Probabilistic modeling for single-photon lidar

Lidar is an increasingly prevalent technology for depth sensing, with applications including scientific measurement and autonomous navigation systems. While conventional systems require hundreds or thousands of photon detections per pixel to form accurate depth and reflectivity images, recent results for single-photon lidar (SPL) systems using single-photon avalanche diode (SPAD) detectors have shown accurate images formed from as little as one photon detection per pixel, even when half of those detections are due to uninformative ambient light. The keys to such photon-efficient image formation are two-fold: (i) a precise model of the probability distribution of photon detection times, and (ii) prior beliefs about the structure of natural scenes. Reducing the number of photons needed for accurate image formation enables faster, farther, and safer acquisition. Still, such photon-efficient systems are often limited to laboratory conditions more favorable than the real-world settings in which they would be deployed. This thesis focuses on expanding the photon detection time models to address challenging imaging scenarios and the effects of non-ideal acquisition equipment. The processing derived from these enhanced models, sometimes modified jointly with the acquisition hardware, surpasses the performance of state-of-the-art photon counting systems. We first address the problem of high levels of ambient light, which causes traditional depth and reflectivity estimators to fail. We achieve robustness to strong ambient light through a rigorously derived window-based censoring method that separates signal and background light detections. Spatial correlations both within and between depth and reflectivity images are encoded in superpixel constructions, which fill in holes caused by the censoring. Accurate depth and reflectivity images can then be formed with an average of 2 signal photons and 50 background photons per pixel, outperforming methods previously demonstrated at a signal-to-background ratio of 1. We next approach the problem of coarse temporal resolution for photon detection time measurements, which limits the precision of depth estimates. To achieve sub-bin depth precision, we propose a subtractively-dithered lidar implementation, which uses changing synchronization delays to shift the time-quantization bin edges. We examine the generic noise model resulting from dithering Gaussian-distributed signals and introduce a generalized Gaussian approximation to the noise distribution and simple order statistics-based depth estimators that take advantage of this model. Additional analysis of the generalized Gaussian approximation yields rules of thumb for determining when and how to apply dither to quantized measurements. We implement a dithered SPL system and propose a modification for non-Gaussian pulse shapes that outperforms the Gaussian assumption in practical experiments. The resulting dithered-lidar architecture could be used to design SPAD array detectors that can form precise depth estimates despite relaxed temporal quantization constraints. Finally, SPAD dead time effects have been considered a major limitation for fast data acquisition in SPL, since a commonly adopted approach for dead time mitigation is to operate in the low-flux regime where dead time effects can be ignored. We show that the empirical distribution of detection times converges to the stationary distribution of a Markov chain and demonstrate improvements in depth estimation and histogram correction using our Markov chain model. An example simulation shows that correctly compensating for dead times in a high-flux measurement can yield a 20-times speed up of data acquisition. The resulting accuracy at high photon flux could enable real-time applications such as autonomous navigation

Modeling and Analysis of Neural Spike Trains

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Fractional Calculus and the Future of Science

Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding

Game Theory Relaunched

The game is on. Do you know how to play? Game theory sets out to explore what can be said about making decisions which go beyond accepting the rules of a game. Since 1942, a well elaborated mathematical apparatus has been developed to do so; but there is more. During the last three decades game theoretic reasoning has popped up in many other fields as well - from engineering to biology and psychology. New simulation tools and network analysis have made game theory omnipresent these days. This book collects recent research papers in game theory, which come from diverse scientific communities all across the world; they combine many different fields like economics, politics, history, engineering, mathematics, physics, and psychology. All of them have as a common denominator some method of game theory. Enjoy

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described