The what, how, and why of wavelet shrinkage denoising

Principles of wavelet shrinkage denoising are reviewed. Both 1-D and 2-D examples are demonstrated. The performance of various ideal and practical Fourier and wavelet based denoising procedures are evaluated and compared in a new Monte Carlo simulation experiment. Finally, recommendations for the practitioner are discussed. 1 Some Opposing Viewpoints Applied scientists and engineers who work with data obtained from the real world know that signals do not exist without noise. Under ideal conditions, this noise may decrease to such negligible levels, while the signal increases to such significant levels, that for all practical purposes denoising is not necessary. Unfortunately, the noise corrupting the signal, more often than not, must be removed in order to recover the signal and proceed with further data analysis. Should this noise removal take place in the original signal (time-space) domain or in a transform domain? If the latter, should it be the time-frequency domain via the Fourier transform or the time-scale domain via the wavelet transform? Enthusiastic supporters have described the development of wavelet transforms as revolutionizing modern signal and image processing over the past two decades. Conservative observers, however, would simply recognize this new field as contributing additional useful tools to a growing toolbox of transforms, in fact, an old toolbox that has had an evolving history over the past two centuries. For a review of available software libraries and an introduction to some of the wavelet literature, refer to the survey by Shearman [7]. Even more zealous advocates have claimed that a particular wavelet method called wavelet shrinkage denoising “offers all that we might desire of a technique, from optimality to generality” [4, page 312]. Inquiring skeptics, however, might be loath to accept these claims based on asymptotic theory without persuasive evidence from real-world experiments. Fortunately, a burgeoning literature is now addressing these concerns, and leading to a more realistic appraisal of the utility of wavelet shrinkage denoising. 2 A Simple Explanation and a 1-D Example But what is wavelet shrinkage denoising? First, it is not smoothing (despite the use by some authors of the term smoothing as a synonym for the term denoising). Whereas smoothing removes high frequencies and retains low frequencies, denoising attempts to remove whatever noise is present and retain whatever signal is present regardless of the frequency content of the signal. For example, ∗Email: [email protected]; Website: www.toolsmiths.com; Tel/Fax: 650-323-4336/5779. Technical Report CT-1998-09: original 10/13/98, revision 1/25/99. 2 Taswell: Wavelet Shrinkage Denoising. when we denoise music corrupted by noise, we would like to preserve both the treble and the bass. Second, it is denoising by shrinking (i.e., nonlinear soft thresholding) in the wavelet transform domain. Third, it consists of three steps: 1) a linear forward wavelet transform, 2) a nonlinear shrinkage denoising, and 3) a linear inverse wavelet transform. Because of the nonlinear shrinking of coefficients in the transform domain, this procedure is distinct from those denoising methods that are entirely linear. Finally, wavelet shrinkage denoising is considered a non-parametric method. Thus, it is distinct from parametric methods [6] in which parameters must be estimated for a particular model that must be assumed a priori. (For example, the most commonly cited parametric method is that of using least squares to estimate the parameters a and b in the model y = ax+ b.) Figure 1 displays a practical 1-D example demonstrating the three steps of wavelet shrinkage denoising with plots of a known test signal with added noise, the wavelet transform (from step 1), the denoised wavelet transform (from step 2), and the denoised signal estimate (from step 3). In the latter, note that the green curve is the estimate and the red curve is the difference between this estimate and the original true signal without noise. All results and figures reported here were generated with WAVB3X 4.5b3 Software [8] using filters from the systematized collection of Daubechies wavelets [11]. We can describe this example and the steps of the procedure more clearly with some mathematical notation (Sections 3 and 4) and software commands (Section 5). 3 A More Precise Definition Assume that the observed data