Exploitation of special functions orthogonal on finite domains of 2D and 3D lattices-Fourier expansions of functions sampled on lattices

The advantages of digital data (signal functions) processing have made this technique a standard for processing real world analog data in many areas of life, from music industry to seismology. Stored digitally, the data also become less sensitive to physical limitations than their analog counterparts. Other advantages include straightforward low frequency filtering procedures (e.g. as required in seismology, oceanography and other environmental monitoring), as well as frequency bounding (e.g. in telecommunications, sending large bandwidth signals over a narrow bandwidth). To transform analog functions to their digital counterpart, a discretization is necessary to obtain their corresponding lattice based versions. In many problems, it is sufficient to use discrete Fourier expansions of data sampled at equidistant points of a line (in 1D), or on a square lattice formed by two orthogonal 1D lattices (in 2D). We propose to develop and evaluate the methodology which will make use of the special symmetry properties of data that will be reflected in more efficient expansions, e.g. data sampled in 2D on a triangular lattice which will significantly improve multidimensional signal processing in terms of computational speed, memory and hardware requirements. The proposed methodology will be developed using semi-simple Lie group theory.

Faculty Supervisor:

Dr. Jiri Patera

Student:

Marzena Szajewska, Lenka Motlochova & Gayane Malkhasyan

Partner:

M-Health Solutions

Discipline:

Mathematics

Sector:

Information and communications technologies

University:

Université de Montréal