Frames in Finite Dimensional Spaces


In recent years, frames in signal processing applications have received more and more attention. Models, data, and operators must be discretized in order to function numerically. As a result, applications and algorithms always work with finite dimensional data. In the finite dimensional case, frames are equivalent to a spanning system. If reconstruction is wanted, frames are the only feasible generalization of bases. In contrast to bases, frames lose their linear independency. This project aims to investigate the properties of frames in finite dimensional spaces.


In this project, we will implement algorithms to work with frames in finite dimensional spaces. We will look at a way to "switch" between different frames, i.e. find a way to bijectively map between their coefficient spaces and provide the corresponding algorithm. This will be done by using the Cross-Gram matrix of the two involved frames. This matrix is a canonical extension of the basic transformation matrix used for orthonormal bases (ONB). The properties of the Gram matrix use a frame and its dual. We will investigate a criterion for finite dimensional spaces using frames. In particular, a space is finite dimensional if and only if Σ||gk||2 < ∞.


Any analysis / synthesis system that allows perfect reconstruction is equivalent to a frame in its discrete version. This can be applied to Gabor, wavelet, or any other such system (e.g. a Gamma tone filter bank).