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Frame Theory

Frames are nowadays a standard methodology in applied mathematics as well as in various application areas when redundant, yet stable expansions are required. Moreover, due to their flexibility, frames often allow much sparser approximations than traditional orthonormal bases. Thus, they are key to the novel area of compressed sensing, which can only be applied if the signal to be recovered admits a sparse approximation usually given by a frame.

Frame theory is a very rich mathematical theory, in particular, due to the variety of systems to be considered. Already finite frames, i.e., frames in a finite dimensional Hilbert sparse, and infinite frames, i.e., frames living in an infinite dimensional Hilbert space, have often to be treated with very different methods. The first requires tool from linear algebra and matrix theory, whereas the second from functional analysis and operator theory. Typical examples of frames are finite frames and frames from applied harmonic analysis such as Gabor frames, curvelets, and shearlets.

However, recently, a number of new applications have emerged which cannot be modeled naturally by one single frame system. They typically share a common property that requires distributed processing such as sensor networks. Fusion frames, which were recently introduced by Casazza and Kutyniok, extend the notion of a frame and provide exactly the mathematical framework not only to model these applications but also to derive efficient and robust algorithms. In particular, fusion frames generalize frame theory by using subspaces in the place of vectors as signal building blocks. Thus signals can be represented as linear combinations of components that lie in particular, and often overlapping, signal subspaces. Such a representation provides significant flexibility in representing signals of interest compared to classical frame representations.

Some of our Research Topics

  • Development of the theory of fusion frames.
  • Analysis of robustness of frames and fusion frames under perturbations.
  • Application of frames to the problem of phase retrieval, i.e., the recovery of signals from the magnitude of their frame coefficients.
  • Notions of redundancy for frames and fusion frames.

Survey Papers

  • P. G. Casazza and G. Kutyniok.
    Fusion Frames.
    Finite Frames: Theory and Applications, 437-478, Birkhäuser Boston, 2012.
    Download (PDF, 332,2 KB)
  • P. G. Casazza, G. Kutyniok, and F. Philipp.
    Introduction to Finite Frame Theory.
    Finite Frames: Theory and Applications, 1-53, Birkhäuser Boston, 2012.
    Download (PDF, 289,9 KB)

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Auxiliary Functions