Abstract
De-Noising with the traditional (orthogonal, maximally-decimated) wavelet transform sometimes exhibits visual artifacts; we attribute some of these—for example, Gibbs phenomena in the neighborhood of discontinuities—to the lack of translation invariance of the wavelet basis. One method to suppress such artifacts, termed “cycle spinning” by Coifman, is to “average out” the translation dependence. For a range of shifts, one shifts the data (right or left as the case may be), De-Noises the shifted data, and then unshifts the de-noised data. Doing this for each of a range of shifts, and averaging the several results so obtained, produces a reconstruction subject to far weaker Gibbs phenomena than thresholding based De-Noising using the traditional orthogonal wavelet transform.
Cycle-Spinning over the range ofall circulant shifts can be accomplished in ordernlog2(n) time; it is equivalent to de-noising using the undecimated or stationary wavelet transform.
Cycle-spinning exhibits benefits outside of wavelet de-noising, for example in cosine packet denoising, where it helps suppress ‘clicks’. It also has a counterpart in frequency domain de-noising, where the goal of translation-invariance is replaced by modulation invariance, and the central shift-De-Noise-unshift operation is replaced by modulate-De-Noise-demodulate.
We illustrate these concepts with extensive computational examples; all figures presented here are reproducible using the WaveLab software
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References
Buckheit, J. and Donoho, D.L. (1995) WaveLab and Reproducible Research, in this Volume.
Berger, J., Coifman, R.R. and Goldberg, M. (1994) Removing Noise from Music using Local Trigonometric Bases and Wavelet Packets.J. Audio Eng. Soc. 42.
Beylkin, G. (1992) On the Representation of Operators in Bases of Compactly Supported Wavelets.SIAM J. Numer. Anal.,291716–1740.
Chen, S. and Donoho, D.L. (1995) Atomic Decomposition by Basis Pursuit. Technical Report, Department of Statistics, Stanford University.
Coifman, R.R. and Meyer, Y. (1991) “Remarques sur l’analyse de Fourier à fenêtre,”Comptes Rendus Acad. Sci. Paris (A)312 259–261.
Coifman, R.R., Meyer, Y. and Wickerhauser, M.V. (1992) “Wavelet analysis and signal processing,” pp. 153–178 inWavelets and Their Applications, M.B. Ruskai et al. (eds.), Jones and Bartlett, Boston.
Coifman, R.R. and Wickerhauser, M.V. (1992) “Entropy-based algorithms for best-basis selection.”IEEE Trans. Info. Theory 38713–718.
Coifman, R.R. and Wickerhauser, M.V. (1993) “Wavelets and Adapted Waveform Analysis: A Toolkit for Signal Processing and Numerical Analysis.” inDifferent Perspectives on Wavelets, in I. Daubechies, ed. pp. 119–153. Providence, RI: American Mathematical Society.
Daubechies, I. (1992)Ten Lectures on Wavelets. Philadelphia: SIAM.
Donoho, D.L. (1992) De-Noising via Soft Thresholding. To appearIEEE Trans. Info. Thry, May, 1995.
Donoho, D.L. (1993) Unconditional bases are optimal bases for data compression and for statistical estimation,Applied and Computational Harmonic Analysis,1, 100–115.
Donoho, D.L. (1993) Nonlinear Wavelet Methods for Recovery of Signals, Images, and Densities from noisy and incomplete data, inDifferent Perspectives on Wavelets, I. Daubechies, ed. pp. 173–205. Providence, RI: American Mathematical Society.
Donoho, D.L. (1993) Wavelet Shrinkage and W.V.D. - A Ten-Minute Tour, inProgress in Wavelet Analysis and Applications. Y. Meyer and S. Roques, eds. pp. 109–128., Gif-sur- Yvette (France): Éditions Frontières.
Donoho, D.L. (1994) On Minimum Entropy Segmentation, in Wavelets:Theory,Algorithms and Applications. C.K. Chui, L. Montefusco and L. Puccio, Eds. San Diego: Academic Press.
Donoho, D.L. and Johnstone, I.M. (1994) Ideal spatial adaptation via wavelet shrinkage.Biometrika,81, 425–455.
Donoho, D.L. and Johnstone, I.M. (1992) Adapting to Unknown Smoothness via Wavelet Shrinkage, to appearJASA, 1995.
Donoho, D.L. and Johnstone, I.M. (1994) Ideal denoising in an orthonormal basis chosen from a library of bases.Comptes Rendus Acad. Sci. Paris A319, 1317–1322.
Donoho, D.L., I.M. Johnstone, G. Kerkyacharian and D. Picard (1995) Wavelet Shrinkage: Asymptopia.J. Roy. Statist. Soc.B 572, 301–369.
Meyer, Y. (1993)Wavelets: Algorithms and ApplicationsPhiladelphia: SIAM, 1993.
Mallat, S. and Zhang, S. (1993) Matching Pursuits with Time-Frequency Dictionaries.IEEE Transactions on Signal Processing,41, 3397–3415.
Nason, Guy, and Silverman, B.W. (1995) The Stationary Wavelet Transform and some Statistical Applications. This Volume.
Saito, N. (1994)Feature Extraction using Local Discriminant Basis. Yale Dissertation, December 1994.
Simoncelli, E.P., Freeman. W.T., Adelson, E.H., and Heeger, D.J. (1992) Shiftable multiscale transforms.IEEE Trans. Info. Theory,38, pp. 587–607.
Wickerhauser, M.V. (1994)Adapted Wavelet Analysis, from Theory to Software. AK Peters: Boston.
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Coifman, R.R., Donoho, D.L. (1995). Translation-Invariant De-Noising. In: Antoniadis, A., Oppenheim, G. (eds) Wavelets and Statistics. Lecture Notes in Statistics, vol 103. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2544-7_9
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DOI: https://doi.org/10.1007/978-1-4612-2544-7_9
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