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Manifold-Valued Signal Proocessing

A manifold is a mathematical concept which generalizes surfaces to higher dimensions. Examples of 2-dimensional manifolds are for instance the surface of a sphere and the the surface of a torus, both being examples of non-linear manifolds. Locally however, manifolds are flat and equivalent to the an Euclidean space. Features found in signals can often be described using manifolds. This is often not stated explicitly, but instead various parameterizations of manifolds are used. In order to describe a quantity which can be seen as a point on a sphere, spherical coordinates are commonly used. This has some drawbacks however which we wish to avoid if possible. We see a need for manifolds in the field of medical image analysis. Medical doctors express a wish to objectively quantify various features in medical images, such as local texture, shape and orientation of organs. We know from previous research that manifolds can do the job, but we lack a generic framework for dealing with manifold-valued signals in signal processing. In fact, we believe that such a framework will be useful in other areas signal processing too. The goal of this project is to explore a specific flavour of signal processing and continue the development of methods 1) to learn or identify manifold-valued representations from examples and 2) apply signal processing on manifold-valued signals which is analogous to filtering and interpolation using convolution operators in classic signal processing.

  • Principal Investigator:
    Hans Knutsson
  • Main Supervisor:
    Hans Knutsson
  • Medical Area:
    Central Nervous System
    Cardiovascular System
  • Technical Area:
    Machine Learning and Signal Representation
    Modelling and Simulation
    Segmentation, Classification and Quantification
    Visualization and Image Enhancement
    User Interfaces and Interaction
  • Modality:
    Magnetic Resonance Imaging
  • Medical Activity:
  • Technical Activity:
  • Grants:
    1860 kSEK
  • Financial Body:
    Vetenskapsrådet, The Swedish Research Council
  • Financial Support:
  • Man Months:
  • Project Duration:
    2005/01/01 - 2007/12/31
  • Former Staff:
  • Project Description:
  • Short History

    The following is a short list of milestones in tensor- and manifold-valued signal processing, related to our group.

    1978The Double angle representation of 2-D line orientation. (Granlund)
    19855-D representation of 3-D line orientations. (Knutsson)
    1989The structure tensor. (Knutsson)
    2000Automated Generation of manifold representations in vision using CCA. (Knutsson, Borga and Andersson)
    2004Quaternion outer product representation of 3-D object orientation. (Brun et al.)

    Dissemination Activities

    • Summer 2005: Magnus Herberthson gave a course on tensor algebra to the participants of the Similar Medical Applications Workshop (WP10) held in Linköping.
    • Spring 2005: Magnus Herberthson gave a course in differential geometry, specifically tailored to increase the understanding of tensors, manifolds and differential geometry within the whole medical informatics group.

    Research Group

    Prof. Hans Knutsson Project manager, Supervisor IMT, CMIV
    Magnus Herberthson, PhDAssociate Professor, Asst. Supervisor MAI
    Carl-Fredrik Westin, PhDAssistant Professor, Asst. SupervisorLMI, Harvard Medical School / Brigham & Womens Hospital
    Anders Brun, MScPhD Student


    A tensor-like representation for averaging, filtering and interpolation of 3-D object orientation data by A. Brun and C.-F. Westin and S. Haker and H. Knutsson, Proceedings of the IEEE-ICIP, Genoa, Italy, September, 2005. [PDF]

    Fast Manifold Learning Based on Riemannian Normal Coordinates by A. Brun and C.-F. Westin and M. Herberthson and H. Knutsson, Proceedings of the 14th Scandinavian conference on image analysis (SCIA'05), Joensuu, Finland, June, 2005. [PDF]

    LOGMAP: Preliminary results using a new method for manifold learning by A. Brun and C.-F. Westin and M. Herberthson and H. Knutsson, Proceedings of the Swedish Symposium on Image Analysis, March, Malmö, Sweden, 2005.

    A novel approach to averaging, filtering and interpolation of 3-D object orientation data by A. Brun, C.-F. Westin, S. Haker, H. Knutsson, Proceedings of the Swedish Symposium on Image Analysis, Feb. 11-12, Uppsala, Sweden, 2004.

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Anders Persson
Phone: +46 (0)10 - 103 8906

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