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Temporal phase unwrapping using deep learning

Wei Yin, Qian Chen, Shijie Feng, Tianyang Tao, Lei Huang, Maciej Trusiak, Anand Asundi, Chao Zuo

arXiv:1903.09836v2eess.IV

TL;DR

Conventional MF-TPU faces a trade-off: high-frequency patterns improve measurement precision, but noise limits reliable unwrapping to about 16 periods, while intermediate patterns lengthen the sequence. This paper uses a trained deep neural network for temporal phase unwrapping and reports reliable direct unwrapping of 64-period phase from a unit-frequency phase.

  • Problem

    MF-TPU generally limits high-frequency fringes to about 16 periods for reliable unwrapping, while intermediate-frequency patterns add to the sequence.

    Source: B. Multi-frequency temporal phase… · p. 3

  • Method

    DL-TPU uses a trained deep neural network to infer fringe order from single-period and high-frequency wrapped phases.

    Source: C. Deep-learning based temporal phase… · p. 4, §IV · p. 7

  • Results

    64-period high-frequency phase was directly and reliably unwrapped from one unit-frequency phase using DL-TPU.

    Source: §IV · p. 8

  • Takeaways & Limitations

    The reported approach achieves reliable high-frequency phase unwrapping with six projected patterns, without prior information or geometric constraints.

    Source: §I · p. 2

    The study focuses on TPU and FPP, so its conclusions are scoped to those applications.

    Source: §IV · p. 8

Abstract

from arXiv · show

The multi-frequency temporal phase unwrapping (MF-TPU) method, as a classical phase unwrapping algorithm for fringe projection profilometry (FPP), is capable of eliminating the phase ambiguities even in the presence of surface discontinuities or spatially isolated objects. For the simplest and most efficient case, two sets of 3-step phase-shifting fringe patterns are used: the high-frequency one is for 3D measurement and the unit-frequency one is for unwrapping the phase obtained from the high-frequency pattern set. The final measurement precision or sensitivity is determined by the number of fringes used within the high-frequency pattern, under the precondition that the phase can be successfully unwrapped without triggering the fringe order error. Consequently, in order to guarantee a reasonable unwrapping success rate, the fringe number (or period number) of the high-frequency fringe patterns is generally restricted to about 16, resulting in limited measurement accuracy. On the other hand, using additional intermediate sets of fringe patterns can unwrap the phase with higher frequency, but at the expense of a prolonged pattern sequence. Inspired by recent successes of deep learning techniques for computer vision and computational imaging, in this work, we report that the deep neural networks can learn to perform TPU after appropriate training, as called deep-learning based temporal phase unwrapping (DL-TPU), which can substantially improve the unwrapping reliability compared with MF-TPU even in the presence of different types of error sources, e.g., intensity noise, low fringe modulation, and projector nonlinearity. We further experimentally demonstrate for the first time, to our knowledge, that the high-frequency phase obtained from 64-period 3-step phase-shifting fringe patterns can be directly and reliably unwrapped from one unit-frequency phase using DL-TPU.

I. INTRODUCTION

The introduction frames temporal phase unwrapping as a way to handle discontinuities that challenge spatial methods, while conventional MF-TPU faces a trade-off between reliability, accuracy, and pattern count. The paper proposes a deep neural network approach and reports reliable unwrapping of 64-period fringes from a unit-frequency phase.

I. INTRODUCTION

  • Temporal phase unwrapping analyzes multiple wrapped phase values separately at each sample, allowing it to handle large discontinuities and spatially isolated regions.Spatial phase unwrapping instead relies on neighboring pixels and is challenging for disjoint regions and phase discontinuities.Source: §I · p. 1
  • MF-TPU generally limits high-frequency patterns to about 16 periods for reasonable success, while intermediate fringe sets enable higher frequencies at the cost of more projected patterns.In the simplest case, two 3-step pattern sets use six patterns total.Source: §I · p. 2
  • The paper positions its DNN approach as more reliable than conventional MF-TPU, including under different error sources.The introduction identifies intensity noise, low fringe modulation, and projector nonlinearity among the tested conditions.Source: §I · p. 2
  • DL-TPU directly unwraps 64-period high-frequency phase from one unit-frequency phase, with six projected patterns and no prior information or geometric constraint.The reported experiments include intensity noise, low fringe modulation, and intensity nonlinearity.Source: §I · p. 2

II. PRINCIPLE

The principle section describes three-step phase-shifting profilometry and how it yields a wrapped phase map. Recovering absolute phase then requires determining each pixel's fringe order.

II. PRINCIPLE

  • Three-step phase-shifting uses the theoretical minimum number of fringe images for PSP, which is desirable for reducing measurement time and motion-induced phase-shifting errors.The projected patterns use a shift offset of 2π/3.Source: A. Phase-shifting profilometry (PSP) · p. 3
  • The captured fringe patterns encode the object's average intensity, intensity modulation, and phase distribution.The standard three-step patterns are projected onto the object and captured as deformed fringe patterns.Source: A. Phase-shifting profilometry (PSP) · p. 3
  • Because of arctangent-function truncation, the obtained phase φ(x, y) is wrapped within (−π, π]; the fringe order k(x, y) for Φ(x, y) ranges from 0 to N −1, with N the fringe-pattern period number.Source: A. Phase-shifting profilometry (PSP) · p. 3

B. Multi-frequency temporal phase unwrapping (MF-TPU)

MF-TPU uses phase maps at different frequencies to infer the high-frequency fringe order. Noise narrows the tolerable phase-error range as fringe frequency rises, limiting reliable frequency and motivating extra intermediate patterns.

B. Multi-frequency temporal phase unwrapping (MF-TPU)

  • MF-TPU uses low- and high-frequency wrapped phases to calculate the high-frequency fringe order, rounding the estimate to an integer.When the low frequency is one, its phase is inherently unwrapped.Source: B. Multi-frequency temporal phase… · p. 3
  • With unit-frequency fringes, higher high-frequency fringe counts narrow the phase-error range for correct order recovery; conventional FPP systems therefore reliably unwrap about 16 periods.Outside the allowable error range, the exact high-frequency fringe order can be determined incorrectly.Source: B. Multi-frequency temporal phase… · p. 3
  • MF-TPU commonly uses more than two frequency sets to unwrap high-frequency phase step by step, but the intermediate patterns add time and hinder high-speed measurement.The method's phase-order error bound is derived from the phase errors in the low- and high-frequency maps.Source: B. Multi-frequency temporal phase… · pp. 3–4

C. Deep-learning based temporal phase unwrapping (DL-TPU)

DL-TPU uses a deep neural network to predict fringe order from single-period and high-frequency wrapped phases. It then reconstructs absolute phase and maps it to 3D results.

C. Deep-learning based temporal phase unwrapping (DL-TPU)

  • The network takes single-period and high-frequency wrapped phases as input and predicts the high-frequency fringe order rather than absolute phase.The absolute phase follows from the predicted order and the high-frequency wrapped phase.Source: C. Deep-learning based temporal phase… · p. 4
  • DL-TPU uses a residual-network skeleton with multi-scale pooling and upsampling, and learns to use spatial and temporal phase information together.The described design also uses shortcuts between residual blocks to stabilize convergence.Source: C. Deep-learning based temporal phase… · p. 4
  • The reconstructed absolute phase is converted into 3D results through phase-to-height mapping, for which camera and projector calibration is needed.The paper suggests GPU implementation or look-up tables to speed up reconstruction.Source: C. Deep-learning based temporal phase… · p. 4

A. Quantitative comparison with MF-TPU

DL-TPU retains high phase-unwrapping reliability as high fringe frequency increases, whereas MF-TPU errors rise sharply. Its remaining low-SNR errors are described as accumulated and readily correctable, including at fh=64.

A. Quantitative comparison with MF-TPU

  • From 0 to 12.71%, MF-TPU's phase-unwrapping error rate rises as fh increases, while DL-TPU correctly unwraps more than 95% of valid pixels and reduces errors by almost an order of magnitude.The comparison uses 200 testing image pairs.Source: A. Quantitative comparison with MF-TPU · p. 4
  • At dark regions and object edges with low fringe quality, DL-TPU reduces low-SNR phase-unwrapping errors; residual errors accumulate and are described as readily correctable, even at fh=64.The paper notes that fringe-order compensation algorithms can further correct these accumulated errors.Source: A. Quantitative comparison with MF-TPU · pp. 4–5
  • The framework predicts a period-order map from unit- and high-frequency phase inputs, then obtains the unwrapped phase for conversion to 3D results.The stated stages are data processing, a deep neural network, and phase-to-height mapping.Source: Figure 1 · p. 5

1) Intensity noise:

As exposure time decreases, phase-measurement quality drops, but DL-TPU provides higher unwrapping success and lower phase-error variance than MF-TPU. The experiment tests a ceramic plate at four exposure times with fh=32.

1) Intensity noise:

  • The reference phase is obtained with a 12-step phase-shifting algorithm and a redundant multi-frequency TPU sequence using frequencies 1, 8, 16, and 32.Error rate and phase-error variance are calculated against this reference at valid points.Source: 1) Intensity noise · p. 5
  • At 39, 20, 15, and 10 ms exposure, DL-TPU provides higher phase-unwrapping success and lower phase-error variance than MF-TPU as exposure time decreases.The test uses a standard ceramic plate with fh=32.Source: 1) Intensity noise · p. 5
  • Figure 3 compares captured images and line-230 intensity across exposure times, then contrasts MF-TPU and DL-TPU 3D reconstructions after unwrapping.The reconstructions are shown in panels (c)–(d).Source: Figure 3 · p. 6

2) Low fringe modulation:

DL-TPU produced smoother results than MF-TPU in a low-modulation region and reduced unwrapping errors under gamma distortion. The gamma-distortion tests covered several high fringe frequencies and degrees of distortion.

2) Low fringe modulation:

3) Intensity nonlinearity:

  • DL-TPU produced a close-to-ideal result at γ = 0.8, while MF-TPU failed even under low-level gamma distortion.At γ = 0.5 and 1.5, both methods produced meaningful results, with success/error rates about fifty-fifty.Source: 3) Intensity nonlinearity · p. 7
  • The gamma-distortion test used an industrial metal workpiece and projector-response values from 0.5 to 1.5.The authors report reduced unwrapping errors at high frequencies 32, 48, and 64.Source: 3) Intensity nonlinearity · p. 7

C. Application to high-speed 3D surface imaging

DL-TPU was tested on translating and rotating objects in a 25 Hz projection-and-capture system. The authors report reliable reconstructions and less motion-artifact sensitivity than conventional approaches.

C. Application to high-speed 3D surface imaging

  • For fast translation, DL-TPU had the highest unwrapping reliability and best noise-robustness among the compared methods.The comparison included MF-TPU and MF-TPU (3f); the latter used three fringe-pattern sets to stabilize high-frequency unwrapping.Source: C. Application to high-speed 3D… · p. 7
  • The ceramic plate moving at 1.25 cm/s and the statue rotating at 3 rpm were reconstructed with high quality.The reported projection-and-capture rate was 25 Hz.Source: C. Application to high-speed 3D… · p. 7
  • For rapid rotation, the experiment yielded results similar to those for fast translation.The statue of David rotated counter-clockwise at 3 rpm.Source: C. Application to high-speed 3D… · p. 7

IV. CONCLUSIONS AND DISCUSSION

The paper presents DL-TPU as a neural-network approach that unwraps high-frequency phase from two wrapped phases. It reports reliable unwrapping at 64 periods and improved performance across error conditions and dynamic scenes.

IV. CONCLUSIONS AND DISCUSSION

  • DL-TPU combines spatial and temporal phase information from two wrapped phases with different frequencies to recover absolute phase.The network is trained to unwrap the high-frequency phase from the unit-frequency and high-frequency inputs.Source: §IV · pp. 7–8
  • 64-period phase from 3-step patterns was directly and reliably unwrapped from one unit-frequency phase using six projected patterns.The authors report this without prior information or geometric constraints.Source: §IV · p. 8
  • Experiments found improved unwrapping reliability under intensity noise, low fringe modulation, and intensity nonlinearity, and less sensitivity to motion artifacts in dynamic measurement.The authors report successful reconstructions of moving and rotating objects at high speed.Source: §IV · p. 8
  • The authors suggest that the deep-learning framework may also apply to stereo vision, DIC, and other 3D surface-imaging modalities.These are proposed potential applications beyond the paper’s main focus on TPU and FPP.Source: §IV · pp. 8–9
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