Numerical algebraic geometry approach to polynomial optimization, The

2017 Summer.Includes bibliographical references.Numerical algebraic geometry (NAG) consists of a collection of numerical algorithms, based on homotopy continuation, to approximate the solution sets of systems of polynomial equations arising from applications in science and engineering. This research focused on finding global solutions to constrained polynomial optimization problems of moderate size using NAG methods. The benefit of employing a NAG approach to nonlinear optimization problems is that every critical point of the objective function is obtained with probability-one. The NAG approach to global optimization aims to reduce computational complexity during path tracking by exploiting structure that arises from the corresponding polynomial systems. This thesis will consider applications to systems biology and life sciences where polynomials solve problems in model compatibility, model selection, and parameter estimation. Furthermore, these techniques produce mathematical models of large data sets on non-euclidean manifolds such as a disjoint union of Grassmannians. These methods will also play a role in analyzing the performance of existing local methods for solving polynomial optimization problems

Nonlinear Filtering based on Log-homotopy Particle Flow : Methodological Clarification and Numerical Evaluation

The state estimation of dynamical systems based on measurements is an ubiquitous problem. This is relevant in applications like robotics, industrial manufacturing, computer vision, target tracking etc. Recursive Bayesian methodology can then be used to estimate the hidden states of a dynamical system. The procedure consists of two steps: a process update based on solving the equations modelling the state evolution, and a measurement update in which the prior knowledge about the system is improved based on the measurements. For most real world systems, both the evolution and the measurement models are nonlinear functions of the system states. Additionally, both models can also be perturbed by random noise sources, which could be non-Gaussian in their nature. Unlike linear Gaussian models, there does not exist any optimal estimation scheme for nonlinear/non-Gaussian scenarios. This thesis investigates a particular method for nonlinear and non-Gaussian data assimilation, termed as the log-homotopy based particle flow. Practical filters based on such flows have been known in the literature as Daum Huang filters (DHF), named after the developers. The key concept behind such filters is the gradual inclusion of measurements to counter a major drawback of single step update schemes like the particle filters i.e. namely the degeneracy. This could refer to a situation where the likelihood function has its probability mass well seperated from the prior density, and/or is peaked in comparison. Conventional sampling or grid based techniques do not perform well under such circumstances and in order to achieve a reasonable accuracy, could incur a high processing cost. DHF is a sampling based scheme, which provides a unique way to tackle this challenge thereby lowering the processing cost. This is achieved by dividing the single measurement update step into multiple sub steps, such that particles originating from their prior locations are graduated incrementally until they reach their final locations. The motion is controlled by a differential equation, which is numerically solved to yield the updated states. DH filters, even though not new in the literature, have not been fully explored in the detail yet. They lack the in-depth analysis that the other contemporary filters have gone through. Especially, the implementation details for the DHF are very application specific. In this work, we have pursued four main objectives. The first objective is the exploration of theoretical concepts behind DHF. Secondly, we build an understanding of the existing implementation framework and highlight its potential shortcomings. As a sub task to this, we carry out a detailed study of important factors that affect the performance of a DHF, and suggest possible improvements for each of those factors. The third objective is to use the improved implementation to derive new filtering algorithms. Finally, we have extended the DHF theory and derived new flow equations and filters to cater for more general scenarios. Improvements in the implementation architecture of a standard DHF is one of the key contributions of this thesis. The scope of the applicability of DHF is expanded by combining it with other schemes like the Sequential Markov chain Monte Carlo and the tensor decomposition based solution of the Fokker Planck equation, resulting in the development of new nonlinear filtering algorithms. The standard DHF, using improved implementation and the newly derived algorithms are tested in challenging simulated test scenarios. Detailed analysis have been carried out, together with the comparison against more established filtering schemes. Estimation error and the processing time are used as important performance parameters. We show that our new filtering algorithms exhibit marked performance improvements over the traditional schemes

Human Motion Trajectory Prediction: A Survey

With growing numbers of intelligent autonomous systems in human environments, the ability of such systems to perceive, understand and anticipate human behavior becomes increasingly important. Specifically, predicting future positions of dynamic agents and planning considering such predictions are key tasks for self-driving vehicles, service robots and advanced surveillance systems. This paper provides a survey of human motion trajectory prediction. We review, analyze and structure a large selection of work from different communities and propose a taxonomy that categorizes existing methods based on the motion modeling approach and level of contextual information used. We provide an overview of the existing datasets and performance metrics. We discuss limitations of the state of the art and outline directions for further research.Comment: Submitted to the International Journal of Robotics Research (IJRR), 37 page

Sparse Modeling for Image and Vision Processing

In recent years, a large amount of multi-disciplinary research has been conducted on sparse models and their applications. In statistics and machine learning, the sparsity principle is used to perform model selection---that is, automatically selecting a simple model among a large collection of them. In signal processing, sparse coding consists of representing data with linear combinations of a few dictionary elements. Subsequently, the corresponding tools have been widely adopted by several scientific communities such as neuroscience, bioinformatics, or computer vision. The goal of this monograph is to offer a self-contained view of sparse modeling for visual recognition and image processing. More specifically, we focus on applications where the dictionary is learned and adapted to data, yielding a compact representation that has been successful in various contexts.Comment: 205 pages, to appear in Foundations and Trends in Computer Graphics and Visio

Geometry of the ergodic quotient reveals coherent structures in flows

Dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach. However, by identifying coherent structures in their state spaces, i.e., regions of uniform and simpler behavior, we could hope to study each of the structures separately and then form the understanding of the system as a whole. The method we present in this paper uses trajectory averages of scalar functions on the state space to: (a) identify invariant sets in the state space, (b) form coherent structures by aggregating invariant sets that are similar across multiple spatial scales. First, we construct the ergodic quotient, the object obtained by mapping trajectories to the space of trajectory averages of a function basis on the state space. Second, we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space. Finally, we parametrize the ergodic quotient using intrinsic diffusion modes on it. By segmenting the ergodic quotient based on the diffusion modes, we extract coherent features in the state space of the dynamical system. The algorithm is validated by analyzing the Arnold-Beltrami-Childress flow, which was the test-bed for alternative approaches: the Ulam's approximation of the transfer operator and the computation of Lagrangian Coherent Structures. Furthermore, we explain how the method extends the Poincar\'e map analysis for periodic flows. As a demonstration, we apply the method to a periodically-driven three-dimensional Hill's vortex flow, discovering unknown coherent structures in its state space. In the end, we discuss differences between the ergodic quotient and alternatives, propose a generalization to analysis of (quasi-)periodic structures, and lay out future research directions.Comment: Submitted to Elsevier Physica D: Nonlinear Phenomen

Improving Sampling-Based Motion Planning Using Library of Trajectories

Plánování pohybu je jedním z podstatných problémů robotiky. Tato práce kombinuje pokroky v plánování pohybu a hodnocení podobnosti objektů za účelem zrychlení plánování ve statických prostředích. První část této práce pojednává o současných metodách používaných pro hodnocení podobnosti objektů a plánování pohybu. Prostřední část popisuje, jak jsou tyto metody použity pro zrychlení plánování s využitím získaných znalostí o prostředí. V poslední části jsou navržené metody porovnány s ostatními plánovači v nezávislém testu. Námi navržené algoritmy se v experimentech ukázaly být často rychlejší v porovnání s ostatními plánovači. Také často nacházely cesty v prostředích, kde ostatní plánovače nebyly schopny cestu nalézt.Motion planning is one of the fundamental problems in robotics. This thesis combines the advances in motion planning and shape matching to improve planning speeds in static environments. The first part of this thesis covers current methods used in object similarity evaluation and motion planning. The middle part describes how these methods are used together to improve planning speeds by utilizing prior knowledge about the environment, along with additional modifications. In the last part, the proposed methods are tested against other state-of-the-art planners in an independent benchmarking facility. The proposed algorithms are shown to be faster than other planners in many cases, often finding paths in environments where the other planners are unable to

Belief State Planning for Autonomous Driving: Planning with Interaction, Uncertain Prediction and Uncertain Perception

This work presents a behavior planning algorithm for automated driving in urban environments with an uncertain and dynamic nature. The algorithm allows to consider the prediction uncertainty (e.g. different intentions), perception uncertainty (e.g. occlusions) as well as the uncertain interactive behavior of the other agents explicitly. Simulating the most likely future scenarios allows to find an optimal policy online that enables non-conservative planning under uncertainty

Belief State Planning for Autonomous Driving: Planning with Interaction, Uncertain Prediction and Uncertain Perception

This thesis presents a behavior planning algorithm for automated driving in urban environments with an uncertain and dynamic nature. The uncertainty in the environment arises by the fact that the intentions as well as the future trajectories of the surrounding drivers cannot be measured directly but can only be estimated in a probabilistic fashion. Even the perception of objects is uncertain due to sensor noise or possible occlusions. When driving in such environments, the autonomous car must predict the behavior of the other drivers and plan safe, comfortable and legal trajectories. Planning such trajectories requires robust decision making when several high-level options are available for the autonomous car. Current planning algorithms for automated driving split the problem into different subproblems, ranging from discrete, high-level decision making to prediction and continuous trajectory planning. This separation of one problem into several subproblems, combined with rule-based decision making, leads to sub-optimal behavior. This thesis presents a global, closed-loop formulation for the motion planning problem which intertwines action selection and corresponding prediction of the other agents in one optimization problem. The global formulation allows the planning algorithm to make the decision for certain high-level options implicitly. Furthermore, the closed-loop manner of the algorithm optimizes the solution for various, future scenarios concerning the future behavior of the other agents. Formulating prediction and planning as an intertwined problem allows for modeling interaction, i.e. the future reaction of the other drivers to the behavior of the autonomous car. The problem is modeled as a partially observable Markov decision process (POMDP) with a discrete action and a continuous state and observation space. The solution to the POMDP is a policy over belief states, which contains different reactive plans for possible future scenarios. Surrounding drivers are modeled with interactive, probabilistic agent models to account for their prediction uncertainty. The field of view of the autonomous car is simulated ahead over the whole planning horizon during the optimization of the policy. Simulating the possible, corresponding, future observations allows the algorithm to select actions that actively reduce the uncertainty of the world state. Depending on the scenario, the behavior of the autonomous car is optimized in (combined lateral and) longitudinal direction. The algorithm is formulated in a generic way and solved online, which allows for applying the algorithm on various road layouts and scenarios. While such a generic problem formulation is intractable to solve exactly, this thesis demonstrates how a sufficiently good approximation to the optimal policy can be found online. The problem is solved by combining state of the art Monte Carlo tree search algorithms with near-optimal, domain specific roll-outs. The algorithm is evaluated in scenarios such as the crossing of intersections under unknown intentions of other crossing vehicles, interactive lane changes in narrow gaps and decision making at intersections with large occluded areas. It is shown that the behavior of the closed-loop planner is less conservative than comparable open-loop planners. More precisely, it is even demonstrated that the policy enables the autonomous car to drive in a similar way as an omniscient planner with full knowledge of the scene. It is also demonstrated how the autonomous car executes actions to actively gather more information about the surrounding and to reduce the uncertainty of its belief state