5D VL53L1 ToF Array

• The paper presents a novel histogram-less LiDAR acquisition method using per-pixel SPAD linearization to eliminate dead-time effects.
• The design integrates a 5×5 pixel array that achieves high-flux immunity, MHz-scale acquisition rates, and significant memory savings compared to traditional methods.
• The approach details methodologies for ToF estimation via averaged timestamps, precise calibration, and scalable system integration for real-time 3D imaging.

A 5D VL53L1 ToF Array denotes a 5×5 pixel tile utilizing VL53L1-style direct-Time-of-Flight (d-ToF) acquisition, with complete per-pixel implementation of histogram-less LiDAR through SPAD response linearization. This design eliminates the need for per-pixel timestamp histograms, instead emulating a dead-time-free SPAD and extracting ToF by averaging photon arrival timestamps. The resulting system realizes high-flux immunity, MHz-scale acquisition rates, and substantial memory savings, facilitating robust, scalable, and low-latency 3D imaging architectures for LiDAR applications (Tontini et al., 2023).

1. Acquisition Architecture and SPAD Linearization

Traditional d-ToF arrays accumulate histograms of first-photon timestamps over numerous laser cycles, consuming $O(\text{\#bins} \times \text{bit-depth})$ memory per pixel and exhibiting significant nonlinear pile-up above approximately a 5% detection rate. The histogram-less method instead emulates a dead-time-free SPAD where photon arrivals are superposed linearly.

Per-pixel acquisition comprises two phases:

• Phase 1 (Background): With the laser OFF, repeatedly arm the SPAD/TDC, record first-photon timestamps $t_i^{(\text{bg})} \in [0, T_{acq}]$, increment $N_{bg}$ and accumulate $S_{bg}$ until reaching a design target or time-out.
• Phase 2 (Signal + Background): With the laser ON, identically arm and record timestamps $t_j^{(\text{tot})}$, increment $N_{tot}$ and accumulate $S_{tot}$.

Real SPADs introduce a dead-time $\tau$ after each detection, which distorts timestamp statistics. Two schemes rectify this:

• Acquire-or-Discard: Track the latest detection timestamp, discard subsequent timestamps within the same cycle that precede it, yielding an ordered sample from a dead-time-free Poisson process.
• Time-Gated Acquisition: After each detection, re-arm the SPAD only after a programmable delay matching the last timestamp, ensuring accepted timestamps are monotonically ordered.

Both mechanisms yield timestamp sets statistically identical to those produced by a linear (dead-time-free) SPAD.

The photon flux $N(t)$ is modeled as an inhomogeneous Poisson process with instantaneous rate:

$$\lambda(t) = \begin{cases} \lambda_B, & 0 \leq t < \text{ToF} \ \lambda_B + \lambda_S(t), & \text{ToF} \leq t \leq \text{ToF} + T_W \ \lambda_B, & \text{ToF} + T_W < t \leq T_{acq} \end{cases}$$where$\lambda_B$denotes background rate,$\lambda_S(t)$the laser-echo envelope, ToF the time-of-flight, and$T_W$ the pulse window.

2. Time-of-Flight Estimation via Averaging

ToF estimation exploits averaged timestamps, eschewing histogram fitting. Let

• $$\alpha := \Lambda_{bg} / \Lambda_{tot} = N_{bg}/N_{tot}$$
• $\overline{t}_{tot} = S_{tot}/N_{tot}$, $\overline{t}_{bg} = S_{bg}/N_{bg}$

From the analytic model, the mean timestamp $\mu$ obeys:

$$\mu = \alpha \cdot E_{\text{bg}}[T] + (1-\alpha) \cdot (\text{ToF} + \overline{t}_l)$$

where $\overline{t}_l$ is the mean intrinsic laser-echo delay within the window.

Solving for ToF yields (see Eq. (estimate1) in the paper):

$$\widehat{\text{ToF}} = \frac{1}{1-\hat{\alpha}}\left(\overline{t}_{tot} - \hat{\alpha} \, \overline{t}_{bg}\right) - \overline{t}_l = \frac{N_{tot}\,\overline{t}_{tot} - N_{bg}\,\overline{t}_{bg}}{N_{tot}-N_{bg}} - \overline{t}_l$$

Empirically, $\overline{t}_{bg}$ is often set to $T_{acq}/2$ for uniform backgrounds, further simplifying computation.

Shot-noise dominates the estimator variance for large $N_{tot}$ and $N_{bg}$, yielding:

$$\mathrm{Var}[\widehat{\text{ToF}}] \approx \frac{1}{(1-\alpha)^2}\left(\frac{\sigma_{tot}^2}{N_{tot}} + \alpha^2\frac{\sigma_{bg}^2}{N_{bg}}\right)$$

with closed-form results for $\sigma_{tot}^2$ and $\sigma_{bg}^2$, particularly under uniform-background assumptions.

3. Pixel Array Design and Data Pipeline

A 5×5 VL53L1-style array integrates 25 pixel tiles. Each pixel incorporates:

• A SPAD photodetector
• A time-to-digital converter (TDC) with ≃100 ps LSB
• Two 16-bit counters ($N_{bg}$, $N_{tot}$)
• One 24-bit accumulator ($S_{tot}$, optionally $S_{bg}$)

Array-level readout leverages row aggregators (5 rows), each aggregating 5 pixel registers post-frame. Global aggregation collects and transmits row data via SPI/I²C in a single burst to the host.

Simplified block diagram:

Component	Role	Scale
[SPAD, TDC]	Timestamp acquisition	per pixel (25×)
Gate Logic	Emulate dead-time-free response	per pixel
Row Aggregator	Offload registers to host bus	per row (5×)
Global Aggregator	SPI/I²C burst to host	system-wide

Pipeline timing consists of host-initiated frame start/reset, phase 1 background acquisition, phase 2 signal+background, and daisy-chained register readout. ToF is computed by firmware or FPGA per Eq. (3).

4. Synchronization, Calibration, and System Integration

Precision synchronization is achieved via a distributed common clock (≥1 MHz ARM pulse) for all pixels. Per-pixel calibration involves:

• Offset calibration: Illuminate with near-zero-range reflector (ToF ≈ 0) to measure $\overline{t}_l$ and pixel-specific TDC offsets, entered into a lookup table.
• Intensity calibration: Measure $N_{bg}$ versus ambient lux with the laser OFF to linearize $\lambda_B(\text{lux})$.

This design supports easy mosaicking in custom VL53L1-block arrangements or host-driven multi-module arrays, with straightforward scaling to higher pixel counts (e.g., 32×32).

5. Experimental Performance Metrics

Single pixel validation reveals:

• Range: measured up to 3.8 m (optics-limited)
• Accuracy: <±0.5% in no-background, ±2% at 15 klux (7.7×10⁶ evt/s), ±6.7% at 75 klux (1.2×10⁸ evt/s); precision 0.25–21% depending on flux
• Pile-up: estimator remains effective up to 90% detection rate (18× above typical 5% histogram pile-up threshold)
• Memory: ~4 Bytes/pixel vs ~8 kB for histogram schemes (≃2000× reduction)
• Sustained flux: limited by TDC, tolerant to ≈1.5×10⁹ evt/s at a 100 ps LSB

For a 5×5 array, predicted performance includes:

• Maximum per-pixel flux: ≃1×10⁹ evt/s (≈3000× above typical pile-up rule)
• Ambient-light resilience: functional up to 85 klux with ≤±2% range bias
• Depth precision: shot-noise-limited, $\sigma_{ToF} \approx T_W/\sqrt{12 N_{sig}}$; for $N_{sig} \sim 10^3$–$10^4$, jitter falls below 1 cm (few-mm depth)
• Memory: ~100 Bytes/frame vs ~200 kB for histogram approaches
• Frame rate: $N_{acq} \approx 3 \times 10^4$ ARM pulses in 33 ms yields 0.9 MHz/pixel, easily distributed via a single FPGA (22.5 MHz total ARM rate)

6. Significance and Implications

The histogram-less ToF acquisition method, utilizing SPAD response linearization and mean-based timestamp estimation, directly addresses traditional d-ToF limitations in memory bandwidth, pile-up nonlinearity, and ambient-light resilience. The resulting rackable, low-power, and low-bandwidth LiDAR array is capable of robust 30 FPS 3D imaging to ~4 m, maintaining operational fidelity in high ambient and high-flux conditions (Tontini et al., 2023). This suggests strong applicability for scalable, real-time 3D sensing in both embedded and distributed sensor systems.

A plausible implication is that the architectural simplicity and scalability of the 5D VL53L1 ToF Array enable rapid deployment in larger arrays (e.g., 32×32), with per-pixel dead-time artifacts fully suppressed and ToF estimation consistently reliable under extreme conditions. The overall approach unifies low-latency, shot-noise-limited ranging performance with minimized digital resource overhead, providing a core design reference for academic and industrial LiDAR development.