Sampling Without Input Constraints: Consistent Reconstruction in Arbitrary Spaces
Part of the
Applied and Numerical Harmonic Analysis
book series (ANHA)
In this chapter we develop a general framework for sampling and reconstruction procedures. The procedures we develop allow for almost arbitrary sampling and reconstruction spaces, as well as arbitrary input signals. The rudimentary constraint we impose on the reconstruction is that if the input lies in the reconstruction space, then the reconstruction will be equal to the original signal, so that our framework includes the more restrictive perfect reconstruction theories as special cases.
In our development we consider both nonredundant sampling and redundant sampling in which the samples constitute an overcomplete representation of the signal. In this case reconstruction is obtained using the oblique dual frame vectors, which lead to frame expansions in which the analysis and synthesis frame vectors are not constrained to lie in the same space as in conventional frame expansions. As we show, the algorithms we develop are consistent, so that the reconstructed signal has the property that although it is not necessarily equal to the original signal, it nonetheless yields the same measurements. Building upon this property of our algorithms, we develop a general procedure for constructing signals with prescribed properties.
KeywordsOblique Projection Perfect Reconstruction Reconstruction Scheme Sampling Vector Input Constraint
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