Andrea Fuster

• PhD
• Professor (Assistant) at Eindhoven University of Technology

We obtain a new family of exact vacuum solutions to quadratic gravity that describe pp-waves with two-dimensional wave surfaces that can have any prescribed constant curvature. When the wave surfaces are flat we recover the Peres waves obtained by Madsen, a subset of which forms precisely the vacuum pp-waves of general relativity. If, on the other...

We present a new family of exact vacuum solutions to Pfeifer and Wohlfarth’s field equation in Finsler gravity, consisting of Finsler metrics that are Landsbergian but not Berwaldian, also known as unicorns due to their rarity. Interestingly, we find that these solutions have a physically viable light cone structure, even though in some cases the s...

We introduce a new class of (α,β)-type exact solutions in Finsler gravity closely related to the well-known pp-waves in general relativity. Our class contains most of the exact solutions currently known in the literature as special cases. The linearized versions of these solutions may be interpretted as Finslerian gravitational waves, and we invest...

We introduce a new class of $(\alpha,\beta)$-type exact solutions in Finsler gravity closely related to the well-known pp-waves in general relativity. Our class contains most of the exact solutions currently known in the literature as special cases. The linearized versions of these solutions may be interpretted as Finslerian gravitational waves, an...

We investigate the local metrizability of Finsler spaces with m-Kropina metric F = α 1+ m β − m , where β is a closed null one-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric α and one-form β have a very specific form in certain coordinates. In particular, when the signature of α is Lorentzian, α bel...

Based on a diffusion tensor image (DTI) and a tentative tractogram of a fiber bundle we propose a filtering method for operationally defining and removing outliers using tractometry. To this end we assign to each track a set of K invariants, i.e. scalars invariant under rigid transformations. The cluster of K-tuples of all tracks in a bundle may be...

We investigate the local metrizability of Finsler spaces with $m$-Kropina metric $F = \alpha^{1+m}\beta^{-m}$, where $\beta$ is a closed null 1-form. We show that such a space is of Berwald type if and only if the (pseudo-)Riemannian metric $\alpha$ and 1-form $\beta$ have a very specific form in certain coordinates. In particular, when the signatu...

Berwald geometries are Finsler geometries close to (pseudo)-Riemannian geometries. We establish a simple first order partial differential equation as necessary and sufficient condition, which a given Finsler Lagrangian has to satisfy to be of Berwald type. Applied to (α,β)-Finsler spaces or spacetimes, respectively, this reduces to a necessary and...

We study theoretical and operational issues of geodesic tractography, a geometric methodology for retrieving biologically plausible neural fibers in the brain from diffusion weighted magnetic resonance imaging. The premise is that true positives are geodesics in a suitably constructed metric space, but unlike traditional first order methods these a...

Geodesic tractographyGeodesic tractography is an elegant, though typically time consuming method for finding connections or ‘tracks’ between given endpoints from diffusion-weighted MRI images, which can be representative of brain white matter fibers. In this work we consider the problem of constructing bundles of tracks between seed and target regi...

In this work, we study Randers spacetimes of Berwald type and analyze Pfeifer and Wohlfarth’s vacuum field equation of Finsler gravity for this class. We show that, in this case, the field equation is equivalent to the vanishing of the Finsler Ricci tensor, analogously to Einstein gravity. This implies that the considered vacuum field equation and...

Based on diffusion tensor imaging (DTI), one can construct a Riemannian manifold in which the dual metric is proportional to the DTI tensor. Geodesic tractography then amounts to solving a coupled system of nonlinear differential equations, either as initial value problem (given seed location and initial direction) or as boundary value problem (giv...

Clinical tractography is a challenging problem in diffusion tensor imaging (DTI) due to persistent validation issues. Geodesic tractography, based on a shortest path principle, is conceptually appealing, but has not produced convincing results so far. A major weakness is its rigidity with respect to candidate tracts it is capable of producing given...

In this work we study Randers spacetimes of Berwald type and analyze Pfeifer and Wohlfarth's vacuum field equation of Finsler gravity for this class. We show that in this case the field equation is equivalent to the vanishing of the Finsler Ricci tensor, analogously to Einstein gravity. This implies that the considered vacuum field equation and Rut...

We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent...

We investigate whether Szabo's metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern-Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent...

We propose a novel connectivity measure between brain regions using diffusion-weighted MRI. This connectivity measure is based on optimal sub-Finslerian geodesic front propagation on the 5D base manifold of (3D) positions and (2D) directions, the so-called sphere bundle. The advantage over spatial front propagations is that it prevents leakage at o...

In this work we demonstrate how Finsler geometry---and specifically the related geodesic tracto-graphy---can be levied to analyze structural connections between different brain regions. We present new theoretical developments which support the definition of a novel Finsler metric and associated connectivity measures, based on closely related works...

In this work, we study Berwald spacetimes and their vacuum dynamics, where the latter are based on a Finsler generalization of Einstein’s equations derived from an action on the unit tangent bundle. In particular, we consider a specific class of spacetimes that are nonflat generalizations of the very special relativity (VSR) line element, which we...

In this work we demonstrate how Finsler geometry -- and specifically the related geodesic tractography -- can be levied to analyze structural connections between different brain regions. We present new theoretical developments which support the definition of a novel Finsler metric and associated connectivity measures, based on closely related works...

Stephan Meesters promoveerde op 29 maart 2018 aan de Technische Universiteit Eindhoven op het proefschrift Functional and structural methods for minimally invasive treatment of epilepsy. Zijn proefschrift beschrijft de ontwikkeling van analysemethoden voor functionele en structurele beeldvorming ter ondersteuning van een minimaal invasieve behandel...

In this work we study Berwald spacetimes and their vacuum dynamics, where the latter are based on a Finsler generalization of the Einstein's equations derived from an action on the unit tangent bundle. In particular, we consider a specific class of spacetimes which are non-flat generalizations of the very special relativity (VSR) line element, to w...

Objective: The interictal epileptic discharges (IEDs) occurring in stereotactic EEG (SEEG) recordings are in general abundant compared to ictal discharges, but difficult to interpret due to complex underlying network interactions. A framework is developed to model these network interactions. Methods: To identify the synchronized neuronal activit...

The hypothesis that brain pathways form 2D sheet-like structures layered in 3D as “pages of a book” has been a topic of debate in the recent literature. This hypothesis was mainly supported by a qualitative evaluation of “path neighborhoods” reconstructed with diffusion MRI (dMRI) tractography. Notwithstanding the potentially important implications...

In this survey we review classical and recently proposed Riemannian metrics and interpolation schemes on the space of symmetric positive definite (SPD) matrices. We perform simulations that illustrate the problem of tensor fattening not only in the usually avoided Frobenius metric, but also in other classical metrics on SPD matrices such as the Was...

This volume offers a valuable starting point for anyone interested in learning computational diffusion MRI and mathematical methods for brain connectivity, while also sharing new perspectives and insights on the latest research challenges for those currently working in the field. Over the last decade, interest in diffusion MRI has virtually explode...

In this work we present Finsler gravitational waves. These are a Finslerian version of the well-known pp-waves, generalizing the very special relativity line element. Our Finsler pp-waves are an exact solution of Finslerian Einstein’s equations in vacuum and describe gravitational waves propagating in an anisotropic background.

The question whether our brain pathways adhere to a geometric grid structure has been a popular topic of debate in the diffusion imaging and neuroscience society. Wedeen et al. (2012a, b) proposed that the brain's white matter is organized like parallel sheets of interwoven pathways. Catani et al. (2012) concluded that this grid pattern is most lik...

Cardiac deformation and changes therein have been linked to pathologies. Both can be extracted in detail from tagged Magnetic Resonance Imaging (tMRI) using harmonic phase (HARP) images. Although point tracking algorithms have shown to have high accuracies on HARP images, these vary with position. Detecting and discarding areas with unreliable resu...

One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor. Such a metric is used for geodesic tractography and connectivity analysis in white matter. We propose a metric tensor given by the adjugate rather than the previously proposed inverse diffusion tensor. The adjugate metric can...

These Proceedings of the 2015 MICCAI Workshop “Computational Diffusion MRI” offer a snapshot of the current state of the art on a broad range of topics within the highly active and growing field of diffusion MRI. The topics vary from fundamental theoretical work on mathematical modeling, to the development and evaluation of robust algorithms, new c...

In this work we present a Finslerian version of the well-known pp-waves, which generalizes the very special relativity (VSR) line element. Our Finsler pp-waves are an exact solution of Finslerian Einstein's equations in vacuum.

In this short note we consider a method of enhancing diffusion MRI data based on analytically deblurring the ensemble average propagator. Because of the Fourier relationship between the normalized signal and the propagator, this technique can be applied in a straightforward manner to a large class of models. In the case of diffusion ten-sor imaging...

Changes in cardiac deformation patterns are correlated with cardiac pathologies. Deformation can be extracted from tagging Magnetic Resonance Imaging (tMRI) using Optical Flow (OF) techniques. For applications of OF in a clinical setting it is important to assess to what extent the performance of a particular OF method is stable across different cl...

By analyzing stochastic processes on a Riemannian manifold, in particular Brownian motion, one can deduce the metric structure of the space. This fact is implicitly used in diffusion tensor imaging of the brain when cast into a Riemannian framework. When modeling the brain white matter as a Riemannian manifold one finds (under some provisions) that...

Diffusion Tensor Imaging (DTI) is a popular model for representing diffusion weighted magnetic resonance images due to its simplicity and the fact that it strikes a good balance between signal fit and robustness. Nevertheless, problematic issues remain. One of these concerns the problem of interpolation. Because the DTI assumption forces Apparent D...

In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, espe...

One of the approaches in diffusion tensor imaging is to consider a Riemannian metric given by the inverse diffusion tensor . Such a metric is used for white matter tractography and connectivity analysis. We propose a modified metric tensor given by the adjugate rather than the inverse diffusion tensor. Tractography experiments on real brain diffusi...

We consider Riemann-Finsler geometry as a potentially powerful mathematical framework in the context of diffusion weighted magnetic resonance imaging. We explain its basic features in heuristic terms, but also provide mathematical details that are essential for practical applications, such as tractography and voxel-based classification. We stipulat...

We propose a new method to analyse deformation of the cardiac left ventricular wall from tagging magnetic resonance images. The method exploits the fact that the time-dependent frequency covector �field representing the tag pattern is tightly coupled to the myocardial deformation and not a�ffected by tag fading. Deformation and strain tensor fi�eld...

Intracranial depth electrodes are commonly used to identify the regions of the brain that are responsible for epileptic seizures. Knowledge of the exact location of the electrodes is important as to properly interpret the EEG in relation to the anatomy. In order to provide fast and accurate identification of these electrodes, a procedure has been d...

We consider a new method to analyse deformation of the myocardial wall from tagging magnetic resonance images. The method exploits the fact that a regular pattern of stripe tags induces a time-dependent frequency covector field tightly coupled to the myocardial tissue and not affected by tag fading. The corresponding local frequency can be disambig...

Diffusion imaging is a noninvasive tool for probing the microstructure of fibrous nerve and muscle tissue. Higher-order tensors provide a powerful mathemat-ical language to model and analyze the large and complex data that is generated by its modern variants such as High Angular Resolution Diffusion Imaging (HARDI) or Diffusional Kurtosis Imaging....

N.B. This is not the camera-ready, but a submitted first version. We investigate saddle points in 3D cardiac images. We do so by improving a critical point detection algorithm, the 3D winding number, or Poincaré index. We consider two different applications. We estimate cardiac motion from 3D tagged MRI data, based on tracking of saddle points. We...

We study a well-known scalar quantity in differential geometry , the Ricci scalar, in the context of Diffusion Tensor Imaging (DTI). We explore the relation between the Ricci scalar and the two most popular scalar measures in DTI: Mean Diffusivity and Fractional Anisotropy. We discuss results of computing the Ricci scalar on synthetic as well as re...

We consider new scalar quantities in the context of High Angular Resolution Diffusion Imaging (HARDI), namely, the principal invariants of fourth-order tensors modeling the diffusion profiles. We present the formalism needed to compute tensor invariants. We show results on the orthogonal basis of fourth-order tensor and on real HARDI phantom and br...

We develop a new formulation, mathematically elegant, to detect critical points of 3D scalar images. It is based on a topological number, which is the generalization to three dimensions of the 2D winding number. We illustrate our method by considering three different biomedical applications, namely, detection and counting of ovarian follicles and n...

We study a well-known scalar quantity in Riemannian geometry, the Ricci scalar, in the context of diffusion tensor imaging (DTI), which is an emerging non-invasive medical imaging modality. We derive a physical interpretation for the Ricci scalar and explore experimentally its significance in DTI. We also extend the definition of the Ricci scalar t...

We study a class of constant scalar invariant (CSI) space–times which belong to the higher-dimensional Kundt class and which are solutions of supergravity. We review the known CSI supergravity solutions in this class and we explicitly present a number of new exact CSI supergravity solutions, some of which are Einstein.

We show that the higher-dimensional vanishing scalar invariant (VSI) spacetimes with fluxes and dilaton are solutions of type IIB supergravity, and we argue that they are exact solutions in string theory. We also discuss the supersymmetry properties of VSI spacetimes.

We present the explicit form of higher dimensional VSI spacetimes in arbitrary number of dimensions. We discuss briefly the VSI's in the context of supergravity/strings.

We present the explicit metric forms for higher dimensional vanishing scalar invariant (VSI) Lorentzian spacetimes. We note that all of the VSI spacetimes belong to the higher dimensional Kundt class. We determine all of the VSI spacetimes which admit a covariantly constant null vector, and we note that in general in higher dimensions these spaceti...

We present new exact solutions of the Einstein-Yang-Mills system. The solutions are described by a null Yang-Mills field in a Kundt spacetime. They generalize a previously known solution for a metric of $pp$ wave type. The solutions are formally of Petrov type III. Comment: Talk presented at the XXVIII Spanish Relativity Meeting E.R.E. 2005, Oviedo...

We construct two distinct classes of exact type III solutions of the D=4 Einstein-Yang-Mills system. The solutions are embeddings of the non-abelian plane waves in spacetimes in Kundt's class. Reduction of the solutions to type N leads to generalized $pp$ and Kundt waves. The geodesic equations are briefly discussed.

The quantization of SU(2) Yang-Mills theory reduced to 0+1 space-time dimensions is performed in the BRST framework. We show that in the unitary gauge $A_0 = 0$ the BRST procedure has difficulties which can be solved by introduction of additional singlet ghost variables. In the Lorenz gauge $\dot{A}_0 = 0$ one has additional unphysical degrees of f...

The quantization of SU(2) Yang-Mills theory reduced to 0+1 space-time dimensions is performed in the BRST framework. We show that in the unitary gauge $A_0 = 0$ the BRST procedure has difficulties which can be solved by introduction of additional singlet ghost variables. In the Lorenz gauge $\dot{A}_0 = 0$ one has additional unphysical degrees of f...