8 research outputs found
Mathematical morphology for tensor data induced by the Loewner orderingin higher dimensions
Positive semidefinite matrix fields are becoming increasingly important in digital imaging. One reason for this tendency consists of the introduction of diffusion tensor magnetic resonance imaging (DTMRI). In order to perform shape analysis, enhancement or segmentation of such tensor fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the matrix-valued setting. We start by presenting novel definitions for the maximum and minimum of a set of matrices since these notions lie at the heart of the morphological operations. In contrast to naive approaches like the component-wise maximum or minimum of the matrix channels, our approach is based on the Loewner ordering for symmetric matrices. The notions of maximum and minimum deduced from this partial ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. We introduce erosion, dilation, opening, closing, top hats, morphological derivatives, shock filters, and mid-range filters for positive semidefinite matrix-valued images. These morphological operations incorporate information simultaneously from all matrix channels rather than treating them independently. Experiments on DT-MRI images with ball- and rod-shaped structuring elements illustrate the properties and performance of our morphological operators for matrix-valued data
Curvature-driven PDE methods for matrix-valued images
Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edgelike structures in tensor fields, we first generalise Di Zenzo\u27s concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts
Median and related local filters for tensor-valued images
We develop a concept for the median filtering of tensor data. The main part of this concept is the definition of median for symmetric matrices. This definition is based on the minimisation of a geometrically motivated objective function which measures the sum of distances of a variable matrix to the given data matrices. This theoretically wellfounded concept fits into a context of similarly defined median filters for other multivariate data. Unlike some other approaches, we do not require by definition that the median has to be one of the given data values. Nevertheless, it happens so in many cases, equipping the matrix-valued median even with root signals similar to the scalar-valued situation. Like their scalar-valued counterparts, matrix-valued median filters show excellent capabilities for structure-preserving denoising. Experiments on diffusion tensor imaging, fluid dynamics and orientation estimation data are shown to demonstrate this. The orientation estimation examples give rise to a new variant of a robust adaptive structure tensor which can be compared to existing concepts. For the efficient computation of matrix medians, we present a convex programming framework. By generalising the idea of the matrix median filters, we design a variety of other local matrix filters. These include matrix-valued mid-range filters and, more generally, M-smoothers but also weighted medians and \alpha-quantiles. Mid-range filters and quantiles allow also interesting cross-links to fundamental concepts of matrix morphology
Adaptive microstructure-informed tractography for accurate brain connectivity analyses
Human brain has been subject of deep interest for centuries, given it's central role in controlling and directing the actions and functions of the body as response to external stimuli. The neural tissue is primarily constituted of neurons and, together with dendrites and the nerve synapses, constitute the gray matter (GM) which plays a major role in cognitive functions. The information processed in the GM travel from one region to the other of the brain along nerve cell projections, called axons. All together they constitute the white matter (WM) whose wiring organization still remains challenging to uncover. The relationship between structure organization of the brain and function has been deeply investigated on humans and animals based on the assumption that the anatomic architecture determine the network dynamics. In response to that, many different imaging techniques raised, among which diffusion-weighted magnetic resonance imaging (DW-MRI) has triggered tremendous hopes and expectations. Diffusion-weighted imaging measures both restricted and unrestricted diffusion, i.e. the degree of movement freedom of the water molecules, allowing to map the tissue fiber architecture in vivo and non-invasively. Based on DW-MRI data, tractography is able to exploit information of the local fiber orientation to recover global fiber pathways, called streamlines, that represent groups of axons. This, in turn, allows to infer the WM structural connectivity, becoming widely used in many different clinical applications as for diagnoses, virtual dissections and surgical planning. However, despite this unique and compelling ability, data acquisition still suffers from technical limitations and recent studies have highlighted the poor anatomical accuracy of the reconstructions obtained with this technique and challenged its effectiveness for studying brain connectivity. The focus of this Ph.D. project is to specifically address these limitations and to improve the anatomical accuracy of the structural connectivity estimates. To this aim, we developed a global optimization algorithm that exploits micro and macro-structure information, introducing an iterative procedure that uses the underlying tissue properties to drive the reconstruction using a semi-global approach. Then, we investigated the possibility to dynamically adapt the position of a set of candidate streamlines while embedding the anatomical prior of trajectories smoothness and adapting the configuration based on the observed data. Finally, we introduced the concept of bundle-o-graphy by implementing a method to model groups of streamlines based on the concept that axons are organized into fascicles, adapting their shape and extent based on the underlying microstructure
Tractographie adaptative basée sur la microstructure pour des analyses précises de la connectivité cérébrale
Le cerveau est un sujet de recherche depuis plusieurs décennies, puisque son rôle
est central dans la compréhension du genre humain. Le cerveau est composé de
neurones, où leurs dendrites et synapses se retrouvent dans la matière grise alors que
les axones en constituent la matière blanche. L’information traitée dans les différentes
régions de la matière grise est ensuite transmise par l’intermédiaire des axones afin
d’accomplir différentes fonctions cognitives.
La matière blanche forme une structure d’interconnections complexe encore dif-
ficile à comprendre et à étudier. La relation entre l’architecture et la fonction du
cerveau a été étudiée chez les humains ainsi que pour d’autres espèces, croyant que
l’architecture des axones déterminait la dynamique du réseau fonctionnel.
Dans ce même objectif, l’Imagerie par résonance (IRM) est un outil formidable
qui nous permet de visualiser les tissus cérébraux de façon non-invasive. Plus partic-
ulièrement, l’IRM de diffusion permet d’estimer et de séparer la diffusion libre de celle
restreinte par la structure des tissus. Cette mesure de restriction peut être utilisée
afin d’inférer l’orientation locale des faisceaux de matière blanche. L’algorithme de
tractographie exploite cette carte d’orientation pour reconstruire plusieurs connexions
de la matière blanche (nommées “streamlines”).
Cette modélisation de la matière blanche permet d’estimer la connectivité cérébrale
dite structurelle entre les différentes régions du cerveau. Ces résultats peuvent être
employés directement pour la planification chirurgicale ou indirectement pour l’analyse
ou une Ă©valuation clinique.
Malgré plusieurs de ses limitations, telles que sa variabilité et son imprécision, la
tractographie reste l’unique moyen d’étudier l’architecture de la matière blanche ainsi
que la connectivité cérébrale de façon non invasive.
L’objectif de ce projet de doctorat est de répondre spécifiquement à ces limitations
et d’améliorer la précision anatomique des estimations de connectivité structurelle.
Dans ce but, nous avons développé un algorithme d’optimisation globale qui exploite
les informations de micro et macrostructure, en introduisant une procédure itéra-
tive qui utilise les propriétés sous-jacentes des tissus pour piloter la reconstruction
en utilisant une approche semi-globale. Ensuite, nous avons étudié la possibilité
d’adapter dynamiquement la position d’un ensemble de lignes de courant candidates
tout en intégrant le préalable anatomique de la douceur des trajectoires et en adap-
tant la configuration en fonction des données observées. Enfin, nous avons introduit
le concept de bundle-o-graphy en mettant en œuvre une méthode pour modéliser des
groupes de lignes de courant basées sur le concept que les axones sont organisés en
fascicules, en adaptant leur forme et leur Ă©tendue en fonction de la microstructure
sous-jacente.Abstract : Human brain has been subject of deep interest for centuries, given it’s central role in controlling and directing the actions and functions of the body as response to external stimuli. The neural tissue is primarily constituted of neurons and, together with dendrites and the nerve synapses, constitute the gray matter (GM) which plays a major role in cognitive functions. The information processed in the GM travel from one region to the other of the brain along nerve cell projections, called axons. All together they constitute the white matter (WM) whose wiring organization still remains challenging to uncover. The relationship between structure organization of the brain and function has been deeply investigated on humans and animals based on the assumption that the anatomic architecture determine the network dynamics. In response to that, many different imaging techniques raised, among which diffusion-weighted magnetic resonance imaging (DW-MRI) has triggered tremendous hopes and expectations. Diffusion-weighted imaging measures both restricted and unrestricted diffusion, i.e. the degree of movement freedom of the water molecules, allowing to map the tissue fiber architecture in vivo and non-invasively. Based on DW-MRI data, tractography is able to exploit information of the local fiber orien- tation to recover global fiber pathways, called streamlines, that represent groups of axons. This, in turn, allows to infer the WM structural connectivity, becoming widely used in many different clinical applications as for diagnoses, virtual dissections and surgical planning. However, despite this unique and compelling ability, data acqui- sition still suffers from technical limitations and recent studies have highlighted the poor anatomical accuracy of the reconstructions obtained with this technique and challenged its effectiveness for studying brain connectivity. The focus of this Ph.D. project is to specifically address these limitations and to improve the anatomical accuracy of the structural connectivity estimates. To this aim, we developed a global optimization algorithm that exploits micro and macro- structure information, introducing an iterative procedure that uses the underlying tissue properties to drive the reconstruction using a semi-global approach. Then, we investigated the possibility to dynamically adapt the position of a set of candidate streamlines while embedding the anatomical prior of trajectories smoothness and adapting the configuration based on the observed data. Finally, we introduced the concept of bundle-o-graphy by implementing a method to model groups of streamlines based on the concept that axons are organized into fascicles, adapting their shape and extent based on the underlying microstructure.Sommario : Il cervello umano è oggetto di profondo interesse da secoli, dato il suo ruolo centrale
nel controllare e dirigere le azioni e le funzioni del corpo in risposta a stimoli
esterno. Il tessuto neurale è costituito principalmente da neuroni che, insieme ai dendriti
e alle sinapsi nervose, costituiscono la materia grigia (GM), la quale riveste un
ruolo centrale nelle funzioni cognitive. Le informazioni processate nella GM viaggiano
da una regione all’altra del cervello lungo estensioni delle cellule nervose, chiamate
assoni. Tutti insieme costituiscono la materia bianca (WM) la cui organizzazione
strutturale rimane tuttora sconosciuta. Il legame tra struttura e funzione del cervello
sono stati studiati a fondo su esseri umani e animali partendo dal presupposto che
l’architettura anatomica determini la dinamica della rete funzionale. In risposta a
ciò, sono emerse diverse tecniche di imaging, tra cui la risonanza magnetica pesata
per diffusione (DW-MRI) ha suscitato enormi speranze e aspettative. Questa tecnica
misura la diffusione sia libera che ristretta, ovvero il grado di libertĂ di movimento
delle molecole d’acqua, consentendo di mappare l’architettura delle fibre neuronali
in vivo e in maniera non invasiva. Basata su dati DW-MRI, la trattografia è in
grado di sfruttare le informazioni sull’orientamento locale delle fibre per ricostruirne
i percorsi a livello globale. Questo, a sua volta, consente di estrarre la connettivitĂ
strutturale della WM, utilizzata in diverse applicazioni cliniche come per diagnosi,
dissezioni virtuali e pianificazione chirurgica. Tuttavia, nonostante questa capacitĂ
unica e promettente, l’acquisizione dei dati soffre ancora di limitazioni tecniche
e recenti studi hanno messo in evidenza la scarsa accuratezza anatomica delle ricostruzioni
ottenute con questa tecnica, mettendone in dubbio l’efficacia per lo studio
della connettività cerebrale. Il focus di questo progetto di dottorato è quello di affrontare in modo specifico
queste limitazioni e di migliorare l’accuratezza anatomica delle stime di connettivitĂ
strutturale. A tal fine, abbiamo sviluppato un algoritmo di ottimizzazione globale che
sfrutta le informazioni sia micro che macrostrutturali, introducendo una procedura
iterativa che utilizza le proprietĂ del tessuto neuronale per guidare la ricostruzione utilizzando
un approccio semi-globale. Successivamente, abbiamo studiato la possibilitĂ
di adattare dinamicamente la posizione di un insieme di streamline candidate incorporando
il prior anatomico per cui devono seguire traiettorie regolari e adattando
la configurazione in base ai dati osservati. Infine, abbiamo introdotto il concetto
di bundle-o-graphy implementando un metodo per modellare gruppi di streamline
basato sul concetto che gli assoni sono organizzati in fasci, adattando la loro forma
ed estensione in base alla microstruttura sottostante