Clifford algebras and image processing

Wednesday, October 7, 2009 - 9:45am - 10:15am
Lind 305
Michel Berthier (Université de La Rochelle)
Keywords: Geometric and Clifford Algebras, Image and Signal Processing, Segmentation, Diffusion, Differential Geometry for Image Processing.

Since the seminal work of Hestenes, Clifford algebras (also called geometric algebras) appeared to be a powerful tool
for geometric modeling in theoretical physics. During the last years, the so-called geometric calculus has found many
applications in computer science, especially in vision and robotics (Lasenby, Bayro Corrochano, Dorst ...). Concerning
image processing, Clifford algebras were already used implicitly by Sangwine through the coding of color with quaternions.

The aim of this talk is first to give basic notions about these algebras and the related spinor groups. I will then detail two
applications : a metric approach for edge detection in nD images and the construction of a Clifford Hodge operator that
generalizes the Beltrami operator of Sochen et al. This latter can be used for diffusion in color images but also for diffusion
of vector and orthonormal frame fields. The geometric framework of these applications is the one of fiber and Clifford bundles.
Very roughly speaking, the basic idea is to take advantage of embedding an acquisition space in a higher dimensional algebra
containing elements of different kinds and related to a specified metric. If time remains, I will also mention in few words applications
for signal processing (Clifford Fourier transform and color monogenic signal).

Two basic references :


Chevalley, C.: The Algebraic Theory of Spinors and Clifford Algebras, new edn. Springer (1995)

Computer Science:

Sommer, G.: Geometric Computing with Clifford Algebras. Theoretical Fundations and Applications in Computer Vision and
Robotics. Springer, Berlin (2001)
MSC Code: