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.