FUSION FRAMES AS CODES

BERNHARD G. BODMANN

A finite frame {fj}j=1n on a k-dimensional Hilbert space H and its analysis operator T can be interpreted as a block code for analog signals. Instead of blocks of bits (or strings of some length), the analysis operator transforms a vector x, which can be characterized by its expansion in a given orthonormal basis of size k, into a sequence of n frame coefficients {⟨x,fj⟩}j=1n. Similarly, the analysis operator of a fusion frame is given by a family of linear maps {Tj}j=1m which transform a vector x ∈H into its image j Tjx ∈ j Kj in the direct sum of spaces Kj containing the range of each Tj. We will refer to each vector Tjx ∈Kj as a component of x.

Frames can be understood as a special case of fusion frames when the rank of each Tj is one. Thus, many insights for frames have an analogue for fusion frames. In finite dimensions, the condition for having a frame or a fusion frame is simply the spanning property of {fj}j=1n or of the ranges {ran T j*} j=1m. This implies for the number of frame vectors, n k, and for the ranks j=1mrank T j k. In both cases, the redundancy which is incorporated by the analysis operator can be exploited to compensate errors which may corrupt the frame coefficients or components in the course of transmission or when they are stored. This is the goal of designing frames or fusion frames as codes. The purpose of optimal design is generally to use the redundancy to suppress the effect of errors maximally.

In the literature on the application of frames for coding, the most frequently discussed sources of errors are additive noise, quantization and erasures. So far, the mathematical treatment of fusion frames for coding has focused on erasures. This means, that one or more of the components of x have been lost. Linear reconstruction without knowledge of this component amounts to setting it to zero. A common measure for the impact of an erasure is the Euclidean error of the vector which has been reconstructed using the canonical dual (fusion) frame.

We know sharp estimates for the Euclidean reconstruction error and corresponding optimal designs for deterministic and random signals. The deterministic, worst-case scenario was examined for one lost component1 and for a higher number of losses.2 Averaged performance has been discussed as well, in the probabilistic treatment for the mean-square error occurring for random input vectors and a small number of erasures.3 4 We are only at the beginning of this fascinating subject and its application to coding. Many results on frames have already found a counterpart in the context of fusion frames, but many more deserve future research.

1P. G. Casazza and G. Kutyniok. Robustness of Fusion Frames under Erasures of Subspaces and of Local Frame Vectors. Radon transforms, geometry, and wavelets (New Orleans, LA, 2006), Contemp. Math., Amer. Math. Soc., Providence, RI, to appear.

2B. G. Bodmann, Optimal linear transmission by loss-insensitive packet encoding, Appl. Comput. Harmon. Anal. 22 (3) (2007) 274-285.

3A. Pezeshki, G. Kutyniok, and A. R. Calderbank. Fusion frames and Robust Dimension Reduction. 42nd Annual Conference on Information Sciences and Systems (CISS) (Princeton University, NJ, 2008)

4G. Kutyniok, A. Pezeshki, R. Calderbank, and T. Liu. Robust Dimension Reduction, Fusion Frames, and Grassmannian Packings. Appl. Comput. Harmon. Anal., to appear.