**Jirí Hrivnák**, University
of Montreal

*(Anti)symmetric multivariate exponential functions and corresponding
Fourier transforms*
We consider recently introduced symmetric or antisymmetric exponential
and trigonometric functions. These are defined as determinants or
antideterminants of matrices whose elements are corresponding functions
of one variable. To each of these multivariate functions correspond
expansion into Fourier series, integral Fourier transform and finite
Fourier transform. We give explicit formulas of these functions,
expansions and Fourier transforms in dimension two. We also present
some examples and discuss possible applications of these functions
and transforms.

**Frederic Lesage**, École Polytechnique
de Montréal

*Compressed sensing in photo-acoustic tomography, a potential
application for Lie Algebra bases.*

Recent work in photo-acoustic tomography indicates that this modality
might bring high resolution imaging at low cost. The technique however
is hampered by long acquisition times. In this work we describe
new image acquisition techniques based on the theory of compressed
sensing are able to partly solve this problem. Here the choice of
basis is crucial in having the compressed sensing work properly
and Lie Algebra bases could provide an avenue to extend to new applications.

**Maryna Nesterenko**, Université
de Montréal

*Computing with almost periodic functions*

Computational Fourier analysis of functions defined on quasicrystals
is developed. A key point is to build the analysis around the emergent

theory of quasicrystals and diffraction based on local hulls and
dynamical systems. Numerically computed approximations arising in
this way are built out of the precise Fourier module of the quasicrystal
in question, and approximate their target functions uniformly on
the entire infinite space. This is in striking contrast with numerical
approximations based on periodization of some finite part of the
crystal. The methods are practical and computable. Examples of functions
based on the standard Fibonacci quasicrystal serve to illustrate
the method.

**Jiri Petera**, University of Montreal

**Morning short course**

*Discrete and continuous multidimensional transforms based on
$C$-, $S$-, and $E$-functions of a compact semisinple Lie group*

References:

J. Patera, {\it Compact simple Lie groups and theirs $C$-, $S$-,
and

$E$-transforms,\/} SIGMA (Symmetry, Integrability and Geometry:
Methods and

Applications) {\bf 1} (2005) 025, 6 pages, math-ph/0512029.

R.V. Moody, J.~Patera, {\it Orthogonality within the families of
\ $C$-,

$S$-, and $E$-functions of any compact semisimple Lie group,\/}
SIGMA

(Symmetry, Integrability and Geometry: Methods and Applications)
{\bf 2} (2006) 076, 14 pages, math-ph/0611020.

A. Klimyk, J. Patera, {\it Orbit functions,\/} SIGMA (Symmetry,

Integrability and Geometry: Methods and Applications) {\bf 2} (2006),
006, 60 pages, math-ph/0601037

A. Klimyk, J. Patera, {\it Antisymmetric orbit functions,\/} SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications)
{\bf 3} (2007), paper 023, 83 pages; math-ph/0702040v1

A. Klimyk, J. Patera, {\it $E$-orbit functions,\/} SIGMA (Symmetry,
Integrability and Geometry: Methods and Applications) {\bf 4} (2008),
002, 57 pages; arXiv:0801.0822

**Matthieu Voorons**, Université
de Montréal

*Interpolation based on Lie group theory and comparison with standard
techniques*

New interpolation algorithms, based on the Lie group theory, were
recently developed by Mr Patera and his team. The Continuous Extension
of the Discrete Orbit Function Transform (CEDOFT) based on the C-functions
of the Lie groups leading to square lattices were considered as
interpolators. More precisely, Lie groups SU(2)xSU(2) and O(5) were
used to interpolate 2-dimensional data, and the cubic lattice SU(2)xSU(2)xSU(2)
for 3-dimensional data. All algorithms presented for 2 and 3-dimensional
data have the advantage to give the exact value of the original
data at the points of the grid lattice, and interpolate well the
data values between the grid points. The quality and speed of the
interpolation are comparable with the most efficient classical interpolation
techniques. Interpolation results for many application are presented,
from simple zooming and filtering of still images to video interpolation
and refinement of volume estimation in medical imagery.

**Yusong Yan** and **Hongmei Zhu**, York
University

*Integer Lie group transforms*

The C- or S-functions derived from the Lie Group C2 form an orthogonal
basis in its corresponding fundamental region F. Discretizing such
a basis results in a class of discrete orthogonal transforms. Using
the lifting schemes, we develop the integer-to-integer transforms
associated to these discrete orthogonal transforms on a discrete
grid FM of F of density defined by a positive integer M. Since these
integer transforms are invertible, it has potential applications
such as lossless image compression and encryption.

**Hongmei Zhu**, York University

*Interpolation using the discrete group transforms*

Interpolation methods are often used in many applications for image
generation and processing, such as image resampling and zooming.

Here, we introduce a new family of interpolation algorithms based
on a compact semisimple Lie groups of rank n, although here we explore
mainly the cases n = 2. The performance of the algorithms is compared
with the other commonly used interpolation methods.