# Focal Plane Wavefront Sensing using Machine Learning: Performance of Convolutional Neural Networks compared to Fundamental Limits

**Authors:** G. Orban de Xivry, M. Quesnel, P. -O. Vanberg, O. Absil, G. Louppe  
**arXiv:** [2106.04456](https://arxiv.org/abs/2106.04456)  
**Date:** 2021-06-08  
**Summary page:** https://www.summarizepaper.com/en/paper/2106.04456v1/

## Key points

- Deep convolutional neural networks (CNNs) can be used to measure non-common path aberrations (NCPA) based on focal plane images in focal plane wavefront sensing (FPWFS).
- FPWFS is an appealing solution for NCPA measurement and could be used as an adaptive optics wavefront sensor.
- Numerically simulated aberrated point spread function (PSF) pairs were used to generate the labelled dataset required for training the CNNs.
- Phase diversity was introduced to ensure uniqueness of the solution while being easy to implement in practice.
- Each individual Zernike mode had a uniform density distribution with minimum and maximum values depending on the Zernike index, desired median rms WFE, and number of Zernike coefficients.
- Two CNN architectures were considered: ResNet-50 and U-Net which are used respectively to estimate Zernike coefficients or directly the phase.
- In idealized simulations, they demonstrated that CNN-based models can reduce root mean squared wavefront error (WFE) to < λ/1500 for 2 x 10^6 photons in one iteration when estimating 20 Zernike modes.
- They also showed that CNN-based models are sufficiently robust to varying signal-to-noise ratio, under the presence of higher-order aberrations, and under different amplitudes of aberrations.
- Additionally, they displayed similar to superior performance compared to iterative phase retrieval algorithms.
- The authors suggest that CNNs represent a compelling way to implement FPWFS which can leverage its high sensitivity over a broad range of conditions.
- Future work could explore ways to exploit real life data's temporal structure and incorporate it into the training process for improved performance.

## Summary

This paper proposes the use of deep convolutional neural networks (CNNs) to measure non-common path aberrations (NCPA) based on focal plane images in focal plane wavefront sensing (FPWFS). FPWFS is an appealing solution for NCPA measurement, a key limitation for high-contrast imaging, and could be used as an adaptive optics wavefront sensor. To generate the labelled dataset required for training the CNNs, numerically simulated aberrated point spread function (PSF) pairs were used: in-focus and out-of-focus. Phase diversity was introduced to ensure uniqueness of the solution while being easy to implement in practice. The generated phase maps reflect typical errors of high-quality optical surfaces with a spatial power spectral density profile S ≈ 1/f^2, where f is the spatial frequency. Each individual Zernike mode had a uniform density distribution with minimum and maximum values depending on the Zernike index, desired median rms WFE, and number of Zernike coefficients. The authors considered two CNN architectures: ResNet-50 and U-Net which are used respectively to estimate Zernike coefficients or directly the phase. The models were evaluated at various flux levels and for two spatial frequency contents (20 and 100 Zernike modes). In idealized simulations, they demonstrated that CNN-based models can reduce root mean squared wavefront error (WFE) to < λ/1500 for 2 x 10^6 photons in one iteration when estimating 20 Zernike modes. They also showed that CNN-based models are sufficiently robust to varying signal-to-noise ratio, under the presence of higher-order aberrations, and under different amplitudes of aberrations. Additionally, they displayed similar to superior performance compared to iterative phase retrieval algorithms. The authors suggest that CNNs represent a compelling way to implement FPWFS which can leverage its high sensitivity over a broad range of conditions. Future work could explore ways to exploit real life data's temporal structure and incorporate it into the training process for improved performance.