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 $(CT(D),\widetilde{\pi},D)$ over the domain $D$ of the image and with fiber $Cl(\mathbb{R}^n,\|\|_2)$. Due to the triviality, any connection $\nabla_1$ on the given bundle is the sum of the trivial connection $\widetilde{\nabla}_0$ 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 $\nabla_1$ 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 $\mu$ is a vector section coding the given color.