Abstract
Conformal Geometric Algebra (CGA) is a high level language commonly used in mathematical, physics and engineering problems. At a top level, CGA is a free coordinate tool for designing and modeling geometric problems; at a low level CGA provides a new coordinate framework for numeric processing in problem solving. In this paper we show how to use quadratic programming and CGA for, given two sets p and q of points in ℝn, construct an optimal separation sphere S such that, all points of p are contained inside of it, and all points of q are outside. To classify an unknown pattern x, an inner product must be applied between x and S. Some numerical and real examples to test the proposal are given.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hildebrand, D., Fontijne, D., Perwass, C., Dorst, L.: Geometric algebra and its application to computer graphics. In: EUROGRAPHICS 2004 Tutorial 3. Eurographics Association, Aire-la-Ville (2004)
Clifford, W.K.: Applications of Grassmann’s Extensive Algebra. American Journal of Mathematics 1(4), 350–358 (1878)
Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric calculus, A Unified language for Mathematics and Physics. D. Reidel Publishing Company (1984)
Dorst, L., Fontijne, D.: 3D Euclidean geometry through conformal geometric algebra (a GAViewer tutorial) (2005), http://www.science.uva.nl/ga/tutorials/CGA/
Hildebrand, D.: Geometric computing in Computer graphics using Conformal Geometric Algebra, Interactive Graphics Systems Group, TU Darmstadt, Germany (2005)
Perwass, C., Hildebrand, D.: Aspects of geometric algebra in Euclidean, projective and conformal space. Technical Report Number 0310, Christian-Albrechts-Universität zu Kiel, Institut für Informatik und Praktische Mathematik (2003)
Cover, T., Hart, P.: Nearest Neighbor pattern classification. IEEE Transactions on Information Theory 13(1), 21–27 (1967)
Barrón, R., et al.: New Improved Algorithm for the Training of a Morphological Associative Memory. Research in Computing Science 21, 49–59 (2006)
Barron, R., Cruz, B., Laguna, G.: Conformal Geometric Algebra for Spherical Convex Hull Optimization. In: AGACSE 2008 (accepted for publication, 2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cruz, B., Barrón, R., Sossa, H. (2008). Pattern Classification Based on Conformal Geometric Algebra and Optimization Techniques. In: Gelbukh, A., Morales, E.F. (eds) MICAI 2008: Advances in Artificial Intelligence. MICAI 2008. Lecture Notes in Computer Science(), vol 5317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88636-5_26
Download citation
DOI: https://doi.org/10.1007/978-3-540-88636-5_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-88635-8
Online ISBN: 978-3-540-88636-5
eBook Packages: Computer ScienceComputer Science (R0)