FUSION FRAMES AND SPARSE REPRESENTATIONS

GITTA KUTYNIOK

The last years have seen a rapidly increasing urgency in the design of special representation systems to derive sparse decompositions of particular classes of signals. We have learned that sparsity can be exploited in numerous ways such as optimal compression of signals and efficient acquisition of data, to name a few. Also it seems to be a property given to us by nature, since surprisingly it turns out that ’real-world’ signals ’usually’ live on lower dimensional manifolds, thus satisfying one main constraint for the existence of a sparsifying dictionary.

The present research in this area now follows two paths: one aiming to generate a deterministic representation system for some given class of signals – as one example we would like to mention the curvelets or shearlets for cartoon-like images –, and the other taking a probabilistic approach by considering random dictionaries – compressed sensing might serve as one example here. The design of sparsifying dictionaries is however far from being complete, and even for finite signals we have only seen the tip of the iceberg so far.

Let us now take a step aside for a moment and review the design of finite representation systems. Beginning from orthonormal bases, the need for redundancy led to the introduction of frames, and recently, the necessity for dealing with more complex signals as well as different processing requirements initiated the study of fusion frames. The other articles on this webpage shall prove enlightening for the interested reader to demonstrate a variety of practical applications of these new objects.

Thus it becomes now essential to study dictionaries of fusion frames with respect to their sparsity promoting properties. We might think, for instance, of a signal contained in the union of the subspaces constituting a fusion frame as being 1-sparse, since we can write it as the linear combination of elements from one subspace. In general, we would call a signal k-sparse with respect to a fusion frame {Wi}i∈I, if there exists J I with J∣≤ k such that

x ∈  〈Wi  : i ∈ J〉.

We realize that this notion coincides with the well-known one in the case of one-dimensional subspaces, whereas higher-dimensional subspaces lead to a weaker notion due to the clustering of special aspects of the signal inside the subspaces where no sparsity constraints are imposed on.

In general, we now face a variety of exciting questions such as, for instance:

Some of these questions are currently under investigation, but again there is far more to say. Concluding, we in fact strongly believe that fusion frames are much more flexible than frames for providing the versatility needed for the design of sparse dictionaries.