Study Of Some Questions For Sparse Of Frames Based On ST Algorithm

The concept of frame was introduced by Dufn and Schaefer when they studiednonharmonic Fourier series in1952. A discrete frame in a separable Hilbert space isalmost as good as an orthonormal basis for expanding arbitrary elements. Frame the-ory as a new research direction developed after the wavelet theory, it’s the combinationof functional analysis, operator theory, the nolinear approximation and the informationtheory. Frames have stable decomposition of a signal for analysis or transmission, whilealso promting sparse expansions in signal processing. However, when the signal dimen-sion is large, the computation of the frame mesaurements of a signal typically requiresa large number of additions and multiplications, and this makes a frame decompositiondifcult in applications with limited computing budget. In order to solve the problem, weintroduce sparsity for such frames as a new feature in fnite-dimensional Hilbert spaces inthis thesis. A sparse frame is a frame whose every element has a sparse representation inan orthonormal basis, thereby enabling low complexity frame decompositions. We takethe sum of the numbers of vectors for this orthonormal basis when expanding each framevector as sparsity measure.This thesis consists of fve chapters.Chapter1introduces the origin of the frame concept, the frame and research contentsof sparse frames, and sketches the structure of the thesis.Chapter2lists some basic theorems to be used throughout the thesis, and states theSpectral Tetris algorithm for constructing unit norm frames. By using the key idea,anew block matrix T (r) of3×3is constructed.Chapter3introduces the background of sparse frame, and gives the defnition ofsparse frame and the concept of optimally sparse frames.Chapter4is one of the main contents. An optimally result for sparse frame isestablished when the eigenvalues of the frame operator equal to2or are greater than2.Chapter5is another main content. The the defnition of prime and divisible tightframes are stated, and the condition which ST algorithm can work is given when the eigenvalues of the frame operator are smaller than2.