Anna Torstensson

A complete characterization and solution to the microphone position self-calibration problem

This paper presents a complete characterization and solution to microphone position self-calibration problem for time-of-arrival (TOA) measurements. This is the problem of determining the positions of receivers and transmitters given all receiver-transmitter distances. Such calibration problems arise in application such as calibration of radio antenna networks, audio or ultra-sound arrays and WiFi transmitter arrays. We show for what cases such calibration problems are well-defined and derive efficient and numerically stable algorithms for the minimal TOA based self-calibration problems. The proposed algorithms are non-iterative and require no assumptions on the sensor positions. Experiments on synthetic data show that the minimal solvers are numerically stable and perform well on noisy data. The solvers are also tested on two real datasets with good results.

Prime Rigid Graphs and Multidimensional Scaling with Missing Data

In this paper we investigate the problem of embedding a number of points given certain (but typically not all) inter-pair distance measurements. This problem is relevant for multi-dimensional scaling problems with missing data, and is applicable within anchor-free sensor network node calibration and anchor-free node localization using radio or sound TOA measurements. There are also applications within chemistry for deducing molecular 3D structure given inter-atom distance measurements and within machine learning and visualization of data, where only similarity measures between sample points are provided. The problem has been studied previously within the field of rigid graph theory. Our aim is here to construct numerically stable and efficient solvers for finding all embeddings of such minimal rigid graphs. The method is based on the observation that all graphs are either irreducibly rigid, here called prime rigid graphs, or contain smaller rigid graphs. By solving the embedding problem for the prime rigid graphs and for ways of assembling such graphs to other minimal rigid graphs, we show how to (i) calculate the number of embeddings and (ii) construct numerically stable and efficient algorithms for obtaining all embeddings given inter-node measurements. The solvers are verified with experiments on simulated data.

Projective linear groups as maximal symmetry groups

A maximal symmetry group is a group of isomorphisms of a three-dimensional hyperbolic manifold of maximal order in relation to the volume of the manifold. In this paper we determine all maximal symmetry groups of the types PSL(2, q) and PGL(2, q). Depending on the prime p there are one or two such groups with q=pk and k always equals 1, 2 or 4.

On Sagbi Bases and Resultants

A resultant-type identity for univariate polynomials is proved and applied to characterization of SAGBI bases of subalgebras, generated by two polynomials. Besides a new condition for polynomials f(x) and g(x) to form a SAGBI basis, expressed in terms of field extensions is derived.

Using resultants for SAGBI basis verification in the univariate polynomial ring

A resultant-type identity for univariate polynomials is proved and used to characterise SAGBI bases of subalgebras generated by two polynomials. A new equivalent condition, expressed in terms of the degree of a field extension, for a pair of univariate polynomials to form a SAGBI basis is derived.