11441 Minimax properties The NLP method and hard thresholding have similar convergence properties

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s THEOREM 113 Let X Bp,q

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s , then RN LP RHT + O(RHT ) 2s + 1

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Local Regularity and Multifractal Methods for Image and Signal Analysis

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Thus, NLP is near-minimax Additionally, If > 1 the estimator is adaptive 2

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11442 Regularity control The advantage of NLP is that it allows a control over the local regularity through the parameter PROPOSITION 113 (Increase of regularity) Let Y (n, t) and X N LP (n, t) denote respectively the regularity of the noisy signal Y and of the estimator X N LP at the point t, estimated by wcr Then at each point t:

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X N LP (n, t) = Y (n, t) + Kn

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In other words, NLP increases the H lder regularity proportionally to the parameter This result sometimes allows us to nd an optimal value for This is particularly the case for the set of functions P ART ( ) (de ned in (114)) PROPOSITION 114 For a signal X P ART ( ), at each t: lim E[ X N LP (n, t)] E[ Y (n, t)] = 1 + (6 1) (2 + 1)3

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ie lim E[ X N LP (n, t)] = 2 2 (6 1) + 1+ (1 + 2 )2 (2 + 1)3

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Using this proposition, we may calculate the value ideal that ensures that the denoised signal will have the same average regularity as the original signal PROPOSITION 115 For a signal X P ART ( ), the optimal parameter is ideal = (1 + 2 )(2 + 3) 2(2 2 + 3 + 3)

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Scaling, Fractals and Wavelets

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Figure 114 Denoising of a lacunary wavelet series: (a) original signal (regularity: 02, lacunarity: 07); (b) noisy version; (c) denoising by the NLP method; (d) denoising by hard thresholding

11443 Numerical experiments Lacunary wavelet series We present an example of denoising with NLP, along with a comparison with hard thresholding, on a lacunary wavelet series [JAF 00] The regularity is equal to 02, and the lacunarity parameter is 07 Figure 114 represents the original signal, the noisy signal and the two denoisings The NLP method provides a reasonable result, while the hard thresholding clearly oversmooths the signal SAR images As a second illustration, we display an original synthetic aperture radar (SAR) image along with its hard thresholding and NLP denoisings in Figure 115 As we can see, the original image appears very noisy, and does not seem to hold any useful information The hard thresholded image is not very readable either However, we can see clearly on the image processed with NLP a river owing from the top of the image and assuming roughly an inverted Y shape Denoising is used in this application as a pre-processing step that enhances the image so that it will be possible to automatically detect the river Such a procedure is used by IRD, a French agency, which, in this particular application, is interested in monitoring water resources in a region of Africa

Local Regularity and Multifractal Methods for Image and Signal Analysis

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Figure 115 Left: original SAR image Middle: denoising by HT Right: denoising by NLP

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