Clifford Algebra Bundles to Multidimensional Image Segmentation

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), (pi) over tilde, D) over the domain D of the image and with fiber Cl(R-n, parallel to parallel to 2). Due to the triviality, any connection del(1) on the given bundle is the sum of the trivial connection (del) over tilde (0) with omega a 1-form on D with values in End(CT(D)). We show that varying omega and derivating well-chosen sections with respect to del(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 del(1)(I) = (del) over tilde (0)(I)+(dx + dy) circle times mu I, where I is the image section and mu is a vector section coding the given color.