In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior.
This chapter describes a framework for storage of tensor array data, useful to describe regularly sampled tensor fields. The main component of the framework, called Similar Tensor Array Core (STAC), is the result of a collaboration between research groups within the SIMILAR network of excellence. It aims to capture the essence of regularly sampled tensor fields using a minimal set of attributes and can therefore be used as a “greatest common divisor” and interface between tensor array processing algorithms. This is potentially useful in applied fields like medical image analysis, in particular in Diffusion Tensor MRI, where misinterpretation of tensor array data is a common source of errors. By promoting a strictly geometric perspective on tensor arrays, with a close resemblance to the terminology used in differential geometry, (STAC) removes ambiguities and guides the user to define all necessary information. In contrast to existing tensor array file formats, it is minimalistic and based on an intrinsic and geometric interpretation of the array itself, without references to other coordinate systems.
We present a one-pass framework for filtering vector-valued images and unordered sets of data points in an N-dimensional feature space. It is based on a local Bayesian framework, previously developed for scalar images, where estimates are computed using expectation values and histograms. In this paper we extended this framework to handle N-dimensional data. To avoid the curse of dimensionality, it uses importance sampling instead of histograms to represent probability density functions. In this novel computational framework we are able to efficiently filter both vector-valued images and data, similar to e.g. the well-known bilateral, median and mean shift filters.
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 Dijkstra’s 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.
For data samples in R^{n}, the mean is a well known estimator. When the data set belongs to an embedded manifold M in R^{n}, e.g. the unit circle in R^{2}, 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
In this paper, the multipole moments of stationary asymptotically flat spacetimes are considered. We show how the tensorial recursion of Geroch and Hansen can be replaced by a scalar recursion on . We also give a bound on the multipole moments. This gives a proof of the 'necessary part' of a long-standing conjecture due to Geroch.
In this paper, we study multipole moments of axisymmetric stationary asymptotically flat spacetimes. We show how the tensorial recursion of Geroch and Hansen can be reduced to a recursion of scalar functions. We also demonstrate how a careful choice of conformal factor collects all moments into one complex-valued function on , where the moments appear as the derivatives at 0. As an application, we calculate the moments of the Kerr solution. We also discuss the freedom in choosing the potential for the moments.
In this paper we develop a method of finding the static axisymmetric spacetime corresponding to any given set ofmultipolemoments. In addition to an implicit algebraic form for the general solution, we also give a power series expression for all finite sets of multipole moments. As conjectured by Geroch we prove in the special case of axisymmetry, that there is a static spacetime for any given set of multipole moments subject to a (specified) convergence criterion. We also use this method to confirm a conjecture of Hernández-Pastora and Martín concerning the monopole–quadrupole solution.
It is well known that, for asymptotically flat spacetimes, one cannot in general have a smooth differentiable structure at spacelike infinity, i(0). Normally, one uses direction dependent structures, whose regularity has to match the regularity of the (rescaled) metric. The standard C->1-structure at i(0) ensures sufficient regularity in spacelike directions, but examples show very low regularity on I+ and I-. The alternative C1+-structure shows that both null and spacelike directions may be treated on an equal footing, at the expense of some manageable logarithmic singularities at i(0). In this paper, we show that the Kerr spacetime may be rescaled and given a structure which is C->1 in both null and spacelike directions from i(0).
The multipole moments of stationary asymptotically flat spacetimes are considered. We demostrate how the tensorial recursion of Geroch and Hansen can be replaced by a scalar recursion on R2. We also give a bound on multipole moments. This confirms the "necessary part" of a long standing conjecture due to Geroch. © 2007 IOP Publishing Ltd.
EM scattering from PEC surfaces are mostly calculated through the induced surface current J. In this paper, we consider PEC surfaces homeomorphic to the sphere, apply Hodge decomposition theorem to a slightly rewritten surface current, and show how this enables us to replace the unknown current with two scalar functions which serve as potentials for the current. Implications of this decomposition are pointed out, and numerical results are demonstrated.
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.
This paper gives sufficient conditions on a sequence of multipole moments for a static spacetime to exist with precisely these moments. We outline the proof, which is constructive in the sense that a metric having prescribed multipole moments up to a given order can be calculated. These sufficient conditions agree with already known necessary conditions, and hence this completes the proof of a long standing conjecture due to Geroch
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.
For positive semi-definite tensors like diffusion tensors in the plane it is possible to calculate several different means or p-averages. These are related to p-norms for functions, but produce mappings rather than numbers as means. We compare these means for various values of the real parameter p. One important future application is the filtering and interpolation of tensor fields in Diffusion Tensor Magnetic Resonance
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.
We present a novel approach to investigate the properties of diffusion weighted magnetic resonance imaging (dMRI). The process of restricted diffusion of spin particles in the presence of a magnetic field is simulated by an iterated complex matrix multiplication approach. The approach is based on first principles and provides a flexible, transparent and fast simulation tool. The experiments carried out reveals fundamental features of the dMRI process. A particularly interesting observation is that the induced speed of the local spatial spin angle rate of change is highly shift variant. Hence, the encoding basis functions are not the complex exponentials associated with the Fourier transform as commonly assumed. Thus, reconstructing the signal using the inverse Fourier transform leads to large compartment estimation errors, which is demonstrated in a number of 1D and 2D examples. In accordance with previous investigations the compartment size is under-estimated. More interestingly, however, we show that the estimated shape is likely to be far from the true shape using state of the art clinical MRI scanners.
Diffusion MRI (dMRI) is a valuable tool in the assessment of tissue microstructure. By fitting a model to the dMRI signal it is possible to derive various quantitative features. Several of the most popular dMRI signal models are expansions in an appropriately chosen basis, where the coefficients are determined using some variation of least-squares. However, such approaches lack any notion of uncertainty, which could be valuable in e.g. group analyses. In this work, we use a probabilistic interpretation of linear least-squares methods to recast popular dMRI models as Bayesian ones. This makes it possible to quantify the uncertainty of any derived quantity. In particular, for quantities that are affine functions of the coefficients, the posterior distribution can be expressed in closed-form. We simulated measurements from single- and double-tensor models where the correct values of several quantities are known, to validate that the theoretically derived quantiles agree with those observed empirically. We included results from residual bootstrap for comparison and found good agreement. The validation employed several different models: Diffusion Tensor Imaging (DTI), Mean Apparent Propagator MRI (MAP-MRI) and Constrained Spherical Deconvolution (CSD). We also used in vivo data to visualize maps of quantitative features and corresponding uncertainties, and to show how our approach can be used in a group analysis to downweight subjects with high uncertainty. In summary, we convert successful linear models for dMRI signal estimation to probabilistic models, capable of accurate uncertainty quantification.
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.
Neuronal and glial projections can be envisioned to be tubes of infinitesimal diameter as far as diffusion magnetic resonance (MR) measurements via clinical scanners are concerned. Recent experimental studies indicate that the decay of the orientationally-averaged signal in white-matter may be characterized by the power-law, Ē(q) ∝ q−1, where q is the wavenumber determined by the parameters of the pulsed field gradient measurements. One particular study by McKinnon et al. [1] reports a distinctively faster decay in gray-matter. Here, we assess the role of the size and curvature of the neurites and glial arborizations in these experimental findings. To this end, we studied the signal decay for diffusion along general curves at all three temporal regimes of the traditional pulsed field gradient measurements. We show that for curvy projections, employment of longer pulse durations leads to a disappearance of the q−1 decay, while such decay is robust when narrow gradient pulses are used. Thus, in clinical acquisitions, the lack of such a decay for a fibrous specimen can be seen as indicative of fibers that are curved. We note that the above discussion is valid for an intermediate range of q-values as the true asymptotic behavior of the signal decay is Ē(q) ∝ q−4 for narrow pulses (through Debye-Porod law) or steeper for longer pulses. This study is expected to provide insights for interpreting the diffusion-weighted images of the central nervous system and aid in the design of acquisition strategies.
The signature of diffusive motion on the NMR signal has been exploited to characterize the mesoscopic structure of specimens in numerous applications. For compartmentalized specimens comprising isolated subdomains, a representation of individual pores is necessary for describing restricted diffusion within them. When gradient waveforms with long pulse durations are employed, a quadratic potential profile is identified as an effective energy landscape for restricted diffusion. The dependence of the stochastic effective force on the center-of-mass position is indeed found to be approximately linear (Hookean) for restricted diffusion even when the walls are sticky. We outline the theoretical basis and practical advantages of our picture involving effective potentials.