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- Brun, Anders, 1976-
(författare)
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Manifolds in Image Science and Visualization
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- 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.
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| 2. |
- Hemmendorff, Magnus, 1972-
(författare)
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Motion estimation and compensation in medical imaging
- 2001
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This dissertation presents a framework for estimation of motion fields in 2D images, 3D volumes and multi-dimensional signal registration. The primary application is motion compensation for sequences of medical images and volumes with contrast agents.The framework implies motion estimation in two steps where the intermediate result is constraints on the local motion vectors. One algorithm generates constraints and a second algorithm computes motion vector fields.We present two methods for generation of local motion constraints. The first method is based on phase from quadrature filters. The second method is based on canonical correlation and scalar products of quadrature filters. In both methods, a local confidence measure produced to increase accuracy and robustness.A mathematical result is a novel method for maximizing canonical correlation. The novel method can handle covariance matrices that are complex and singular.Parametric models, such as affine or finite elements, are used to estimate motion fields from local motion constraints and confidence measures. In order to control smoothness, the model is extended to incorporate stiffness and cost of deformations.Multiple layers of motion fields are estimated using implicit or explicit clustering of motion constraints. We also discuss some philosophical issues in the analysis of multiple motions. An extension of the known EM algorithm is presented together with experimental results on multiple layers for 2D images and 3D volumes. As an alternative to the EM algorithm, this thesis also introduces a method based on higher order outer products. In addition, we present a back projection algorithm for reconstruction of transparent layers.Clinical evaluation shows good results for 2D X-ray angiography images. Experimental results also show accurate motion estimates for 3D MRI mammograms and simulated images in 3D angiography.
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| 3. |
- Linderhed, Anna, 1961-
(författare)
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Adaptive image compression with wavelet packets and empirical mode decomposition
- 2004
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis addresses the problem of using wavelet packets and empirical mode decomposition (EMD) for image compression. The wavelet packet basis selection algorithm is studied through an extensive experimental survey of the generated decomposition trees. We formulate the "triplet problem" for image compression as follows: How is the decomposition tree related to the image content, filter set and cost function? Our aim is to find an optimal basis for compression of images. Results are presented using test images from the Brodatz texture set. We also present a method to analytically calculate the cost of splitting a node, for a given signal model and filter, without actually performing the split.A totally different approach to signal decomposition is the EMD. This is an adaptive decomposition scheme with which any complicated signal is decomposed into its intrinsic mode functions (IMF). The concept of EMD is extended to two dimensions to make it useful for image processing. The EMD and the sifting process to generate the IMFs are described. Different known and newly found difficulties with implementation of the method in two dimensions are highlighted and solutions are proposed. The method of variable sampling of the EMO, using overlapping blocks, is presented and the concept of empiquency is introduced to describe spatialfrequency since the traditional Fourier-based frequency concept is not applicable.Several ways to use EMD for image compression are examined and presented. The two-dimensional extension of the EMD is original as well as its application for image compression.
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