On the expected diameter, width, and complexity of a stochastic convex-hull

We investigate several computational problems related to the stochastic convex hull (SCH). Given a stochastic dataset consisting of $n$ points in $\mathbb{R}^d$ each of which has an existence probability, a SCH refers to the convex hull of a realization of the dataset, i.e., a random sample including each point with its existence probability. We are interested in computing certain expected statistics of a SCH, including diameter, width, and combinatorial complexity. For diameter, we establish the first deterministic 1.633-approximation algorithm with a time complexity polynomial in both $n$ and $d$. For width, two approximation algorithms are provided: a deterministic $O(1)$-approximation running in $O(n^{d+1} \log n)$ time, and a fully polynomial-time randomized approximation scheme (FPRAS). For combinatorial complexity, we propose an exact $O(n^d)$-time algorithm. Our solutions exploit many geometric insights in Euclidean space, some of which might be of independent interest

Continuous Three-Dimensional Control of a Virtual Helicopter Using a Motor Imagery Based Brain-Computer Interface

Brain-computer interfaces (BCIs) allow a user to interact with a computer system using thought. However, only recently have devices capable of providing sophisticated multi-dimensional control been achieved non-invasively. A major goal for non-invasive BCI systems has been to provide continuous, intuitive, and accurate control, while retaining a high level of user autonomy. By employing electroencephalography (EEG) to record and decode sensorimotor rhythms (SMRs) induced from motor imaginations, a consistent, user-specific control signal may be characterized. Utilizing a novel method of interactive and continuous control, we trained three normal subjects to modulate their SMRs to achieve three-dimensional movement of a virtual helicopter that is fast, accurate, and continuous. In this system, the virtual helicopter's forward-backward translation and elevation controls were actuated through the modulation of sensorimotor rhythms that were converted to forces applied to the virtual helicopter at every simulation time step, and the helicopter's angle of left or right rotation was linearly mapped, with higher resolution, from sensorimotor rhythms associated with other motor imaginations. These different resolutions of control allow for interplay between general intent actuation and fine control as is seen in the gross and fine movements of the arm and hand. Subjects controlled the helicopter with the goal of flying through rings (targets) randomly positioned and oriented in a three-dimensional space. The subjects flew through rings continuously, acquiring as many as 11 consecutive rings within a five-minute period. In total, the study group successfully acquired over 85% of presented targets. These results affirm the effective, three-dimensional control of our motor imagery based BCI system, and suggest its potential applications in biological navigation, neuroprosthetics, and other applications

A Tutorial on EEG Signal Processing Techniques for Mental State Recognition in Brain-Computer Interfaces

International audienceThis chapter presents an introductory overview and a tutorial of signal processing techniques that can be used to recognize mental states from electroencephalographic (EEG) signals in Brain-Computer Interfaces. More particularly, this chapter presents how to extract relevant and robust spectral, spatial and temporal information from noisy EEG signals (e.g., Band Power features, spatial filters such as Common Spatial Patterns or xDAWN, etc.), as well as a few classification algorithms (e.g., Linear Discriminant Analysis) used to classify this information into a class of mental state. It also briefly touches on alternative, but currently less used approaches. The overall objective of this chapter is to provide the reader with practical knowledge about how to analyse EEG signals as well as to stress the key points to understand when performing such an analysis

On the most likely Voronoi diagram and nearest neighbor searching

We consider the problem of nearest-neighbor searching among a set of stochastic sites, where a stochastic site is a tuple (si,pi) consisting of a point si in a d-dimensional space and a probability pi determining its existence. The problem is interesting and non-trivial even in 1-dimension, where the Most Likely Voronoi Diagram (LVD) is shown to have worst-case complexity O(n2). We then show that under more natural and less adversarial conditions, the size of the 1-dimensional LVD is significantly smaller: (1) T(kn) if the input has only k distinct probability values, (2) O(nlogn) on average, and (3) O(nnv) under smoothed analysis. We also present an alternative approach to the most likely nearest neighbor (LNN) search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models, as well as worst-case with a bounded number of distinct probabilities. Using the Pareto-set approach, we can also reduce the multi-dimensional LNN search to a sequence of nearest neighbor and spherical range queries