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1145 Denoising using exponent between scales 11451 Introduction In [ECH 07], Echelard presents a denoising method that is similar in spirit to that just described, and is thus also well tted to the processing of irregular signals The proposed approach consists of extrapolating the unknown, small, coef cients by imposing a local regularity constraint More precisely, the small coef cients are reconstructed in such a way that the local regularity at each point of the denoised signal matches the regularity of the original signal Of course, since the original signal is unknown, so is its regularity Thus, we rst need to estimate the local regularity of the original signal from the noisy observations As in the previous section, a dif culty arises from working on discrete signals Indeed, the very de nition of H lder exponents requires us to let the resolution tend to , which cannot be done here We require an adapted de nition of that both makes sense at nite resolution and allows us to capture the visual impression of regularity on sampled signals In the previous section, a regression of the wavelet coef cients was used for this purpose Here, a different path is taken In view of the fact that the perceived regularity depends on the considered range of scales, an exponent between two scales is de ned as follows: log |xj,k | 1/2 g (j1 , j2 , X) = min min j j [j1 ,j2 ] k Z In order to maintain some information at small scale, the proposed method follows the steps below: Estimate the critical scale cn , de ned as the scale where the coef cients of the white noise become predominant as compared to the ones of the signal Estimate the regularity sn of the original signal at the considered point, using coef cients at scales larger than cn.

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