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Manifolds in Image Science and Visualization
Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
2007 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

A Riemannian manifold is a mathematical concept that generalizes curved surfaces to higher dimensions, giving a precise meaning to concepts like angle, length, area, volume and curvature. A glimpse of the consequences of a non-flat geometry is given on the sphere, where the shortest path between two points – a geodesic – is along a great circle. Different from Euclidean space, the angle sum of geodesic triangles on the sphere is always larger than 180 degrees.

Signals and data found in applied research are sometimes naturally described by such curved spaces. This dissertation presents basic research and tools for the analysis, processing and visualization of such manifold-valued data, with a particular emphasis on future applications in medical imaging and visualization.

Two-dimensional manifolds, i.e. surfaces, enter naturally into the geometric modelling of anatomical entities, such as the human brain cortex and the colon. In advanced algorithms for processing of images obtained from computed tomography (CT) and ultrasound imaging (US), images themselves and derived local structure tensor fields may be interpreted as two- or three-dimensional manifolds. In diffusion tensor magnetic resonance imaging (DT-MRI), the natural description of diffusion in the human body is a second-order tensor field, which can be related to the metric of a manifold. A final example is the analysis of shape variations of anatomical entities, e.g. the lateral ventricles in the brain, within a population by describing the set of all possible shapes as a manifold.

Work presented in this dissertation include: Probabilistic interpretation of intrinsic and extrinsic means in manifolds. A Bayesian approach to filtering of vector data, removing noise from sampled manifolds and signals. Principles for the storage of tensor field data and learning a natural metric for empirical data.

The main contribution is a novel class of algorithms called LogMaps, for the numerical estimation of logp (x) from empirical data sampled from a low-dimensional manifold or geometric model embedded in Euclidean space. The logp (x) function has been used extensively in the literature for processing data in manifolds, including applications in medical imaging such as shape analysis. However, previous approaches have been limited to manifolds where closed form expressions of logp (x) have been known. The introduction of the LogMap framework allows for a generalization of the previous methods. The application of LogMaps to texture mapping, tensor field visualization, medial locus estimation and exploratory data analysis is also presented.

Place, publisher, year, edition, pages
Institutionen för medicinsk teknik , 2007. , 151 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1157
Keyword [en]
manifold learning, image analysis, signal processing, diffusion tensor mri
National Category
Medical Laboratory and Measurements Technologies
Identifiers
URN: urn:nbn:se:liu:diva-10475ISBN: 978-91-85715-02-2 (print)OAI: oai:DiVA.org:liu-10475DiVA: diva2:17224
Public defence
2008-01-25, Linden, 421, Universitetssjukhuset i Linköping, Linköping, 10:15 (English)
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Note
The electronic version is corrected for grammatical and spelling errors.Available from: 2008-03-17 Created: 2008-03-17 Last updated: 2013-08-28

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Brun, Anders

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