Presentation by Julián Tachella at EUSIPCO 2019, La Coruña, Spain. Paper: "Fast surface detection in single-photon Lidar waveforms"

1. Fast surface detection in
single-photon Lidar waveforms
J. Tachella1, Y. Altmann1, J.Y. Tourneret2 and S. McLaughlin1
1School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, UK
2INP-ENSEEHIT-IRIT-TeSA, University of Toulouse, Toulouse, France

2. Outline
The single-photon Lidar data 3D detection problem
• Challenges
New Bayesian detection approach
• Priors
• Computation of marginals
• Operation curves
• Spatial regularization
Experiments
• Real lidar data
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3. Single-photon Lidar
1. Time-of-flight-based imaging
raw data 3D surface
𝑧𝑡 ∼ 𝒫 𝑟ℎ 𝑡 − 𝑡0 + 𝑏
∀ 𝑡 = 1, … , 𝑇
2. Classical observation model
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4. State-of-the-art
Surfaces per pixel
≤1 𝟏 ≥1
Single-depth algorithms
Multi-depth algorithms
Most common cases in practice
Target detection +
single-depth algorithm
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5. Target detection problem
Target
𝑟 > 0
No target
𝑟 = 0
Large background
𝑟 ≪ 𝑏𝑇
Few photons
𝑟 + 𝑏𝑇 ≪ 1
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Additional challenge: unknown 𝒕 𝟎, 𝒓, 𝒃
Strong signal
𝑟 > 𝑏𝑇

6. Observation model
Classical model:
𝑧𝑡 ∼ 𝒫 𝑟ℎ 𝑡 − 𝑡0 + 𝑏
Alternative representation: Signal-to-background ratio 𝒘 = 𝒓/𝒃𝑻
𝑧𝑡 ∼ 𝒫 𝑏 𝑤𝑇ℎ 𝑡 − 𝑡0 + 1
Conjugate prior for 𝑏!
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7. Prior independence: 𝑝(𝑡0, 𝑏, 𝑟) = 𝑝 𝑡0 𝑝(𝑏)𝑝(𝑟)
Background level: 𝑝 𝑏 𝛼 𝑏, 𝛼 𝑏 = 𝐺 𝑏; 𝛼 𝑏, 𝛽 𝑏
Intensity: 𝑝 𝑟 𝑢, 𝛼 𝑟, 𝛽𝑟 = 𝑢𝐺 𝑟; 𝛼 𝑟, 𝛽𝑟 + 1 − 𝑢 𝛿 𝑟
Depth: 𝑝 𝑡0 = 1/𝑇
Target presence 𝑝 𝑢 = 1 = 𝑝 𝑢 = 0 = 0.5
Prior model
𝑤 =
𝑟
𝑏𝑇
, 𝝓 = 𝛼 𝑏, 𝛼 𝑏, 𝛼 𝑟, 𝛽𝑟 𝑝 𝑤, 𝑏 𝑢, 𝝓
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8. Decision rule
𝑡0, 𝑏, 𝑤 : nuisance parameters
Marginal posterior presence probability
𝑝 𝑢|𝒛, 𝝓 = ∑ 𝑝(𝑢, 𝑏, 𝑤, 𝑡0|𝒛, 𝝓) 𝑑𝑏𝑑𝑤
𝑝 𝑢 = 1|𝒛, 𝝓 ≶ 𝑝 𝑢 = 0|𝒛, 𝝓
No targetTarget
𝒛
𝑡0𝑏, 𝑤
𝝓𝑢
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9. Marginalization
How do we compute the integrals?
𝑝 𝑢|𝒛, 𝝓 = 𝑝(𝑢) ∑ 𝑝(𝑡0)[ 𝑝 𝒛 𝑡0, 𝑤, 𝑏 𝑝 𝑤, 𝑏 𝑢, 𝝓 𝑑𝑏] 𝑑𝑤
closed form, conjugacy 𝑂(1)
1D integral with quadrature method 𝑂(𝐾)
convolution form with FFT 𝑂(𝑇log𝑇)
Overall complexity 𝑂(𝐾𝑇log𝑇)
𝒛
𝑡0𝑏, 𝑤
𝝓𝑢
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10. Region of operation
• Detection probability (MC simulation)
• Guideline to practitioners
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11. Region of operation
• False detection probability
• Red curve: 𝑡0 estimated by
ML (not marginalized)
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But … pixel-wise test

12. Spatial regularization
Total variation (TV) approach: Post-process the marginal per-pixel posterior
probabilities
1. Image of log-ratios:
𝑦𝑖,𝑗 = log 𝑝 𝑢𝑖,𝑗 = 1 𝒛𝑖,𝑗, 𝝓 − log 𝑝 𝑢𝑖,𝑗 = 0 𝒛𝑖,𝑗, 𝝓
2. TV denoising, then threshold
𝒖 = 𝑓𝑡ℎ(argmin 𝒗 𝒗 − 𝒚 + 𝜏 𝒗 𝑇𝑉
)
Rother et al. "Grabcut: Interactive foreground extraction using iterated graph cuts."
ACM transactions on graphics (TOG). 2004.
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13. Experiments
Data size: 200x200x2700
Stand-off distance: 325 m
SBR interval: [0.05,0.67]
Acquisition times: 30, 3, 1 and 0.1 ms
Competing methods:
• Cross-correlation (MLE) + thresholding
• Altmann et al. (detection algorithm)
• ManiPoP (multi-depth)
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14. Experiments
30 ms
3 ms
1 ms
0.3 ms
proposed proposed + TV
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Execution
time
1 ms 12 hs 416 s 80 ms50 ms
Acq.
time

15. Conclusions and future work
• New fast target detection algorithm for single-photon Lidar
– Spike-and-slab priors – model selection
– Efficient marginalization
– Complexity similar to standard MLE 𝑂(𝐾𝑇log𝑇) vs 𝑂(𝑇log𝑇)
– Highly parallelizable
• Future/ongoing work
– Fast/adaptive reconstruction of complex scenes
– Efficient multispectral single-photon data processing
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16. Contact: jat3@hw.ac.uk
Online codes: https://gitlab.com/Tachella/LidarDetection
Thanks for your attention!