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The advent of single molecule microscopy has rev- olutionized biological investigations by providing a powerful tool for the study of intercellular and intracellular trafficking processes of protein molecules which was not available before through conventional microscopy. In practice, pixelated detectors are used to acquire the images of fluorescently labeled objects moving in cellular environments. Then, the acquired fluorescence microscopy images contain the numbers of the photons detected in each pixel, during an exposure time interval. Moreover, instead of having the exact locations of detection of the photons, we only know the pixel areas in which the photons impact the detector. These challenges make the analysis of single molecule trajectories, from pixelated images, a complex problem. Here, we investigate the effect of pixelation on the parameter estimation of single molecule trajectories. In particular, we develop a stochastic framework to calculate the maximum likelihood estimates of the parameters of a stochastic differential equation that describes the motion of the molecule in living cells. We also calculate the Fisher information matrix for this parameter estimation problem. The analytical results are complicated through the fact that the observation process in a microscope prohibits the use of standard Kalman filter type approaches. The analytical framework presented here is illustrated with examples of low photon count scenarios for which we rely on Monte Carlo methods to compute the associated probability distributions

In [1] a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In [2], this structure is also derived from a LC ladder. In this paper, a connection is established between Ober?s canonical form and a Schur algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober?s form as interpolation values at infinity and a recursive construction of the balanced realization.

The problem of finding the global minimum of a multivariate polynomial can be approached by the matrix method of Stetter-Moller, which reformulates it as a large eigenvalue problem. The linear operators involved in this approach are studied using the theory of nD-systems. This supports the efficient application of iterative methods for solving eigenvalue problems such as Arnoldi methods and Jacobi-Davidson methods. This approach is demonstrated by an example which addresses optimal H 2 -model reduction of a linear dynamical model of order 10 to order 9.

International audience; Symmetric inner rational functions naturally arise in the description of physical systems which satisfy the conservation and reciprocity laws. Inner matrix functions can be parametrized by a sequence of interpolation vectors obtained from a tangential Schur algorithm. In this paper, we present a Schur type algorithm which allows to describe symmetric inner functions, based on a two-sided Nudelman interpolation problem. This Schur algorithm gives rise to an interesting interpretation in the context of surface acoustic wave filters.

This paper deals with the problem of computing a best stable rational L/sup 2/ approximation of specified order to a given multivariable transfer function. The problem is equivalently formulated as a minimization problem over the manifold of stable all-pass (or lossless) transfer functions of fixed order. Some special Schur parameters are used to describe this manifold. Such a description presents numerous advantages: it takes into account the stability constraint, possesses a good numerical behavior and provides a model in state-space form, which is very useful in practice. A rigorous and convergent algorithm is proposed to compute local minima which has been implemented using standard MATLAB subroutines. The effectiveness of our approach to solve model reduction problems as well as identification problems in frequency domain is demonstrated through several examples, including real-data simulations.

A new approach to the question of parametrizing state-space systems is considered. The approach consists of using input-normal state-space representations. These representations are unique up to an isometric state isomorphism. By decomposing the tangent space of the set of normalized controllable matrix pairs into the tangent space of the equivalence class and the orthogonal complement one obtains an interesting set of local coordinates with a number of remarkable properties. Here the approach is presented in the context of a separable least squares approach to maximum likelihood estimation of a linear system.

A state-space approach to H 2 model reduction of stable systems is pursued, using techniques from global rational optimization, and aimed at finding global optima. First, a concentrated version of the H 2 criterion is derived, by optimizing analytically with respect to a subset of parameters. Next, a bound is developed, which is shown to be sharp for reduction to models of order 1. The choice of parameterization is such as to keep all these criteria rational. To illustrate the approach, a nontrivial example of a multivariable stable system of order 4 was designed, together with an approximation of order 2. Detailed analysis shows the approximation to be the global optimum among all stable system of order 2, constituting the first such system described in the literature. The fact that it also yields a global optimum to the bound, which happens to be sharp at the optimum, leads to a couple of questions for further research.

Two new theorems are presented concerning a class of homogeneous systems. The first result shows that the state isomorphism for a pair of indistinguishable systems is, under certain conditions, homogeneous of degree one. The second result shows that, under certain conditions, this state isomorphism is linear in a case when the observability rank condition holds at the origin of the both systems, and for each system the observability rank condition and the controllability rank condition hold in an open set along the origin.