Iwasawa Decomposition and Computational Riemannian Geometry
2010 (English)In: 20th International Conference on Pattern Recognition: ICPR 2010, Los Alamitos, CA, USA: IEEE Computer Society, 2010, 4472-4475 p.Conference paper (Refereed)
We investigate several topics related to manifoldtechniquesfor signal processing. On the most general levelwe consider manifolds with a Riemannian Geometry. Thesemanifolds are characterized by their inner products on thetangent spaces. We describe the connection between the symmetricpositive-definite matrices defining these inner productsand the Cartan and the Iwasawa decomposition of the generallinear matrix groups. This decomposition gives rise to thedecomposition of the inner product matrices into diagonal matricesand orthonormal and into diagonal and upper triangularmatrices. Next we describe the estimation of the inner productmatrices from measured data as an optimization process onthe homogeneous space of upper triangular matrices. Weshow that the decomposition leads to simple forms of partialderivatives that are commonly used in optimization algorithms.Using the group theoretical parametrization ensures also thatall intermediate estimates of the inner product matrix aresymmetric and positive definite. Finally we apply the methodto a problem from psychophysics where the color perceptionproperties of an observer are characterized with the help ofcolor matching experiments. We will show that measurementsfrom color weak observers require the enforcement of thepositive-definiteness of the matrix with the help of the manifoldoptimization technique.
Place, publisher, year, edition, pages
Los Alamitos, CA, USA: IEEE Computer Society, 2010. 4472-4475 p.
, International Conference on Pattern Recognition, ISSN 1051-4651
Engineering and Technology
IdentifiersURN: urn:nbn:se:liu:diva-58854DOI: 10.1109/ICPR.2010.1086ISBN: 978-1-4244-7542-1OAI: oai:DiVA.org:liu-58854DiVA: diva2:346071
20th International Conference on Pattern Recognition,Istanbul, Turkey, 23-26 Aug. 2010