Generalized hardi invariants by method of tensor contraction

pre-printWe propose a 3D object recognition technique to construct rotation invariant feature vectors for high angular resolution diffusion imaging (HARDI). This method uses the spherical harmonics (SH) expansion and is based on generating rank-1 contravariant tensors using the SH coefficients, and contracting them with covariant tensors to obtain invariants. The proposed technique enables the systematic construction of invariants for SH expansions of any order using simple mathematical operations. In addition, it allows construction of a large set of invariants, even for low order expansions, thus providing rich feature vectors for image analysis tasks such as classification and segmentation. In this paper, we use this technique to construct feature vectors for eighth-order fiber orientation distributions (FODs) reconstructed using constrained spherical deconvolution (CSD). Using simulated and in vivo brain data, we show that these invariants are robust to noise, enable voxel-wise classification, and capture meaningful information on the underlying white matter structure

Visualizing High-Order Symmetric Tensor Field Structure with Differential Operators

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionally and spatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic data sets of order 2, 3 and 4

Anisotropy Across Fields and Scales

This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018

Anisotropy Across Fields and Scales

This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain. The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas. This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28–November 2, 2018

Teisenemiseeskirjad ja invariandid skalaar-tensortüüpi gravitatsiooniteooriates

Einsteini üldrelatiivsusteooria, tänane gravitatsiooni standardteooria on tensorteooria. Keskseks objektiks selles on meetriline tensor ehk sisuliselt juhend lõikude pikkuste ja nendevaheliste nurkade arvutamiseks. Termin tensor rõhutab, et eelmainitud juhend, mida võib ette kujutada sisendiga ja väljundiga kastina, seab täpselt kahele lõigule (pikkuse korral sama lõik kaks korda) vastavusse ühe arvu, mille väärtuses erinevad vaatlejad nõustuvad. Einsteini teooria ideeks on uurida situatsiooni, kus lisaks välisele informatsioonile ehk lõikude omadustele sõltub väljund, arv, ka juhendi enda sisemisest informatsioonist. Just eelnev, pikkuste ja nurkade arvutamise juhendi sisemine informatsioon, on üldrelatiivsusteoorias dünaamiline ja aegruumi punktist sõltuv, lubades seega kirjeldada erinevaid kõveraid aegruume. Skalaar-tensortüüpi gravitatsiooniteooriates üldistatakse Einsteini teooriat, lisades sellele skalaarse suuruse, tensori sugulase, mis lõikudest sõltumatuna ehk puhtalt sisemise informatsiooni najal väljastab vaatlejast sõltumatu, kuid aegruumi punktist sõltuva arvu. Väitekirjas uurin teisendusi, mis segavad meetrilise tensori ja skalaari sisemist informatsiooni, jättes sealjuures aegruumi põhjusliku struktuuri muutumatuks. Täpsemalt uurin erinevate avaldiste teisenemist sellise segamise tagajärjel ning kirjutan välja muutumatud kombinatsioonid ehk niinimetatud invariandid. Viimaseid kasutan sama teooriate pere erinevate formuleeringute ja erinevate abstraktsuse tasemete vaheliste suhete selgitamiseks. Töö tulemusena järeldan, et mugavaim ja sirgjooneliseim formuleering teisenemiseeskirjade uurimiseks on kõige abstraktsem, sest siis igal avaldisel on kindel ja ühene teisenemiseeskiri.Einstein’s general relativity, current standard theory of gravity is a tensor theory. Its central object is the metric tensor, i.e., the prescription for calculating the length of line segments and angles between two of them. The term ‘tensor’ emphasizes that the prescription, pictured as a box with input and output, takes exactly two of those line segments (or twice the same in case of length) and produces one number which is the same for different observers. The key idea behind Einstein’s theory is to consider the situation where in addition to extrinsic information, i.e., the properties of the line segments, the output depends also on the intrinsic information of the prescription. It is the latter, the intrinsic information of the prescription for calculating lengths and angles that is dynamical and spacetime-point-dependent in general relativity, thus permitting to describe different curved spacetime geometries. In scalar-tensor theories of gravity Einstein’s theory is extended by adding a scalar, a sibling of a tensor that receives no line segments and thus produces an observer-independent number solely based on spacetime-point-dependent intrinsic information. In the thesis I study transformations that mix the intrinsic information of the metric tensor and the scalar in a manner that preserves the causal structure. In particular I consider how different expressions change under the transformations, and identify those which remain unchanged, i.e., the so-called invariants. The latter are used to clarify relations between different formulations of the same family of theories on different levels of abstraction. I conclude that the most convenient level for studying the transformation properties under such transformations is the most abstract one as in that case each expression has an explicit and unique transformation rule

Visualization of Tensor Fields in Mechanics

Tensors are used to describe complex physical processes in many applications. Examples include the distribution of stresses in technical materials, acting forces during seismic events, or remodeling of biological tissues. While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging. Visualization can be beneficial here and is frequently used by domain experts. Typical strategies include the use of glyphs, color plots, lines, and isosurfaces. However, data complexity is nowadays accompanied by the sheer amount of data produced by large-scale simulations and adds another level of obstruction between user and data. Given the limitations of traditional methods, and the extra cognitive effort of simple methods, more advanced tensor field visualization approaches have been the focus of this work. This survey aims to provide an overview of recent research results with a strong application-oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio-, and geomechanics. As such, the survey is complementing and extending previously published surveys. Its utility is twofold: (i) It serves as basis for the visualization community to get an overview of recent visualization techniques. (ii) It emphasizes and explains the necessity for further research for visualizations in this context

Topological visualization of tensor fields using a generalized Helmholtz decomposition

Analysis and visualization of fluid flow datasets has become increasing important with the development of computer graphics. Even though many direct visualization methods have been applied in the tensor fields, those methods may result in much visual clutter. The Helmholtz decomposition has been widely used to analyze and visualize the vector fields, and it is also a useful application in the topological analysis of vector fields. However, there has been no previous work employing the Helmholtz decomposition of tensor fields. We present a method for computing the Helmholtz decomposition of tensor fields of arbitrary order and demonstrate its application. The Helmholtz decomposition can split a tensor field into divergence-free and curl-free parts. The curl-free part is irrotational, and it is useful to isolate the local maxima and minima of divergence (foci of sources and sinks) in the tensor field without interference from curl-based features. And divergence-free part is solenoidal, and it is useful to isolate centers of vortices in the tensor field. Topological visualization using this decomposition can classify critical points of two-dimensional tensor fields and critical lines of 3D tensor fields. Compared with several other methods, this approach is not dependent on computing eigenvectors, tensor invariants, or hyperstreamlines, but it can be computed by solving a sparse linear system of equations based on finite difference approximation operators. Our approach is an indirect visualization method, unlike the direct visualization which may result in the visual clutter. The topological analysis approach also generates a single separating contour to roughly partition the tensor field into irrotational and solenoidal regions. Our approach will make use of the 2nd order and the 4th order tensor fields. This approach can provide a concise representation of the global structure of the field, and provide intuitive and useful information about the structure of tensor fields. However, this method does not extract the exact locations of critical points and lines

Joint Spatial-Angular Sparse Coding, Compressed Sensing, and Dictionary Learning for Diffusion MRI

Neuroimaging provides a window into the inner workings of the human brain to diagnose and prevent neurological diseases and understand biological brain function, anatomy, and psychology. Diffusion Magnetic Resonance Imaging (dMRI) is an emerging medical imaging modality used to study the anatomical network of neurons in the brain, which form cohesive bundles, or fiber tracts, that connect various parts of the brain. Since about 73% of the brain is water, measuring the flow, or diffusion of water molecules in the presence of fiber bundles, allows researchers to estimate the orientation of fiber tracts and reconstruct the internal wiring of the brain, in vivo. Diffusion MRI signals can be modeled within two domains: the spatial domain consisting of voxels in a brain volume and the diffusion or angular domain, where fiber orientation is estimated in each voxel. Researchers aim to estimate the probability distribution of fiber orientation in every voxel of a brain volume in order to trace paths of fiber tracts from voxel to voxel over the entire brain. Therefore, the traditional framework for dMRI processing and analysis has been from a voxel-wise vantage point with added spatial regularization considered post-hoc. In contrast, we propose a new joint spatial-angular representation of dMRI data which pairs signals in each voxel with the global spatial environment, jointly. This has the ability to improve many aspects of dMRI processing and analysis and re-envision the core representation of dMRI data from a local perspective to a global one. In this thesis, we propose three main contributions which take advantage of such joint spatial-angular representations to improve major machine learning tasks applied to dMRI: sparse coding, compressed sensing, and dictionary learning. First, we will show that we can achieve sparser representations of dMRI by utilizing a global spatial-angular dictionary instead of a purely voxel-wise angular dictionary. As dMRI data is very large in size, we provide a number of novel extensions to popular spare coding algorithms that perform efficient optimization on a global-scale by exploiting the separability of our dictionaries over the spatial and angular domains. Next, compressed sensing is used to accelerate signal acquisition based on an underlying sparse representation of the data. We will show that our proposed representation has the potential to push the limits of the current state of scanner acceleration within a new compressed sensing model for dMRI. Finally, sparsity can be further increased by learning dictionaries directly from datasets of interest. Prior dictionary learning for dMRI learn angular dictionaries alone. Our third contribution is to learn spatial-angular dictionaries jointly from dMRI data directly to better represent the global structure. Traditionally, the problem of dictionary learning is non-convex with no guarantees of finding a globally optimal solution. We derive the first theoretical results of global optimality for this class of dictionary learning problems. We hope the core foundation of a joint spatial-angular representation will open a new perspective on dMRI with respect to many other processing tasks and analyses. In addition, our contributions are applicable to any general signal types that can benefit from separable dictionaries. We hope the contributions in this thesis may be adopted in the larger signal processing, computer vision, and machine learning communities. dMRI signals can be modeled within two domains: the spatial domain consisting of voxels in a brain volume and the diffusion or angular domain, where fiber orientation is estimated in each voxel. Computationally speaking, researchers aim to estimate the probability distribution of fiber orientation in every voxel of a brain volume in order to trace paths of fiber tracts from voxel to voxel over the entire brain. Therefore, the traditional framework for dMRI processing and analysis is from a voxel-wise, or angular, vantage point with post-hoc consideration of their local spatial neighborhoods. In contrast, we propose a new global spatial-angular representation of dMRI data which pairs signals in each voxel with the global spatial environment, jointly, to improve many aspects of dMRI processing and analysis, including the important need for accelerating the otherwise time-consuming acquisition of advanced dMRI protocols. In this thesis, we propose three main contributions which utilize our joint spatial-angular representation to improve major machine learning tasks applied to dMRI: sparse coding, compressed sensing, and dictionary learning. We will show that sparser codes are possible by utilizing a global dictionary instead of a voxel-wise angular dictionary. This allows for a reduction of the number of measurements needed to reconstruct a dMRI signal to increase acceleration using compressed sensing. Finally, instead of learning angular dictionaries alone, we learn spatial-angular dictionaries jointly from dMRI data directly to better represent the global structure. In addition, this problem is non-convex and so we derive the first theories to guarantee convergence to a global minimum. As dMRI data is very large in size, we provide a number of novel extensions to popular algorithms that perform efficient optimization on a global-scale by exploiting the separability of our global dictionaries over the spatial and angular domains. We hope the core foundation of a joint spatial-angular representation will open a new perspective on dMRI with respect to many other processing tasks and analyses. In addition, our contributions are applicable to any separable dictionary setting which we hope may be adopted in the larger image processing, computer vision, and machine learning communities

Anizotropna radna okruženja za dinamičke sisteme i obradu slika

The research topic of this PhD thesis is a comparative analysis of classical specic geometric frameworks and of their anisotropic extensions; the construction of three different types of Finsler frameworks, which are suitable for the analysis of the cancer cells population dynamical system; the development of the anisotropic Beltrami framework theory with the derivation of the evolution ow equations corresponding to different classes of anisotropic metrics, and tentative applications in image processing.Predmet istraživanja doktorske disertacije je uporedna analiza klasičnih i specifičnih geometrijskih radnih okruženja i njihovih anizotropnih proširenja; konstrukcija  tri Finslerova radna okruženja različitog tipa koja su pogodna za analizu dinamičkog  sistema populacije kanceroznih ćelija; razvoj teorije anizotropnog Beltramijevog radnog okruženja i formiranje jednačina evolutivnog toka za različite klase anizotropnih metrika, kao i mogućnost primene dobijenih teorijskih rezultata u digitalnoj obradi slika

Väändel baseeruvad gravitatsiooniteooriad: teoreetilised ja vaatluslikud piirangud

Kuigi üldrelatiivsusteooria (ÜRT) on väga erinevate nähtuste kirjeldamisel olnud märkimisväärselt edukas, ei anna see head seletust mitmetele vaatlustele kaasaegses kosmoloogias: kosmilise taustkiirguse homogeensus, universumi kiirenev paisumine, struktuuri päritolu ja galaktikate pöörlemine. Nimetatud vaatlusi saab seletada inflatsiooni, tumeenergia ja tumeaine sissetoomisega, kuid nonde täpne olemus on seni teadmata. Antud põhjused on motiveerinud uurima suurt hulka võimalikke ÜRT modifikatsioone. Selles töös ei vaata me aga ÜRT enda modifikatsioone, vaid mitmeid teooriaid, mis erinevad sellest aluseks oleva geomeetrilise kirjelduse poolest. Kõveruse asemel kasutavad need teleparalleelsed teooriad gravitatsiooni vahendajana väänet või mittemeetrilisust. Töö eesmärk on kontrollida antud teooriate vastavust vaatlustele. Mainitud geomeetriates on oluline teoreetiline küsimus lokaalne Lorentzi invariantsus. Väändepõhistes teooriates käsitletakse seda küsimust kovariantsel kujul, mis taastab Lorenzi invariantsuse vastava kalibratsioonivälja sissetoomisega. Nõnda esitame esmalt skalaar-väändegravitatsiooni kovariantse formulatsiooni. Seejärel uurime gravitatsioonilainete levikut teleparalleelsete gravitatsiooniteooriate kahes üldises perekonnas, mis põhinevad gravitatsiooni erinevatel geomeetrilistel tõlgendustel. Meie tulemused näitavad, et kõik lainete režiimid levivad valguse kiirusel ja omavad kuni 6 võimalikku polarisatsiooni. Tuletame ka teleparalleelsete gravitatsiooniteooriate üldklassi post-Newtoni piiri. Meie tulemused näitavad, et vaadeldav teooriate klass on täielikult konservatiivne, mis tähendab, et ei lokaalset Lorentzi ja asukoha invariantsust ega energiaimpulsside täielikku jäävust ei rikuta. Lõpuks kasutame dünaamilisi süsteeme kosmoloogialiste mudelite mitmete aspektide kirjeldamiseks nagu kiirendus, fantoomne tume energia ja lõpliku aja singulaarsused f(T) teooriates.Although General Relativity (GR) has been very successful in describing a wide range of phenomena, by itself does not provide any explanation for a number of observations in modern cosmology: the homogeneity of the cosmic microwave background, the accelerating expansion of the universe, the origin of structure and the motion of galaxies. These observations can be explained by introducing inflation, dark energy and dark matter, but the precise nature of these has remained unknown. These reasons have motivated the study of a large number of possible modifications of GR. However, in this thesis we do not study the modifications of GR itself, but a number of theories that differ from it by the underlying geometric description. Instead of curvature, these teleparallel theories employ torsion or nonmetricity as the mediator of gravity. The aim of the thesis is to check the viability of these theories. An important theoretical issue in the mentioned geometries is local Lorentz invariance. In torsion-based theories, this issue is addressed in their covariant formulation, which restores the Lorenz invariance by introducing a corresponding gauge field. In particular, we present the covariant formulation of scalartorsion gravity. Then we study the gravitational wave (GW) propagation in two general families of teleparallel gravity theories, which are based on different geometric interpretations of gravity. Our results show that all GW modes propagate at the speed of light and there are up to 6 possible polarizations. We also derive the post-Newtonian limit of a general class of teleparallel gravity theories. Our results show that the class of theories we consider is fully conservative, which means that there are no violation of local Lorentz invariance, local position invariance or total energy-momentum conservation. Finally, we use the dynamical systems to describe a wide range of phenomena in cosmology, like acceleration, phantom dark energy and finite time singularities in f(T) theories.https://www.ester.ee/record=b545040