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A Few Photons Among Many: Unmixing Signal and Noise for Photon-Efficient Active Imaging

Joshua Rapp, Vivek K Goyal

arXiv:1609.07407v1stat.APphysics.ins-det

TL;DR

Photon-efficient imaging must separate signal from substantial background noise without relying on longer acquisition or higher illumination power. The paper uses adaptive temporal windowing and spatially adaptive aggregation to estimate depth and reflectivity, demonstrating effective imaging at SBR = 0.04 and improved performance over prior methods at low SBR.

  • Problem

    Photon-efficient imaging methods need accurate depth and reflectivity estimates when background noise substantially exceeds signal, including noise levels 25 times higher than signal.

  • Method

    The method approximately separates signal and noise using adaptive short-duration range windows, then aggregates detections within spatially adaptive superpixels when too few signal detections are available.

  • Results

    SBR = 0.04: the proposed method is considerably more effective than prior photon-efficient imaging at handling high background, with improved reflectivity and depth estimation performance as SBR decreases.

  • Takeaways & Limitations

    The results support photon-efficient LIDAR imaging for rapid depth acquisition under high ambient light without increasing acquisition time or illumination power.

  • Takeaways & Limitations

    Windowing alone can leave too many pixels without depth estimates when signal detections are sparse, especially in low-SBR data.

Abstract

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Conventional LIDAR systems require hundreds or thousands of photon detections to form accurate depth and reflectivity images. Recent photon-efficient computational imaging methods are remarkably effective with only 1.0 to 3.0 detected photons per pixel, but they are not demonstrated at signal-to-background ratio (SBR) below 1.0 because their imaging accuracies degrade significantly in the presence of high background noise. We introduce a new approach to depth and reflectivity estimation that focuses on unmixing contributions from signal and noise sources. At each pixel in an image, short-duration range gates are adaptively determined and applied to remove detections likely to be due to noise. For pixels with too few detections to perform this censoring accurately, we borrow data from neighboring pixels to improve depth estimates, where the neighborhood formation is also adaptive to scene content. Algorithm performance is demonstrated on experimental data at varying levels of noise. Results show improved performance of both reflectivity and depth estimates over state-of-the-art methods, especially at low signal-to-background ratios. In particular, accurate imaging is demonstrated with SBR as low as 0.04. This validation of a photon-efficient, noise-tolerant method demonstrates the viability of rapid, long-range, and low-power LIDAR imaging.

I. INTRODUCTION

Photon-counting LIDAR uses timed photon detections to estimate depth and return counts to estimate reflectivity. This paper targets photon-efficient imaging under substantially higher background noise by separating signal and noise contributions and using spatial adaptivity.

  • Prior photon efficiency: Photon-efficient LIDAR has formed accurate depth and reflectivity images from a single detected photon per pixel or about 1.0 photon per pixel on average.
  • Research gap: Existing photon-efficient methods degrade significantly under high background noise, motivating imaging at noise levels 25 times higher than signal.
  • Proposed direction: The proposed approach approximately unmixed signal and noise using detection times, then applied spatial adaptivity when too few signal photons made pixelwise depth estimates unreliable.
  • Measurement model: The modeled system raster-scans scene patches with a pulsed laser and records picosecond-resolution photon timestamps, while detector counts can include signal, ambient, and dark-count events.
  • Measurement model: SBR is defined as mean backreflected-light detections divided by mean ambient-light and dark-count detections, excluding Poisson variation in signal counts from noise.
  • Measurement setup: Photon detections correlated with laser pulses provide time-of-flight information, while return counts indicate scene reflectivity.

C. Binomial vs. Poisson Modeling

The binomial and Poisson formulations are approximately equivalent in the low-flux regime, but the Poisson formulation preserves detection-time information for separating signal and background.

  • Binomial model: At most one detection is recorded per pulse-repetition period, motivating a Bernoulli model per period and a binomial model for total detections.
  • Model equivalence: Under low flux, the binomial and Poisson count models are approximately equal because detector dead-time approximations introduce negligible error.
  • Model choice: The Poisson model retains detection times and separates total counts into signal and background constituents, improving reflectivity estimation from signal counts at low SBR.

D. Parameter Estimation

Parameter estimation becomes simpler when noise is removed, but the binomial reflectivity estimator becomes unreliable when background is large relative to signal. The Poisson formulation instead uses photon times for joint estimation.

  • Reflectivity estimation: The binomial constrained maximum-likelihood reflectivity estimator counts detections across illumination intervals and subtracts an expected noise offset.
  • Reflectivity estimation: When signal remains small and background is significantly larger, increased background variance reduces the reliability of the binomial noise-count estimate.
  • Poisson estimation: The Poisson-process formulation uses detection times in reflectivity estimation, modeling the signal as the illumination pulse modulated by reflectivity.
  • Poisson estimation: The Poisson constrained reflectivity optimum is unique because the relevant likelihood expression is monotonically decreasing in reflectivity, although it requires the true depth.
  • Noise-free estimation: When noise is eliminated, reflectivity estimation reduces to a closed-form normalized photon count that does not require detection times or the true depth.

3) Depth Estimation:

Depth estimation from noisy detection times is nonconvex and depends on reflectivity, whereas removing noise yields a simpler convex depth solution independent of reflectivity.

  • Depth estimation: The noisy depth estimator requires the true reflectivity and includes a noise-induced nonconvexity; in practice, depth is found by maximizing a log-matched-filter output.
  • Depth estimation: With zero noise, the depth estimate is greatly simplified and becomes convex without dependence on reflectivity.
  • Parameter coupling: The coupling of reflectivity and depth estimates is complicated by noise but greatly simplified when noise is removed.

A. Pixelwise Unmixing

The method separates clustered signal detections from uniformly distributed background using short windows, then borrows detections from similar neighboring pixels when pixelwise evidence is insufficient.

  • Pixelwise Unmixing: Signal detections cluster near true depth, whereas homogeneous background detections are generally dispersed across time.The difference in temporal concentration motivates window-based unmixing.
  • Pixelwise Unmixing: A window of duration Twind satisfying Tp < Twind ≪ Tr captures most signal detections without accepting excessive noise.The window is shifted across detections to identify the densest local cluster.
  • Pixelwise Unmixing: The largest cluster provides depth and signal-count estimates, while a minimum cluster size allows the algorithm to withhold unreliable depth estimates.The threshold is deliberately used to produce missing estimates rather than accept an unreliable cluster.
  • Pixelwise Unmixing: Twind = 2Tp, with Tp = 270 ps, covers more than 95% of Gaussian-approximated signal-detection probability mass.The pulse repetition period used in the experiments is Tr = 100 ns.
  • Pixelwise Unmixing: The noise-based Ncl rule selects the smallest cluster size whose false acceptance probability is at most τFA.The rule uses calibrated background information because the signal rate is unknown.
  • Pixelwise Unmixing: Low-α, low-SBR scenes can leave too many pixels without depth estimates because stringent thresholds require clusters that may not exist.This limitation persists even with no background when pixels have no signal detections.
  • Pixelwise Unmixing: Similar neighboring pixels can supply detections that strengthen signal clusters, while adaptive scene grouping avoids treating object boundaries as homogeneous regions.Similarity is based on transverse position and reflectivity, which are associated with similar depth.

IV. ALGORITHM

The algorithm applies calibrated, adaptive windowing to raw photon detections, retaining a dominant cluster only when its size meets a noise-controlled acceptance criterion.

  • IV. ALGORITHM: The procedure starts from photon detection times for each patch, calibrated background counts, and an estimated unattenuated signal count.These quantities define the raw inputs and calibration assumptions.
  • IV. ALGORITHM: Adaptive windowing uses window length Twind and target false-acceptance probability τFA to censor detections likely to be background noise.The procedure initially considers each patch independently.
  • IV. ALGORITHM: For each detection, the algorithm forms a Twind interval and selects the interval containing the largest number of detections.Ties among maximally populated intervals are resolved uniformly at random.
  • IV. ALGORITHM: Ncl is computed as the smallest integer satisfying the calibrated noise-cluster probability constraint associated with τFA and the superpixel size.For single-pixel windowing, the contributing superpixel count is Nsp_i,j = 1.
  • IV. ALGORITHM: If kmax_i,j ≥ Ncl, only detections inside the selected window are retained and the remaining detections are censored.The retained set becomes the input to subsequent estimation.
  • IV. ALGORITHM: With superpixels, the algorithm repeats windowing on augmented detection vectors and adjusts Ncl and reflectivity calculations for the number of contributing pixels.Spatial borrowing therefore changes both available detections and noise-rate accounting.

B. Reflectivity Estimation

Reflectivity is estimated from detections retained within the selected window, with noise contribution modeled using the effective background rate and spatial regularization enforcing scene smoothness.

  • B. Reflectivity Estimation: The expected noise count in a superpixel window is Nsp_i,j Nr B Twind/Tr, which remains small even at low SBR.Because background detections form a homogeneous Poisson process, this expected count also estimates its variance contribution.
  • B. Reflectivity Estimation: The window-based noise estimate modifies the reflectivity estimator using the effective number of contributing pixels and the retained detections.This replaces whole-acquisition-interval counting with the shorter selected window.
  • B. Reflectivity Estimation: When no cluster meets Ncl, the temporary reflectivity estimate can be slightly high because the selected window may contain the largest noise cluster.The estimate is updated after windowing augmented superpixel data.
  • B. Reflectivity Estimation: Regularized maximum-likelihood estimation produces the reflectivity image by penalizing deviations from spatial smoothness.The penalty is controlled by βα and applied to the reflectivity field.

C. Superpixel Formation

Superpixels adaptively combine detections from nearby pixels with similar reflectivity to improve windowing where individual pixels lack reliable signal clusters, followed by depth estimation and inpainting.

  • C. Superpixel Formation: After initial windowing, only pixels with kmax_i,j ≥ Ncl are assigned reliable depth estimates.Pixels below the threshold require spatial support.
  • C. Superpixel Formation: Superpixels borrow detections from neighboring pixels within distance dsp and reflectivity tolerance τsp.Typical settings are dsp = 1, 2, or 3 and τsp is about 5% of the full reflectivity range.
  • C. Superpixel Formation: The superpixel detection set is formed from pixels satisfying the local spatial and reflectivity similarity bounds.The construction defines an augmented detection vector for the target pixel.
  • C. Superpixel Formation: Adaptive borrowing searches a small local area and selects pixels similar in both transverse position and reflectivity.The resulting superpixel has a new detection count ˜ki,j.
  • C. Superpixel Formation: Windowing and reflectivity estimation are repeated on each superpixel vector, with Ncl adjusted for its contributing pixel count.The resulting kmax_i,j is usually larger than the single-pixel value.
  • C. Superpixel Formation: The algorithm increases dsp iteratively until reliable depths are obtained or dmax_sp is reached.This balances local support against the need to preserve spatial specificity.
  • C. Superpixel Formation: Remaining pixels without reliable depth estimates are filled by inpainting during depth estimation.Depth estimation assumes retained detections are signal, with optional ROM censoring for excessive false acceptance.

V. RESULTS

Simulations evaluate reflectivity and depth estimation under severe background noise, including SBR = 0.04 with 2.0 signal photons per pixel. The unmixing method improves over the comparison method and approaches signal-only oracle performance in many cases.

  • Simulation Setup: The simulation uses Poisson-generated signal and background detections, with 1000 or 1500 illumination periods for 2.0 or 3.0 signal photons per pixel.At SBR = 0.04, the average photon detection rate is one detection in approximately 5% of illumination periods.
  • Simulated Results: At SBR = 0.04 and 2.0 signal photons per pixel, Figure 6 compares reflectivity and depth estimates for the Art and Bowling scenes.The comparison includes the proposed unmixing method, Shin et al., and an ideal signal oracle.
  • Simulated Results: The unmixing method estimates reflectivity more accurately than detection counts alone under high background levels.Superpixel formation enables precise fractional estimates of signal photon counts, especially in dark regions.
  • Simulated Results: Shin et al. depth estimates fail as noise detections pull them toward the 7.5-meter mean scene depth.Figure 6 notes that these estimates are outside the actual scene range and displays them from 6 to 8 meters.
  • Simulated Results: Windowing handles high-variance noise more effectively for the unmixing method, while its largest remaining depth errors occur in dark regions and at object boundaries.Superpixels can overcome low signal counts in dark regions, but boundary neighborhoods may contain insufficient or mixed-depth support.
  • Simulated Results: At SBR = 0.04, unmixing is more robust to decreasing SBR than Shin et al. and produces depth estimates almost as good as the signal oracle in many cases.Figure 7 averages MSE and RMSE over 10 trials for 2.0 and 3.0 signal detections per pixel across SBR values.

B. Experimental Results

On the 1000 × 1000 Mannequin dataset at SBR about 0.04, the unmixing method produced clearer reflectivity and more effective depth estimates than the comparison method, while remaining imperfect in difficult regions. The approach uses adaptive detection clustering and spatial aggregation, but computational cost and residual missing estimates remain limitations.

  • Reflectivity: At SBR about 0.04, the unmixing method produced clearer reflectivity images than, with results comparing favorably to oracle and baseline estimates.Approximate reflectivity MSE showed little distinction among methods, but produced excessive smoothing that obscured text and facial features.
  • Depth: [12] produced depth estimates completely outside the true scene range, whereas unmixing handled high background more effectively.The estimate was dominated by noise; unmixing had low absolute error in bright regions, with largest errors at object boundaries and dark regions.
  • Experimental setup: 4.05 signal detections per pixel were combined with synthetic noise to set the Mannequin data at SBR about 0.04.The baseline used conventional processing on long-acquisition data at SBR = 1, while the figure’s error metrics were approximate because baseline LIDAR data was not exact ground truth.
  • Limitations: Mannequin performance was worse than in simulations because strong reflectivity contrast made the average signal count unrepresentative of distinct regions.The large, low-reflectivity shirt region required higher dmax sp and higher average signal detection counts for improved estimates.
  • Method: Adaptive windowing limits falsely accepted background detections, while spatially adaptive superpixels aggregate detections to fill gaps with too few signal photons.The method is built around unmixing signal and background processes before forming estimates.
  • Limitations: The algorithm still requires inpainting for some pixels, and processing many large detection vectors makes iterative refinement computationally costly.The authors identify distributed or GPU-accelerated computation and improved component blocks as possible directions.

APPENDIX

The appendix approximates cluster probabilities for homogeneous Poisson noise and derives a lower bound for signal-cluster probability using a centered window around true depth. The noise approximation assumes independence among overlapping candidate windows.

  • Noise clusters: Noise detections are modeled as order statistics of independent uniform arrival times over the repetition period.After rescaling to [0, 1], the kth order statistic has beta distribution β(k, n + 1 − k).
  • Noise clusters: The gap S(k) between detections k and ℓ follows β((ℓ−k), n + 1 − (ℓ−k)), depending only on their index difference.This beta-distribution result supports the cluster-probability calculation for noise detections.
  • Noise clusters: A noise cluster requires at least Ncl detections within a window of duration Twind, and candidate windows begin at the first n − (Ncl − 1) detections.The appendix combines these candidate-window events to approximate the probability of no clusters.
  • Approximation: The noise-cluster probability approximation assumes overlapping candidate-window events are independent.The appendix states that numerical evaluations support this simplifying approximation.

Signal Clusters

For signal detections modeled with a Gaussian pulse, the appendix evaluates clusters only in a window centered at true depth and consequently derives a lower bound on cluster probability.

  • Signal clusters: The signal-cluster model assumes a Gaussian pulse and considers a cluster only when Ncl detections fall within Twind centered at true depth.Because other window positions are omitted, the resulting probability is a lower bound.
  • Signal clusters: Signal detection times shifted by true depth and scaled by pulse duration follow a standard normal distribution.The standard normal CDF Φ gives the probability that a detection lands in the centered window.
  • Signal clusters: Given m signal detections, the derivation computes the probability that exactly k fall inside the centered window and that fewer than Ncl do so.Poisson-distributed signal counts then yield a bound on the unconditional signal-cluster probability.
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