Point spread function wavefront recovery from in-focus stellar observations

Technical Summary: Point Spread Function Wavefront Recovery from In-Focus Stellar Observations

Problem Statement
Accurate modeling of the Point Spread Function (PSF) is critical for precision cosmology, particularly for weak gravitational lensing studies conducted by wide-field space telescopes (e.g., Euclid, Vera C. Rubin Observatory, Nancy Grace Roman). The PSF varies spatially across the field of view (FOV), with wavelength, and over time. While non-parametric models have shown success in pixel-space PSF reconstruction, they often fail to recover the underlying Wavefront Error (WFE) field with high fidelity.

The specific challenge addressed in this work is the recovery of the WFE field from in-focus, polychromatic, and undersampled stellar observations. This is an ill-posed inverse problem because in-focus PSFs collapse into small regions in the focal plane, losing significant phase information compared to out-of-focus (phase-diverse) observations. Previous attempts using the semi-parametric WaveDiff model achieved excellent pixel-space PSF recovery (<1% error) but resulted in poor WFE recovery (~30% relative error), as the non-parametric features learned directly from data did not align with the physical ground truth.

Methodology
The authors propose a new optimization scenario for the WaveDiff semi-parametric PSF model. WaveDiff represents the PSF in wavefront space by combining a parametric component (based on Zernike polynomials) and a learnable non-parametric (NP) component. The core innovation lies in bridging these two components through wavefront feature projection.

The proposed methodology introduces five key modifications to the standard WaveDiff optimization process:

1. Wavefront Feature Projection: At the end of each optimization cycle, the NP wavefront features are projected onto the parametric Zernike basis. This involves an iterative projection algorithm (Algorithm 1) designed to handle the non-orthogonality of Zernike polynomials caused by the telescope's obscuration mask. This step transfers information from the flexible NP model to the physically grounded parametric model.
2. Increased Optimization Cycles: The number of optimization cycles is increased from 2 to 12 to allow the algorithm to escape local minima and converge toward the global minimum.
3. Non-Parametric Reinitialization: After each projection, the NP model is reinitialized. This prevents the model from stagnating in a local minimum and allows it to explore new regions of the optimization landscape in the subsequent cycle.
4. Convergence Assurance: The number of epochs per cycle is maintained at original levels to ensure the NP model converges before projection, despite the increased computational cost.
5. Optimization Strategy: The parametric component is not directly optimized via gradient descent on the in-focus data (due to high degeneracy). Instead, it serves as a repository for the projected information from the NP component, which is optimized for pixel-space accuracy.

Key Contributions

• Novel Optimization Framework: The paper presents a method to constrain the underlying WFE field using only in-focus observations by leveraging the spatial diversity of the PSF across the FOV.
• Iterative Projection Algorithm: A specific algorithm is developed to project arbitrary wavefront maps onto an obscured Zernike basis, correcting for the loss of orthogonality due to telescope obscuration.
• Bridging Parametric and Non-Parametric Models: The work demonstrates how to effectively combine the physical interpretability of parametric models with the flexibility of non-parametric learning.

Experimental Results
The authors evaluated the new framework using simulations based on the Euclid mission specifications, generating 2,400 in-focus stellar observations with varying signal-to-noise ratios.

• WFE Recovery: The new method reduced the relative WFE recovery error from approximately 30% (original WaveDiff) to approximately 3.4% (using only the parametric part of the new model) and 6.4% (using the full model). This represents a tenfold improvement.
• Pixel-Space Performance: The pixel-space PSF recovery remained highly accurate, with low-resolution RMSE around 0.3–0.4% and super-resolution RMSE around 0.7%, comparable to or slightly better than the original model.
• Convergence: The relative WFE error was observed to decrease steadily over the 12 optimization cycles, confirming the effectiveness of the projection and reinitialization strategy.
• Single-Star Limitation: A control experiment attempting to recover WFE from a single in-focus star resulted in ~100% relative error, reinforcing the necessity of using multiple stars across the FOV to break degeneracies.

Significance
The paper claims that this work demonstrates the first method to combine wide-field WFE recovery, in-focus-only polychromatic observations, and non-parametric wavefront features within a single framework. By achieving ~3% WFE error without requiring out-of-focus observations, the method enables precise PSF modeling and direct diagnosis of telescope optical states (e.g., detecting malfunctions or mirror deformations) using standard survey data. The authors note that while the WFE recovery is significantly improved, the parametric model alone yields the best WFE accuracy, while the full model (including NP features) offers the best pixel-space representation. The code for reproducing these results is made publicly available.