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For general results about adaptivity, see [LEP 90, LEP 91, LEP 92, BIR 97] 1143 Wavelet based denoising A popular set of denoising methods is based on decomposing the corrupted signal in a wavelet basis, processing the wavelet coef cients, and then going back to the time domain In the case of additive white noise, this is justi ed by two fundamental facts: rst, many real-world signals have a sparse structure in the wavelet domain, ie a few coef cients are signi cant, and most are small or zero Second, for an orthonormal wavelet transform, all wavelet coef cients of a white noise are iid random variables Denoising in the wavelet domain thus allows us to separate in an easy way large , signi cant, coef cients, from small coef cients due mainly to noise Throughout this section, the wavelet coef cients of a signal X are denoted by xj,k where j is scale and k is location X is the original signal, Y the observed noisy signal and X an estimator of X We assume that Y = X +B, where B is a centered Gaussian white noise with variance 2 , independent from the original signal X Thus, we have yj,k = xj,k + bj,k Since the wavelet basis is supposed to be orthonormal, the bj,k are also Gaussian and iid The rst and simplest methods for denoising based on the above principles are the so-called hard and soft thresholding [DEV 92, DON 94] Since the time these methods were introduced, a huge number of improvements have been proposed, ranging from block thresholding [HAR 98] to Bayesian approaches [VID 99] and many more We brie y recall the basics of hard thresholding, and show why a different method is needed for the processing of irregular signals DEFINITION 117 Let Yn be a sample of Y on 2n points The estimator of X by hard thresholding is X HT , a signal with the following wavelet coef cients: { HT }j,k = {yj,k 1|yj,k | n }j,k xj,k where n is a given threshold.

Traditional choices for n include the so-called universal, sure and Bayesian n thresholds [VID 99] In this section, the universal threshold n = 2 2 2n will be used throughout

Figure 76 Neighbor location where the node estimates its location as the centroid of the locations of anchors that it hears

s THEOREM 112 (Risk for hard thresholding (D Donoho)) Let X Bp,q and Xn its n sampled version on 2 points

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In other organizations, projects, and circumstances, price decisions are not sufficient Boeing Corporation, the world s largest manufacturer of airframes, has a highly sophisticated outsourcing operation that involves establishing long-term linkages with relatively few, high-quality subcontractors These subcontractors must prove their commitment to the Boeing quality model by allowing the aircraft manufacturer s quality staff onsite, in order to instruct the subcontractor in the expected process for developing the subassembly parts Price matters, but of equal importance to the Boeing personnel is the evidence of commitment to their quality model In another example, consider the operations in the oil industry where an oil company contracts with a firm to deliver an offshore facility worth hundreds of millions of dollars This kind of contract is backed up with an extensive amount of preparation work and documentation including detailed definitions of scope, price, schedule, and a detailed project execution plan You would assume that such preparation would take care of all customer concerns However, even here the commitment from the contractor team to meet the jointly agreed goals is still an important factor in making the project happen This is often emphasized in team-building sessions and commitment statements signed by all project team members

RHT := E[(Xn X HT )2 ] Cn2 2s+1 Thus, hard thresholding is near-minimax A limitation of hard thresholding, as well as of most wavelet-based methods, is that they are not well adapted to denoise highly textured or everywhere irregular signals, in particular (multi)fractal or multifractional signals, with potentially rapid variations in local regularity It is particularly well-known that, when the original signal is itself irregular, most wavelet-based denoising methods will typically produce an oversmoothed signal and/or so-called ringing effects Indeed, as recalled above, the basic idea behind wavelet thresholding is that many real-world signals have a sparse wavelet representation, with few large wavelet coef cients Putting small coef cients to 0 in the noisy signal will then in general do no harm, since these are mainly due to noise Everywhere irregular signals, on the other hand, have signi cant coef cients scattered all other the time-frequency plane At high frequencies, these signi cant but relatively small coef cients in the signal crucially determine the local irregularity Zeroing small coef cients will thus typically destroy the regularity information As a consequence, it is no surprise that a speci c method has to be designed for such signals Figure 114 illustrates some of the drawbacks just mentioned A theoretical result on a particular class of signals also allows us to measure precisely the over-smoothing effect of hard thresholding De ne the set P ART ( ) as follows: PART( ) := X, {xj,k }j,k = { j,k 2 j( + 2 ) }j,k , j,k iid in { 1, 1}

(114)

PROPOSITION 112 Let X HT (n, t) denote the regularity of the signal after hard thresholding, estimated by the wcr method Then, for a signal X P ART ( ), at each point t, 6 + 1 1 8 3 + 12 2 + 12 + p lim E[ X HT (n, t)] = n 2 2(2 + 1)2 (2 + 1)3 where p is the number of vanishing moments of the wavelet This result means that the regularity of the denoised signal is essentially controlled by p In particular, if we use a wavelet with an in nite number of vanishing moments, the estimated regularity of the hard thresholded signal will be equal to in nity

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