Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Uncertainty Control for Reliable Video Understanding on Complex Environments

International audienceThe most popular applications for video understanding are those related to video-surveillance (e.g. alarms, abnormal behaviours, expected events, access control). Video understanding has several other applications of high impact to the society as medical supervision, traffic control, violent acts detection, crowd behaviour analysis, among many others. We propose a new generic video understanding approach able to extract and learn valuable information from noisy video scenes for real-time applications. This approach comprises motion segmentation, object classification, tracking and event learning phases. This work is focused on building the first fundamental blocks allowing a proper management of uncertainty of data in every phase of the video understanding process. The main contributions of the proposed approach are: (i) a new algorithm for tracking multiple objects in noisy environments, (ii) the utilisation of reliability measures for modelling uncertainty in data and for proper selection of valuable information extracted from noisy data, (iii) the improved capability of tracking to manage multiple visual evidence-target associations, (iv) the combination of 2D image data with 3D information in a dynamics model governed by reliability measures for proper control of uncertainty in data, and (v) a new approach for event recognition through incremental event learning, driven by reliability measures for selecting the most stable and relevant data

Dynamic movement primitives: volumetric obstacle avoidance

Dynamic Movement Primitives (DMPs) are a framework for learning a trajectory from a demonstration. The trajectory can be learned efficiently after only one demonstration, and it is immediate to adapt it to new goal positions and time duration. Moreover, the trajectory is also robust against perturbations. However, obstacle avoidance for DMPs is still an open problem. In this work, we propose an extension of DMPs to support volumetric obstacle avoidance based on the use of superquadric potentials. We show the advantages of this approach when obstacles have known shape, and we extend it to unknown objects using minimal enclosing ellipsoids. A simulation and experiments with a real robot validate the framework, and we make freely available our implementation

Simulating Hard Rigid Bodies

Several physical systems in condensed matter have been modeled approximating their constituent particles as hard objects. The hard spheres model has been indeed one of the cornerstones of the computational and theoretical description in condensed matter. The next level of description is to consider particles as rigid objects of generic shape, which would enrich the possible phenomenology enormously. This kind of modeling will prove to be interesting in all those situations in which steric effects play a relevant role. These include biology, soft matter, granular materials and molecular systems. With a view to developing a general recipe for event-driven Molecular Dynamics simulations of hard rigid bodies, two algorithms for calculating the distance between two convex hard rigid bodies and the contact time of two colliding hard rigid bodies solving a non-linear set of equations will be described. Building on these two methods, an event-driven molecular dynamics algorithm for simulating systems of convex hard rigid bodies will be developed and illustrated in details. In order to optimize the collision detection between very elongated hard rigid bodies, a novel nearest-neighbor list method based on an oriented bounding box will be introduced and fully explained. Efficiency and performance of the new algorithm proposed will be extensively tested for uniaxial hard ellipsoids and superquadrics. Finally applications in various scientific fields will be reported and discussed.Comment: 36 pages, 17 figure

An investigation into the Gustafsson limit for small planar antennas using optimisation

The fundamental limit for small antennas provides a guide to the effectiveness of designs. Gustafsson et al, Yaghjian et al, and Mohammadpour-Aghdam et al independently deduced a variation of the Chu-Harrington limit for planar antennas in different forms. Using a multi-parameter optimisation technique based on the ant colony algorithm, planar, meander dipole antenna designs were selected on the basis of lowest resonant frequency and maximum radiation efficiency. The optimal antenna designs across the spectrum from 570 to 1750 MHz occupying an area of $56mm \times 25mm$ were compared with these limits calculated using the polarizability tensor. The results were compared with Sievenpiper's comparison of published planar antenna properties. The optimised antennas have greater than 90% polarizability compared to the containing conductive box in the range $0.3<ka<1.1$, so verifying the optimisation algorithm. The generalized absorption efficiency of the small meander line antennas is less than 50%, and results are the same for both PEC and copper designs.Comment: 6 pages, 10 figures, in press article. IEEE Transactions on Antennas and Propagation (2014

MINVO Basis: Finding Simplexes with Minimum Volume Enclosing Polynomial Curves

This paper studies the problem of finding the smallest $n$-simplex enclosing a given $n^{\text{th}}$-degree polynomial curve. Although the Bernstein and B-Spline polynomial bases provide feasible solutions to this problem, the simplexes obtained by these bases are not the smallest possible, which leads to undesirably conservative results in many applications. We first prove that the polynomial basis that solves this problem (MINVO basis) also solves for the $n^\text{th}$-degree polynomial curve with largest convex hull enclosed in a given $n$-simplex. Then, we present a formulation that is \emph{independent} of the $n$-simplex or $n^{\text{th}}$-degree polynomial curve given. By using Sum-Of-Squares (SOS) programming, branch and bound, and moment relaxations, we obtain high-quality feasible solutions for any $n\in\mathbb{N}$ and prove numerical global optimality for $n=1,2,3$. The results obtained for $n=3$ show that, for any given $3^{\text{rd}}$-degree polynomial curve, the MINVO basis is able to obtain an enclosing simplex whose volume is $2.36$ and $254.9$ times smaller than the ones obtained by the Bernstein and B-Spline bases, respectively. When $n=7$, these ratios increase to $902.7$ and $2.997\cdot10^{21}$, respectively.Comment: 25 pages, 16 figure