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Fringe pattern analysis using deep learning
Shijie Feng, Qian Chen, Guohua Gu, Tianyang Tao, Liang Zhang, Yan Hu, Wei Yin, Chao Zuo
TL;DR
Accurate fringe analysis must recover phase information from as few patterns as possible, but single-pattern methods face accuracy and edge-preservation challenges. The paper trains cascaded deep neural networks to demodulate phase from one fringe pattern, and reports higher accuracy and better edge preservation than Fourier transform and Windowed Fourier profilometry in fringe projection experiments.
Problem
Fringe analysis seeks highly accurate phase recovery from the minimum number of fringe patterns, while single-pattern demodulation remains challenging.
Method
Two cascaded convolutional neural networks estimate background intensity, then predict the arctangent numerator and denominator for phase recovery from a raw fringe pattern.
Results
The deep learning method produced the lowest mean absolute phase error, 0.08 rad, versus 0.21 rad for FT and 0.18 rad for WFT, while reducing boundary errors.
Takeaways & Limitations
The framework enables high-accuracy, edge-preserving phase reconstruction from a single fringe pattern without user-specified parameters in fringe projection profilometry.
Abstract
from arXiv · showhide
In many optical metrology techniques, fringe pattern analysis is the central algorithm for recovering the underlying phase distribution from the recorded fringe patterns. Despite extensive research efforts for decades, how to extract the desired phase information, with the highest possible accuracy, from the minimum number of fringe patterns remains one of the most challenging open problems. Inspired by recent successes of deep learning techniques for computer vision and other applications, here, we demonstrate for the first time, to our knowledge, that the deep neural networks can be trained to perform fringe analysis, which substantially enhances the accuracy of phase demodulation from a single fringe pattern. The effectiveness of the proposed method is experimentally verified using carrier fringe patterns under the scenario of fringe projection profilometry. Experimental results demonstrate its superior performance in terms of high accuracy and edge-preserving over two representative single-frame techniques: Fourier transform profilometry and Windowed Fourier profilometry.
Introduction
Optical metrology stores measured physical information in fringe-pattern phase, making accurate phase demodulation fundamental. The central challenge is recovering phase from as few fringe patterns as possible while balancing the robustness of multi-shot methods against the practicality of single-shot analysis.
- Optical metrology techniques encode quantities such as displacement, strain, surface profile, and refractive index in the phase of two-dimensional fringe patterns.
- Measurement accuracy fundamentally depends on accurately demodulating the underlying phase distribution.
- Phase-shifting methods require multiple fringe patterns, whereas spatial demodulation methods retrieve phase from a single pattern using FT, WFT, or WT techniques.
- Although multi-shot phase-shifting is generally more robust and accurate, its multi-shot nature makes it vulnerable to dynamic-scene constraints, external disturbance, and vibrations.
Principle
The method emulates conventional phase demodulation with two cascaded convolutional neural networks. CNN1 estimates the fringe background, while CNN2 predicts phase-demodulation terms from the original fringe and estimated background.
- Principle: Fringe patterns are modeled using intensity, background intensity, fringe amplitude, and desired phase variables.The background intensity is treated as a disturbance that is removed before recovering the wrapped phase.
- Principle: The phase is determined modulo 2π using numerator and denominator terms whose signs identify the quadrant in four-quadrant phasor space.A constant c depends on the phase-demodulation algorithm, while M(x,y) and D(x,y) denote the numerator and denominator terms.
- Network design: CNN1 takes the raw fringe pattern I(x,y) as input and estimates its background intensity A(x,y).The first network is one stage of a cascaded architecture containing two convolutional neural networks.
- Network design: CNN2 uses the estimated background image together with the original fringe image to predict the numerator and denominator terms.This design emulates the conventional phase-demodulation process rather than directly mapping the fringe pattern to phase.
- Training data: Training ground truth is generated by retrieving phase with an N-step phase-shifting method using N phase-shifted fringe patterns.The phase shifts are indexed by n and defined through δn = 2πn/N.
- Network design: CNN1 processes a raw W × H fringe pattern through convolutional layers, four residual blocks, and additional convolutional layers.The final layer estimates output gray values; the network structure is illustrated in Figure 2.
Experiment
The method was evaluated in fringe projection profilometry using trained networks and unseen test scenes. It achieved lower phase error, higher-quality 3-D reconstructions, and micrometer-scale accuracy on calibrated spheres.
- Training and testing: 960 fringe patterns were used for training, with 150 additional images held out for validation.The networks were trained using 12-step phase-shifting ground truth data.
- Training and testing: The trained networks were tested on a previously unseen scene containing two isolated plaster models with complex surface details.CNN1 predicted the background intensity from the input fringe image before CNN2 estimated phase-related quantities.
- Phase-error comparison: 0.08 rad MAE was obtained by the proposed method, compared with 0.21 rad for FT and 0.18 rad for WFT.The proposed method also reduced errors concentrated at boundaries and abrupt depth-changing regions.
- 3-D reconstruction: The deep learning method produced the highest-quality 3-D reconstruction, visually almost reproducing the 12-step phase-shifting ground truth.FT showed grainy distortions, while WFT produced smoother surfaces but failed to preserve details such as eyes and skirt wrinkles.
- Quantitative accuracy: 17 µm and 21 µm deviations were measured for reconstructed sphere radii, while the center-to-center distance error was -26 µm.The reconstructed sphere radii were 25.415 mm and 25.424 mm, compared with calibrated radii of 25.398 mm and 25.403 mm.
Conclusion
The study concludes that deep learning improves single-fringe phase demodulation accuracy. Its framework predicts intermediate quantities for arctangent-based reconstruction and achieves high-accuracy, edge-preserving results without user-specified parameters.
- Deep learning significantly improves the accuracy of phase demodulation from a single fringe pattern.
- The framework predicts the background image, numerator, and denominator for arctangent-based phase reconstruction.
- The technique provides high-accuracy, edge-preserving phase reconstruction without user-specified parameters.
- The demonstrated application uses carrier fringe patterns under fringe projection profilometry.
Funding Information
The work was supported by Chinese national and provincial research funding programs.
- The research received support from the National Key R&D Program of China, the National Natural Science Fund of China, and postdoctoral research funding programs.