Clifford algebras bundles to multidimensional image segmentation
Thomas Batard, Michel Berthier · 2008
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), eπ, D) over the domain D of the image and with fiber Cl(R n, ‖‖2). Due to the triviality, any connection ∇1 on the given bundle is the sum of the trivial connection e ∇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 ∇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 ∇1(I) = e∇0(I)+(dx+dy) ⊗µI, where I is the image section and µ is a vector section coding the given color.