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Point Spread Function Engineering Using Implicit Neural Representations

This paper proposes a neural field-based pupil design method that treats PSF engineering as a phase retrieval problem, enabling the optimization of flexible, user-defined 3D PSF distributions with robustness to initialization compared to traditional pixel-wise approaches.

Original authors: Suet Ying Chan, Mitchell Gilmore, Qilin Deng, Guorong Hu, Joseph Greene, Ruipeng Guo, Lei Tian

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Suet Ying Chan, Mitchell Gilmore, Qilin Deng, Guorong Hu, Joseph Greene, Ruipeng Guo, Lei Tian

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

In the world of high-powered microscopy, seeing clearly is only half the battle; understanding where things are in three dimensions is the other. When a microscope looks at a tiny point of light, the laws of physics cause that point to blur into a small, fuzzy disk. This blur is known as the point spread function. For decades, scientists have tried to manipulate this blur to their advantage. By placing special filters in the path of the light, they can stretch this disk into a spiral to track depth, flatten it to keep everything in focus at once, or split it into multiple dots to illuminate different layers simultaneously. However, designing these filters has traditionally been a rigid process. It required experts to choose from a limited menu of pre-made mathematical shapes, much like trying to build a custom house using only a few standard brick patterns. If the desired shape didn't fit the available patterns, the design failed, or the result was a messy, noisy image that was difficult to interpret.

A team of researchers at Boston University and the Georgia Tech Research Institute has developed a new way to solve this problem, one that treats the design of these optical filters as a learning task rather than a puzzle of fixed pieces. Instead of forcing the filter to fit a pre-existing mathematical category, they used a type of artificial intelligence known as a neural field. This system acts like a continuous, smooth map that can be drawn at any level of detail. The researchers fed this system a description of the exact 3D pattern of light they wanted to create—whether it was a rotating spiral, a flat sheet of focus, or a scattered set of glowing dots. The computer then figured out the precise shape of the filter needed to turn a standard beam of light into that specific pattern. The result is a flexible method that can generate custom optical filters for almost any imaging need without requiring the deep, specialized knowledge previously necessary to choose the right mathematical building blocks.

The core of this work lies in how the researchers approached the problem of finding the right filter shape. In the past, scientists would try to solve this by adjusting the filter pixel by pixel, like trying to paint a masterpiece by changing the color of every single square on a grid one at a time. This approach often got stuck in local errors, producing filters that were jagged and full of high-frequency noise, which would ruin the final image with grainy artifacts. The new method avoids this by using a neural network to describe the filter as a smooth, continuous function. This means the computer learns a single, flowing rule for the entire filter rather than a collection of disconnected dots. This smoothness acts as a natural guardrail, preventing the design from becoming noisy and ensuring that the resulting filter is physically realistic and easy to manufacture.

To test their idea, the team built a tabletop optical system equipped with a spatial light modulator, a device that can act as a programmable filter. They programmed the device with the shapes generated by their neural network and then shone light through it to see what happened. They successfully created several complex patterns, including a single-helix point spread function that rotates a full 360 degrees over an 80-micrometer range, and a double-helix version that rotates 180 degrees over 100 micrometers. They also designed an extended depth-of-field pattern that stays sharp across a 400-micrometer range. In every case, the experimental results matched the computer simulations closely, proving that the smooth, continuous designs learned by the neural network could be physically realized and would perform exactly as predicted.

One of the most practical advantages of this new method is its ability to control the brightness of different parts of the image independently. In deep tissue imaging, light often gets dimmer as it travels further away due to scattering and absorption. The researchers showed that their system could be instructed to make the focal points at greater depths brighter to compensate for this loss. By simply adjusting the weights in the computer's target description, the system learned a filter that pushed more energy to the deeper layers, effectively evening out the intensity across the entire volume. This kind of fine-tuned control is difficult to achieve with older methods, which often struggle to balance the energy distribution across different depths without introducing errors.

The researchers also compared their neural field approach against traditional pixel-by-pixel optimization techniques. The traditional methods required many more attempts to find a solution and often needed a specific starting guess to work at all. Even when they succeeded, the resulting filters were often noisy and less accurate. The neural field method, by contrast, converged to a solution much faster and without needing any special starting point. The filters it produced were smooth and free of the grainy speckle noise that plagues other approaches. This suggests that the method is not only more efficient but also more robust, capable of finding high-quality solutions even when the desired pattern is complex or unusual.

This work opens the door to a new era of custom microscopy where the limitations of pre-defined shapes are removed. Scientists can now specify exactly how they want light to behave in three dimensions, and the system will generate the necessary optical filter to make it happen. The method is general enough to be applied to various tasks, from tracking the movement of single molecules to illuminating multiple planes of a biological sample at once. Because the neural network learns a continuous rule, the resulting filter can be scaled to any resolution without losing quality, making it adaptable to different microscopes and experimental setups. The ability to design these filters automatically and reliably means that researchers can focus more on the biological questions they are trying to answer rather than the mathematical hurdles of designing the tools to see them.

Technical Summary: Point Spread Function Engineering Using Implicit Neural Representations

Problem Statement
Point spread function (PSF) engineering via pupil plane modulation is critical for advanced microscopy applications requiring specific imaging properties, such as depth encoding or extended depth of field (EDOF). Traditional design methods rely heavily on extensive domain knowledge and task-specific basis functions (e.g., Laguerre–Gaussian modes for helical PSFs, cubic phase plates for EDOF, or Fresnel/Walsh zone plates for multifocal imaging). This reliance makes generalization across different applications difficult, as selecting the correct basis a priori is a substantial part of the design problem.

Alternatively, treating PSF engineering as a phase retrieval problem (similar to computer-generated holography) allows for inverse design. However, existing pixel-wise optimization methods suffer from several limitations:

Susceptibility to Stagnation: Optimization often gets trapped in local minima, requiring careful initialization (e.g., Gerchberg-Saxton algorithms).
Lack of Continuity: Pixel-wise approaches lack continuity between pixels, leading to high-frequency speckle noise in the learned phase profiles.
Physical Interpretability: The resulting phase masks are often non-physically intuitive and lack the micron-scale accuracy required for PSF engineering.
Generalization: These methods are often tailored to specific tasks and do not easily generalize to arbitrary 3D PSF distributions without redefining the basis.

Methodology
The authors propose a Neural Field Pupil Design method that treats the PSF engineering task as a phase retrieval problem using Implicit Neural Representations (INRs).

Implicit Neural Representation (INR): Instead of optimizing a discrete pixel grid, the pupil phase mask is parameterized by a Multi-Layer Perceptron (MLP). The network maps normalized pupil coordinates $\rho$ to the phase value $\phi_m(\rho)$ . This coordinate-based approach provides a continuous, smooth representation of the phase mask, inherently regularizing the solution and removing the need to define a basis function a priori.
Fourier Feature Mapping: To address the "spectral bias" of MLPs (which tend to learn low-frequency functions more easily), the input coordinates are encoded using Fourier feature mapping. This projects coordinates into a higher-dimensional space using a sinusoidal basis, enabling the network to learn high-frequency variations required for complex phase masks.
Phase Representation: To resolve phase wrapping ambiguity ( $2\pi$ shifts), the MLP outputs the real and imaginary components of the complex exponential $e^{i\phi_m}$ , which are normalized to unit magnitude. This ensures the output is a phase-only mask with unit amplitude.
Forward Model: The system utilizes a differentiable Fourier optics model based on the angular spectrum method. The pupil function is computed as the product of the neural field-learned mask and a defocus kernel.
Target Preprocessing & Loss: A key innovation is the preprocessing of the target PSF stack. The authors apply a binary support mask derived from the system's Optical Transfer Function (OTF) to the target spectrum. This mimics the band-limited nature of the optical system, easing convergence. Additionally, a user-defined weight factor ( $\alpha_k$ ) is applied to each axial slice in the loss function to control relative intensities, compensating for intensity falloff due to scattering or absorption in 3D imaging.
Optimization: The MLP is trained using gradient-based algorithms (e.g., Adam, L-BFGS) to minimize the Mean Squared Error (MSE) between the predicted and target 3D PSF stacks.

Key Contributions

Basis-Free Design: The method eliminates the need for application-specific basis functions, allowing users to specify arbitrary 3D PSF distributions.
Robustness and Smoothness: By leveraging INRs, the method produces smooth, continuous phase profiles that are robust to initialization and free from the high-frequency speckle noise common in pixel-wise optimizations.
Flexible Intensity Control: The framework incorporates a preprocessing step and weighted loss function to manage relative intensities across axial slices, addressing practical issues like depth-dependent signal loss.
Experimental Validation: The method was validated on a tabletop optical system using a Spatial Light Modulator (SLM), demonstrating successful deployment of various PSF designs.

Results
The authors demonstrated the method's efficacy through several experimental and simulation results:

Classical PSFs: The method successfully designed single-helix (360° rotation), double-helix (180° rotation), and EDOF (invariant) PSFs. Experimental results matched simulations closely.
Intensity Control: The method successfully controlled relative intensities between focal points. In one experiment, weighting deeper axial slices compensated for intensity falloff, resulting in more uniform intensity across the depth range.
Arbitrary Designs: The framework generated complex patterns, including alternating 3-foci PSFs with staggered spacing and 2×2 multifoci PSFs.
Regularization: Adding a regularization term to enforce energy concentration in the central lobe yielded phase masks that broke rotational symmetry, suggesting the potential to counter non-ideal effects like higher-order aberrations.
Comparison: Compared to pixel-wise MSE and pixel-NOVO optimization methods, the neural field approach converged faster, required no initialization, and produced significantly smoother phase profiles with less high-frequency noise.

Significance and Claims
The paper claims that the proposed neural field pupil engineering method provides a flexible and generalizable framework for 3D PSF engineering. Its primary significance lies in reducing the dependency on extensive domain knowledge and specific basis functions, thereby making task-specific PSF design more accessible.

The authors emphasize that the method's inherent regularization leads to stable training and physically interpretable phase profiles, overcoming the stagnation and noise issues of traditional pixel-wise optimization. They suggest this stability and flexibility can be extended to more sophisticated, end-to-end pipelines for co-designing PSFs for specific experimental conditions (e.g., in-vivo imaging with severe scattering). Furthermore, the compact nature of the neural field representation naturally supports high-dimensional inputs, making it suitable for future multi-wavelength or spatially variant PSF designs. The paper maintains a modest tone, noting that while the method is robust, it is part of an evolving field where further integration with experimental conditions (like sensorless wavefront correction) is possible.

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