NDLI: Clifford Algebra Bundles to Multidimensional Image Segmentation

Content Provider	Springer Nature Link
Author	Batard, Thomas Berthier, Michel
Copyright Year	2010
Abstract	We present a new theoretical framework for multidimensional image processing using Clifford algebras. The aim of the paper is to detect edges by computing the first fundamental form of a surface associated to an image. We propose to construct this metric in the Clifford bundles setting. A nD image, i.e. an image of dimension n, is considered as a section of a trivial Clifford bundle $$(CT(D), \widetilde{\pi}, D)$$ over the domain D of the image and with fiber $$Cl({\mathbb{R}}^n, \parallel\parallel_2)$$ . Due to the triviality, any connection $$\nabla_1$$ on the given bundle is the sum of the trivial connection $$\widetilde{\nabla}_0$$ with ω, a 1-form on D with values in End(CT(D)). We show that varying ω and derivating well-chosen sections with respect to $$\nabla_1$$ provides all the information needed to perform various kind of segmentation. We present several illustrations of our results, dealing with color (n=3) and color/infrared (n=4) images. As an example, let us mention the problem of detecting regions of a given color with constraints on temperature; the segmentation results from the computation of $$\nabla_1(I) = \widetilde{\nabla}_0(I) + (dx+dy)\otimes\mu{I}$$ , where I is the image section and μ is a vector section coding the given color.
Ending Page	516
Page Count	28
Starting Page	489
File Format	PDF
ISSN	01887009
e-ISSN	16614909
Journal	Advances in Applied Clifford Algebras
Issue Number	3
Volume Number	20
Language	English
Publisher	Birkhäuser-Verlag
Publisher Date	2010-03-12
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Connections, general theory Fiber bundles Mathematical Methods in Physics Image processing Clifford bundles Clifford algebras, spinors Theoretical, Mathematical and Computational Physics Geometric algebra Covariant derivatives Applications of Mathematics Physics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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