Clifford Algebra Bundles to Multidimensional Image Segmentation

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.