The unique sparsest solution to an under-determined system of linear equations can be recovered for suitable coefficient matrices and admissible sparsity levels. The focus of this thesis is therefore on two subjects.
Firstly, on the used coefficient matrices:
A common property of such matrices is the mutual coherence among their columns. With the help of Best Spherical Codes, it is possible to optimize this coherence property. This method should be further investigated.
Additionally it should be extended to scenarios, where the under-determined system is built by a multiplication of a so called measurement matrix with a dictionary matrix. For such scenarios, other optimization criteria do exist as well, which need to be considered and compared.
Secondly, admissible sparsity levels should be investigated for which a successful recovery is possible. The bounds typically given for optimal recovery are strict guarantees. However, it should be shown in this thesis that for general under-determined systems, a sparse solution is unique with high probability and may contain much more non-zero values than predicted by the strict bound.