Signal Reconstruction from Frame and Sampling Erasures

We give some new methods for perfect reconstruction from frame and sampling erasures in a small number of steps. By bridging an erasure set we mean replacing the erased Fourier coefficients of a function with respect to a frame by appropriate linear combinations of the non-erased coefficients. We prove that if a minimal redundancy condition is satisfied bridging can always be done to make the reduced error operator nilpotent of index 2 using a bridge set of indices no larger than the cardinality of the erasure set. This results in perfect reconstruction of the erased coefficients. We also obtain a new formula for the inverse of an invertible partial reconstruction operator. This leads to a second method of perfect reconstruction from frame and sampling erasures in a small number of steps. This gives an alternative to the bridging method for many (but not all) cases. The methods we use employ matrix techniques only of the order of the cardinality of the erasure set, and are applicable to rather large finite erasure sets for infinite frames and sampling schemes as well as for finite frame theory. These methods are usually more efficient than inverting the frame operator for the remaining coefficients because the size of the erasure set is usually much smaller than the dimension of the underlying Hilbert space. Some new classification theorems for frames are obtained and some new methods of measuring redundancy are introduced based on our bridging theory.