Special Seminar in Applied Mathematics

In modern signal processing, basis-like systems are applied to derive stable and redundant signal representations. Frames are basis-like systems that span a vector space but allow for linear dependency, that can be used to reduce noise, find sparse representations, or obtain other desirable features unavailable with orthonormal bases. Fusion frames enable signal decompositions into weighted linear subspace components, and tight fusion frames provide a direct reconstruction formula. If all the subspaces are equiangular, then fusion frames are maximally robust against two subspace erasures.

We first derive an upper bound on the maximal number of mutually equiangular subspaces. We circumvent this limitation for robustness against erasures by extending the concept of fusion frames and by changing the measure of robustness. Our main contribution is the introduction of the notion of tight p-fusion frames, a sharpening of the notion of tight fusion frames, that is closely related to the classical notions of designs and cubature formulas in Grassmann spaces. We analyze tight p-fusion frames with methods from harmonic analysis in the Grassmannians.

Finally, we verify that tight p-fusion frames are robust against up to p subspace erasures.