Lapped Tight Frame Transforms

We propose a new class of equal-norm tight frames termed lapped tight frame transforms (LTFTs). These can be seen as a redundant counterpart to bases known as lapped orthogonal transforms (LOTs) introduced by Malvar and Cassereau, as well as an infinite-dimensional counterpart to harmonic tight frames (HTFs). To construct LTFTs, we seed them from LOTs and show that, in a specific case, the process preserves the equal norm. As both their basis counterpart LOTs as well as their finite-dimensional one HTFs, LTFTs possess many desirable properties, such as equal norm and efficient implementation.

[1]  John Princen,et al.  Subband/Transform coding using filter bank designs based on time domain aliasing cancellation , 1987, ICASSP '87. IEEE International Conference on Acoustics, Speech, and Signal Processing.

[2]  P. Casazza.  THE ART OF FRAME THEORY , 1999, math/9910168.

[3]  Helmut Bölcskei,et al.  Frame-theoretic analysis of oversampled filter banks , 1998, IEEE Trans. Signal Process..

[4]  N.G. Kingsbury,et al.  Frequency-domain motion estimation using a complex lapped transform , 1993, IEEE Trans. Image Process..

[5]  Martin Vetterli,et al.  Oversampled filter banks , 1998, IEEE Trans. Signal Process..

[6]  Henrique S. Malvar,et al.  Signal processing with lapped transforms , 1992 .

[7]  Jelena Kovačević,et al.  Life Beyond Bases: The Advent of Frames , 2006 .

[8]  Jean-Christophe Pesquet,et al.  Low Redundancy Oversampled Lapped Transforms and Application to 3D Seismic Data Filtering , 2006, 2006 IEEE International Conference on Acoustics Speech and Signal Processing Proceedings.

[9]  Philippe Michel Cassereau.  A new class of optimal unitary transforms for image processing , 1985 .

[10]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[11]  I. M. Glazman,et al.  Theory of linear operators in Hilbert space , 1961 .

[12]  J. Kovacevic,et al.  Life Beyond Bases: The Advent of Frames (Part II) , 2007, IEEE Signal Processing Magazine.

[13]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[14]  Jelena Kovacevic,et al.  Real, tight frames with maximal robustness to erasures , 2005, Data Compression Conference.

[15]  I. Daubechies,et al.  PAINLESS NONORTHOGONAL EXPANSIONS , 1986 .