An optimal polynomial approximation of Brownian motion

In this paper, we will present a strong (or pathwise) approximation of standard Brownian motion by a class of orthogonal polynomials. The coefficients that are obtained from the expansion of Brownian motion in this polynomial basis are independent Gaussian random variables. Therefore it is practical (requires $N$ independent Gaussian coefficients) to generate an approximate sample path of Brownian motion that respects integration of polynomials with degree less than $N$. Moreover, since these orthogonal polynomials appear naturally as eigenfunctions of an integral operator defined by the Brownian bridge covariance function, the proposed approximation is optimal in a certain weighted $L^{2}(\mathbb{P})$ sense. In addition, discretizing Brownian paths as piecewise parabolas gives a locally higher order numerical method for stochastic differential equations (SDEs) when compared to the standard piecewise linear approach. We shall demonstrate these ideas by simulating Inhomogeneous Geometric Brownian Motion (IGBM). This numerical example will also illustrate the deficiencies of the piecewise parabola approximation when compared to a new version of the asymptotically efficient log-ODE (or Castell-Gaines) method.Comment: 27 pages, 8 figure

Imaging of Magnetic Nanoparticles using Magnetoelectric Sensors

Imaging of magnetic nanoparticles offers a variety of promising medical applications for therapeutics and diagnostics. Using magnetic nanoparticles as tracer material for imaging allows for the non-invasive detection of spatial distributions of nanoparticles that can give information about diseases or can be used in preventive medicine. Imaging biodistributions of magnetically labeled cells offers applicability for tissue engineering, as a means to monitor cell growth within artificial scaffolds non-destructively. In the presented work, the capabilities of an imaging system for magnetic nanoparticles via magnetoelectric sensors are investigated. The investigated technique, called Magnetic Particle Mapping, is based on the detection of the nonlinear magnetic response of magnetic nanoparticles. A resonant magnetoelectric sensor is used for frequency selective measurements of the nanoparticles magnetic response. Extensive modeling was performed that enabled proper imaging of magnetic nanoparticle distributions. Fundamental limitations of the imaging system were derived to describe resolution in correspondence to signal-to-noise ratios. Incorporation of additional parameters in the imaging system for the data analysis resulted in an algorithm for a more robust reconstruction of spatial particle distributions, increasing its imaging capabilities. Experimental investigations of the imaging system show the capabilities for imaging of cell densities using magnetically labeled cells. Furthermore, resolution limitations were investigated and differentiation of different particle types in imaging was shown, referred to as ”colored” imaging. The imaging of biodistributions of magnetically labeled cells thus enable exciting perspectives on further research and possible applications in tissue engineering

Convex Mathematical Programs for Relational Matching of Object Views

Automatic recognition of objects in images is a difficult and challenging task in computer vision which has been tackled in many different ways. Based on the powerful and widely used concept to represent objects and scenes as relational structures, the problem of graph matching, i.e. to find correspondences between two graphs is a part of the object recognition problem. Belonging to the field of combinatorial optimization graph matching is considered to be one of the most complex problems in computer vision: It is known to be NP-complete in the general case. In this thesis, two novel approaches to the graph matching problem are proposed and investigated. They are based on recent progress in the mathematical literature on convex programming. Starting out from describing the desired matchings by suitable objective functions in terms of binary variables, relaxations of combinatorial constraints and an adequate adaption of the objective function lead to continuous convex optimization problems which can be solved without parameter tuning and in polynomial time. A subsequent post-processing step results in feasible, sub-optimal combinatorial solutions to the original decision problem. In the first part of this thesis, the connection between specific graph-matching problems and the quadratic assignment problem is explored. In this case, the convex relaxation leads to a convex quadratic program , which is combined with a linear program for post-processing. Conditions under which the quadratic assignment representation is adequate from the computer vision point of view are investigated, along with attempts to relax these conditions by modifying the approach accordingly. The second part of this work focuses directly on the matching of subgraphs -- representing a model -- to a considerably larger scene graph. A bipartite matching is extended with a quadratic regularization term to take into account relations within each set of vertices. Based on this convex relaxation, post-processing and the application to computer vision are investigated and discussed. Numerical experiments reveal both the power and the limitations of the approach. For problems of sizes which occur in applications the approach is quite reasonable and often the combinatorial optimal solution is found. For larger instances the intrinsic combinatorial nature of the problem comes out and leads to sub-optimal solutions which, however, are still good