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Single-photon Light Detection and Ranging (LiDAR) systems are often equipped with an array of detectors for improved spatial resolution and sensing speed. However, given a fixed amount of flux produced by the laser transmitter across the scene, the per-pixel Signal-to-Noise Ratio (SNR) will decrease when more pixels are packed in a unit space. This presents a fundamental trade-off between the spatial resolution of the sensor array and the SNR received at each pixel. Theoretical characterization of this fundamental limit is explored. By deriving the photon arrival statistics and introducing a series of new approximation techniques, the Mean Squared Error (MSE) of the maximum-likelihood estimator of the time delay is derived. The theoretical predictions align well with simulations and real data.

3D depth sensors using single-photon avalanche diodes (SPADs) are becoming increasingly common in applications such as autonomous navigation and object detection. Recent designs implement on-chip histogramming time-to-digital converters (TDCs) to compress the photon timestamps and reduce the bottleneck in the read-out and processing of large volumes of photon data. However, the use of full histogramming with large SPAD arrays poses significant challenges due to the associated demands in silicon area and power consumption. We propose a TDC-less dToF sensor which uses Spiking Neural Networks (SNN) to process the SPAD events directly. The proposed SNN is trained and tested on synthetic SPAD events, and while it offers five times lower precision in depth prediction than a classic centre-of-mass (CoM) algorithm (applied to histograms of the events), it achieves similar Mean Absolute Error (MAE) with faster processing speeds and significantly lower power consumption is anticipated., Comment: 7 Pages, 9 Figures

3D Time-of-Flight (ToF) image sensors are used widely in applications such as self-driving cars, Augmented Reality (AR) and robotics. When implemented with Single-Photon Avalanche Diodes (SPADs), compact, array format sensors can be made that offer accurate depth maps over long distances, without the need for mechanical scanning. However, array sizes tend to be small, leading to low lateral resolution, which combined with low Signal-to-Noise Ratio (SNR) levels under high ambient illumination, may lead to difficulties in scene interpretation. In this paper, we use synthetic depth sequences to train a 3D Convolutional Neural Network (CNN) for denoising and upscaling (x4) depth data. Experimental results, based on synthetic as well as real ToF data, are used to demonstrate the effectiveness of the scheme. With GPU acceleration, frames are processed at >30 frames per second, making the approach suitable for low-latency imaging, as required for obstacle avoidance., Comment: 18 pages, 10 figures, 3 tables

Single-Photon Avalanche Detector (SPAD) arrays are a rapidly emerging technology. These multi-pixel sensors have single-photon sensitivities and pico-second temporal resolutions thus they can rapidly generate depth images with millimeter precision. Such sensors are a key enabling technology for future autonomous systems as they provide guidance and situational awareness. However, to fully exploit the capabilities of SPAD array sensors, it is crucial to establish the quality of depth images they are able to generate in a wide range of scenarios. Given a particular optical system and a finite image acquisition time, what is the best-case depth resolution and what are realistic images generated by SPAD arrays? In this work, we establish a robust yet simple numerical procedure that rapidly establishes the fundamental limits to depth imaging with SPAD arrays under real world conditions. Our approach accurately generates realistic depth images in a wide range of scenarios, allowing the performance of an optical depth imaging system to be established without the need for costly and laborious field testing. This procedure has applications in object detection and tracking for autonomous systems and could be easily extended to systems for underwater imaging or for imaging around corners.

3D flash LIDAR is an alternative to the traditional scanning LIDAR systems, promising precise depth imaging in a compact form factor, and free of moving parts, for applications such as self-driving cars, robotics and augmented reality (AR). Typically implemented using single-photon, direct time-of-flight (dToF) receivers in image sensor format, the operation of the devices can be hindered by the large number of photon events needing to be processed and compressed in outdoor scenarios, limiting frame rates and scalability to larger arrays. We here present a 64x32 pixel (256x128 SPAD) dToF imager that overcomes these limitations by using pixels with embedded histogramming, which lock onto and track the return signal. This reduces the size of output data frames considerably, enabling maximum frame rates in the 10 kFPS range or 100 kFPS for direct depth readings. The sensor offers selective readout of pixels detecting surfaces, or those sensing motion, leading to reduced power consumption and off-chip processing requirements. We demonstrate the application of the sensor in mid-range LIDAR., Comment: 24 pages, 16 figures. The visualisations may be viewed by clicking on the hyperlinks in the text

The growing ubiquity of drones has raised concerns over the ability of traditional air-space monitoring technologies to accurately characterise such vehicles. Here, we present a CNN using a decision tree and ensemble structure to fully characterise drones in flight. Our system determines the drone type, orientation (in terms of pitch, roll, and yaw), and performs segmentation to classify different body parts (engines, body, and camera). We also provide a computer model for the rapid generation of large quantities of accurately labelled photo-realistic training data and demonstrate that this data is of sufficient fidelity to allow the system to accurately characterise real drones in flight. Our network will provide a valuable tool in the image processing chain where it may build upon existing drone detection technologies to provide complete drone characterisation over wide areas.

The process of tracking human anatomy in computer vision is referred to pose estimation, and it is used in fields ranging from gaming to surveillance. Three-dimensional pose estimation traditionally requires advanced equipment, such as multiple linked intensity cameras or high-resolution time-of-flight cameras to produce depth images. However, there are applications, e.g.~consumer electronics, where significant constraints are placed on the size, power consumption, weight and cost of the usable technology. Here, we demonstrate that computational imaging methods can achieve accurate pose estimation and overcome the apparent limitations of time-of-flight sensors designed for much simpler tasks. The sensor we use is already widely integrated in consumer-grade mobile devices, and despite its low spatial resolution, only 4$\times$4 pixels, our proposed Pixels2Pose system transforms its data into accurate depth maps and 3D pose data of multiple people up to a distance of 3 m from the sensor. We are able to generate depth maps at a resolution of 32$\times$32 and 3D localization of a body parts with an error of only $\approx$10 cm at a frame rate of 7 fps. This work opens up promising real-life applications in scenarios that were previously restricted by the advanced hardware requirements and cost of time-of-flight technology.

3D time-of-flight (ToF) imaging is used in a variety of applications such as augmented reality (AR), computer interfaces, robotics and autonomous systems. Single-photon avalanche diodes (SPADs) are one of the enabling technologies providing accurate depth data even over long ranges. By developing SPADs in array format with integrated processing combined with pulsed, flood-type illumination, high-speed 3D capture is possible. However, array sizes tend to be relatively small, limiting the lateral resolution of the resulting depth maps, and, consequently, the information that can be extracted from the image for applications such as object detection. In this paper, we demonstrate that these limitations can be overcome through the use of convolutional neural networks (CNNs) for high-performance object detection. We present outdoor results from a portable SPAD camera system that outputs 16-bin photon timing histograms with 64x32 spatial resolution. The results, obtained with exposure times down to 2 ms (equivalent to 500 FPS) and in signal-to-background (SBR) ratios as low as 0.05, point to the advantages of providing the CNN with full histogram data rather than point clouds alone. Alternatively, a combination of point cloud and active intensity data may be used as input, for a similar level of performance. In either case, the GPU-accelerated processing time is less than 1 ms per frame, leading to an overall latency (image acquisition plus processing) in the millisecond range, making the results relevant for safety-critical computer vision applications which would benefit from faster than human reaction times., Comment: 13 pages, 5 figures, 3 tables

Light-in-flight (LIF) imaging is the measurement and reconstruction of light's path as it moves and interacts with objects. It is well known that relativistic effects can result in apparent velocities that differ significantly from the speed of light. However, less well known is that Rayleigh scattering and the effects of imaging optics can lead to observed intensities changing by several orders of magnitude along light's path. We develop a model that enables us to correct for all of these effects, thus we can accurately invert the observed data and reconstruct the true intensity-corrected optical path of a laser pulse as it travels in air. We demonstrate the validity of our model by observing the photon arrival time and intensity distribution obtained from single-photon avalanche detector (SPAD) array data for a laser pulse propagating towards and away from the camera. We can then reconstruct the true intensity-corrected path of the light in four dimensions (three spatial dimensions and time)., Comment: 10 pages, 9 figures

Three-dimensional imaging plays an important role in imaging applications where it is necessary to record depth. The number of applications that use depth imaging is increasing rapidly, and examples include self-driving autonomous vehicles and auto-focus assist on smartphone cameras. Light detection and ranging (LIDAR) via single-photon sensitive detector (SPAD) arrays is an emerging technology that enables the acquisition of depth images at high frame rates. However, the spatial resolution of this technology is typically low in comparison to the intensity images recorded by conventional cameras. To increase the native resolution of depth images from a SPAD camera, we develop a deep network built specifically to take advantage of the multiple features that can be extracted from a camera's histogram data. The network is designed for a SPAD camera operating in a dual-mode such that it captures alternate low resolution depth and high resolution intensity images at high frame rates, thus the system does not require any additional sensor to provide intensity images. The network then uses the intensity images and multiple features extracted from downsampled histograms to guide the upsampling of the depth. Our network provides significant image resolution enhancement and image denoising across a wide range of signal-to-noise ratios and photon levels. We apply the network to a range of 3D data, demonstrating denoising and a four-fold resolution enhancement of depth.

The ability to measure and record high-resolution depth images at long stand-off distances is important for a wide range of applications, including connected and automotive vehicles, defense and security, and agriculture and mining. In LIDAR (light detection and ranging) applications, single-photon sensitive detection is an emerging approach, offering high sensitivity to light and picosecond temporal resolution, and consequently excellent surface-to-surface resolution. The use of large format CMOS single-photon detector arrays provides high spatial resolution and allows the timing information to be acquired simultaneously across many pixels. In this work, we combine state-of-the-art single-photon detector array technology with non-local data fusion to generate high resolution three-dimensional depth information of long-range targets. The system is based on a visible pulsed illumination system at 670~nm and a 240~$\times$ 320 pixel array sensor, achieving sub-centimeter precision in all three spatial dimensions at a distance of 150 meters. The non-local data fusion combines information from an optical image with sparse sampling of the single-photon array data, providing accurate depth information at low signature regions of the target.

SPAD cameras offer single photon detection sensitivity, high frame rates and zero readout noise. They are a core technology for widefield FLIM, but have further potential in ultra-fast imaging applications. However, in practice sensitivity falls behind that of EMCCD and sCMOS devices due to the lower photon detection efficiency of SPAD arrays. This paper considers the application of a binary SPAD camera to the capture of blinking molecules as part of a superresolution (dSTORM/PALM) experiment. Simulation and experimental studies demonstrate that the effective sensitivity of the camera can be improved significantly by aggregation of signal only binary frames. The simulations also indicate that with future advances in SPAD camera technology, SPAD devices could in time outperform existing scientific cameras when capturing fast temporal dynamics.

Finite difference schemes in the spatial variable for degenerate stochastic parabolic PDEs are investigated. Sharp results on the rate of $L_p$ and almost sure convergence of the finite difference approximations are presented and results on Richardson extrapolation are established for stochastic parabolic schemes under smoothness assumptions.

A comparison principle for stochastic integro-differential equations driven by Levy processes is proved. This result is obtained via an extension of an Ito formula from [11] for the square of the norm of the positive part of $L_2-$valued, continuous semimartingales, to the case of discontinuous semimartingales., Comment: 20 pages

We give sufficient conditions under which the convergence of finite difference approximations in the space variable of the solution to the Cauchy problem for linear stochastic PDEs of parabolic type can be accelerated to any given order of convergence by Richardson's method., Comment: 24 pages

A general result on the method of randomized stopping is proved. It is applied to optimal stopping of controlled diffusion processes with unbounded coefficients to reduce it to an optimal control problem without stopping. This is motivated by recent results of Krylov on numerical solutions to the Bellman equation., Comment: Published in at http://dx.doi.org/10.3150/07-BEJ108 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)