Window-Dependent Bases for Efficient Representations of the Stockwell Transform

Since its appearing in 1996, the Stockwell transform (S-transform) has been applied to medical imaging, geophysics and signal processing in general. In this paper, we prove that the system of functions (so-called DOST basis) is indeed an orthonormal basis of L^2([0,1]), which is time-frequency localized, in the sense of Donoho-Stark Theorem (1989). Our approach provides a unified setting in which to study the Stockwell transform (associated to different admissible windows) and its orthogonal decomposition. Finally, we introduce a fast -- O(N log N) -- algorithm to compute the Stockwell coefficients for an admissible window. Our algorithm extends the one proposed by Y. Wang and J. Orchard (2009).

[1]  M. Fornasier,et al.  Generalized coorbit theory, Banach frames, and the relation to α‐modulation spaces , 2008 .

[2]  Hongmei Zhu,et al.  The generalization of discrete Stockwell transforms , 2011, 2011 19th European Signal Processing Conference.

[3]  Nicki Holighaus,et al.  Theory, implementation and applications of nonstationary Gabor frames , 2011, J. Comput. Appl. Math..

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

[5]  D. Donoho,et al.  Uncertainty principles and signal recovery , 1989 .

[6]  A. Galbis,et al.  Atrial fibrillation subtypes classification using the General Fourier-family Transform. , 2014, Medical engineering & physics.

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

[8]  Dusmanta Kumar Mohanta,et al.  Power quality analysis using Discrete Orthogonal S-transform (DOST) , 2013, Digit. Signal Process..

[9]  M. W. Wong,et al.  Continuous inversion formulas for multi-dimensional modified Stockwell transforms , 2015 .

[10]  Sylvia Drabycz,et al.  Image Texture Characterization Using the Discrete Orthonormal S-Transform , 2008, Journal of Digital Imaging.

[11]  Yanwei Wang,et al.  Efficient Stockwell Transform with Applications to Image Processing , 2011 .

[12]  Hongmei Zhu,et al.  A Characterization of Stockwell Spectra , 2006 .

[13]  A. Galbis,et al.  A Group Representation Related to the Stockwell Transform , 2009 .

[14]  Karlheinz Gröchenig,et al.  Foundations of Time-Frequency Analysis , 2000, Applied and numerical harmonic analysis.

[15]  M. W. Wong,et al.  Continuous inversion formulas for multi-dimensional stockwell transforms , 2013 .

[16]  Pradipta Kishore Dash,et al.  Detection and characterization of multiple power quality disturbances with a fast S-transform and decision tree based classifier , 2013, Digit. Signal Process..

[17]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[18]  Jeff Orchard,et al.  Symmetric Discrete Orthonormal Stockwell Transform , 2008 .

[19]  Qiang Guo,et al.  Modified Stockwell Transforms and Time-Frequency Analysis , 2008 .

[20]  Zhenhua He,et al.  Seismic data denoising based on mixed time-frequency methods , 2011 .

[21]  Yanwei Wang,et al.  Fast Discrete Orthonormal Stockwell Transform , 2009, SIAM J. Sci. Comput..

[22]  John Ladan.  An Analysis of Stockwell Transforms, with Applications to Image Processing , 2014 .

[23]  B. Torrésani,et al.  N-dimensional affine Weyl-Heisenberg wavelets , 1993 .

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

[25]  Bradley G Goodyear,et al.  Removal of phase artifacts from fMRI data using a Stockwell transform filter improves brain activity detection , 2004, Magnetic resonance in medicine.

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

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

[28]  Massimo Fornasier,et al.  Banach frames for α-modulation spaces , 2007 .

[29]  J. R. Mitchell,et al.  A new local multiscale Fourier analysis for medical imaging. , 2003, Medical physics.

[30]  Ryszard S. Choras,et al.  Time-Frequency Analysis of Image Based on Stockwell Transform , 2013, IP&C.

[31]  J. Benedetto,et al.  The Theory of Multiresolution Analysis Frames and Applications to Filter Banks , 1998 .

[32]  Jin Jiang,et al.  Frequency-based window width optimization for S-transform , 2008 .

[33]  H. Feichtinger,et al.  Gabor Frames and Time-Frequency Analysis of Distributions* , 1997 .

[34]  P. Boggiatto,et al.  Two Aspects of the Donoho-Stark Uncertainty Principle , 2015, 1510.02621.