# Clifford algebras bundles to multidimensional image segmentation

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1 MIA - Mathématiques, Image et Applications

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 $CTD,\widetilde{\pi},D$ over the domain $D$ of the image and with fiber $Cl\mathbb{R}^n,\|\| 2$. Due to the triviality, any connection $abla 1$ on the given bundle is the sum of the trivial connection $\widetilde{ abla} 0$ with $\omega$, a 1-form on $D$ with values in $EndCTD$. We show that varying $\omega$ and derivating well-chosen sections with respect to $abla 1$ 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 $abla 1I=\widetilde{ abla} 0I+ dx + dy \otimes \mu \, I$, where $I$ is the image section and $\mu$ is a vector section coding the given color.

Keywords : Geometric algebra Image processing Clifford bundles Covariant derivatives Geometric algebra.

Autor: Thomas Batard - Michel Berthier -

Fuente: https://hal.archives-ouvertes.fr/

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