Guaranteed bounds on the Kullback-Leibler divergence of univariate mixtures using piecewise log-sum-exp inequalities

Information-theoretic measures such as the entropy, cross-entropy and the Kullback-Leibler divergence between two mixture models is a core primitive in many signal processing tasks. Since the Kullback-Leibler divergence of mixtures provably does not admit a closed-form formula, it is in practice either estimated using costly Monte-Carlo stochastic integration, approximated, or bounded using various techniques. We present a fast and generic method that builds algorithmically closed-form lower and upper bounds on the entropy, the cross-entropy and the Kullback-Leibler divergence of mixtures. We illustrate the versatile method by reporting on our experiments for approximating the Kullback-Leibler divergence between univariate exponential mixtures, Gaussian mixtures, Rayleigh mixtures, and Gamma mixtures.Comment: 20 pages, 3 figure

Generalized offsetting of planar structures using skeletons

We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams

Network Farthest-Point Diagrams

Consider the continuum of points along the edges of a network, i.e., an undirected graph with positive edge weights. We measure distance between these points in terms of the shortest path distance along the network, known as the network distance. Within this metric space, we study farthest points. We introduce network farthest-point diagrams, which capture how the farthest points---and the distance to them---change as we traverse the network. We preprocess a network G such that, when given a query point q on G, we can quickly determine the farthest point(s) from q in G as well as the farthest distance from q in G. Furthermore, we introduce a data structure supporting queries for the parts of the network that are farther away from q than some threshold R > 0, where R is part of the query. We also introduce the minimum eccentricity feed-link problem defined as follows. Given a network G with geometric edge weights and a point p that is not on G, connect p to a point q on G with a straight line segment pq, called a feed-link, such that the largest network distance from p to any point in the resulting network is minimized. We solve the minimum eccentricity feed-link problem using eccentricity diagrams. In addition, we provide a data structure for the query version, where the network G is fixed and a query consists of the point p.Comment: A preliminary version of this work was presented at the 24th Canadian Conference on Computational Geometr

Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space

We present a general framework for computing two-dimensional Voronoi diagrams of different site classes under various distance functions. The computation of the diagrams employs the Cgal software for constructing envelopes of surfaces in 3-space, which implements a divide-and-conquer algorithm. A straightforward application of the divide-andconquer approach for Voronoi diagrams yields highly inefficient algorithms. We show that through randomization, the expected running time is near-optimal (in a worst-case sense). We believe this result, which also holds for general envelopes, to be of independent interest. We describe the interface between the construction of the diagrams and the underlying construction of the envelopes, together with methods we have applied to speed up the (exact) computation. We then present results, where a variety of diagrams are constructed with our implementation, including power diagrams, Apollonius diagrams, diagrams of line segments, Voronoi diagrams on a sphere, and more. In all cases the implementation is exact and can handle degenerate input

Wideband Autonomous Cognitive Radios: Spectrum Awareness and PHY/MAC Decision Making

The cognitive radios (CRs) have opened up new ways of better utilizing the scarce wireless spectrum resources. The CRs have been made feasible by recent advances in software-defined radios (SDRs), smart antennas, reconfigurable radio frequency (RF) front-ends, and full-duplex RF front-end architectures, to name a few. Generally, a CR is considered as a dynamically reconfigurable radio capable of adapting its operating parameters to the surrounding environment. Recent developments in spectrum policy and regulatory domains also allow more flexible and efficient utilization of wider RF spectrum range in the future. In line with the future directions of CRs, a new vision of a future autonomous CR device, called Radiobots, was previously proposed. The goals of the proposed Radiobot surpass the dynamic spectrum access (DSA) to achieve wideband operability and the main features of cognition. In order to ensure the practicality and robust operation of the Radiobot structure, the research focus of this dissertation includes the following aspects: 1) robust spectrum sensing and operability in a centralized CR network setup; 2) robust multivariate non-parametric quickest detection for dynamic spectrum usage tracking in an alien RF environment; 3) joint physical layer and medium access control layer (PHY/MAC) decision-making for wideband bandwidth aggregation (simultaneous operation over multiple modes/networks); and 4) autonomous spectrum sensing scheduling solutions in an alien ultra wideband RF environment. The major contribution of this dissertation is to investigate the feasibility of the autonomous CR operation in heterogeneous RF environments, and to provide novel solutions to the fundamental and crucial problems/challenges, including spectrum sensing, spectrum awareness, wideband operability, and autonomous PHY/MAC protocols, thus bringing the autonomous Radiobot one step closer to reality

Constraint-based navigation for safe, shared control of ground vehicles

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 138-147).Human error in machine operation is common and costly. This thesis introduces, develops, and experimentally demonstrates a new paradigm for shared-adaptive control of human-machine systems that mitigates the effects of human error without removing humans from the control loop. Motivated by observed human proclivity toward navigation in fields of safe travel rather than along specific trajectories, the planning and control framework developed in this thesis is rooted in the design and enforcement of constraints rather than the more traditional use of reference paths. Two constraint-planning methods are introduced. The first uses a constrained Delaunay triangulation of the environment to identify, cumulatively evaluate, and succinctly circumscribe the paths belonging to a particular homotopy with a set of semi autonomously enforceable constraints on the vehicle's position. The second identifies a desired homotopy by planning - and then laterally expanding - the optimal path that traverses it. Simulated results show both of these constraint-planning methods capable of improving the performance of one or multiple agents traversing an environment with obstacles. A method for predicting the threat posed to the vehicle given the current driver action, present state of the environment, and modeled vehicle dynamics is also presented. This threat assessment method, and the shared control approach it facilitates, are shown in simulation to prevent constraint violation or vehicular loss of control with minimal control intervention. Visual and haptic driver feedback mechanisms facilitated by this constraint-based control and threat-based intervention are also introduced. Finally, a large-scale, repeated measures study is presented to evaluate this control framework's effect on the performance, confidence, and cognitive workload of 20 drivers teleoperating an unmanned ground vehicle through an outdoor obstacle course. In 1,200 trials, the constraint-based framework developed in this thesis is shown to increase vehicle velocity by 26% while reducing the occurrence of collisions by 78%, improving driver reaction time to a secondary task by 8.7%, and increasing overall user confidence and sense of control by 44% and 12%, respectively. These performance improvements were realized with the autonomous controller usurping less than 43% of available vehicle control authority, on average.by Sterling J. Anderson.Ph.D

Contributions to mobile robot navigation based embedded systems and grid mapping

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Escuela Politécnica Superior, Departamento de Tecnología Electrónica y de las Comunicaciones. Fecha de lectura: 13-07-2015Path planning is a problem as old as humankind. The necessity of optimizing the resources to reach a location has been a concern since prehistory. Technology has allowed approaching this problematic using new resources. However, it has also introduced new requirements. This thesis is focused on path planning from the perspective of an embedded system using grid maps. For battery-dependent robots, path length is very relevant because it is directly related to motor consumption and the autonomy of the system. Nevertheless, a second aspect to be considered when using embedded systems is the HW requirements. These requirements comprise floating point units or storage capacity. When computer-based path planning algorithms are directly ported to these embedded systems, their HW requirements become a limitation. This thesis presents two novel path planning algorithms which take into account both the search of the shortest path and the optimization of HW resources. These algorithms are HCTNav and NafisNav. The HCTNav algorithm was developed using the intuitive approach as trying to reach the goal in a straight trajectory until an obstacle is found. When an obstacle is found, it must be surrounded until the straight path to the goal can be continued, reaching this goal or another obstacle. Considering HCTNav as a path planning algorithm, both possible surrounding trajectories can be explored and then choose the best solution. Therefore, for each obstacle the algorithm finds, there is a branch in the search of the solution. Finally, the algorithm includes an optimization procedure which reduces the length of the obtained paths if it is possible to go between nonconsecutive waypoints in straight line. The NafisNav algorithm evolves from a depth-first search. For each iteration of the algorithm, the straight trajectory to the goal position is verified. If this trajectory is not available, the algorithm selects from the unexplored neighbor cells the closest one to the target. If two neighbors were at the same distance, the algorithm would branch evaluating both alternatives. This algorithm includes a backtracking procedure just in case it finds a dead end. Finally, from every possible solution, the algorithm proposes the one that, after optimization, provides the shortest path. The new algorithms have been evaluated and compared with the most extended algorithms of the state of the art: Dijkstra and A*. The two chosen evaluation metrics have been final path length and required dynamic memory. HCTNav provides an average penalization in the path length of 2.1% and NafisNav has this penalization increased to 4.5%. However, these algorithms present a decrease of the memory requirements of a 19% for HCTNav and of a 49% for the NafisNav algorithmLa planificación de una ruta es un problema casi tan antiguo como la humanidad. La necesidad de optimizar esfuerzos para alcanzar un objetivo ha sido una gran preocupación desde la prehistoria. La tecnología ha permitido abordar la solución de esta problemática con nuevos medios, pero también ha planteado otros requisitos distintos. Esta tesis aborda el problema de navegación desde la perspectiva de los sistemas empotrados en entornos de mapas de rejilla. En todo robot dependiente de batería, la longitud final es un factor relevante porque se traduce directamente en el consumo de los motores y repercute en la autonomía del sistema. No obstante, un segundo factor que aparece al utilizar sistemas empotrados es el uso de recursos HW, ya sean unidades de coma flotante o capacidad de almacenamiento. Cuando se intenta adaptar los algoritmos diseñados para ser ejecutados en un ordenador nos enfrentamos a una gran demanda de estos recursos. La tesis plantea dos algoritmos novedosos que tienen en cuenta tanto la búsqueda de un camino lo más corto posible como la optimización de recursos HW: HCTNav y NafisNav. El algoritmo HCTNav se desarrolló siguiendo el movimiento intuitivo de quien trata de ir en línea recta hasta que encuentra un obstáculo y lo rodea hasta que puede continuar en línea recta hasta el destino, o en caso contrario hasta otro obstáculo. Dado que se trata un algoritmo de planificación, se puede plantear rodear el obstáculo por ambos lados y elegir cuál es la mejor opción. Por lo tanto, cada obstáculo genera una bifurcación en la búsqueda de solución. Este algoritmo incluye un proceso de optimización por el que se reduce el recorrido final si se pueden saltar puntos intermedios viajando en línea recta. El algoritmo NafisNav plantea una búsqueda en profundidad modificada. En cada iteración se intenta alcanzar el destino verificando si se puede alcanzar en línea recta. En caso de no poder alcanzarlo, se avanza al vecino, de entre los contiguos no explorados, aplicando un criterio de mínima distancia al objetivo. Si hubiera dos candidatos posibles, la búsqueda se bifurca, evaluando ambas opciones. Por último, se incluye un proceso de retroceso para el caso en el que se llegara a un punto sin salida. De entre las soluciones posibles se presenta aquella que, tras la optimización, obtiene el mínimo recorrido. Los nuevos algoritmos han sido evaluados y comparados con los algoritmos más extendidos en el estado del arte: Dijkstra y A*. Los dos criterios utilizados han sido la longitud final del camino y el espacio de memoria que se necesita. HCTNav tiene una penalización promedio del 2,1 % en la longitud de la solución, mientras que NafisNav aplica una penalización promedio del 4,5 %. HCTNav obtiene una reducción del consumo de memoria del 19 % comparado con la mejor solución entre Dijkstra y A*. NafisNav mejora estos resultados con una reducción del 49