Gray-Code-Inspired Measurement Strategy

• The paper introduces a Gray-code-inspired approach that encodes spatial data via binary masks to achieve real-time, high-accuracy 3D depth measurement.
• It integrates event cameras and tripartite phase unwrapping to mitigate noise and jump errors, enhancing robustness in dynamic or low-SNR environments.
• The method dramatically reduces acquisition time using logarithmic pattern scaling, achieving up to 57× speed-up and sub-0.1 mm geometric accuracy.

A Gray-Code-Inspired Measurement Strategy is a structured light computational approach that leverages the robust properties of Gray code, a binary sequence wherein consecutive values differ by only one bit, to encode spatial or phase information for high-speed, high-accuracy three-dimensional shape and depth measurement. Gray code’s single-bit transition principle mitigates jump errors and improves noise resilience, particularly when combined with techniques like tripartite phase unwrapping and time-overlapping pattern projection. This strategy has been further advanced by its integration with event cameras, which respond asynchronously to intensity changes. Such systems attain ultra-fast, real-time, and high-precision dense depth reconstruction while fully utilizing optical and sensor bandwidth.

1. Gray-Code Pattern Generation and Spatial Encoding

Gray code is utilized to systematically encode stripe or disparity information for structured light systems. For a projector with horizontal resolution $C$, the number of Gray-code bits is chosen as $N = \lceil \log_2 C \rceil$. The $i$-th stripe’s Gray code is generated as $g(i) = i \oplus (i \gg 1)$, where $\oplus$ is a bitwise XOR and $\gg 1$ is a right-shift by one bit. For each bit-plane $k = 0\ldots N-1$, an instantaneous binary mask $P_k(u,v)$ is formed by extracting the $k$-th bit of $g(u)$, with projector coordinates $(u,v)$ spanning the hardware’s spatial domain (Lu et al., 2024).

Projection is performed at the maximum bit-rate of the DLP system, with typical one-bit pattern durations $\Delta T_{\mathrm{proj}}$ (e.g., $402\,\mu$s per bit) and short dark intervals $\delta T$ separating patterns, resulting in a total cycle time $$T_{\mathrm{cycle}} = N(\Delta T_{\mathrm{proj}} + \delta T)$$.

For fringe projection profilometry, Gray-code patterns can be dithered and slightly defocused to yield pseudo-sinusoidal fringes, maximizing high-speed true-tone projection rates (up to $\sim2$ kHz) (Wu et al., 2020).

2. Event Camera Model and Timestamp-Noise Immunity

When an event camera is integrated, each optical event is described by $e = (x,y,\tau,p)$, where $(x,y)$ are sensor pixel coordinates, $\tau = t_\mathrm{true} + \sigma$ is the timestamp with noise $\sigma$ (latency/jitter of a few $\mu$s), and $p = \pm1$ reflects event polarity. A logarithmic intensity change is thresholded:

• $p = +1$ if $\log[I(t)/I(t-\Delta t)] > C_\mathrm{pos}$
• $p = -1$ if $\log[I(t)/I(t-\Delta t)] < C_\mathrm{neg}$

Decoding does not depend on precise timestamp matching, but instead segments event streams by dark intervals (no-event windows) between projected patterns. This segmentation method renders Gray-code event-based decoding effectively immune to timestamp noise (Lu et al., 2024).

3. Pixelwise Gray-Code Decoding and Depth Retrieval

A continuous event stream is sorted by recorded timestamps to identify dark intervals, segmenting the data into distinct bit-plane slices. For each slice and camera pixel $(x,y)$, binarization is achieved by:

$$b_k(x,y) = \begin{cases} 1 & \text{if}~\sum_{e \in E_k(x,y)} [p_e = +1] \geq 1 \ 0 & \text{otherwise} \end{cases}$$

where $E_k(x,y)$ denotes all events at pixel $(x,y)$ for bit $k$. The $N$-bit Gray code vector $G(x,y) = [b_{N-1},...,b_0]$ is constructed and decoded back to binary via the reverse mapping:

• $B_{N-1} = b_{N-1}$
• For $j = N-2$ down to $0$, $B_j = B_{j+1} \oplus b_j$
• $i(x,y) = \sum_{j=0}^{N-1} B_j \cdot 2^j$

This produces the projector coordinate $u = i(x,y)$ at each pixel. After camera and projector calibration—typically by the Zhang method and multi-plane fitting (Wu et al., 2020)—the disparity $d = x_{\text{cr}} - x_{\text{pr}}$ is computed using rectified camera pixel $x_{\text{cr}}$ and decoded projector coordinate $x_{\text{pr}}$. Metric depth follows the pinhole triangulation formula:

$$Z(x_{\mathrm{cr}}, y_{\mathrm{cr}}) = \frac{f \cdot b}{d}$$

where $f$ is camera focal length (pixels), $b$ is the baseline (mm).

4. Advanced Phase Unwrapping and Time-Overlapping Strategies

In dynamic or low-SNR settings, robust 3D reconstruction requires avoidance of jump errors at Gray-code transitions. The tripartite phase unwrapping (Tri-PU) method (Wu et al., 2020) projects three sinusoidal phase-shifted patterns (offsets $0, 2\pi/3, 4\pi/3$), generating wrapped phases $\phi_1$, $\phi_2$, $\phi_3$ at each pixel via permutations of the intensity triple $(I_1, I_2, I_3)$. Gray-code order extraction binarizes each bit, sums bits to form the decimal codeword $V(x,y)$ and looks up the stripe index $k(x,y)$. Regional partitioning uses reference phase measurements to segment each stripe into $k_{\text{low}}$, $k_{\text{mid}}$, and $k_{\text{high}}$ bands, each assigned a safe wrapped phase, thereby eliminating jump errors.

Time-overlapping Gray-code strategies further enhance efficiency: classical methods require $3 + N$ patterns/frame (three phase-shifts plus $N$ Gray codes); interleaved schemes reduce this to four (three phase-shifts plus one Gray pattern), maintaining unambiguous absolute phase recovery with fewer acquisitions (e.g., 75% coding efficiency gain for $N=4$), without loss from motion-induced shifts (Wu et al., 2020).

5. Bandwidth Utilization, Redundancy Reduction, and Efficiency

Gray-code structured light encoding offers substantial reductions in data acquisition time and storage by virtue of logarithmic pattern scaling. Whereas conventional point-scanning systems need $C$ projections, Gray code requires only $N \approx \log_2 C$, for a speed-up factor $\approx C/\log_2 C$ (e.g., $512/9 \approx 57\times$ fewer patterns for $C=512$). The binary nature of both event data and Gray-code representations maximizes information throughput per pixel ($\log_2 C$ bits from $N$ projections, averaging $1$ bit/pattern) and saturates sensor bandwidth only at bit transitions. This enables real-time operation at full sensor rates with minimal redundancy (Lu et al., 2024).

6. Performance Metrics and Practical Demonstrations

Key metrics for Gray-code-inspired measurement strategies include root-mean-square error (RMSE) of depth,

$$\mathrm{RMSE} = \sqrt{ \frac{1}{M} \sum_{i=1}^{M} \left[ Z_{\mathrm{est}}(i) - Z_{\mathrm{ref}}(i) \right]^2 }$$

and fill-rate (FR) defined as the fraction of valid SGE pixels with error below threshold $\epsilon_{\text{depth}}$ (typically $1\%$ of mean scene depth), compared to classical systems.

Reported implementations (Lu et al., 2024) can achieve accuracy indistinguishable from state-of-the-art point-scanning while attaining 41× higher data-acquisition speed (up to $2487\,\mathrm{Hz}$ depth maps). The total speed-up is a product of projection reduction, CPU lookup (GX-map: $\approx1000\times$ faster than epipolar search), and time-overlapping decoding ($N\times$ frame rate). In empirical studies (Wu et al., 2020), the Gray+Tri-PU method registered 0% unwrapping/jump errors under strong defocus and motion, outperforming two-frequency and two-wavelength phase schemes, which exhibited $\sim1.5 - 2\%$ jump errors. Achievable geometric accuracy is sub-0.1~mm RMS over arbitrary shaped objects, with millimeter-level repeatability.

7. Implementation Pipeline and Application Scope

The Gray-Code-Inspired Measurement Strategy proceeds as follows (Lu et al., 2024, Wu et al., 2020):

1. Calibrate camera and projector, obtain intrinsic/extrinsic parameters and rectification.
2. Precompute the Gray→binary lookup map (GX-map).
3. Project $N$ Gray-code bit-planes in a sequence, with dark intervals.
4. Continuously record event data, segment into binary slices via dark gaps.
5. For each pixel, assemble the Gray codeword, decode to stripe or disparity index.
6. Use phase unwrapping (Tri-PU) as needed for high-noise/motion scenes.
7. Compute disparity and map to metric 3D via the triangulation or phase-to-height model.
8. In dynamic scenarios, implement time-overlapping decoding to exploit hardware and scene temporal redundancy.

This framework is applicable to dense depth sensing, high-frequency dynamic 3D profiling, industrial inspection, biometric capture, and robotics. Its demonstrated robustness against noise, defocus, and motion-induced artifacts, plus its efficiency, position it as a foundational paradigm for next-generation real-time structured light systems.