Need for speed: Fast Stockwell transform (FST) with O(N) complexity

In this paper, we propose two fast, spline based, algorithms for computing the Stockwell Transform or the S-transform. It is a redundant, time-frequency representation that has certain desirable features which make it an attractive choice for signal analysis in different areas and motivated by its diverse applications, we seek to reduce its computational complexity. The S-transform bears an acute resemblance with the Gabor transform and can also be associated to the Continuous Wavelet Transform (CWT). Our formulation is based on the above mentioned connectivity with the two classical time-frequency tools. What singles out our approach is that it is recursive in nature and leads to a complexity of O(N) - for arbitrary scales, independent of scale of window.

[1]  Michael Unser,et al.  On the asymptotic convergence of B-spline wavelets to Gabor functions , 1992, IEEE Trans. Inf. Theory.

[2]  M. Benedicks.  On Fourier transforms of functions supported on sets of finite Lebesgue measure , 1985 .

[3]  Jin Jiang,et al.  Instantaneous Frequency Estimation Using the ${\rm S}$ -Transform , 2008, IEEE Signal Processing Letters.

[4]  M. Unser.  Fast Gabor-like windowed Fourier and continuous wavelet transforms , 1994, IEEE Signal Processing Letters.

[5]  Michael Unser,et al.  B-spline signal processing. I. Theory , 1993, IEEE Trans. Signal Process..

[6]  C. Robert Pinnegar,et al.  The Bi-Gaussian S-Transform , 2002, SIAM J. Sci. Comput..

[7]  James H. McClellan,et al.  Efficient approximation of Gaussian filters , 1997, IEEE Trans. Signal Process..

[8]  Akram Aldroubi,et al.  B-SPLINE SIGNAL PROCESSING: PART I-THEORY , 1993 .

[9]  Michael Unser,et al.  Continuous wavelet transform with arbitrary scales and O(N) complexity , 2002, Signal Process..

[10]  Kok-Kiong Poh,et al.  Analysis of Neonatal EEG Signals using Stockwell Transform , 2007, 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[11]  Gary F. Margrave,et al.  Letter to the Editor: Stockwell and Wavelet Transforms , 2006 .

[12]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[13]  Jingang Zhong,et al.  Multiscale windowed Fourier transform for phase extraction of fringe patterns. , 2007, Applied optics.

[14]  Jin Jiang,et al.  A Window Width Optimized S-Transform , 2008, EURASIP J. Adv. Signal Process..

[15]  Juan José Dañobeitia,et al.  The S-Transform and Its Inverses: Side Effects of Discretizing and Filtering , 2007, IEEE Transactions on Signal Processing.

[16]  Michael Unser,et al.  Recursive Regularization Filters: Design, Properties, and Applications , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[17]  Lalu Mansinha,et al.  Localization of the complex spectrum: the S transform , 1996, IEEE Trans. Signal Process..

[18]  Robert Glenn Stockwell,et al.  A basis for efficient representation of the S-transform , 2007, Digit. Signal Process..

[19]  Juan José Dañobeitia,et al.  The $S$-Transform From a Wavelet Point of View , 2008, IEEE Transactions on Signal Processing.

[20]  C. Robert Pinnegar,et al.  The S-transform with windows of arbitrary and varying shape , 2003 .

[21]  Michael Unser,et al.  Fast B-spline Transforms for Continuous Image Representation and Interpolation , 1991, IEEE Trans. Pattern Anal. Mach. Intell..