Codomain scale space and regularization for high angular resolution diffusion imaging

Regularization is an important aspect in high angular resolution diffusion imaging (HARDI), since, unlike with classical diffusion tensor imaging (DTI), there is no a priori regularity of raw data in the co-domain, i.e. considered as a multispectral signal for fixed spatial position. HARDI preprocessing is therefore a crucial step prior to any subsequent analysis, and some insight in regularization paradigms and their interrelations is compulsory. In this paper we posit a codomain scale space regularization paradigm that has hitherto not been applied in the context of HARDI. Unlike previous (first and second order) schemes it is based on infinite order regularization, yet can be fully operationalized. We furthermore establish a closed-form relation with first order Tikhonov regularization via the Laplace transform

Two canonical representations for regularized high angular resolution diffusion imaging

Two canonical representations for regularization of unit spherefunctions encountered in the context of high angular resolution diffusionimaging (HARDI) are discussed. One of these is based on spherical harmonicdecomposition, and its one-parameter extension via Tikhonov regularization.This case is well-established, and is mainly reviewed for thesake of completeness. The second one is new, and is based on a higherorder diffusion tensor decomposition. A homogeneous representation ofthis type has been proposed in the literature, but we show that thisis inconvenient for the purpose of regularization. We instead construct aheterogeneous representation that can be regarded as "canonical", to theextent that its behaviour under regularization mimics that of sphericalharmonics

Finsler geometry on higher order tensor fields and applications to high angular resolution diffusion imaging.

We study 3D-multidirectional images, using Finsler geometry. The application considered here is in medical image analysis, specifically in High Angular Resolution Diffusion Imaging (HARDI) (Tuch et al. in Magn. Reson. Med. 48(6):1358–1372, 2004) of the brain. The goal is to reveal the architecture of the neural fibers in brain white matter. To the variety of existing techniques, we wish to add novel approaches that exploit differential geometry and tensor calculus. In Diffusion Tensor Imaging (DTI), the diffusion of water is modeled by a symmetric positive definite second order tensor, leading naturally to a Riemannian geometric framework. A limitation is that it is based on the assumption that there exists a single dominant direction of fibers restricting the thermal motion of water molecules. Using HARDI data and higher order tensor models, we can extract multiple relevant directions, and Finsler geometry provides the natural geometric generalization appropriate for multi-fiber analysis. In this paper we provide an exact criterion to determine whether a spherical function satisfies the strong convexity criterion essential for a Finsler norm. We also show a novel fiber tracking method in Finsler setting. Our model incorporates a scale parameter, which can be beneficial in view of the noisy nature of the data. We demonstrate our methods on analytic as well as simulated and real HARDI data

Diffusion on the 3D Euclidean motion group for enhancement of HARDI data

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this paper we study left-invariant diffusion on the 3D Euclidean motion group SE(3), which is useful for processing three-dimensional data. In particular, it is useful for the processing of High Angular Resolution Diffusion Imaging (HARDI) data, since these data can be considered as orientation scores directly, without the need to transform the HARDI data to a different form. In principle, all theory of the 2D case can be mapped to the 3D case. However, one of the complicating factors is that all practical 3D orientation scores are not functions on the entire group SE(3), but rather on a coset space of the group. We will show how we can still conceptually apply processing on the entire group by requiring the operations to preserve the introduced notion of alpha-right-invariance of such functions on SE(3). We introduce left-invariant derivatives and describe how to estimate tangent vectors that locally fit best to the elongated structures in the 3D orientation score. We propose generally applicable techniques for smoothing and enhancing functions on SE(3) using left-invariant diffusion on the group. Finally, we will discuss implementational issues and show a number of results for linear diffusion on artificial HARDI data

Diffusion, convection and erosion on R3 x S2 and their application to the enhancement of crossing fibers

In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R3 x S2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3D-rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on R3 x S2 and can be solved by R3 x S2-convolution with the corresponding Green’s functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on R3 x S2 and they are solved by a morphological R3 x S2-convolution with the corresponding Green’s functions. We will reveal the remarkable analogy between these erosions/dilations and diffusions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution. In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on R3 x S2 we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes. We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques (HARDI and DTI) for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. We provide experiments of our crossing-preserving (non-linear) left-invariant evolutions on neural images of a human brain containing crossing fibers

Homogeneity based segmentation and enhancement of Diffusion Tensor Images : a white matter processing framework

In diffusion magnetic resonance imaging (DMRI) the Brownian motion of the water molecules, within biological tissue, is measured through a series of images. In diffusion tensor imaging (DTI) this diffusion is represented using tensors. DTI describes, in a non-invasive way, the local anisotropy pattern enabling the reconstruction of the nervous fibers - dubbed tractography. DMRI constitutes a powerful tool to analyse the structure of the white matter within a voxel, but also to investigate the anatomy of the brain and its connectivity. DMRI has been proved useful to characterize brain disorders, to analyse the differences on white matter and consequences in brain function. These procedures usually involve the virtual dissection of white matters tracts of interest. The manual isolation of these bundles requires a great deal of neuroanatomical knowledge and can take up to several hours of work. This thesis focuses on the development of techniques able to automatically perform the identification of white matter structures. To segment such structures in a tensor field, the similarity of diffusion tensors must be assessed for partitioning data into regions, which are homogeneous in terms of tensor characteristics. This concept of tensor homogeneity is explored in order to achieve new methods for segmenting, filtering and enhancing diffusion images. First, this thesis presents a novel approach to semi-automatically define the similarity measures that better suit the data. Following, a multi-resolution watershed framework is presented, where the tensor field’s homogeneity is used to automatically achieve a hierarchical representation of white matter structures in the brain, allowing the simultaneous segmentation of different structures with different sizes. The stochastic process of water diffusion within tissues can be modeled, inferring the homogeneity characteristics of the diffusion field. This thesis presents an accelerated convolution method of diffusion images, where these models enable the contextual processing of diffusion images for noise reduction, regularization and enhancement of structures. These new methods are analysed and compared on the basis of their accuracy, robustness, speed and usability - key points for their application in a clinical setting. The described methods enrich the visualization and exploration of white matter structures, fostering the understanding of the human brain

Reconstruction and rendering of time-varying natural phenomena

While computer performance increases and computer generated images get ever more realistic, the need for modeling computer graphics content is becoming stronger. To achieve photo-realism detailed scenes have to be modeled often with a significant amount of manual labour. Interdisciplinary research combining the fields of Computer Graphics, Computer Vision and Scientific Computing has led to the development of (semi-)automatic modeling tools freeing the user of labour-intensive modeling tasks. The modeling of animated content is especially challenging. Realistic motion is necessary to convince the audience of computer games, movies with mixed reality content and augmented reality applications. The goal of this thesis is to investigate automated modeling techniques for time-varying natural phenomena. The results of the presented methods are animated, three-dimensional computer models of fire, smoke and fluid flows.Durch die steigende Rechenkapazität moderner Computer besteht die Möglichkeit immer realistischere Bilder virtuell zu erzeugen. Dadurch entsteht ein größerer Bedarf an Modellierungsarbeit um die nötigen Objekte virtuell zu beschreiben. Um photorealistische Bilder erzeugen zu können müssen sehr detaillierte Szenen, oft in mühsamer Handarbeit, modelliert werden. Ein interdisziplinärer Forschungszweig, der Computergrafik, Bildverarbeitung und Wissenschaftliches Rechnen verbindet, hat in den letzten Jahren die Entwicklung von (semi-)automatischen Methoden zur Modellierung von Computergrafikinhalten vorangetrieben. Die Modellierung dynamischer Inhalte ist dabei eine besonders anspruchsvolle Aufgabe, da realistische Bewegungsabläufe sehr wichtig für eine überzeugende Darstellung von Computergrafikinhalten in Filmen, Computerspielen oder Augmented-Reality Anwendungen sind. Das Ziel dieser Arbeit ist es automatische Modellierungsmethoden für dynamische Naturerscheinungen wie Wasserfluss, Feuer, Rauch und die Bewegung erhitzter Luft zu entwickeln. Das Resultat der entwickelten Methoden sind dabei dynamische, dreidimensionale Computergrafikmodelle