## Denoising via Recursive Wavelet Thresholding

Alyson Fletcher
(Professor Kannan Ramchandran)
Intel and (NSF) CCR 0096071

Wavelet thresholding has been a popular technique for denoising since the seminal work by Donoho and Johnstone [1]. The basic principle of wavelet thresholding is to identify and zero out wavelet coefficients of a signal which are likely to contain mostly noise. By preserving the most significant coefficients, wavelet thresholding preserves important highpass features of a signal such as discontinuities. This property is useful, for example, in image denoising to maintain the sharpness of the edges in an image [2].

It has long been recognized that the periodic shift-invariance of the wavelet transform implies that denoising by wavelet thresholding is itself periodically shift-invariant. Thus, various choices of shifting, denoising, and shifting back give different denoised estimates, and there is no simple way to choose the best from among these estimates. Coifman and Donoho [3] introduced cycle spinning, a technique estimating the true signal as the linear average of individual estimates derived from wavelet-thresholded translated versions of the noisy signal.

In this work, it is demonstrated that such an average can be dramatically improved upon [4]. In the conceptually simplest form, the proposed algorithm is to apply wavelet thresholding recursively by repeatedly translating, denoising the input via basic wavelet denoising, and then translating back. (Significant computational improvements are possible.) Exploiting the convergence properties of projections, the proposed algorithm can be regarded as a sequence of denoising projections that converge to the projection of the original noisy signal to a small subspace containing the true signal. Certain local and global converge properties are proven, and simulations on piecewise polynomial signals show marked improvement over both basic wavelet thresholding and standard cycle spinning. Figure 1 shows the improvement in denoising typically provided by our algorithm when the true signal is a piecewise polynomial. See [4] for details. Further work involves investigating various extensions to sets of wavelet bases and sets of observations.

Figure 1: Piecewise polynomial denoising results

[1]
D. L. Donoho and I. M. Johnstone, "Ideal Spatial Adaptation via Wavelet Shrinkage," Biometrika, Vol. 81, 1994.
[2]
Y. Xu, J. B. Weaver, D. M. Healy, Jr., and J. Lu, "Wavelet Transform Domain Filters: A Spatially Selective Noise Filtration Technique," IEEE Trans. Image Processing, Vol. 3, No. 6, November 1994.
[3]
R. R. Coifman and D. L. Donoho, "Translation-Invariant De-Noising," ed. A. Antoniadis and G. Oppenheim, Wavelets and Statistics, Springer Lecture Notes in Statistics, Vol. 103, Springer-Verlag, New York, 1995.
[4]
A. K. Fletcher, K. Ramchandran, and V. K. Goyal, "Wavelet Denoising by Recursive Cycle Spinning," Proc. IEEE Int. Conf. Image Processing, Rochester, NY, September 2002.