Uncertainty-aware Fourier ptychography

Abstract

Fourier ptychography (FP) offers both wide field-of-view and high-resolution holographic imaging, making it valuable for applications ranging from microscopy and X-ray imaging to remote sensing. However, its practical implementation remains challenging due to the requirement for precise numerical forward models that accurately represent real-world imaging systems. This sensitivity to model-reality mismatches makes FP vulnerable to physical uncertainties, including misalignment, optical element aberrations, and data quality limitations. Conventional approaches address these challenges through separate methods: manual calibration or digital correction for misalignment; pupil or probe reconstruction to mitigate aberrations; or data quality enhancement through exposure adjustments or high dynamic range (HDR) techniques. Critically, these methods cannot simultaneously address the interconnected uncertainties that collectively degrade imaging performance. We introduce Uncertainty-Aware FP (UA-FP), a comprehensive framework that simultaneously addresses multiple system uncertainties without requiring complex calibration and data collection procedures. Our approach develops a fully differentiable forward imaging model that incorporates deterministic uncertainties (misalignment and optical aberrations) as optimizable parameters, while leveraging differentiable optimization with domain-specific priors to address stochastic uncertainties (noise and data quality limitations). Experimental results demonstrate that UA-FP achieves superior reconstruction quality under challenging conditions. The method maintains robust performance with reduced sub-spectrum overlap requirements and retains high-quality reconstructions even with low bit sensor data. Beyond improving image reconstruction, our approach enhances system reconfigurability and extends FP's capabilities as a measurement tool suitable for operation in environments where precise alignment and calibration are impractical.

Fig. 1. Imaging results in comparison with state-of-the-art techniques.

a Shows our FPM setup along with the captured 15 × 15 low-resolution images. b Presents the reconstructed amplitude and phase, while c displays the reconstructed Fourier spectrum. The solid and dashed boxes represent objective lens NA and the maximum illumination NA_illum

Fig. 2. Reconstruction of system parameters.

Reconstruction of system parameters including a aberrations and b, c misalignment. b Shows stereograms of the LED locations in both two dimensional (2D) and three dimensional (3D) views, before and after misalignment correction, while c presents the evolution of the misalignment parameter set θ_illum during the optimization

Fig. 3. Recovery of human blood smear results.

a Raw low-resolution image with reconstruction amplitude and phase. b Reconstruction aberrations and Zernike coefficients of modes. c Fourier spectrum, with solid and dashed boxes showing objective lens NA and maximum illumination NA_illum. d 2D and 3D views of the LED array after misalignment correction. e Evolution of θ_illum parameters during optimization

Fig. 4. Extended imaging resolution with UA-FP.

a Recovery full Fourier spectrum and reconstructed amplitude, and b Fourier spectrum by eliminating extended information and corresponding reconstructed amplitude. c Comparison of resolution capabilities of (a and b)

Fig. 5. Numerical results compared with the ground truth.

a Shows the ground truth target image alongside one of the simulated low-resolution images. b Depicts the introduced misalignment in the LED array. c and d Display the recovered amplitude and reconstructed phase using ePIE and UA-FP, respectively. e Provides a zoomed-in view of c and d, with f presenting a plot comparison of the lines in e. g shows the recovered Fourier spectrum of the target

Fig. 6. Quantitative analysis on reconstruction of the uncertainty parameters.

Reconstruction of system parameters including a, b misalignment and c aberrations. a 3D and 2D views of LED array location after correction. b Evolution of θ_illum parameter during the optimization. c Reconstructed aberrations by ePIE and UA-FP method and corresponding Zernike coefficients

Fig. 7. Quantitative analysis on the robustness to noise.

Robustness to low-quality data was tested through anti-noise evaluations a, b and imaging quality assessments under varying sensor quantization bits c

Fig. 8. Quantitative analysis on the required measurement amount.

Imaging quality assessed using the metrics PSNR a and SSIM b at different light source-to-sample distances, corresponding to varying sub-spectrum overlap ratios

Fig. 9. FP imaging process and the computational graph for the reconstruction.

a FP implementation by LED array. Issues that make FP vulnerable to uncertainties include misalignment, optical element aberrations, poor quality data, and etc. b Proposed UA-FP, which includes differentiable FP forward model and differentiable optimization. c Both the imaging target and system characteristics can be recovered simultaneously