A Metric Approach to nD Images Edge Detection With Clifford Algebras

Thomas Batard (1) , Christophe Saint-Jean (1) , Michel Berthier (1)

1 MIA - Mathématiques, Image et Applications - EA 3165

The aim of this paper is to perform edge detection in color-infrared images from the point of view of Clifford algebras. The main idea is that such an image can be seen as a section of a Clifford bundle associated to the RGBT-space (Red, Green, Blue, Temperature) of acquisition. Dealing with geometric calculus and covariant derivatives of appropriate sections with respect to well-chosen connections allows to get various color and temperature information needed for the segmentation. We show in particular how to recover the first fundamental form of the image embedded in a LSHT-space (Luminance, Saturation , Hue, Temperature) equipped with a metric tensor. We propose applications to color edge detection with respect to a given hue interval and to edge detection in color-infrared images with constraints on temperature. Others applications related to different choices of connections, sections and embedding spaces for nD images may be considered from this general theorical framework.

Mots clés: Edge detection; Clifford algebras bundle; Geometric calculus; Color; Infrared; Di Zenzo gradient

Domaines: Traitement du signal et de l'image [eess.SP]; Géométrie différentielle [math.DG]; Traitement des images [eess.IV]; Traitement du signal et de l'image [eess.SP]

Thomas Batard, Christophe Saint-Jean, Michel Berthier. A Metric Approach to nD Images Edge Detection With Clifford Algebras. Journal of Mathematical Imaging and Vision, 2009, 33 (3), pp.296-312. ⟨10.1007/s10851-008-0115-0⟩. ⟨hal-00267629v2⟩