Studying brain connectivity: a new multimodal approach for structure and function integration \u200b

Il cervello \ue8 un sistema che integra organizzazioni anatomiche e funzionali. Negli ultimi dieci anni, la comunit\ue0 neuroscientifica si \ue8 posta la domanda sulla relazione struttura-funzione. Essa pu\uf2 essere esplorata attraverso lo studio della connettivit\ue0. Nello specifico, la connettivit\ue0 strutturale pu\uf2 essere definita dal segnale di risonanza magnetica pesato in diffusione seguito dalla computazione della trattografia; mentre la correlazione funzionale del cervello pu\uf2 essere calcolata a partire da diversi segnali, come la risonanza magnetica funzionale o l\u2019elettro-/magneto-encefalografia, che consente la cattura del segnale di attivazione cerebrale a una risoluzione temporale pi\uf9 elevata. Recentemente, la relazione struttura-funzione \ue8 stata esplorata utilizzando strumenti di elaborazione del segnale sui grafi, che estendono e generalizzano le operazioni di elaborazione del segnale ai grafi. In specifico, alcuni studi utilizzano la trasformata di Fourier applicata alla connettivit\ue0 strutturale per misurare la decomposizione del segnale funzionale in porzioni che si allineano (\u201caligned\u201d) e non si allineano (\u201cliberal\u201d) con la sottostante rete di materia bianca. Il relativo allineamento funzionale con l\u2019anatomia \ue8 stato associato alla flessibilit\ue0 cognitiva, sottolineando forti allineamenti di attivit\ue0 corticali, e suggerendo che i sistemi sottocorticali contengono pi\uf9 segnali liberi rispetto alla corteccia. Queste relazioni multimodali non sono, per\uf2, ancora chiare per segnali con elevata risoluzione temporale, oltre ad essere ristretti a specifiche zone cerebrali. Oltretutto, al giorno d'oggi la ricostruzione della trattografia \ue8 ancora un argomento impegnativo, soprattutto se utilizzata per l'estrazione della connettivit\ue0 strutturale. Nel corso dell'ultimo decennio si \ue8 vista una proliferazione di nuovi modelli per ricostruire la trattografia, ma il loro conseguente effetto sullo strumento di connettivit\ue0 non \ue8 ancora chiaro. In questa tesi, ho districato i dubbi sulla variabilit\ue0 dei trattogrammi derivati da diversi metodi di trattografia, confrontandoli con un paradigma di test-retest, che consente di definire la specificit\ue0 e la sensibilit\ue0 di ciascun modello. Ho cercato di trovare un compromesso tra queste, per definire un miglior metodo trattografico. Inoltre, ho affrontato il problema dei grafi pesati confrontando alcune possibili stime, evidenziando la sufficienza della connettivit\ue0 binaria e la potenza delle propriet\ue0 microstrutturali di nuova generazione nelle applicazioni cliniche. Qui, ho sviluppato un modello di proiezione che consente l'uso dei filtri aligned e liberal per i segnali di encefalografia. Il modello estende i vincoli strutturali per considerare le connessioni indirette, che recentemente si sono dimostrate utili nella relazione struttura-funzione. I risultati preliminari del nuovo modello indicano un\u2019implicazione dinamica di momenti pi\uf9 aligned e momenti pi\uf9 liberal, evidenziando le fluttuazioni presenti nello stato di riposo. Inoltre, viene presentata una relazione specifica di periodi pi\uf9 allineati e liberali per il paradigma motorio. Questo modello apre la prospettiva alla definizione di nuovi biomarcatori. Considerando che l\u2019encefalografia \ue8 spesso usata nelle applicazioni cliniche, questa integrazione multimodale applicata su dati di Parkinson o di ictus potrebbe combinare le informazioni dei cambiamenti strutturali e funzionali nelle connessioni cerebrali, che al momento sono state dimostrate individualmente.The brain is a complex system of which anatomical and functional organization is both segregated and integrated. A longstanding question for the neuroscience community has been to elucidate the mutual influences between structure and function. To that aim, first, structural and functional connectivity need to be explored individually. Structural connectivity can be measured by the Diffusion Magnetic Resonance signal followed by successive computational steps up to virtual tractography. Functional connectivity can be established by correlation between the brain activity time courses measured by different modalities, such as functional Magnetic Resonance Imaging or Electro/Magneto Encephalography. Recently, the Graph Signal Processing (GSP) framework has provided a new way to jointly analyse structure and function. In particular, this framework extends and generalizes many classical signal-processing operations to graphs (e.g., spectral analysis, filtering, and so on). The graph here is built by the structural connectome; i.e., the anatomical backbone of the brain where nodes represent brain regions and edge weights strength of structural connectivity. The functional signals are considered as time-dependent graph signals; i.e., measures associated to the nodes of the graph. The concept of the Graph Fourier Transform then allows decomposing regional functional signals into, on one side, a portion that strongly aligned with the underlying structural network (\u201caligned"), and, on the other side, a portion that is not well aligned with structure (\u201cliberal"). The proportion of aligned-vs-liberal energy in functional signals has been associated with cognitive flexibility. However, the interpretation of these multimodal relationships is still limited and unexplored for higher temporal resolution functional signals such as M/EEG. Moreover, the construction of the structural connectome itself using tractography is still a challenging topic, for which, in the last decade, many new advanced models were proposed, but their impact on the connectome remains unclear. In the first part of this thesis, I disentangled the variability of tractograms derived from different tractography methods, comparing them with a test-retest paradigm, which allows to define specificity and sensitivity of each model. I want to find the best trade-off between specificity and sensitivity to define the best model that can be deployed for analysis of functional signals. Moreover, I addressed the issue of weighing the graph comparing few estimates, highlighting the sufficiency of binary connectivity, and the power of the latest-generation microstructural properties in clinical applications. In the second part, I developed a GSP method that allows applying the aligned and liberal filters to M/EEG signals. The model extends the structural constraints to consider indirect connections, which recently demonstrated to be powerful in the structure/function link. I then show that it is possible to identify dynamic changes in aligned-vs-liberal energy, highlighting fluctuations present motor task and resting state. This model opens the perspective of novel biomarkers. Indeed, M/EEG are often used in clinical applications; e.g., multimodal integration in data from Parkinson\u2019s disease or stroke could combine changes of both structural and functional connectivity

Tractographie de la matière blanche orientée par a priori anatomiques et microstructurels

Diffusion-weighted magnetic resonance imaging is a unique imaging modality sensitive to the microscopic movement of water molecules in biological tissues. By characterizing the movement of water molecules, it is possible to infer the macroscopic neuronal pathways of the brain. The technique, so-called tractography, had become the tool of choice to study non-invasively the human brain's white matter in vivo. For instance, it has been used in neurosurgical intervention planning and in neurodegenerative diseases monitoring. In this thesis, we report biases from current tractography reconstruction and suggest methods to reduce them. We first use anatomical priors, derived from a high resolution T1-weighted image, to guide tractography. We show that knowledge of the nature of biological tissue helps tractography to reconstruct anatomically valid neuronal pathways, and reduces biases in the estimation of complex white matter regions. We then use microstructural priors, derived from the state-of-the-art diffusionweighted magnetic resonance imaging protocol, in the tractography reconstruction process. This allows tractography to follow the movement of water molecules not only along neuronal pathways, but also in a microstructurally specific environment. Thus, the tractography distinguishes more accurately neuronal pathways and reduces reconstruction errors. Moreover, it provides the mean to study white matter microstructure characteristics along neuronal pathways. Altogether, we show that anatomical and microstructural priors used during the tractography process improve brain’s white matter reconstruction.L’imagerie par résonance magnétique pondérée en diffusion est une modalité unique sensible aux mouvements microscopiques des molécules d’eau dans les tissus biologiques. Il est possible d’utiliser les caractéristiques de ce mouvement pour inférer la structure macroscopique des faisceaux de la matière blanche du cerveau. La technique, appelée tractographie, est devenue l’outil de choix pour étudier cette structure de façon non invasive. Par exemple, la tractographie est utilisée en planification neurochirurgicale et pour le suivi du développement de maladies neurodégénératives.Dans cette thèse, nous exposons certains des biais introduits lors de reconstructions par tractographie, et des méthodes sont proposées pour les réduire. D’abord, nous utilisons des connaissances anatomiques a priori pour orienter la reconstruction. Ainsi, nous montrons que l’information anatomique sur la nature des tissus permet d'estimer des faisceaux anatomiquement plausibles et de réduire les biais dans l’estimation de structures complexes de la matière blanche. Ensuite, nous utilisons des connnaissances microstructurelles a priori dans la reconstruction, afin de permettre à la tractographie de suivre le mouvement des molécules d’eau non seulement le long des faisceaux, mais aussi dans des milieux microstructurels spécifiques. La tractographie peut ainsi distinguer différents faisceaux, réduire les erreurs de reconstruction et permettre l’étude de la microstructure le long de la matière blanche. Somme toute, nous montrons que l’utilisation de connaissances anatomiques et microstructurelles a priori, en tractographie, augmente l’exactitude des reconstructions de la matière blanche du cerveau

Fast diffusion MRI based on sparse acquisition and reconstruction for long-term population imaging

Diffusion weighted magnetic resonance imaging (dMRI) is a unique MRI modality to probe the diffusive molecular transport in biological tissue. Due to its noninvasiveness and its ability to investigate the living human brain at submillimeter scale, dMRI is frequently performed in clinical and biomedical research to study the brain’s complex microstructural architecture. Over the last decades large prospective cohort studies have been set up with the aim to gain new insights into the development and progression of brain diseases across the life span and to discover biomarkers for disease prediction and potentially prevention. To allow for diverse brain imaging using different MRI modalities, stringent scan time limits are typically imposed in population imaging. Nevertheless, population studies aim to apply advanced and thereby time consuming dMRI protocols that deliver high quality data with great potential for future analysis. To allow for time-efficient but also versatile diffusion imaging, this thesis contributes to the investigation of accelerating diffusion spectrum imaging (DSI), an advanced dMRI technique that acquires imaging data with high intra-voxel resolution of tissue microstructure. Combining state-of-the-art parallel imaging and the theory of compressed sensing (CS) enables the acceleration of spatial encoding and diffusion encoding in dMRI. In this way, the otherwise long acquisition times in DSI can be reduced significantly. In this thesis, first, suitable q-space sampling strategies and basis functions are explored that fulfill the requirements of CS theory for accurate sparse DSI reconstruction. Novel 3D q-space sample distributions are investigated for CS-DSI. Moreover, conventional CS-DSI based on the discrete Fourier transform is compared for the first time to CS-DSI based on the continuous SHORE (simple harmonic oscillator based reconstruction and estimation) basis functions. Based on these findings, a CS-DSI protocol is proposed for application in a prospective cohort study, the Rhineland Study. A pilot study was designed and conducted to evaluate the CS-DSI protocol in comparison with state-of-the-art 3-shell dMRI and dedicated protocols for diffusion tensor imaging (DTI) and for the combined hindered and restricted model of diffusion (CHARMED). Population imaging requires processing techniques preferably with low computational cost to process and analyze the acquired big data within a reasonable time frame. Therefore, a pipeline for automated processing of CS-DSI acquisitions was implemented including both in-house developed and existing state-of-the-art processing tools. The last contribution of this thesis is a novel method for automatic detection and imputation of signal dropout due to fast bulk motion during the diffusion encoding in dMRI. Subject motion is a common source of artifacts, especially when conducting clinical or population studies with children, the elderly or patients. Related artifacts degrade image quality and adversely affect data analysis. It is, thus, highly desired to detect and then exclude or potentially impute defective measurements prior to dMRI analysis. Our proposed method applies dMRI signal modeling in the SHORE basis and determines outliers based on the weighted model residuals. Signal imputation reconstructs corrupted and therefore discarded measurements from the sparse set of inliers. This approach allows for fast and robust correction of imaging artifacts in dMRI which is essential to estimate accurate and precise model parameters that reflect the diffusive transport of water molecules and the underlying microstructural environment in brain tissue.Die diffusionsgewichtete Magnetresonanztomographie (dMRT) ist ein einzigartiges MRTBildgebungsverfahren, um die Diffusionsbewegung von Wassermolekülen in biologischem Gewebe zu messen. Aufgrund der Möglichkeit Schichtbilder nicht invasiv aufzunehmen und das lebende menschliche Gehirn im Submillimeter-Bereich zu untersuchen, ist die dMRT ein häufig verwendetes Bildgebungsverfahren in klinischen und biomedizinischen Studien zur Erforschung der komplexen mikrostrukturellen Architektur des Gehirns. In den letzten Jahrzehnten wurden große prospektive Kohortenstudien angelegt, um neue Einblicke in die Entwicklung und den Verlauf von Gehirnkrankheiten über die Lebenspanne zu erhalten und um Biomarker zur Krankheitserkennung und -vorbeugung zu bestimmen. Um durch die Verwendung unterschiedlicher MRT-Verfahren verschiedenartige Schichtbildaufnahmen des Gehirns zu ermöglich, müssen Scanzeiten typischerweise stark begrenzt werden. Dennoch streben Populationsstudien die Anwendung von fortschrittlichen und daher zeitintensiven dMRT-Protokollen an, um Bilddaten in hoher Qualität und mit großem Potential für zukünftige Analysen zu akquirieren. Um eine zeiteffizente und gleichzeitig vielseitige Diffusionsbildgebung zu ermöglichen, leistet diese Dissertation Beiträge zur Untersuchung von Beschleunigungsverfahren für die Bildgebung mittels diffusion spectrum imaging (DSI). DSI ist ein fortschrittliches dMRT-Verfahren, das Bilddaten mit hoher intra-voxel Auflösung der Gewebestruktur erhebt. Werden modernste Verfahren zur parallelen MRT-Bildgebung mit der compressed sensing (CS) Theorie kombiniert, ermöglicht dies eine Beschleunigung der räumliche Kodierung und der Diffusionskodierung in der dMRT. Dadurch können die ansonsten langen Aufnahmezeiten für DSI erheblich reduziert werden. In dieser Arbeit werden zuerst geeigenete Strategien zur Abtastung des q-space sowie Basisfunktionen untersucht, welche die Anforderungen der CS-Theorie für eine korrekte Signalrekonstruktion der dünnbesetzten DSI-Daten erfüllen. Neue 3D-Verteilungen von Messpunkten im q-space werden für die Verwendung in CS-DSI untersucht. Außerdem wird konventionell auf der diskreten Fourier-Transformation basierendes CS-DSI zum ersten Mal mit einem CS-DSI Verfahren verglichen, welches kontinuierliche SHORE (simple harmonic oscillator based reconstruction and estimation) Basisfunktionen verwendet. Aufbauend auf diesen Ergebnissen wird ein CS-DSI-Protokoll zur Anwendung in einer prospektiven Kohortenstudie, der Rheinland Studie, vorgestellt. Eine Pilotstudie wurde entworfen und durchgeführt, um das CS-DSI-Protokoll im Vergleich mit modernster 3-shell-dMRT und mit dedizierten Protokollen für diffusion tensor imaging (DTI) und für das combined hindered and restricted model of diffusion (CHARMED) zu evaluieren. Populationsbildgebung erfordert Prozessierungsverfahren mit möglichst geringem Rechenaufwand, um große akquirierte Datenmengen in einem angemessenen Zeitrahmen zu verarbeiten und zu analysieren. Dafür wurde eine Pipeline zur automatisierten Verarbeitung von CS-DSI-Daten implementiert, welche sowohl eigenentwickelte als auch bereits existierende moderene Verarbeitungsprogramme enthält. Der letzte Beitrag dieser Arbeit ist eine neue Methode zur automatischen Detektion und Imputation von Signalabfall, welcher durch schnelle Bewegungen während der Diffusionskodierung in der dMRT entsteht. Bewegungen der Probanden während der dMRT-Aufnahme sind eine häufige Ursache für Bildfehler, vor allem in klinischen oder Populationsstudien mit Kindern, alten Menschen oder Patienten. Diese Artefakte vermindern die Datenqualität und haben einen negativen Einfluss auf die Datenanalyse. Daher ist es das Ziel, fehlerhafte Messungen vor der dMRI-Analyse zu erkennen und dann auszuschließen oder wenn möglich zu ersetzen. Die vorgestellte Methode verwendet die SHORE-Basis zur dMRT-Signalmodellierung und bestimmt Ausreißer mit Hilfe von gewichteten Modellresidualen. Die Datenimputation rekonstruiert die unbrauchbaren und daher verworfenen Messungen mit Hilfe der verbleibenden, dünnbesetzten Menge an Messungen. Dieser Ansatz ermöglicht eine schnelle und robuste Korrektur von Bildartefakten in der dMRT, welche erforderlich ist, um korrekte und präzise Modellparameter zu schätzen, die die Diffusionsbewegung von Wassermolekülen und die zugrundeliegende Mikrostruktur des Gehirngewebes reflektieren

Tractographie de la matière blanche orientée par a priori anatomiques et microstructurels = White matter tractography guided by anatomical and microstructural priors

Résumé : L'imagerie par résonance magnétique pondérée en diffusion est une modalité unique sensible aux mouvements microscopiques des molécules d'eau dans les tissus biologiques. Il est possible d'utiliser les caractéristiques de ce mouvement pour inférer la structure macroscopique des faisceaux de la matière blanche du cerveau. La technique, appelée tractographie, est devenue l'outil de choix pour étudier cette structure de façon non invasive. Par exemple, la tractographie est utilisée en planification neurochirurgicale et pour le suivi du développement de maladies neurodégénératives. Dans cette thèse, nous exposons certains des biais introduits lors de reconstructions par tractographie, et des méthodes sont proposées pour les réduire. D'abord, nous utilisons des connaissances anatomiques a priori pour orienter la reconstruction. Ainsi, nous montrons que l'information anatomique sur la nature des tissus permet d'estimer des faisceaux anatomiquement plausibles et de réduire les biais dans l'estimation de structures complexes de la matière blanche. Ensuite, nous utilisons des connaissances microstructurelles a priori dans la reconstruction, afin de permettre à la tractographie de suivre le mouvement des molécules d'eau non seulement le long des faisceaux, mais aussi dans des milieux microstructurels spécifiques. La tractographie peut ainsi distinguer différents faisceaux, réduire les erreurs de reconstruction et permettre l'étude de la microstructure le long de la matière blanche. Somme toute, nous montrons que l'utilisation de connaissances anatomiques et microstructurelles a priori, en tractographie, augmente l'exactitude des reconstructions de la matière blanche du cerveau.Abstract : Diffusion-weighted magnetic resonance imaging is a unique imaging modality sensitive to the microscopic movement of water molecules in biological tissues. By characterizing the movement of water molecules, it is possible to infer the macroscopic neuronal pathways of the brain. The technique, so-called tractography, had become the tool of choice to study non-invasively the human brain's white matter in vivo. For instance, it has been used in neurosurgical intervention planning and in neurodegenerative diseases monitoring. In this thesis, we report biases from current tractography reconstruction and suggest methods to reduce them. We first use anatomical priors, derived from a high resolution T1-weighted image, to guide tractography. We show that knowledge of the nature of biological tissue helps tractography to reconstruct anatomically valid neuronal pathways, and reduces biases in the estimation of complex white matter regions. We then use microstructural priors, derived from the state-of-the-art diffusion-weighted magnetic resonance imaging protocol, in the tractography reconstruction process. This allows tractography to follow the movement of water molecules not only along neuronal pathways, but also in a microstructurally specific environment. Thus, the tractography distinguishes more accurately neuronal pathways and reduces reconstruction errors. Moreover, it provides the means to study white matter microstructure characteristics along neuronal pathways. Altogether, we show that anatomical and microstructural priors used during the tractography process improve brain's white matter reconstruction

EXplainable Artificial Intelligence: enabling AI in neurosciences and beyond

The adoption of AI models in medicine and neurosciences has the potential to play a significant role not only in bringing scientific advancements but also in clinical decision-making. However, concerns mounts due to the eventual biases AI could have which could result in far-reaching consequences particularly in a critical field like biomedicine. It is challenging to achieve usable intelligence because not only it is fundamental to learn from prior data, extract knowledge and guarantee generalization capabilities, but also to disentangle the underlying explanatory factors in order to deeply understand the variables leading to the final decisions. There hence has been a call for approaches to open the AI `black box' to increase trust and reliability on the decision-making capabilities of AI algorithms. Such approaches are commonly referred to as XAI and are starting to be applied in medical fields even if not yet fully exploited. With this thesis we aim at contributing to enabling the use of AI in medicine and neurosciences by taking two fundamental steps: (i) practically pervade AI models with XAI (ii) Strongly validate XAI models. The first step was achieved on one hand by focusing on XAI taxonomy and proposing some guidelines specific for the AI and XAI applications in the neuroscience domain. On the other hand, we faced concrete issues proposing XAI solutions to decode the brain modulations in neurodegeneration relying on the morphological, microstructural and functional changes occurring at different disease stages as well as their connections with the genotype substrate. The second step was as well achieved by firstly defining four attributes related to XAI validation, namely stability, consistency, understandability and plausibility. Each attribute refers to a different aspect of XAI ranging from the assessment of explanations stability across different XAI methods, or highly collinear inputs, to the alignment of the obtained explanations with the state-of-the-art literature. We then proposed different validation techniques aiming at practically fulfilling such requirements. With this thesis, we contributed to the advancement of the research into XAI aiming at increasing awareness and critical use of AI methods opening the way to real-life applications enabling the development of personalized medicine and treatment by taking a data-driven and objective approach to healthcare

Metaheuristic design of feedforward neural networks: a review of two decades of research

Over the past two decades, the feedforward neural network (FNN) optimization has been a key interest among the researchers and practitioners of multiple disciplines. The FNN optimization is often viewed from the various perspectives: the optimization of weights, network architecture, activation nodes, learning parameters, learning environment, etc. Researchers adopted such different viewpoints mainly to improve the FNN's generalization ability. The gradient-descent algorithm such as backpropagation has been widely applied to optimize the FNNs. Its success is evident from the FNN's application to numerous real-world problems. However, due to the limitations of the gradient-based optimization methods, the metaheuristic algorithms including the evolutionary algorithms, swarm intelligence, etc., are still being widely explored by the researchers aiming to obtain generalized FNN for a given problem. This article attempts to summarize a broad spectrum of FNN optimization methodologies including conventional and metaheuristic approaches. This article also tries to connect various research directions emerged out of the FNN optimization practices, such as evolving neural network (NN), cooperative coevolution NN, complex-valued NN, deep learning, extreme learning machine, quantum NN, etc. Additionally, it provides interesting research challenges for future research to cope-up with the present information processing era

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

Two-compartment neuronal spiking model expressing brain-state specific apical-amplification, -isolation and -drive regimes

There is mounting experimental evidence that brain-state specific neural mechanisms supported by connectomic architectures serve to combine past and contextual knowledge with current, incoming flow of evidence (e.g. from sensory systems). Such mechanisms are distributed across multiple spatial and temporal scales and require dedicated support at the levels of individual neurons and synapses. A prominent feature in the neocortex is the structure of large, deep pyramidal neurons which show a peculiar separation between an apical dendritic compartment and a basal dentritic/peri-somatic compartment, with distinctive patterns of incoming connections and brain-state specific activation mechanisms, namely apical-amplification, -isolation and -drive associated to the wakefulness, deeper NREM sleep stages and REM sleep. The cognitive roles of apical mechanisms have been demonstrated in behaving animals. In contrast, classical models of learning spiking networks are based on single compartment neurons that miss the description of mechanisms to combine apical and basal/somatic information. This work aims to provide the computational community with a two-compartment spiking neuron model which includes features that are essential for supporting brain-state specific learning and with a piece-wise linear transfer function (ThetaPlanes) at highest abstraction level to be used in large scale bio-inspired artificial intelligence systems. A machine learning algorithm, constrained by a set of fitness functions, selected the parameters defining neurons expressing the desired apical mechanisms.Comment: 19 pages, 38 figures, pape

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