Joint optical-digital design strategy for adaptive optics systems: application to wavelength selection for satellite imaging

Adaptive optics can be used to mitigate the effects of atmospheric turbulence on imaging systems, but the correction is only partial, and deconvolution is often required to improve the resolution. This results in entire optical/digital systems, which are traditionally designed sequentially, i.e. , the adaptive optics system is optimised first, and the restoration algorithms are designed a second time. Studies on optical/digital systems have shown that jointly optimizing the whole system is a better alternative. We propose to extend these co-design strategies to the design of an adaptive optics-assisted imaging system. We derive a simple criterion that takes into account the source properties and the entire optical/ digital system performance. To illustrate its interest, we use it to optimize the wavelength distribution between the wavefront sensor and the imaging camera. In addition, we explore the potential of using multiple imaging channels operating at different wavelengths as a means of making an imaging system robust to turbulence strength and source magnitude variations. Later, any parameter of the optical/digital system, if not the entire system itself, could be optimized this way.

💡 Research Summary

This paper addresses the problem of designing an adaptive‑optics (AO) assisted imaging system for ground‑based observation of low‑Earth‑orbit satellites. Conventional practice treats the optical AO subsystem and the digital image‑restoration stage as separate design problems: first the AO system is optimized (often using Strehl ratio, residual wave‑front error, or PSF peak intensity), and only afterwards a deconvolution algorithm is selected. Such a sequential approach neglects two crucial aspects: the signal‑to‑noise ratio (SNR) of the science camera and the impact of the restoration algorithm on the final image quality. Consequently, the overall performance of the combined optical‑digital chain can be far from optimal.

The authors propose a joint optical‑digital co‑design framework that simultaneously optimizes the distribution of photons and wavelengths between the wave‑front sensor (WFS) and the scientific imaging channel. The key idea is to define an optimisation criterion that directly measures the quality of the restored image rather than an intermediate optical metric. They adopt the mean‑square error (MSE) between the true object and its estimate as the distance function, and introduce a normalised version, the root‑normalised MSE (RNMSE), which scales the error by the total photon count of the object. This criterion can be evaluated analytically, without generating any noisy images, provided that (i) the optical transfer function (OTF) of the AO‑corrected system is known, (ii) a statistical model of the object is available, and (iii) the noise is modelled as white Gaussian.

The object prior is taken as a spatially homogeneous Gaussian field with a power‑spectral density (PSD) of the form
(S_o(f)=A^2/(k^p+f^p)),
which captures the typical 1/f‑like fall‑off of natural scenes and satellite imagery. The noise PSD is assumed constant and equal to the sum of photon‑noise variance and read‑out‑noise variance. Under these assumptions the minimum‑mean‑square‑error (MMSE) estimator reduces to a linear Wiener filter, whose frequency‑domain expression is given in the paper. Substituting the Wiener filter into the analytical MSE expression yields a closed‑form sum over spatial frequencies (Eq. 7). Consequently, the RNMSE can be computed directly from the OTF, object PSD, and noise PSD, dramatically reducing the computational load of the optimisation.

The simulation platform models a 2.5 m ground‑based telescope (the “PROVIDENCE‑like” system under development at ONERA) equipped with a Shack‑Hartmann WFS and a science camera. The AO correction is simulated using a fast Fourier‑based method that incorporates both photon allocation and wavelength selection for the two channels. Table 1 (in the paper) lists the key AO parameters (e.g., number of sub‑apertures, loop gain, latency) and the atmospheric seeing values considered (1″–2″). The reference source for the WFS is the satellite itself, assumed to be a 10 m object at 800 km altitude observed at 60° elevation.

Single‑channel optimisation – By scanning the imaging wavelength and the fraction of photons sent to the science camera, the authors compute RNMSE curves for various seeing conditions and object magnitudes (photon counts ranging from 10⁴ to 10⁶ ph). The results show a clear trade‑off: shorter wavelengths reduce the diffraction limit but increase the residual phase error, leading to a higher OTF attenuation; longer wavelengths improve AO correction but suffer from a larger diffraction blur. The RNMSE minima typically lie between 600 nm and 800 nm, depending on the specific atmospheric and flux conditions. This demonstrates that a Strehl‑based optimisation would select a different wavelength, potentially far from the true optimum for the restored image.

Multi‑spectral channel extension – To mitigate the sensitivity of the optimal wavelength to changing conditions, the authors propose using two (or more) imaging channels each operating at a distinct wavelength. The optimisation now targets the weighted average RNMSE across channels, allowing the system to retain good performance even when seeing deteriorates or the satellite’s apparent magnitude changes. Simulations reveal that a dual‑channel configuration (e.g., one channel at 550 nm, another at 850 nm) reduces the average RNMSE by roughly 10–15 % compared with any single‑wavelength design, and provides a more uniform performance envelope across the tested parameter space.

The paper also evaluates alternative distance metrics (L₁ norm, structural similarity index, SSIM). While the absolute RNMSE values differ, the qualitative ordering of optimal wavelengths remains consistent, confirming the robustness of the proposed criterion.

Limitations and future work – The analysis relies on several simplifying assumptions: (1) the object prior is Gaussian and spatially homogeneous, whereas real satellite images consist of a bright compact target against a dark background, leading to non‑Gaussian statistics; (2) noise is assumed white and uniform, ignoring the spatially varying photon‑noise that arises from the high contrast of the scene; (3) the MMSE estimator is linear, whereas practical deconvolution often employs non‑linear iterative algorithms or deep‑learning approaches. Extending the framework to incorporate more realistic priors, non‑uniform noise models, and alternative restoration algorithms would require either numerical Monte‑Carlo evaluation of the criterion or the derivation of new analytical expressions. Additionally, the current study does not consider laser guide stars or other external references for the WFS, which could alter the photon budget and the optimal flux split.

Conclusions – The authors demonstrate that a joint optical‑digital co‑design, grounded in an analytically tractable image‑quality metric, can systematically determine the optimal wavelength distribution and photon allocation for AO‑assisted satellite imaging. The framework is generic: any system parameter (e.g., detector pixel size, AO loop gain, number of WFS sub‑apertures) can be inserted into the analytical RNMSE expression and optimised jointly with the imaging wavelength. By moving beyond PSF‑centric metrics, the approach yields designs that are more resilient to variations in atmospheric turbulence and target brightness, and it opens the door to robust multi‑spectral imaging architectures for future ground‑based space‑situational‑awareness systems.