Homogenization of a locally-periodic medium with areas of low and high diffusivity

We aim at understanding transport in porous materials including regions with both high and low diffusivities. For such scenarios, the transport becomes structured (here: micro- macro). The geometry we have in mind includes regions of low diffusivity arranged in a locally-periodic fashion. We choose a prototypical advection-diffusion system (of minimal size), discuss its formal homogenization (the heterogenous medium being now assumed to be made of zones with circular areas of low diffusivity of x-varying sizes), and prove the weak solvability of the limit two-scale reaction-diffusion model. A special feature of our analysis is that most of the basic estimates (positivity, L^inf-bounds, uniqueness, energy inequality) are obtained in x-dependent Bochner spaces. Keywords: Heterogeneous porous materials, homogenization, micro-macro transport, two-scale model, reaction-diffusion system, weak solvability

New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)

We consider hypo-elliptic diffusion and convection-diffusion on $\mathbb{R}^3 \rtimes S^2$, the quotient of the Lie group of rigid body motions SE(3) in which group elements are equivalent if they are equal up to a rotation around the reference axis. We show that we can derive expressions for the convolution kernels in terms of eigenfunctions of the PDE, by extending the approach for the SE(2) case. This goes via application of the Fourier transform of the PDE in the spatial variables, yielding a second order differential operator. We show that the eigenfunctions of this operator can be expressed as (generalized) spheroidal wave functions. The same exact formulas are derived via the Fourier transform on SE(3). We solve both the evolution itself, as well as the time-integrated process that corresponds to the resolvent operator. Furthermore, we have extended a standard numerical procedure from SE(2) to SE(3) for the computation of the solution kernels that is directly related to the exact solutions. Finally, we provide a novel analytic approximation of the kernels that we briefly compare to the exact kernels.Comment: Revised and restructure

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

Evolution equations on Gabor transforms and their applications

We introduce a systematic approach to the design, implementation and analysis of left-invariant evolution schemes acting on Gabor transform, primarily for applications in signal and image analysis. Within this approach we relate operators on signals to operators on Gabor transforms. In order to obtain a translation and modulation invariant operator on the space of signals, the corresponding operator on the reproducing kernel space of Gabor transforms must be left invariant, i.e. it should commute with the left regular action of the reduced Heisenberg group H_r. By using the left-invariant vector fields on H_r in the generators of our evolution equations on Gabor transforms, we naturally employ the essential group structure on the domain of a Gabor transform. Here we distinguish between two tasks. Firstly, we consider non-linear adaptive left-invariant convection (reassignment) to sharpen Gabor transforms, while maintaining the original signal. Secondly, we consider signal enhancement via left-invariant diffusion on the corresponding Gabor transform. We provide numerical experiments and analytical evidence for our methods and we consider an explicit medical imaging application

Accelerated diffusion operators for enhancing DW-MRI

High angular resolution diffusion imaging (HARDI) is a MRI imaging technique that is able to better capture the intra-voxel diffusion pattern compared to its simpler predecessor diffusion tensor imaging (DTI). However, HARDI in general produces very noisy diffusion patterns due to the low SNR from the scanners at high b-values. Furthermore, it still exhibits limitations in areas where the diffusion pattern is asymmetrical (bifurcations, splaying fibers, etc.). To overcome these limitations, enhancement and denoising of the data based on context information is a crucial step. In order to achieve it, convolutions are performed in the coupled spatial and angular domain. Therefore the kernels applied become also HARDI data. However, these approaches have high computational complexity of an already complex HARDI data processing. In this work, we present an accelerated framework for HARDI data regularizaton and enhancement. The convolution operators are optimized by: pre-calculating the kernels, analysing kernels shape and utilizing look-up-tables. We provide an increase of speed, compared to previous brute force approaches of simpler kernels. These methods can be used as a preprocessing for tractography and lead to new ways for investigation of brain white matter

Accelerated diffusion operators for enhancing DW-MRI

High angular resolution diffusion imaging (HARDI) is a MRI imaging technique that is able to better capture the intra-voxel diffusion pattern compared to its simpler predecessor diffusion tensor imaging (DTI). However, HARDI in general produces very noisy diffusion patterns due to the low SNR from the scanners at high b-values. Furthermore, it still exhibits limitations in areas where the diffusion pattern is asymmetrical (bifurcations, splaying fibers, etc.). To overcome these limitations, enhancement and denoising of the data based on context information is a crucial step. In order to achieve it, convolutions are performed in the coupled spatial and angular domain. Therefore the kernels applied become also HARDI data. However, these approaches have high computational complexity of an already complex HARDI data processing. In this work, we present an accelerated framework for HARDI data regularizaton and enhancement. The convolution operators are optimized by: pre-calculating the kernels, analysing kernels shape and utilizing look-up-tables. We provide an increase of speed, compared to previous brute force approaches of simpler kernels. These methods can be used as a preprocessing for tractography and lead to new ways for investigation of brain white matter