Catálogo de publicaciones - libros

This chapter is a short introduction into tensor fields, some basic techniques from linear algebra, differential geometry and the mathematical concept of tensor fields are presented. The main goal of this chapter is to give readers from different backgrounds some fundamentals to access the research papers in the following chapters.

The structure tensor, also known as second moment matrix or Förstner interest operator, is a very popular tool in image processing. Its purpose is the estimation of orientation and the local analysis of structure in general. It is based on the integration of data from a local neighborhood. Normally, this neighborhood is defined by a Gaussian window function and the structure tensor is computed by the weighted sum within this window. Some recently proposed methods, however, adapt the computation of the structure tensor to the image data. There are several ways how to do that. This chapter wants to give an overview of the different approaches, whereas the focus lies on the methods based on robust statistics and nonlinear diffusion. Furthermore, the data-adaptive structure tensors are evaluated in some applications. Here the main focus lies on optic flow estimation, but also texture analysis and corner detection are considered.

As a step towards a analysis of local image features, the position, peak value, and covariance matrix of an isolated, noise-free multivariate Gaussian are determined from four ‘observables’, computed by gaussian-weighted averaging first and second powers of (up to second order) partial derivatives of a digitized greyvalue distribution.

Tensors are a useful tool for the detection of low-level features such as edges, lines, corners, and junctions because they can represent feature strength and orientation in a way that is easy to work with. However, traditional approaches to define feature tensors have a number of disadvantages. By means of the first and second order , we propose a new approach called the . Using quadratic convolution equations, we show that the boundary tensor overcomes some problems of the older tensor definitions. When the Riesz transform is combined with the Laplacian of Gaussian, the boundary tensor can be efficiently computed in the spatial domain. The usefulness of the new method is demonstrated for a number of application examples.

This chapter gives an introduction to the principles of diffusion magnetic resonance imaging (MRI) with emphasis on the computational aspects. It introduces the philosophies underlying the technique and shows how to sensitize MRI measurements to the motion of particles within a sample material. The main body of the chapter is a technical review of diffusion MRI reconstruction algorithms, which determine features of the material microstructure from diffusion MRI measurements. The focus is on techniques developed for biomedical diffusion MRI, but most of the methods discussed are applicable beyond this domain. The review begins by showing how the standard reconstruction algorithms in biomedical diffusion MRI, diffusion-tensor MRI and diffusion spectrum imaging, arise from the principles of the measurement process. The discussion highlights the weaknesses of the standard approaches to motivate the development of a new generation of reconstruction algorithms and reviews the current state-of-the-art. The chapter concludes with a brief discussion of diffusion MRI applications, in particular fibre tracking, followed by a summary and a glimpse into the future of diffusion MRI acquisition and reconstruction.

The empirical origin of random noise is described, its influence on DTI variables is illustrated by a review of numerical and in vivo studies supplemented by new simulations investigating high noise levels. A stochastic model of noise propagation is presented to structure noise impact in DTI. Finally, basics of voxelwise and spatial denoising procedures are presented. Recent denoising procedures are reviewed and consequences of the stochastic model for convenient denoising strategies are discussed.

Water diffusion is anisotropic in organized tissues such as white matter and muscle. Diffusion tensor imaging (DTI), a non-invasive MR technique, measures water self-diffusion rates and thus gives an indication of the underlying tissue microstructure. The diffusion rate is often expressed by a second-order tensor. Insightful DTI visualization is challenging because of the multivariate nature and the complex spatial relationships in a diffusion tensor field. This chapter surveys the different visualization techniques that have been developed for DTI and compares their main characteristics and drawbacks. We also discuss some of the many biomedical applications in which DTI helps extend our understanding or improve clinical procedures. We conclude with an overview of open problems and research directions.

In this chapter, we consider the task of anatomically labeling diffusion tensor images of cerebral white matter to facilitate visualization as well as quantitative comparison of these data. The analogous labeling problem for structural magnetic resonance images of the brain has been extensively studied and we propose that advances in atlas-based techniques may be leveraged to anatomically segment the fiber-tractographic maps derived from diffusion tensor data. The feasibility of the approach is demonstrated with data acquired of the corpus callosum, and implications of the results for callosal morphometry are discussed.

Diffusion Tensor Imaging (DTI) became a popular tool for white matter tract visualization in the brain. It provides quantitative measures of water molecule diffusion anisotropy and the ability to delineate major white matter bundles. The diffusion model of DTI was found to be inappropriate in cases of partial volume effect, such as Multiple Fiber Orientations (MFO) ambiguity. Recently, a variety of image processing methods were proposed to enhance DTI results by reducing noise and correcting artifacts, but most techniques were not designed to resolve MFO ambiguity. In this Chapter we describe variational based DTI processing techniques, and show how such techniques can be adapted to the Multiple Tensor (MT) diffusion model via the Multiple Tensor Variational (MTV) framework. We show how the MTV framework can be used in separating differently oriented white matter fiber bundles.

In this work we review how the diffusivity profiles obtained from diffusion MRI can be expressed in terms of Cartesian tensors of ranks higher than 2. When the rank of the tensor being used is 2, one recovers traditional diffusion tensor imaging (DTI). Therefore our approach can be seen as a generalization of DTI. The properties of generalized diffusion tensors are discussed. The shortcomings of DTI experienced in the presence of orientational heterogeneity may cause inaccurate anisotropy values and incorrect fiber orientations. Employment of higher rank tensors is helpful in overcoming these difficulties.