Exact Search Directions for Optimization of Linear and Nonlinear Models Based on Generalized Orthonormal Functions

A novel technique for selecting the poles of orthonormal basis functions (OBF) in Volterra models of any order is presented. It is well-known that the usual large number of parameters required to describe the Volterra kernels can be significantly reduced by representing each kernel using an appropriate basis of orthonormal functions. Such a representation results in the so-called OBF Volterra model, which has a Wiener structure consisting of a linear dynamic generated by the orthonormal basis followed by a nonlinear static mapping given by the Volterra polynomial series. Aiming at optimizing the poles that fully parameterize the orthonormal bases, the exact gradients of the outputs of the orthonormal filters with respect to their poles are computed analytically by using a back-propagation-through-time technique. The expressions relative to the Kautz basis and to generalized orthonormal bases of functions (GOBF) are addressed; the ones related to the Laguerre basis follow straightforwardly as a particular case. The main innovation here is that the dynamic nature of the OBF filters is fully considered in the gradient computations. These gradients provide exact search directions for optimizing the poles of a given orthonormal basis. Such search directions can, in turn, be used as part of an optimization procedure to locate the minimum of a cost-function that takes into account the error of estimation of the system output. The Levenberg-Marquardt algorithm is adopted here as the optimization procedure. Unlike previous related work, the proposed approach relies solely on input-output data measured from the system to be modeled, i.e., no information about the Volterra kernels is required. Examples are presented to illustrate the application of this approach to the modeling of dynamic systems, including a real magnetic levitation system with nonlinear oscillatory behavior.

Automatic camera control in virtual environments augmented using multiple sparse videos

Automated virtual camera control has been widely used in animation and interactive virtual environments. We have developed a multiple sparse camera based free view video system prototype that allows users to control the position and orientation of a virtual camera, enabling the observation of a real scene in three dimensions (3D) from any desired viewpoint. Automatic camera control can be activated to follow selected objects by the user. Our method combines a simple geometric model of the scene composed of planes (virtual environment), augmented with visual information from the cameras and pre-computed tracking information of moving targets to generate novel perspective corrected 3D views of the virtual camera and moving objects. To achieve real-time rendering performance, view-dependent textured mapped billboards are used to render the moving objects at their correct locations and foreground masks are used to remove the moving objects from the projected video streams. The current prototype runs on a PC with a common graphics card and can generate virtual 2D views from three cameras of resolution 768 x 576 with several moving objects at about 11 fps. (C)2011 Elsevier Ltd. All rights reserved.