Source-linked AI summary
Fourier ptychographic reconstruction using Wirtinger flow optimization
Liheng Bian, Jinli Suo, Guoan Zheng, KaiKai Guo, Feng Chen, Qionghai Dai
TL;DR
Fourier Ptychography needs long exposures to obtain high-SNR inputs because conventional AP is sensitive to noise. The paper proposes WFP, which combines Wirtinger-flow phase retrieval with noise relaxation for low-SNR reconstruction, and reports comparable accuracy with substantially shorter exposure. WFP is effective on synthetic and real data but requires more running time than AP.
Problem
Conventional AP is sensitive to input noise, so FP requires long exposures; an FPM setup needs about 3 minutes for 137 illuminations.
Method
WFP applies Wirtinger flow to FP and incorporates a noise relaxation constraint into an iterative gradient-descent optimization framework.
Results
80% exposure-time savings yield comparable resolution to AP using 1 ms exposure versus 5 ms for AP.
Takeaways & Limitations
Experiments on synthetic and real captured data validate WFP's effectiveness for FP reconstruction from noisy inputs.
Takeaways & Limitations
WFP requires more running time than AP, making computational-efficiency improvements future work.
Abstract
from arXiv · showhide
Recently Fourier Ptychography (FP) has attracted great attention, due to its marked effectiveness in leveraging snapshot numbers for spatial resolution in large field-of-view imaging. To acquire high signal-to-noise-ratio (SNR) images under angularly varying illuminations for subsequent reconstruction, FP requires long exposure time, which largely limits its practical applications. In this paper, based on the recently reported Wirtinger flow algorithm, we propose an iterative optimization framework incorporating phase retrieval and noise relaxation together, to realize FP reconstruction using low SNR images captured under short exposure time. Experiments on both synthetic and real captured data validate the effectiveness of the proposed reconstruction method. Specifically, the proposed technique could save around 80% exposure time to achieve similar retrieval accuracy compared to the conventional FP. Besides, we have released our source code for non-commercial use.
1. Introduction
Fourier Ptychography reconstructs high-resolution, large-field-of-view images from low-resolution spectral measurements, but conventional alternating projection is noise-sensitive and requires long exposures. The paper introduces WFP to use low-SNR inputs while reducing exposure and computation costs relative to existing approaches.
- Fourier Ptychography: FP captures low-resolution images containing different spatial-spectrum bands and stitches them in the Fourier domain to recover the high-resolution image.The reconstruction retrieves both amplitude and phase information from magnitude measurements.
- Motivation: Conventional AP is sensitive to input noise, requiring long exposures; an FPM setup reportedly needs about 3 minutes for 137 illuminations.This acquisition burden limits practical applications.
- Proposed approach: WFP applies Wirtinger flow to FP and adds a noise relaxation constraint to reconstruct from low-SNR images.The framework formulates phase retrieval using a non-convex optimization approach solved with gradient descent.
- Contributions: Compared with AP, WFP better handles detector noise and can reduce the required exposure time.The paper also positions WFP as an alternative to matrix-lifting methods such as PhaseLift and PhaseCut.
- Contributions: WFP is presented as a flexible optimization framework that can incorporate additional priors and constraints to reduce costs or improve retrieval accuracy.The paper identifies sparsity and other constraints as possible extensions.
2. Optimization framework
The framework treats phase retrieval as quadratic optimization and solves it iteratively with Wirtinger-flow gradient descent. Its algorithm takes sampling measurements and an initialization, repeatedly updates the recovered signal, and stops when converged.
- Problem formulation: Wirtinger flow retrieves a complex signal from real magnitude measurements represented by b = |Ax|^2.A is the linear sampling matrix and ⊙ denotes the dot product.
- Optimization objective: The phase-retrieval task is transformed into minimizing a quadratic loss over the recovered signal.The objective measures the squared Frobenius-norm discrepancy between predicted and measured quantities.
- Optimization procedure: Gradient descent updates the complex signal using the derivative of the quadratic cost with respect to x*.The update is applied iteratively according to the derived gradient expressions.
- Algorithm: The summarized algorithm takes A, b, and x(0) as inputs and returns the retrieved signal x.Each iteration updates x and increments k while convergence has not been reached.
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The method models Fourier ptychographic measurements as noisy phase-retrieval observations and solves reconstruction through a Wirtinger-flow optimization with explicit noise relaxation. It alternates updates of the reconstruction, noise, and relaxation variables from an initialization based on an up-sampled normal-incidence image.
- Measurement model: The HR spatial spectrum is recovered from LR observations modeled with a combined inverse-Fourier and down-sampling operator A = FS.The operator combines inverse Fourier transform F with down-sampling S.
- Noise model: Capturing noise n is modeled as Gaussian with standard deviation σ, and its constraint is relaxed using ε ∈ R^m.The formulation uses the three-sigma rule to approximate the noise constraint.
- Optimization model: A weighting parameter µ incorporates the noise constraint into the reconstruction objective.The resulting model combines the measurement formation with the noise constraint.
- Optimization procedure: The optimization sequentially updates x, n, and ε using gradient descent or closed-form rules while holding the other variables fixed.The x update uses gradient descent, while n and ε receive their corresponding update rules.
- Algorithm: The WFP algorithm initializes x(0) from the spatial spectrum of an up-sampled normal-incidence LR image and iterates until convergence.Each iteration updates x and n before incrementing the iteration index.
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The algorithm uses settings inherited from Wirtinger flow for its step-size schedule and provides source code for non-commercial use. The reported settings specify k0 = 330 and θmax = 0.4.
- Parameter settings: The WFP step-size settings use k0 = 330 and θmax = 0.4, following settings reported for Wirtinger flow.The paper states that these settings work well and adopts them in WFP.
- Implementation: The authors released the WFP source code for non-commercial use.The code is described as downloadable from the paper.
3. Experiments
Experiments on synthetic and real captured data evaluate WFP against conventional and denoising-based reconstruction methods. WFP remains robust across noise levels, jointly suppresses noise during reconstruction, and achieves comparable resolution with substantially shorter exposure.
- Experimental setup: WFP experiments used both synthetic data with controlled noise and real low-resolution images captured through a 15×15 LED arrangement.The real-data setup used 1 ms exposure per LED and compared WFP with AP.
- Synthetic data: WFP reconstructed intensity and phase information successfully as additive-noise standard deviation increased from 0.002 to 0.008.The reported results indicate that reconstruction quality did not degenerate substantially across the tested noise levels.
- Synthetic data: WFP largely outperformed AP, BM3D+AP, and AP+BM3D on visual and quantitative metrics at noise level σ = 0.004.Pre- or post-reconstruction denoising either removed high-frequency information or filtered crucial image details, whereas WFP jointly incorporated noise suppression into reconstruction.
- Real captured data: On real captured data, WFP produced higher resolution, stronger noise suppression, and more accurate phase reconstruction than conventional AP.Some WFP phase jumps occurred where magnitudes were close to zero, where phase assignments do not affect successful magnitude recovery.
- Real captured data: 1 ms exposure with WFP achieved comparable resolution to conventional AP using 5 ms exposure, indicating around 80% exposure-time savings at similar reconstruction accuracy.The comparison was made using amplitude reconstructions under different exposure times.
- Discussion: WFP offers a feasible FP reconstruction approach for non-ignorable capturing noise, including low-exposure conditions or imprecise hardware, but requires longer computation time than the other tested methods.The computational-cost comparison was implemented in Matlab on an Intel i7 3.6 GHz CPU system with 16G RAM.
4. Conclusions and discussions
WFP combines Wirtinger-flow phase retrieval with noise priors to reconstruct Fourier ptychography data acquired with shorter exposure. It achieves substantial exposure-time savings, while extending to additional noise models, priors, constraints, and FP variants, at the cost of lower running efficiency than AP.
- Conclusions and discussions: Around 80% exposure time can be saved without obvious performance degeneration compared with conventional FP.The framework incorporates priors on capturing noise while maintaining retrieval performance.
- Conclusions and discussions: WFP can accommodate non-uniform noise by modeling the noise parameter spatially rather than as a scalar.The paper also describes extensions using sparsity and total-variation priors to reduce snapshots or suppress reconstruction noise.
- Conclusions and discussions: WFP's sampling matrix can use different linear operations, enabling application to multiplexed FP and extended FP for fluorescence imaging.The framework is presented as flexible to additional priors and constraints.
- Conclusions and discussions: WFP requires more running time than AP, making computational efficiency a remaining limitation.The paper identifies accelerated gradient descent and parallel computation as possible ways to shorten runtime.