Abstract
Due to resilience to background noise, stability of sparse reconstruction, and ability to capture local time-frequency information, the frame theory is becoming a dynamic forefront topic in data science. In this study, we overcome the disadvantages in the construction of traditional framelet packets derived by frame multiresolution analysis and square iterative matrices. We propose two novel approaches: One is to directly split known framelets again and again; the other approach is based on a generalized scaling function whose shifts are not a frame of some space. In these two approaches, the iterative matrices used are not square and the number of rows in the iterative matrix can be any integer number.
Highlights
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As a generalization of wavelet packets, framelet packets was first constructed from frame multiresolution analyses (FMRA) [4,5,6]
The other approach is to remove the use of scaling function in FMRA, i.e., it starts from a generalized scaling function whose shifts are not a frame of some space
Summary
As a generalization of wavelet packets, framelet packets was first constructed from frame multiresolution analyses (FMRA) [4,5,6]. Since generally the scaling function of FMRA is discontinuous in frequency domain, the derived framelet packets cannot possess nice time-domain localization. The other approach is to remove the use of scaling function in FMRA, i.e., it starts from a generalized scaling function whose shifts are not a frame of some space. In these two approaches, all the iterative matrices are not square and the number of rows in iterative matrix can be any integer number.
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