A description of the Manifold-Valued Signal Processing project

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.

Research Group

	Prof. Hans Knutsson	Project manager, Supervisor	IMT, CMIV
	Magnus Herberthson, PhD	Associate Professor, Asst. Supervisor	MAI
	Carl-Fredrik Westin, PhD	Associate prof., Asst. Supervisor	LMI, Harvard Medical School / Brigham & Womens Hospital
	Anders Brun, MSc	PhD Student	IMT, CMIV

Short History

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

1978	The Double angle representation of 2-D line orientation. (Granlund)
1985	5-D representation of 3-D line orientations. (Knutsson)
1989	The structure tensor. (Knutsson)
2000	Automated Generation of manifold representations in vision using CCA. (Knutsson, Borga and Andersson)
2004	Quaternion 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.

Publications

Manifold Learning and Representations for Image Analysis and Visualization A. Brun Lic Thesis March 2006. [PDF]

Sample Logmaps - Intrinsic Processing of Empirical Manifold Data by A. Brun, C.-F. Westin, M. Herberthsson, H. Knutsson, Proceedings of the SSBA Symposium on Image Analysis, Umeå, Sweden March 2006 [Bibtex]

Pairs of Orientation in the Plane by M. Herberthsson, A. Brun, H. Knutsson, Proceedings of the SSBA Symposium on Image Analysis, Umeå, Sweden March 2006.

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.