Systematic Construction of Real Lapped Tight Frame Transforms

Abstract

We present a constructive algorithm for the design of real lapped equal-norm tight frame transforms. These transforms can be efficiently implemented through filter banks and have recently been proposed as a redundant counterpart to lapped orthogonal transforms, as well as an infinitedimensional counterpart to harmonic tight frames. The proposed construction consists of two parts: First, we design a large class of new real lapped orthogonal transforms derived from submatrices of the discrete Fourier transform. Then, we seed these to obtain real lapped tight frame transforms corresponding to tight, equal-norm frames. We identify those frames that are maximally robust to erasures, and show that our construction leads to a large class of new lapped orthogonal transforms as well as new lapped tight frame transforms.

Discrete Signal Processing on Graphs: Sampling Theory

“…The frame is maximally robust to erasures when every K × K submatrix (obtained by deleting N − K rows) is invertible [39]. In [39], the authors show that a polynomial transform matrix is one example of a frame maximally robust to erasures; in [40], the authors show that many lapped orthogonal transforms and lapped tight frame transforms are also maximally robust to erasures. It is clear that if the inverse graph Fourier transform matrix V as in ( 2) is maximally robust to erasures, any sampling operator that samples at least K signal coefficients guarantees perfect recovery; in other words, when a graph Fourier transform matrix happens to be a polynomial transform matrix, sampling any K signal coefficients leads to perfect recovery.…”

Lossless Image Compression Using New Biorthogonal Wavelets

“…This transform is known by the name of the wavelet transform. The wavelet transform is in use to represent different kinds of signals including multi-dimensional signals [5] [6]. Depending on specific application, a number of wavelets are proposed in the literature.…”