Parametrizing Finite Frames and Optimal Frame Completions
AIR FORCE INST OF TECH WRIGHT-PATTERSON AFB OH GRADUATE SCHOOL OF ENGINEERING AND MANAGEMENT
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Frames are used in many signal processing applications. We consider the problem of constructing every frame whose frame operator has a given spectrum and whose vectors have prescribed lengths. For a given spectrum and set of lengths, we know when such a frame exists by the Schur-Horn Theorem it exists if and only if its spectrum majorizes its squared lengths. We provide a more constructive proof of Horns original result. This proof is based on a new method for constructing any and all frames whose frame operator has a prescribed spectrum and whose vectors have prescribed lengths. Constructing all such frames requires one choose eigensteps--a sequence of interlacing spectra--which transform the trivial spectrum into the desired one. We give a complete characterization of the convex set of all eigensteps. Taken together, these results permit us, for the first time, to explicitly parametrize the set of all frames whose frame operator has a given spectrum and whose elements have a given set of lengths. Moreover, we generalize this theory to the problem of constructing optimal frame completions. That is, given a preexisting set of measurements, we add new measurements so that the final frame operator has a given spectrum and whose added vectors have prescribed lengths. We introduce a new matrix notation for representing the final spectrum with respect to the initial spectrum and prove that existence of such a frame relies upon a majorization constraint involving the final spectrum and the frames matrix representation. In a special case, we provide a formula for constructing the optimal frame completion with respect to fusion metrics such as the mean square error MSE and frame potential FP. Such fusion metrics provide a means of evaluating the e cacy of reconstructing signals which have been distorted by noise.
- Numerical Mathematics
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