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  • 1. Björnemo, M.
    et al.
    Brun, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics.
    Kikinis, R.
    Westin, C.-F.
    Regularized stochastic white matter tractography using diffusion tensor MRI2002In: Medical Image Computing and Computer-Assisted Intervention MICCAI02,2002, 2002Conference paper (Refereed)
  • 2.
    Brun, Anders
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Manifold learning and representations for image analysis and visualization2006Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    We present a novel method for manifold learning, i.e. identification of the low-dimensional manifold-like structure present in a set of data points in a possibly high-dimensional space. The main idea is derived from the concept of Riemannian normal coordinates. This coordinate system is in a way a generalization of Cartesian coordinates in Euclidean space. We translate this idea to a cloud of data points in order to perform dimension reduction. Our implementation currently uses Dijkstra's algorithm for shortest paths in graphs and some basic concepts from differential geometry. We expect this approach to open up new possibilities for analysis of e.g. shape in medical imaging and signal processing of manifold-valued signals, where the coordinate system is “learned” from experimental high-dimensional data rather than defined analytically using e.g. models based on Lie-groups.

    We propose a novel post processing method for visualization of fiber traces from DT-MRI data. Using a recently proposed non-linear dimensionality reduction technique, Laplacian eigenmaps (Belkin and Niyogi, 2002), we create a mapping from a set of fiber traces to a low dimensional Euclidean space. Laplacian eigenmaps constructs this mapping so that similar traces are mapped to similar points, given a custom made pairwise similarity measure for fiber traces. We demonstrate that when the low-dimensional space is the RGB color space, this can be used to visualize fiber traces in a way which enhances the perception of fiber bundles and connectivity in the human brain.

  • 3.
    Brun, Anders
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Manifolds in Image Science and Visualization2007Doctoral 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.

  • 4.
    Brun, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics.
    Björnemo, M.
    Kikinis, R.
    Westin, C.-F.
    White matter tractography using sequental importance sampling2002In: ISMRM 02,2002, 2002Conference paper (Refereed)
  • 5.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Tensor Glyph Warping - Visualizing Metric Tensor Fields using Riemannian Exponential Maps2009In: Visualization and Processing of Tensor Fields: Advances and Perspectives / [ed] Laidlaw, David H.; Weickert, Joachim, Springer Berlin/Heidelberg, 2009, p. 139-160Chapter in book (Refereed)
    Abstract [en]

    The Riemannian exponential map, and its inverse the Riemannian logarithm map, can be used to visualize metric tensor fields. In this chapter we first derive the well-known metric sphere glyph from the geodesic equations, where the tensor field to be visualized is regarded as the metric of a manifold. These glyphs capture the appearance of the tensors relative to the coordinate system of the human observer. We then introduce two new concepts for metric tensor field visualization: geodesic spheres and geodesically warped glyphs. These additions make it possible not only to visualize tensor anisotropy, but also the curvature and change in tensorshape in a local neighborhood. The framework is based on the exp maps, which can be computed by solving a second order Ordinary Differential Equation (ODE) or by manipulating the geodesic distance function. The latter can be found by solving the eikonal equation, a non-linear Partial Differential Equation (PDE), or it can be derived analytically for some manifolds. To avoid heavy calculations, we also include first and second order Taylor approximations to exp and log. In our experiments, these are shown to be sufficiently accurate to produce glyphs that visually characterize anisotropy, curvature and shape-derivatives in smooth tensor fields. 

  • 6.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Park, Hae-Jeong
    Clinical Neuroscience Division, Laboratory of Neuroscience, Boston VA, USA Health Care System-Brockton Division, Department of Psychiatry, Harvard Medical School, Boston, MA, USA.
    Shenton, Martha E.
    Clinical Neuroscience Division, Laboratory of Neuroscience, Boston VA, USA Health Care System-Brockton Division, Department of Psychiatry, Harvard Medical School, Boston, MA, USA.
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Clustering Fiber Traces Using Normalized Cuts2004In: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2004: 7th International Conference, Saint-Malo, France, September 26-29, 2004. Proceedings, Part I, Springer Berlin/Heidelberg, 2004, p. 368-375Conference paper (Refereed)
    Abstract [en]

    In this paper we present a framework for unsupervised segmentation of white matter fiber traces obtained from diffusion weighted MRI data. Fiber traces are compared pairwise to create a weighted undirected graph which is partitioned into coherent sets using the normalized cut (Ncut) criterion. A simple and yet effective method for pairwise comparison of fiber traces is presented which in combination with the Ncut criterion is shown to produce plausible segmentations of both synthetic and real fiber trace data. Segmentations are visualized as colored stream-tubes or transformed to a segmentation of voxel space, revealing structures in a way that looks promising for future explorative studies of diffusion weighted MRI data.

  • 7.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV). Centre for Image Analysis, SLU, Uppsala, Sweden.
    Martin-Fernandez, Marcos
    Universidad de Valladolid Laboratorio de Procesado de Imagen (LPI), Dept. Teoría de la Señal y Comunicaciones e Ingeniería Telemática Spain.
    Acar, Burac
    Boğaziçi University 5 Electrical & Electronics Engineering Department Istanbul Turkey.
    Munoz-Moreno, Emma
    Universidad de Valladolid Laboratorio de Procesado de Imagen (LPI), Dept. Teoría de la Señal y Comunicaciones e Ingeniería Telemática Spain.
    Cammoun, Leila
    Signal Processing Institute (ITS), Ecole Polytechnique Fédérale Lausanne (EPFL) Lausanne Switzerland.
    Sigfridsson, Andreas
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV). Center for Technology in Medicine, Dept. Señales y Comunicaciones, University of Las Palmas de Gran Canaria, Spain.
    Sosa-Cabrera, Dario
    Center for Technology in Medicine, Dept. Señales y Comunicaciones, University of Las Palmas de Gran Canaria, Spain.
    Svensson, Björn
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Similar Tensor Arrays - A Framework for Storage of Tensor Array Data2009In: Tensors in Image Processing and Computer Vision / [ed] Santiago Aja-Fern´andez, Rodrigo de Luis Garc´ıa, Dacheng Tao, Xuelong Li, Springer Science+Business Media B.V., 2009, 1, p. 407-428Chapter in book (Refereed)
    Abstract [en]

    This chapter describes a framework for storage of tensor array data, useful to describe regularly sampled tensor fields. The main component of the framework, called Similar Tensor Array Core (STAC), is the result of a collaboration between research groups within the SIMILAR network of excellence. It aims to capture the essence of regularly sampled tensor fields using a minimal set of attributes and can therefore be used as a “greatest common divisor” and interface between tensor array processing algorithms. This is potentially useful in applied fields like medical image analysis, in particular in Diffusion Tensor MRI, where misinterpretation of tensor array data is a common source of errors. By promoting a strictly geometric perspective on tensor arrays, with a close resemblance to the terminology used in differential geometry, (STAC) removes ambiguities and guides the user to define all necessary information. In contrast to existing tensor array file formats, it is minimalistic and based on an intrinsic and geometric interpretation of the array itself, without references to other coordinate systems.

  • 8.
    Brun, Anders
    et al.
    Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston, USA .
    Park, Hae-Jeong
    Clinical Neuroscience Div., Lab. of Neuroscience, Boston VA Health Care System-Brockton Division, Dep. of Psychiatry, Harvard Medical School, and Surgical Planning Laboratory, Brigham and Women’s Hospital, Harvard Medical School, .
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston, USA .
    Coloring of DT-MRI fiber traces using Laplacian Eigenmaps2003In: Computer Aided Systems Theory - EUROCAST 2003: 9th International Workshop on Computer Aided Systems Theory Las Palmas de Gran Canaria, Spain, February 24-28, 2003 Revised Selected Papers / [ed] Roberto Moreno-Díaz and Franz Pichler, Springer Berlin/Heidelberg, 2003, Vol. 2809, p. 518-529Conference paper (Refereed)
    Abstract [en]

    We propose a novel post processing method for visualization of fiber traces from DT-MRI data. Using a recently proposed non-linear dimensionality reduction technique, Laplacian eigenmaps [3], we create a mapping from a set of fiber traces to a low dimensional Euclidean space. Laplacian eigenmaps constructs this mapping so that similar traces are mapped to similar points, given a custom made pairwise similarity measure for fiber traces. We demonstrate that when the low-dimensional space is the RGB color space, this can be used to visualize fiber traces in a way which enhances the perception of fiber bundles and connectivity in the human brain.

  • 9.
    Brun, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Svensson, Björn
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Herberthson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Wrangsjö, Andreas
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Filtering Vector-Valued Images using Importance Sampling2007In: Proceedings of the {SSBA} Symposium on Image Analysis,2007, 2007Conference paper (Other academic)
  • 10.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Svensson, Björn
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Westin, Carl-Fredrik
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering. Linköping University, The Institute of Technology.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Wrangsjö, Andreas
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Using Importance Sampling for Bayesian Feature Space Filtering2007In: Proceedings of the 15th Scandinavian conference on image analysis / [ed] Kjær Bjarne Ersbøll and Kim Steenstrup Pedersen, Berlin, Heidelberg: Springer-Verlag , 2007, p. 818-827Conference paper (Refereed)
    Abstract [en]

    We present a one-pass framework for filtering vector-valued images and unordered sets of data points in an N-dimensional feature space. It is based on a local Bayesian framework, previously developed for scalar images, where estimates are computed using expectation values and histograms. In this paper we extended this framework to handle N-dimensional data. To avoid the curse of dimensionality, it uses importance sampling instead of histograms to represent probability density functions. In this novel computational framework we are able to efficiently filter both vector-valued images and data, similar to e.g. the well-known bilateral, median and mean shift filters.

  • 11.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl_Fredrik
    Haker, S.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    A novel approach to averaging, filtering and interpolation of 3-D object orientation data2004In: Proceedings of the Swedish Symposium on Image Analysis (2004), 2004, p. 5-8Conference paper (Other academic)
  • 12.
    Brun, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Haker, Steven
    Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    A tensor-like representation for averaging, filtering and interpolation of 3D object orientation data2005In: Image Processing, 2005. ICIP 2005. IEEE International Conference on  (Volume:3 ), 2005, p. 1092-1095Conference paper (Refereed)
    Abstract [en]

    Averaging, filtering and interpolation of 3-D object orientation data is important in both computer vision and computer graphics, for instance to smooth estimates of object orientation and interpolate between keyframes in computer animation. In this paper we present a novel framework in which the non-linear nature of these problems is avoided by embedding the manifold of 3-D orientations into a 16-dimensional Euclidean space. Linear operations performed in the new representation can be shown to be rotation invariant, and defining a projection back to the orientation manifold results in optimal estimates with respect to the Euclidean metric. In other words, standard linear filters, interpolators and estimators may be applied to orientation data, without the need for an additional machinery to handle the non-linear nature of the problems. This novel representation also provides a way to express uncertainty in 3-D orientation, analogous to the well known tensor representation for lines and hyperplanes.

  • 13.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging Harvard Medical School, Boston, USA.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Fast manifold learning based on Riemannian normal coordinates2005In: Image Analysis: 14th Scandinavian Conference, SCIA 2005, Joensuu, Finland, June 19-22, 2005. Proceedings / [ed] Heikki Kalviainen, Jussi Parkkinen, Arto Kaarna., Springer Berlin/Heidelberg, 2005, p. 920-Conference paper (Refereed)
    Abstract [en]

    We present a novel method for manifold learning, i.e. identification of the low-dimensional manifold-like structure present in a set of data points in a possibly high-dimensional space. The main idea is derived from the concept of Riemannian normal coordinates. This coordinate system is in a way a generalization of Cartesian coordinates in Euclidean space. We translate this idea to a cloud of data points in order to perform dimension reduction. Our implementation currently uses Dijkstra’s algorithm for shortest paths in graphs and some basic concepts from differential geometry. We expect this approach to open up new possibilities for analysis of e.g. shape in medical imaging and signal processing of manifold-valued signals, where the coordinate system is “learned” from experimental high-dimensional data rather than defined analytically using e.g. models based on Lie-groups.

  • 14.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Intrinsic and Extrinsic Means on the Circle -- a Maximum Likelihood Interpretation2007In: ICASSP 2007. IEEE International Conference on Acoustics, Speech and Signal Processing, 2007, New York, USA: IEEE , 2007, p. III-1053-III-1056Conference paper (Refereed)
    Abstract [en]

    For data samples in Rn, the mean is a well known estimator. When the data set belongs to an embedded manifold M in Rn, e.g. the unit circle in R2, the definition of a mean can be extended and constrained to M by choosing either the intrinsic Riemannian metric of the manifold or the extrinsic metric of the embedding space. A common view has been that extrinsic means are approximate solutions to the intrinsic mean problem. This paper study both means on the unit circle and reveal how they are related to the ML estimate of independent samples generated from a Brownian distribution. The conclusion is that on the circle, intrinsic and extrinsic means are maximum likelihood estimators in the limits of high SNR and low SNR respectively

  • 15.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Lab of Mathematics in Imaging Harvard Medical School, Boston, USA.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    LOGMAP: Preliminary results using a new method for manifold learning2005In: Symposium on Image Analysis SSBA,2005, 2005, p. 101-105Conference paper (Other academic)
  • 16.
    Brun, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Harvard Medical School Boston.
    Herberthson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Knutsson, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Sample Logmaps - Intrinsic processing of empirical manifold data2006In: SSBA Symposium on Image Analysis,2006, 2006, p. 13-16Conference paper (Other academic)
  • 17.
    Cammoun, L.
    et al.
    Signal Processing Institute Ecole Polytechnigue Fédérale de Lausanne Switserland.
    Castano-Moraga, C.A.
    University of Las Palmas de Gran Canaria.
    Munoz-Moreno, E.
    Univ. de Valladolid, Spain.
    Sosa-Cabrera, D.
    University of Las Palmas de Gran Canaria.
    Acar, B.
    University Istanbul.
    Rodriguez-Florido, M.A.
    University of Las Palmas de Gran Canaria.
    Brun, Anders
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Thiran, J.P.
    Signal Processing Institute Ecole Polytechnigue Fédérale de Lausanne Switserland.
    A Review of Tensors and Tensor Signal Processing2009In: Tensors in Image Processing and Computer Vision / [ed] S. Aja-Fernandez, R. de Luis Garcia, D. Tao, and X. Li, Springer London, 2009, p. 1-32Chapter in book (Refereed)
    Abstract [en]

    Tensors have been broadly used in mathematics and physics, since they are a generalization of scalars or vectors and allow to represent more complex properties. In this chapter we present an overview of some tensor applications, especially those focused on the image processing field. From a mathematical point of view, a lot of work has been developed about tensor calculus, which obviously is more complex than scalar or vectorial calculus. Moreover, tensors can represent the metric of a vector space, which is very useful in the field of differential geometry. In physics, tensors have been used to describe several magnitudes, such as the strain or stress of materials. In solid mechanics, tensors are used to define the generalized Hooke’s law, where a fourth order tensor relates the strain and stress tensors. In fluid dynamics, the velocity gradient tensor provides information about the vorticity and the strain of the fluids. Also an electromagnetic tensor is defined, that simplifies the notation of the Maxwell equations. But tensors are not constrained to physics and mathematics. They have been used, for instance, in medical imaging, where we can highlight two applications: the diffusion tensor image, which represents how molecules diffuse inside the tissues and is broadly used for brain imaging; and the tensorial elastography, which computes the strain and vorticity tensor to analyze the tissues properties. Tensors have also been used in computer vision to provide information about the local structure or to define anisotropic image filters.

  • 18.
    Herberthson, Magnus
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Brun, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Pairs of orientation in the plane2006In: SSBA Symposium on Image Analysis,2006, 2006, p. 97-100Conference paper (Other academic)
  • 19.
    Herberthson, Magnus
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Brun, Anders
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    P-averages of diffusion tensors2007In: SSBA 2007, Symposium on image analysis,2007, Linköping, 2007Conference paper (Other academic)
    Abstract [en]

    For positive semi-definite tensors like diffusion tensors in the plane it is possible to calculate several different means or p-averages. These are related to p-norms for functions, but produce mappings rather than numbers as means. We compare these means for various values of the real parameter p. One important future application is the filtering and interpolation of tensor fields in Diffusion Tensor Magnetic Resonance

  • 20.
    Herberthson, Magnus
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Brun, Anders
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Representing Pairs of Orientations in the Plane2007In: Image Analysis: 15th Scandinavian Conference, SCIA 2007, Aalborg, Denmark, June 10-14, 2007 / [ed] Bjarne Kjær Ersbøll, Kim Steenstrup Pedersen, Springer Berlin/Heidelberg, 2007, p. 661-670Conference paper (Refereed)
    Abstract [en]

    In this article we present a way of representing pairs of orientations in the plane. This is an extension of the familiar way of representing single orientations in the plane. Using this framework, pairs of lines can be added, scaled and averaged over in a sense which is to be described. In particular, single lines can be incorporated and handled simultaneously.

  • 21.
    Langer, Max
    et al.
    Linköping University, Department of Biomedical Engineering. Linköping University, The Institute of Technology.
    Svensson, Björn
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Brun, Anders
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Andersson, Mats
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Design of fast multidimensional filters using genetic algorithms2005In: Applications of Evolutionary Computing: EvoWorkkshops 2005: EvoBIO, EvoCOMNET, EvoHOT, EvoIASP, EvoMUSART, and EvoSTOC Lausanne, Switzerland, March 30 - April 1, 2005 Proceedings, Springer Berlin/Heidelberg, 2005, p. 366-375Conference paper (Refereed)
    Abstract [en]

    A method for designing fast multidimensional filters using genetic algorithms is described. The filter is decomposed into component filters where coefficients can be sparsely scattered using filter networks. Placement of coefficients in the filters is done by genetic algorithms and the resulting filters are optimized using an alternating least squares approach. The method is tested on a 2-D quadrature filter and the method yields a higher quality filter in terms of weighted distortion compared to other efficient implementations that require the same ammount of computations to apply. The resulting filter also yields lower weighted distortion than the full implementation.

  • 22.
    Ohlsson, Henrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Roll, Jacob
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Brun, Anders
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Andersson, Mats
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Ljung, Lennart
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Direct Weight Optimization Applied to Discontinuous Functions2008In: 47th IEEE Conference on Decision and Control, 2008. CDC 2008, IEEE , 2008, p. 117-122Conference paper (Refereed)
    Abstract [en]

    The Direct Weight Optimization (DWO) approach is a nonparametric estimation approach that has appeared in recent years within the field of nonlinear system identification. In previous work, all function classes for which DWO has been studied have included only continuous functions. However, in many applications it would be desirable also to be able to handle discontinuous functions. Inspired by the bilateral filter method from image processing, such an extension of the DWO framework is proposed for the smoothing problem. Examples show that the properties of the new approach regarding the handling of discontinuities are similar to the bilateral filter, while at the same time DWO offers a greater flexibility with respect to different function classes handled.

  • 23.
    Ohlsson, Henrik
    et al.
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Rydell, Joakim
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Brun, Anders
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Roll, Jacob
    Linköping University, Department of Electrical Engineering, Automatic Control. Linköping University, The Institute of Technology.
    Andersson, Mats
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Ynnerman, Anders
    Linköping University, Department of Science and Technology, Visual Information Technology and Applications (VITA). Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Enabling Bio-Feedback using Real-Time fMRI2008In: 47th IEEE Conference on Decision and Control, 2008, CDC 2008, IEEE , 2008, p. 3336-3341Conference paper (Refereed)
    Abstract [en]

    Despite the enormous complexity of the human mind, fMRI techniques are able to partially observe the state of a brain in action. In this paper we describe an experimental setup for real-time fMRI in a bio-feedback loop. One of the main challenges in the project is to reach a detection speed, accuracy and spatial resolution necessary to attain sufficient bandwidth of communication to close the bio-feedback loop. To this end we have banked on our previous work on real-time filtering for fMRI and system identification, which has been tailored for use in the experiment setup. In the experiments presented the system is trained to estimate where a person in the MRI scanner is looking from signals derived from the visual cortex only. We have been able to demonstrate that the user can induce an action and perform simple tasks with her mind sensed using real-time fMRI. The technique may have several clinical applications, for instance to allow paralyzed and "locked in" people to communicate with the outside world. In the meanwhile, the need for improved fMRI performance and brain state detection poses a challenge to the signal processing community. We also expect that the setup will serve as an invaluable tool for neuro science research in general.

  • 24. Park, H.C.
    et al.
    MacCarley, R.W.
    Westin, C.-F.
    Kubicki, M.
    Talos, I.F.
    Brun, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics.
    Pieper, S.
    Kikinis, R.
    Jolesz, F.A.
    Shenton, M.E.
    Visualization of white matter fibers and cortical structures derived from diffusion tensor magnetic resonance imaging and cortical parcellations2004In: American Journal of Neuroradiology, ISSN 0195-6108, E-ISSN 1936-959X, Vol. 25, p. 1318-1324Article in journal (Refereed)
  • 25. Park, H.J.
    et al.
    Westin, C.-F.
    Kubicki, M.
    Brun, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics.
    McCarley, R.W.
    Shenton, M.E.
    A method for hemispheric asymmetri of white matters using diffusion tensor MRI2003In: International Conference on Functional Mapping of the Human Brain,2003, New York: Elsevier Science , 2003Conference paper (Refereed)
  • 26. San-Jose Estepar, R.
    et al.
    Brun, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics.
    Westin, C.-F.
    Robust generalized total least squares iterative closest point registration2004In: Medical Image Computing and Computer-Assisted Intervention MICCAI 04,2004, Berlin, Heidelberg: Springer Verlag , 2004, p. 235-Conference paper (Refereed)
  • 27.
    Svensson, Björn
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Brun, Anders
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Andersson, Mats
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Estimation of Non-Cartesian Local Structure Tensor Fields2007In: Image Analysis: 15th Scandinavian Conference, SCIA 2007, Aalborg, Denmark, June 10-14, 2007 / [ed] Bjarne Kjær Ersbøll, Kim Steenstrup Pedersen, Springer Berlin/Heidelberg, 2007, Vol. 4522/2007, p. 948-957Conference paper (Refereed)
    Abstract [en]

    In medical imaging, signals acquired in non-Cartesian coordinate systems are common. For instance, CT and MRI often produce significantly higher resolution within scan planes, compared to the distance between two adjacent planes. Even oblique sampling occurs, by the use of gantry tilt. In ultrasound imaging, samples are acquired in a polar coordinate system, which implies a spatially varying metric.

    In order to produce a geometrically correct image, signals are generally resampled to a Cartesian coordinate system. This paper concerns estimation of local structure tensors directly from the non-Cartesian coordinate system, thus avoiding deteriorated signal and noise characteristics caused by resampling. In many cases processing directly in the warped coordinate system is also less time-consuming. A geometrically correct tensor must obey certain transformation rules originating from fundamental differential geometry. Subsequently, this fact also affects the tensor estimation. As the local structure tensor is estimated using filters, a change of coordinate system also change the shape of the spatial support of these filters. Implications and limitations brought on by sampling require the filter design criteria to be adapted to the coordinate system.

  • 28.
    Svensson, Björn
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Brun, Anders
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Andersson, Mats
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Geometric Transformations of Local Structure Tensors2006In: Similar NoE Tensor Workshop,2006, 2006Conference paper (Other academic)
  • 29.
    Svensson, Björn
    et al.
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Brun, Anders
    Centre for Image Analysis, SLU, Uppsala, Sweden.
    Andersson, Mats
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    On Geometric Transformations of Local Structure Tensors2009In: Tensors in Image Processing and Computer Vision: Part 2 / [ed] S. Aja-Fernandez, R. de Luis Garcia, D. Tao, X. Li, Springer London, 2009, p. 179-193Chapter in book (Refereed)
    Abstract [en]

    The structure of images has been studied for decades and the use of local structure tensor fields appeared during the eighties [3, 14, 6, 9, 11]. Since then numerous varieties of tensors and estimation schemes have been developed. Tensors have for instance been used to represent orientation [7], velocity, curvature [2] and diffusion [19] with applications to adaptive filtering [8], motion analysis [10] and segmentation [17]. Even though sampling in non-Cartesian coordinate system are common, analysis and processing of local structure tensor fields in such systems is less developed. Previous work on local structure in non-Cartesian coordinate systems include [21, 16, 1, 18].

1 - 29 of 29
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