair of crystals.

Exact Model: They define the non-normalized PDF based on the "hit length" of a ray originating from $T$ intersecting the two crystals. The support of this PDF is rectangular. The exact mathematical expression involves complex logarithmic and arctangent functions derived from integrating the PDF over a full $2\pi$ rotation.
Approximation: Recognizing that the exact closed-form solution is too complex for practical implementation, they propose a triangular approximation.
- They model the crystal response as a Dirac line rotated around the origin, weighted by a triangular function representing the crystal length.
- They prove analytically that this triangular approximation yields a negligible Root Mean Square Error (RMSE) compared to the exact solution, especially when the crystal length ( $L_0$ ) is small relative to the scanner radius ( $R_0$ ).
- A simpler rectangular window approximation is also discussed but found to be less accurate.

B. The White Image Model

The "White Image" represents the probability of detecting a uniformly distributed point source across the entire measurement area, accounting for the rotation of all crystal pairs.

Generation: Instead of simulating millions of events (MC) or measuring a physical phantom for hours, the white image is generated analytically by summing the rotated, approximated crystal-to-crystal responses for all possible crystal pairs.
Weighting: Each pair's contribution is weighted by the square of the effective crystal length ( $L_{ij}^2$ ), derived from geometric considerations of the scanner's geometry (e.g., the Raytest ClearPET with two rings).
Validation: The analytical white image was validated against both Monte Carlo simulations and real-world calibration measurements using a NEMA NU 4-2008 phantom, showing indistinguishable results.

C. Modified MLEM Algorithm

The authors integrate the white image model into the MLEM algorithm to replace the traditional system matrix approach.

Ray-Driven Projections: Instead of using a pre-computed system matrix $A$ , the algorithm uses a Radon transform operator ( $R$ ) for forward projection and its adjoint ( $R^*$ ) for back-projection. This is "ray-driven," meaning projections are calculated on-the-fly.
Compensation: The standard MLEM update step is modified. The term $A^* \cdot \mathbf{1}$ (which usually represents the system sensitivity) is replaced by the pre-calculated White Image ( $I_{WI}$ ).
Update Formula:
$I^{(n+1)} = I^{(n)} \odot R^* \left( S \oslash R(I^{(n)}) \right) \oslash I_{WI}$
Where $S$ is the sinogram data, $\odot$ and $\oslash$ are element-wise multiplication and division, and $I_{WI}$ acts as the normalization/compensation factor.
Dithering: To mitigate quantization artifacts caused by discrete crystal positions, a "dithering" technique is applied to the center-of-crystal connecting lines during projection.

3. Key Contributions

Exact Analytical Derivation: The paper provides a rigorous, closed-form mathematical derivation of the crystal-to-crystal response PDF for a rotating PET scanner, including the exact integral solution for the rotated response.
High-Accuracy Approximation: They introduce a simplified triangular approximation of the crystal response that is computationally efficient yet maintains high accuracy (proven via RMSE analysis and Taylor expansion).
Analytical White Image Generation: A method to generate the system sensitivity map (White Image) on-the-fly using only a few geometric parameters (crystal radius, length, and distance), eliminating the need for MC simulations or long physical acquisitions.
System Matrix-Free Reconstruction: A modified MLEM algorithm that achieves high-quality reconstruction without storing the massive system matrix, significantly reducing memory requirements to the size of the image itself.

4. Results

The approach was validated using both synthetic data and real-world acquisitions from a Raytest ClearPET small animal scanner.

Synthetic Data: Reconstructions of NEMA phantoms (with cylindrical holes and varying cylinder sizes) showed that the proposed method successfully recovered object details in both 8-sector and 4-sector configurations.
Real Data Comparison:
- Filtered Back-Projection (FBP): Produced distorted images with significant artifacts.
- Standard MLEM (No Compensation): Suffered from a prominent central artifact (due to the "white image" effect where the center has the highest detection probability) and ring artifacts.
- Proposed Method: Eliminated the central and ring artifacts. The reconstructed images clearly showed the radioactive cylinders and holes with high fidelity.
Performance: The algorithm is highly efficient. Reconstructing a $256 \times 256$ image with ~50,000 coincidences took approximately 1.3 seconds on an Intel i9-11900K. A full high-quality reconstruction of a standard phantom (48 axial intersections) took just over one minute.

5. Significance

Practicality: The method bridges the gap between the accuracy of complex physical models and the speed of analytical methods. It allows for high-quality PET reconstruction on hardware with limited memory (no need for $10^{12}$ element matrices).
Artifact Reduction: By explicitly modeling the system's geometric response via the White Image, the method effectively corrects for non-uniform sensitivity, a common issue in PET scanners that leads to central artifacts.
Scalability: While currently focused on 2D, the authors note that the ray-driven approach and the derived response models pave the way for extending this efficient compensation strategy to full 3D reconstruction schemes.
Open Source: The implementation is made available, facilitating further research and adoption in the PET imaging community.

In conclusion, the paper presents a mathematically robust and computationally efficient framework for PET image reconstruction that outperforms uncompensated iterative methods and traditional analytical techniques by leveraging a precise, analytically derived compensation model.