Beyond binomial and negative binomial: adaptation in Bernoulli parameter estimation

Estimating the parameter of a Bernoulli process arises in many applications, including photon-efficient active imaging where each illumination period is regarded as a single Bernoulli trial. Motivated by acquisition efficiency when multiple Bernoulli processes (e.g., multiple pixels) are of interest, we formulate the allocation of trials under a constraint on the mean as an optimal resource allocation problem. An oracle-aided trial allocation demonstrates that there can be a significant advantage from varying the allocation for different processes and inspires the introduction of a simple trial allocation gain quantity. Motivated by achieving this gain without an oracle, we present a trellis-based framework for representing and optimizing stopping rules. Considering the convenient case of Beta priors, three implementable stopping rules with similar performances are explored, and the simplest of these is shown to asymptotically achieve the oracle-aided trial allocation. These approaches are further extended to estimating functions of a Bernoulli parameter. In simulations inspired by realistic active imaging scenarios, we demonstrate significant mean-squared error improvements up to 4.36 dB for the estimation of p and up to 1.86 dB for the estimation of log p.https://arxiv.org/abs/1809.08801https://arxiv.org/abs/1809.08801First author draf

Beyond Binomial and Negative Binomial: Adaptation in Bernoulli Parameter Estimation

Estimating the parameter of a Bernoulli process arises in many applications, including photon-efficient active imaging where each illumination period is regarded as a single Bernoulli trial. Motivated by acquisition efficiency when multiple Bernoulli processes are of interest, we formulate the allocation of trials under a constraint on the mean as an optimal resource allocation problem. An oracle-aided trial allocation demonstrates that there can be a significant advantage from varying the allocation for different processes and inspires a simple trial allocation gain quantity. Motivated by realizing this gain without an oracle, we present a trellis-based framework for representing and optimizing stopping rules. Considering the convenient case of Beta priors, three implementable stopping rules with similar performances are explored, and the simplest of these is shown to asymptotically achieve the oracle-aided trial allocation. These approaches are further extended to estimating functions of a Bernoulli parameter. In simulations inspired by realistic active imaging scenarios, we demonstrate significant mean-squared error improvements: up to 4.36 dB for the estimation of p and up to 1.80 dB for the estimation of log p.Comment: 13 pages, 16 figure

A Sketching Framework for Reduced Data Transfer in Photon Counting Lidar

Single-photon lidar has become a prominent tool for depth imaging in recent years. At the core of the technique, the depth of a target is measured by constructing a histogram of time delays between emitted light pulses and detected photon arrivals. A major data processing bottleneck arises on the device when either the number of photons per pixel is large or the resolution of the time stamp is fine, as both the space requirement and the complexity of the image reconstruction algorithms scale with these parameters. We solve this limiting bottleneck of existing lidar techniques by sampling the characteristic function of the time of flight (ToF) model to build a compressive statistic, a so-called sketch of the time delay distribution, which is sufficient to infer the spatial distance and intensity of the object. The size of the sketch scales with the degrees of freedom of the ToF model (number of objects) and not, fundamentally, with the number of photons or the time stamp resolution. Moreover, the sketch is highly amenable for on-chip online processing. We show theoretically that the loss of information for compression is controlled and the mean squared error of the inference quickly converges towards the optimal Cram\'er-Rao bound (i.e. no loss of information) for modest sketch sizes. The proposed compressed single-photon lidar framework is tested and evaluated on real life datasets of complex scenes where it is shown that a compression rate of up-to 150 is achievable in practice without sacrificing the overall resolution of the reconstructed image.Comment: 16 pages, 20 figure

Estimation from quantized Gaussian measurements: when and how to use dither

Subtractive dither is a powerful method for removing the signal dependence of quantization noise for coarsely quantized signals. However, estimation from dithered measurements often naively applies the sample mean or midrange, even when the total noise is not well described with a Gaussian or uniform distribution. We show that the generalized Gaussian distribution approximately describes subtractively dithered, quantized samples of a Gaussian signal. Furthermore, a generalized Gaussian fit leads to simple estimators based on order statistics that match the performance of more complicated maximum likelihood estimators requiring iterative solvers. The order statistics-based estimators outperform both the sample mean and midrange for nontrivial sums of Gaussian and uniform noise. Additional analysis of the generalized Gaussian approximation yields rules of thumb for determining when and how to apply dither to quantized measurements. Specifically, we find subtractive dither to be beneficial when the ratio between the Gaussian standard deviation and quantization interval length is roughly less than one-third. When that ratio is also greater than 0.822/K^0.930 for the number of measurements K > 20, estimators we present are more efficient than the midrange.https://arxiv.org/abs/1811.06856Accepted manuscrip

Fast online 3D reconstruction of dynamic scenes from individual single-photon detection events

In this paper, we present an algorithm for online 3D reconstruction of dynamic scenes using individual times of arrival (ToA) of photons recorded by single-photon detector arrays. One of the main challenges in 3D imaging using single-photon Lidar is the integration time required to build ToA histograms and reconstruct reliable 3D profiles in the presence of non-negligible ambient illumination. This long integration time also prevents the analysis of rapid dynamic scenes using existing techniques. We propose a new method which does not rely on the construction of ToA histograms but allows, for the first time, individual detection events to be processed online, in a parallel manner in different pixels, while accounting for the intrinsic spatiotemporal structure of dynamic scenes. Adopting a Bayesian approach, a Bayesian model is constructed to capture the dynamics of the 3D profile and an approximate inference scheme based on assumed density filtering is proposed, yielding a fast and robust reconstruction algorithm able to process efficiently thousands to millions of frames, as usually recorded using single-photon detectors. The performance of the proposed method, able to process hundreds of frames per second, is assessed using a series of experiments conducted with static and dynamic 3D scenes and the results obtained pave the way to a new family of real-time 3D reconstruction solutions

Robust 3D Reconstruction of Dynamic Scenes From Single-Photon Lidar Using Beta-Divergences

In this paper, we present a new algorithm for fast, online 3D reconstruction of dynamic scenes using times of arrival of photons recorded by single-photon detector arrays. One of the main challenges in 3D imaging using single-photon lidar in practical applications is the presence of strong ambient illumination which corrupts the data and can jeopardize the detection of peaks/surface in the signals. This background noise not only complicates the observation model classically used for 3D reconstruction but also the estimation procedure which requires iterative methods. In this work, we consider a new similarity measure for robust depth estimation, which allows us to use a simple observation model and a non-iterative estimation procedure while being robust to mis-specification of the background illumination model. This choice leads to a computationally attractive depth estimation procedure without significant degradation of the reconstruction performance. This new depth estimation procedure is coupled with a spatio-temporal model to capture the natural correlation between neighboring pixels and successive frames for dynamic scene analysis. The resulting online inference process is scalable and well suited for parallel implementation. The benefits of the proposed method are demonstrated through a series of experiments conducted with simulated and real single-photon lidar videos, allowing the analysis of dynamic scenes at 325 m observed under extreme ambient illumination conditions.Comment: 12 page