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Biometrika 2004 91(4):955-973; doi:10.1093/biomet/91.4.955
© 2004 by Biometrika Trust
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‘Analytic’ wavelet thresholding

Sofia C. Olhede and Andrew T. Walden

Department of Mathematics, Imperial College London, London SW7 2AZ, U.K. s.olhede{at}imperial.ac.uk, a.walden{at}imperial.ac.uk

We introduce so-called analytic stationary wavelet transform thresholding where, using the discrete Hilbert transform, we create a complex-valued ‘analytic’ vector from which an amplitude vector is defined. Thresholding of a real-valued wavelet coefficient at some transform level is carried out according to the corresponding value in this amplitude vector; relevant statistical results follow from properties of the discrete Hilbert transform. Analytic stationary wavelet transform thresholding is found to produce consistently a reduced mean squared error compared to using standard stationary wavelet transform, or ‘cycle spinning’, thresholding. For signals with extensive oscillations at some transform levels, this improvement is very marked. Furthermore we show that our thresholding test is invariant to phase shifts in the data, whereas, if complex wavelet filters are being used, the filters must be analytic or anti-analytic at each level of the wavelet transform.

Key Words: Cycle spinning; Denoising; Hilbert transform; Phase; Thresholding; Wavelet transform

Received February 2003. Revised March 2004.


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