The demand for efficient communication and data storage is continuously increasing and signal representation and compression are important factors in digital communication and storage systems.

This work deals with Frame based signal representation and compression. The emphasis is on the design of frames suited for efficient representation, or for low bit rate compression, of classes of signals.

Traditional signal decompositions such as transforms, wavelets, and filter banks, generate expansions using an analysis-synthesis setting. In this thesis we concentrate on the synthesis or reconstruction part of the signal expansion, having a system with no explicit analysis stage. We want to investigate the use of an overcomplete set of vectors, a frame or an overcomplete dictionary, for signal representations and allow sparse representations. Effective signal representations are desirable in many applications, where signal compression is one example. Others can be signal analysis for different purposes, reconstruction of signals from a limited observation set, feature extraction in pattern recognition and so forth.

The lack of an explicit analysis stage originates some questions on finding the optimal representation. Finding an optimal sparse representation from an overcomplete set of vectors is NP-complete, and suboptimal vector selection methods are more practical. We have used some existing methods like different variations of the Matching Pursuit (MP) [52] algorithm, and we developed a robust regularized FOCUSS to be able to use FOCUSS (FOCal Underdetermined System Solver [29]) under lossy conditions.

In this work we develop techniques for frame design, the Method of Optimal Directions (MOD), and propose methods by which such frames can successfully be used in frame based signal representation and in compression schemes. A Multi Frame Compression (MFC) scheme is presented and experiments with several signal classes show that the MFC scheme works well at low bit rates using MOD designed frames. Reconstruction experiments provides complimentary evidence of the good properties of the MOD algorithm.