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Memory- and Bandwidth-Efficient SPAD-LiDAR Ranging via Coarse-to-Fine Spline Sketching

Zhenya Zang, Istvan Gyongy, Mike Davies

arXiv:2609.35126v2eess.IVeess.SP

TL;DR

SPAD-LiDAR data throughput grows quadratically with lateral resolution, creating a bandwidth bottleneck. The paper introduces coarse-to-fine spline sketches that allocate finer temporal encoding around a likely peak, then evaluates the approach in single- and two-peak settings. In synthetic single-peak tests, fine sketches track ground truth more closely than coarse sketches at M = 4; the conclusion also identifies stronger-scattering media as a boundary for further optimization.

  • Problem

    SPAD-LiDAR data throughput theoretically grows quadratically with lateral resolution, contributing to a high-data-rate bottleneck.

    Source: §I · p. 1

  • Method

    CFSS first uses coarse sketches to localize likely peaks, then reallocates fine sketches within selected regions; its two-peak strategy uses shared or independently allocated fine sketches depending on peak locations.

    Source: §I · pp. 1–2, B. Coarse Sketch (CS) · p. 2

  • Results

    At M = 4, fine-sketch estimates track ground truth more closely than coarse-sketch estimates, while no substantial difference is depicted for M > 4.

    Source: §III · p. 5

  • Takeaways & Limitations

    CFSS is designed to improve temporal resolution without increasing per-photon arithmetic complexity and is evaluated on single- and two-peak returns.

    Source: §I · p. 2, §V · p. 9

    Stronger-scattering media that broaden the echo pulse may fall outside the performance regime considered, motivating further optimization.

    Source: §V · p. 9

Abstract

from arXiv · show

This work presents an optimized compression framework for direct time-of-flight (dToF) light detection and ranging (LiDAR), using an accurate, compact timestamp-encoding strategy to address the high data-rate bottleneck from single-photon timing information in single-photon avalanche diode (SPAD) arrays. We propose a hardware-friendly timestamp-to-depth framework in which a sparse single-photon encoding strategy, namely coarse-to-fine spline sketches (CFSS), projects photon timestamps into fine-grained, low-dimensional representations, called sketch values. CFSS converts timestamps on-the-fly to sketch values and accumulates them with lower memory consumption, without constructing histograms, thereby saving on-chip/device memory. CFSS's closed-form solution for ToF retrieval ensures iteration-free reconstruction. Compared with the conventional sketched-LiDAR framework, the proposed method retains the same compression ratio in the single-peak case, ranging from hundreds$\times$ to thousands$\times$ depending on the accuracy-complexity trade-off, while improving depth-estimation accuracy. Both synthetic and real datasets are used to validate the performance. The CFSS framework is further extended to two-peak scenarios, enabling recovery of objects obscured by partially occluding camouflage and uniformly scattering semi-transparent media. Firmware implementation is also investigated by clearly separating online hardware processing from offline software workloads.

I. INTRODUCTION

SPAD-LiDAR throughput grows quadratically with lateral resolution, while prior compression approaches include histogram-based, learned, and histogram-free methods. This work introduces coarse-to-fine spline sketches for finer temporal encoding and extends the framework to two-peak returns.

I. INTRODUCTION

  • SPAD-LiDAR data throughput theoretically increases quadratically with lateral resolution, creating a high-data-rate bottleneck.The passage also identifies LiDAR as the largest bandwidth share in multi-sensor and single-photon LiDAR fusion systems.Source: §I · p. 1
  • Prior approaches include partial histogramming, deep-learning compression, and histogram-free sketches; histogram-based methods report approximately 10–100× bandwidth reduction and primarily target single-depth estimation.The cited deep-learning approaches are described as setup-dependent and limited in mathematical interpretability.Source: §I · p. 1
  • Table I lists coarse- and fine-sketch variables, while Table II presents the three-scenario closed-form depth estimator used in both schemes.The estimator is attributed to [18].Source: Table I · p. 2, Table II · p. 2
  • CFSS first localizes a peak with coarse sketches, then applies fine sketches within that region for finer timing without increasing per-photon arithmetic complexity.Compared with the original coarse-sketch version, this adds modest LUT/BRAM storage on the firmware.Source: §I · pp. 1–2
  • For two peaks, CFSS uses shared coarse sketches when both fall in one region and independently allocated fine sketches when they fall in different regions.The workload is divided between runtime firmware and post-processing software.Source: §I · p. 2
  • Evaluation uses synthetic data across SBR and IRF FWHM conditions and real single- and two-peak data from partially occluding and semitransparent scenes.The real-data evaluations vary photon-counting levels and the number of sketch coefficients.Source: §I · p. 2

A. Observation Model

The observation model treats a SPAD's photon timestamps as samples from a return signal plus background. It estimates the target time of flight from those timestamps under an idealized detector model.

A. Observation Model

  • A SPAD records n photon timestamps in [0, T) over N laser pulses, with T denoting the number of time bins.The single-surface target has depth d* and ToF t* = 2d*/c.Source: A. Observation Model · p. 2
  • The echo is modeled as a target-delay impulse convolved with the system IRF, while the detector model neglects effects such as dead time and pile-up.The IRF is modeled as Gaussian with a specified FWHM.Source: A. Observation Model · p. 2
  • The objective is to estimate the target ToF t* from the observed timestamps.Source: A. Observation Model · p. 2

B. Coarse Sketch (CS)

Coarse sketches (CS) cover the measurement range to localize a likely return; fine sketches then concentrate representation capacity within the selected region. The closed-form decoder estimates ToF from the sketch, and each exposure allocates photons between the two stages.

B. Coarse Sketch (CS)

  • CS projects timestamps into M linear-spline coefficients over the full measurement range, providing a low-dimensional representation without constructing histograms.The knots are equispaced with spacing Δ = T/M.Source: B. Coarse Sketch (CS) · p. 2
  • The closed-form decoder finds the largest sketch coefficient, estimates the signal fraction, and selects among three candidate ToFs to reconstruct depth.CFSS uses a CS acquisition to locate the approximate peak, then an FS acquisition to refine the estimate within the same exposure.Source: B. Coarse Sketch (CS) · p. 2
  • For the illustrated example, M = 8 sketches cover T = 4096 time bins with 500 photons and an IRF FWHM of 1 ns.The caption reports an 80 ps histogram bin width and SBR = 5.Source: Figure 1 · p. 3
  • The photon budget is split between CS and FS, with ρ denoting the fraction assigned to CS; experiments use ρ = 0.1.That setting allocates 10% of photons to CS and 90% to FS.Source: B. Coarse Sketch (CS) · p. 3

C. Fine Sketch (FS)

Fine sketches (FS) reallocate the same number of bases into a local window centered on the coarse estimate, increasing temporal resolution. Their placement and reconstruction depend on the window-local spacing and IRF width.

C. Fine Sketch (FS)

  • The FS window is centered on the winning CS basis peak and has width W = 2Δ, covering the full support of that basis.If the window crosses the measurement boundary, it wraps between T and 0.Source: C. Fine Sketch (FS) · p. 3
  • FS reallocates M bases within the local window at spacing ˜Δ = W/M, and processes only timestamps that fall inside that window.The in-window background density is 1/W, and its signal fraction is estimated using the CS background-bin construction.Source: C. Fine Sketch (FS) · p. 3
  • Because FS processes only the W/T fraction of background, it increases in-window SBR and sketch-domain resolution.The closed-form reconstruction applies with the fine spacing and local knots, while the decoder returns estimates in absolute coordinates using the CS offset.Source: C. Fine Sketch (FS) · p. 3
  • FS placement requires FWHM ≤ ˜Δ = whT/M², so the allowable sketch count becomes more restrictive as M grows.The stated criterion is FWHM × M²/(whT) ≲ 1.Source: C. Fine Sketch (FS) · p. 3

D. Hardware Consideration

The hardware design uses precomputed lookup tables to convert timestamps into sketch coefficients, while runtime processing selects the local window and accumulates the sketches. The two-peak workflow also allocates separate or shared fine windows according to peak placement.

D. Hardware Consideration

  • Precomputed CS and FS lookup tables map timestamp-derived addresses to sketch coefficients, avoiding explicit on-chip computation of the sketch function.The example uses 0.25 KB for CS and 0.5 KB total with FS; storage scales proportionally with M.Source: D. Hardware Consideration · p. 3
  • The two-peak illustration contrasts independent fine windows for returns in separate CS regions with one shared window when both returns occupy a coarse region.Its panels show photon histograms, reallocated FS bases, and accumulated coefficients with a detection threshold.Source: Figure 2 · p. 4, D. Hardware Consideration · p. 4
  • Table III is identified as a hardware–software workload partition for FS processing and labels the decoder as the three-scenario closed-form (CF) decoder.Source: Table III · p. 4

E. Two Peak Scenario

The two-peak extension addresses returns from partially occluding or semi-transparent objects and models two signal components alongside background. It handles returns in distinct coarse regions independently, while a shared-window procedure estimates and subtracts the dominant return to recover a second close peak.

E. Two Peak Scenario

  • The two-return model represents each return with its own ToF and received strength, plus a background component; synthetic experiments assume unequal return amplitudes.Source: E. Two Peak Scenario · p. 4
  • To search for a second peak, the method masks the dominant CS peak's neighborhood to exclude nearby coefficients that may reflect its leakage.The overall method imposes no minimum separation: distinct CS regions handle sufficiently separated returns, while closer returns use shared-window FS.Source: E. Two Peak Scenario · p. 4
  • The second peak is accepted only when its unmasked sketch coefficient passes a background-based threshold.The threshold factor is adapted to first-order sketches because each photon spreads across two adjacent coefficients, whose background fluctuations are smaller than in histogram bins.Source: E. Two Peak Scenario · p. 5
  • When a second return is detected in a different CS region, the method allocates two independent fine windows and applies the CF estimator to each.The second-window offset is added to its estimate while retaining M sketches per peak.Source: E. Two Peak Scenario · p. 5
  • For two returns in one coarse region, a shared FS window estimates the dominant return, subtracts its sketch, and applies the estimator to the residual.If the residual amplitude falls below threshold, the measurement remains a single-peak estimate.Source: E. Two Peak Scenario · p. 5

III. SYNTHETIC DATASETS EVALUATION

The synthetic evaluations show that finer first-order sketches improve single-peak depth estimation over coarse sketches, while two-peak accuracy depends strongly on return separation and photon conditions. Across tested settings, the second return is harder to estimate and detect than the first.

Single-peak accuracy

  • At M = 4, FS tracks the true depth more closely than CS, while no substantial difference is depicted for M > 4.The synthetic comparison uses 500 photons per trial and averages 50 trials at each of 200 true-depth positions.Source: §III · p. 5, Figure 3 · p. 5
  • Up to ∼10× lower median RMSE over CS at M ≥8, with FS approaching its CRB; the FS CRB is roughly an order of magnitude below the CS CRB for all M.The gain is attributed to finer FS window spacing, which increases Fisher information about the delay.Source: §III · p. 5

Two-peak accuracy

  • Peak-2 RMSE is 2–3× higher than peak-1, and reaching a comparable operating point requires roughly 3× the photon budget and SBR.Within the tested range, accuracy is more affected by photon count and SBR than by IRF width.Source: §III · p. 6
  • For peak-2, FS achieves approximately 1–2 cm RMSE across tested depths at M ≥8; reliable detection requires M ≥8.At M = 4, the disjoint-CS branch is not invoked and peak-2 is estimated from a noise-dominated residual.Source: §III · p. 6

Real-data reference point

  • On the polystyrene-head dataset, the face geometry is recovered at M = 8 and visually matches the matched-filter baseline by M = 16 for both sketch orders.The caption reports compression ratios from ≈1153× at M = 4 to ≈144× at M = 32.Source: Figure 8 · p. 7

Two-peak separability

  • Same-CS two-peak separability remains above 95% down to a separation of ∼7 histogram bins; below that, the peaks can no longer be distinguished.The separability test uses M = 8 with peak-1 centered in its coarse basis.Source: Figure 7 · p. 6, §III · p. 7

IV. REAL DATASETS EVALUATION

Real-data tests show that CFSS sketches recover single- and two-return scene geometry at compact coefficient counts. Reconstruction improves with more coefficients, with the second return revealing surfaces behind camouflage or scattering layers.

Polystyrene-head scene

  • On the polystyrene-head reconstruction, first-order FS at M=4 recovers a more coherent face surface with only minor residual deviations from the matched-filter reference; from M=8 to M=32, both passes are visually comparable to MF.Source: §IV · p. 7

Camouflage scene

  • As M increases from 4 to 16, both sketch orders recover smoother, more consistent camouflage-scene surfaces; peak-1 captures the nearer layer and peak-2 reveals the target behind it.At M = 4, the zero-order sketch has layered artifacts and a sparser peak-2 cloud, while first-order produces a denser, more continuous cloud.Source: §IV · p. 8, Figure 10 · p. 8

Scattering scene

  • In the low-count scattering scene, M = 8 clarifies the mannequin outline in peak-2, and M = 16 sharpens its facial geometry while peak-1 isolates foreground scatterers.The full-histogram reference resolves a near wall, mannequin, and rear wall as three depth layers.Source: §IV · p. 9, Figure 11 · p. 9

V. CONCLUSION

The conclusion presents CFSS as a hardware-conscious refinement of sketched LiDAR for single- and two-peak ToF estimation, while identifying stronger-scattering media as a boundary for future work.

Conclusion

  • CFSS uses finer sketch intervals to alleviate the SPAD data-rate bottleneck and improve temporal resolution without increasing per-photon arithmetic complexity.The work evaluates single- and two-peak solutions on synthetic and real datasets and leaves stronger-scattering media for future optimization.Source: §V · p. 9
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