The fusion of algebra, analysis and geometry, and their application
to real world problems, have been dominant themes underlying
mathematics for over a century. Geometric algebras, introduced and
classified by Clifford in the late 19th century, have played a
prominent role in this effort, as seen in the mathematical work of
Cartan, Brauer, Weyl, Chevelley, Atiyah, and Bott, and in
applications to physics in the work of Pauli, Dirac and others. One
of the most important applications of geometric algebras to
geometry is to the representation of groups of Euclidean and
Minkowski rotations. This aspect and its direct relation to
robotics and vision will be discussed in several chapters of this
multi-authored textbook, which resulted from the ASI meeting.
Moreover, group theory, beginning with the work of Burnside,
Frobenius and Schur, has been influenced by even more general
problems. As a result, general group actions have provided the
setting for powerful methods within group theory and for the use of
groups in applications to physics, chemistry, molecular biology,
and signal processing. These aspects, too, will be covered in
detail.
With the rapidly growing importance of, and ever expanding
conceptual and computational demands on signal and image processing
in remote sensing, computer vision, medical image processing, and
biological signal processing, and on neural and quantum computing,
geometric algebras, and computational group harmonic analysis, the
topics of the book have emerged as key tools. The list of authors
includes many of the world's leading experts in the development of
new algebraic modeling and signal representation methodologies,
novel Fourier-based andgeometric transforms, and computational
algorithms required for realizing the potential of these new
application fields.
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