High-speed and high-efficiency three-dimensional shape measurement based on Gray-coded light

1 Introduction

• With projection and capture already fast enough for dynamic scenes, the introduction frames robust, efficient measurement as the remaining methodological challenge.It notes that multi-level Gray coding can improve coding efficiency but reduces signal-to-noise ratio.Source: §1 · pp. 2, 5
• Transform-based fringe methods can suit high-speed measurement, but band-pass filtering makes them difficult to use with sharp edges, abrupt changes, or non-uniform reflectivity.They also require careful parameter selection, hindering automatic processing of complex scenes with time-varying object shapes.Source: §1 · pp. 2–3
• Gray-coded approaches use N binary patterns to uniquely label 2N stripe periods and offer robustness and anti-noise ability for static-scene 3D measurement, but high-speed measurement remains challenging, with jump errors at adjacent-word boundaries and low coding.Source: §1 · p. 4
• Earlier Gray-code corrections had limitations: spatial unwrapping worked well only for simple, smooth objects, while defocus-based binarization reduced but did not eliminate jump errors.The latter required further correction, such as a median filter; ternary and quaternary coding improved efficiency at the cost of signal-to-noise ratio.Source: §1 · p. 5
• The proposed tripartite phase-unwrapping method targets jump errors at Gray-code word boundaries, mainly caused by projector defocus and object motion.It acquires three staggered wrapped phases with a 2π/3 phase difference.Source: §1 · pp. 5–6

• The tripartite unwrapping divides decoding orders into low, middle, and high regions and avoids using wrapped phase at jumps, pre-avoiding jump errors without extra projected patterns.Source: §1 · p. 6
• The techniques use every 4 projected patterns to reconstruct pixel-wise, unambiguous 3D geometry of dynamic scenes, preserve Gray-coded methods’ high anti-noise ability, and provide robust, high-efficiency measurement in high-speed scenes even under strong noise.Source: §1 · p. 6

2 Principle.

• A DLP projector displays binary pseudo-sinusoidal patterns with slight defocus, a synchronized high-speed camera captures the deformed fringes, and phase analysis plus calibration yields 3D shape.Defocusing the projection lens generates high-quality sinusoidal fringes on the object surface.Source: §2 · pp. 6–7

2.1 Integration of Phase-shifting Algorithm with Gray-coded Method.

• Combining phase-shifting with Gray coding brings phase-shifting spatial resolution together with Gray-code unambiguity and robustness, but defocus and object motion can cause boundary jump errors.Defocused Gray-code edges blur black-white transitions, while motion shifts patterns and can mismatch wrapped phase with phase order.Source: §2.1 · pp. 7, 9

2.3 Regional Division Using Reference Wrapped Phase.

• A calibrated reference wrapped phase with discontinuities aligned to the phase order supports its three-way division, while the middle region is identified from ϕ2.The reference is formed from the absolute phase of a measured reference plane determined during system calibration.Source: §2.3 · pp. 11–12
• Reference wrapped-phase values on a critical line, located where ϕ2 has minimum absolute value along y, define the threshold used to assign lower and higher regions.The threshold varies across periods because the reference plane’s absolute phase can lie outside (-π, π].Source: §2.3 · pp. 12–13
• Incorrect regional division can occur in narrow-fringe areas near object shadows and edges, so a correction algorithm is introduced to improve robustness.The division generally performs well, with errors confined to a few specific areas.Source: §2.3 · p. 13

2.4 Time-overlapping Gray-coded Coding Strategy.

• 4 projected patterns per group (3 + 1), down from 7 (3 + 4), are achieved by reusing each Gray-coded pattern four times across groups of sinusoidal patterns.The adjacent four Gray-coded patterns are used to resolve phase ambiguity for each sinusoidal-pattern group.Source: §2.4 · p. 14
• The time-overlapping strategy relies on high-speed projection and capture, under which adjacent-pattern motion disparity remains small relative to the Tri-PU method’s 2/3-period threshold.The described setup operates at speeds over several kHz; Tri-PU uses only middle phase-order regions, limiting the effect of motion-induced mismatch.Source: §2.4 · pp. 14–15

2.5 System Calibration.

• Four measured planes with known height distributions determine parameters u(x, y), v(x, y), and w(x, y) for future phase-to-height mapping.The system also uses Zhang’s camera calibration technique [55].Source: §2.5 · p. 15

3.1 Measurement on the Static Scene.

• On static scenes, Tri-PU avoided traditional Gray-code jump errors without additional projected patterns and corrected errors in shadow and edge regions.The experiment measured a CPU cooling fan and a portrait sculpture.Source: §3.1 · p. 16

3.2 Accuracy Analysis.

• Measurement accuracy was better than 0.1 mm in a 180 mm × 250 mm × 90 mm volume.Plane RMS errors were 0.0999 mm (Plane 1), 0.0685 mm (Plane 2), 0.0594 mm (Plane 3), and 0.0762 mm (Plane 4); the ceramic-ball RMS error was 0.0767 mm. The measured step-plane heights were 0.0002 mm, 30.0054 mm, 59.9867 mm, and 89.9003 mm for Planes 1–4, respectively.Source: §3.2 · p. 18, Figure 7 · p. 19

3.3.1 Collapsing Building Blocks.

• No phase-unwrapping errors occurred with the proposed method, versus 1.46% for two-frequency and 2.02% for two-wavelength methods in the low-SNR comparison.The comparison used binary defocusing; the paper reports higher anti-noise ability for the proposed method.Source: §3.3.1 · p. 20, Figure 8 · p. 21
• The proposed method’s error rate is independent of fringe frequency, and Tri-PU avoids errors at black–white code transitions in low-SNR scenes.Gray codes eliminate phase ambiguity, while tripartite phase unwrapping addresses transition-area errors.Source: §3.3.1 · p. 20
• 542 fps was reconstructed using four projected patterns per 3D frame during the dynamic collapsing-blocks measurement.The scene contained three layers of blocks torn down by hand.Source: §3.3.1 · pp. 21–22
• The proposed method achieved robust, high-efficiency measurement in the complex, dynamic collapsing-building-block scene.The experiment presents captured pattern sequences, representative collapsing scenes, and corresponding 3D frames.Source: §3.3.1 · p. 22, Figure 9 · p. 22

3.3.2 Rotating Fan Blades.

• 542 fps (2170/4) reconstruction and 60 fps replay accompanied the fan’s acceleration, with quarter-turn intervals decreasing from 265.5 to 243.3 to 206.4 ms and average speed about 64 rotations per minute.The quarter-turn intervals indicate the fan was in a speed-up phase.Source: Figure 10 · p. 23, §3.3.2 · p. 23
• Despite dense speckle causing strong noise and low-SNR fringe images, the method reconstructed fan frames and profiles showing blade-height variation over time.The profiles were sampled at 31.3 ms intervals as a blade rotated across the line.Source: §3.3.2 · p. 22, Figure 10 · p. 23

4 Discussion

• Gray-code phase-unwrapping error rate is independent of sinusoidal-fringe frequency, and noise is not amplified during unwrapping.The discussion attributes Gray coding’s stronger anti-noise ability, relative to two-frequency and two-wavelength methods, to this property.Source: §4 · pp. 23–24
• Tri-PU targets errors at black–white code transitions and is proposed as applicable to other fringe-projection techniques.It is combined with Gray-code measurement to support robust phase unwrapping in dynamic scenes.Source: §4 · p. 24
• The method reduces projected patterns from 7 to 4 in every sequence, restoring pixel-wise, unambiguous 3D geometry; separating Gray codes does not decrease phase-unwrapping success rate.Source: §4 · p. 24
• The combined method is reported to preserve Gray coding’s anti-noise ability while addressing jump errors and low coding efficiency in complex dynamic scenes with low SNR.The time-overlapping strategy is paired with high-speed hardware and Tri-PU; the reported scope includes high-speed dynamic measurement.Source: §4 · pp. 24–25

5. Conclusion

• Tri-PU uses staggered wrapped phases and regional decoding to pre-avoid code-boundary jump errors caused mainly by defocusing and motion, without an extra pattern.Three wrapped phases are acquired with 2π/3 phase difference; the method unwraps regions using different decoding orders.Source: §5 · p. 25
• The time-overlapping coding strategy increases coding efficiency, decreasing the projected number in each group from 7 to 4.Source: §5 · p. 25
• Together, the techniques restore pixel-wise, unambiguous 3D geometry in dynamic scenes with strong noise using four patterns while preserving Gray code's anti-noise ability.The jump errors are pre-avoided rather than post-eliminated without additional pattern projection.Source: §5 · p. 25
• 542 fps was achieved for high-speed 3D measurement of dynamic scenes with low SNR.The experiments report the method as high-robust and high-efficiency under this condition.Source: §5 · p. 25

Appendix

• If shadow or edge areas exceed the region where |ϕ2(x, y)| < π/3, a critical line may fall on a phase-order edge and cause incorrect regional division.Source: Appendix · p. 26
• Correction replaces critical-line points inside a phase order with the point where |ϕ2(x, y)| reaches its local minimum outside that order.Residual errors occur in edge or shade regions spanning only several pixels; median filtering or erosion can remove them at the cost of some-pixel accuracy.Source: Appendix · pp. 26–27
• The correction algorithm can avoid some measurement errors at the edge of a rotating fan.Fig. 12 compares measurement results before and after correction.Source: Figure 12 · p. 27, Appendix · p. 27