Magnus Herberthson

Radar Detection of Moving Targets Behind Corners

Detection of moving objects concealed behind a concrete wall corner has been demonstrated, using Doppler-based techniques with a stepped-frequency radar centered at 10 GHz, in a reduced-scale model of a street scenario. Micro-Doppler signatures have been traced in the return from a human target, both for walking and for breathing. Separate material measurements of the reflection and transmission of the concrete in the wall have showed that wall reflections are the dominating wave propagation mechanism for producing target detections, while wave components transmitted through the walls could be neglected. Weaker detections have been made of target returns via diffraction in the wall corner. A simple and fast algorithm for the detection and generation of detection tracks in down range has been developed, based on moving target indication technique.

Fast manifold learning based on riemannian normal coordinates

We present a novel method for manifold learning, i.e. identification of the low-dimensional manifold-like structure present in a set of data points in a possibly high-dimensional space. The main idea is derived from the concept of Riemannian normal coordinates. This coordinate system is in a way a generalization of Cartesian coordinates in Euclidean space. We translate this idea to a cloud of data points in order to perform dimension reduction. Our implementation currently uses Dijkstras algorithm for shortest paths in graphs and some basic concepts from differential geometry. We expect this approach to open up new possibilities for analysis of e.g. shape in medical imaging and signal processing of manifold-valued signals, where the coordinate system is “learned” from experimental high-dimensional data rather than defined analytically using e.g. models based on Lie-groups.

Static axisymmetric spacetimes with prescribed multipole moments

In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case. Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series.

Explicit multipole moments of stationary axisymmetric spacetimes

In this thesis we study multipole moments of axisymmetric spacetimes. Using the recursive definition of the multipole moments of Geroch and Hansen we develop a method for computing all multipole moments of a stationary axisymmetric spacetime without the use of a recursion. This is a generalisation of a method developed by Herberthson for the static case. Using Herberthson’s method we also develop a method for finding a static axisymmetric spacetime with arbitrary prescribed multipole moments, subject to a specified convergence criteria. This method has, in general, a step where one has to find an explicit expression for an implicitly defined function. However, if the number of multipole moments are finite we give an explicit expression in terms of power series.

The gravitational dipole and explicit multipole moments of static axisymmetric spacetimes

The multipole moments of static axisymmetric asymptotically flat spacetimes are considered. The usual set of recursively defined tensors is replaced with one real-valued function m defined on R+ ∪ {0}, where the moments are given by the derivatives of m at 0. As examples of applications, we show that the Schwarzschild spacetime is the gravitational monopole and derive the metric for the gravitational dipole.

ON THE DIFFERENTIABILITY CONDITIONS AT SPACELIKE INFINITY

We consider spacetimes which are asymptotically flat at spacelike infinity, . It is well known that, in general, one cannot have a smooth differentiable structure at , but rather one has to use direction-dependent structures there. Instead of the usual -differentiable structure, we suggest a weaker differential structure, a structure. The reason for this is that there do not appear to be any completions of the Schwarzschild spacetime which is in both spacelike and null directions at . In a structure all directions can be treated on an equal footing, at the expense of logarithmic singularities at . We show that, in general, the relevant part of the curvature tensor, the Weyl part, is free from these singularities, and that the (rescaled) Weyl tensor has a certain symmetry property.

A relationship between future and past null infinity

By the addition of a single pointi∘ at space-like infinity, null infinity,I =I+ ⋃I-, may be regarded as the null cone with vertexi∘. We show that, given suitable regularity conditions ati∘, there exist natural Bondi scalings on the whole ofI which may be used to relate quantities onI+ andI-. In particular, we show that, in terms of such a scaling, the mass aspect Ψ2o onI satisfies limx→°,x∈I+ Ψ2° = limx→°,x∈I- Ψ2° where the limits are taken along the same generator ofI.

Representing pairs of orientations in the plane

In this article we present a way of representing pairs of orientations in the plane. This is an extension of the familiar way of representing single orientations in the plane. Using this framework, pairs of lines can be added, scaled and averaged over in a sense which is to be described. In particular, single lines can be incorporated and handled simultaneously.

Intrinsic and Extrinsic Means on the Circle - A Maximum Likelihood Interpretation

For data samples in Rn, the mean is a well known estimator. When the data set belongs to an embedded manifold M in Rn, e.g. the unit circle in R2, the definition of a mean can be extended and constrained to M by choosing either the intrinsic Riemannian metric of the manifold or the extrinsic metric of the embedding space. A common view has been that extrinsic means are approximate solutions to the intrinsic mean problem. This paper study both means on the unit circle and reveal how they are related to the ML estimate of independent samples generated from a Brownian distribution. The conclusion is that on the circle, intrinsic and extrinsic means are maximum likelihood estimators in the limits of high SNR and low SNR respectively.

Static spacetimes with prescribed multipole moments: a proof of a conjecture by Geroch

In this paper we give sufficient conditions on a sequence of multipole moments for a static spacetime to exist with precisely these moments. The proof is constructive in the sense that a metric having prescribed multipole moments up to a given order can be calculated. Since these sufficient conditions agree with already known necessary conditions, this completes the proof of a long standing conjecture due to Geroch.