Ideal denoising within a family of tree-structured wavelet estimators

Abstract

International audienceWe focus on the performances of tree-structured wavelet estimators belonging to a large family of keep-or-kill rules, namely the Vertical Block Thresholding family. For each estimator, we provide the maximal functional space (maxiset) for which the quadratic risk reaches a given rate of convergence. Following a discussion on the maxiset embeddings, we identify the ideal estimator of this family, that is the one associated with the largest maxiset. We emphasize the importance of such a result since the ideal estimator is different from the usual (plug-in) estimator used to mimic the performances of the Oracle. Finally, we confirm the good performances of the ideal estimator compared to the other elements of that family through extensive numerical experiments

Combining thresholding rules: a new way to improve the performance of wavelet estimators

“…We recall that estimators induced by these rules are, respectively, the Blockshrink estimator studied by Cai (1997) and Autin et al (2011b) and the Hard Tree estimator studied by Autin (2004Autin ( , 2008a and Autin et al (2011a). Our numerical results clearly illustrate the need to use the combination of the previous estimators, called the Block Tree estimator, rather than the Blockshrink estimator or the Hard Tree estimator, since this Block Tree estimator behaves well over all the 12 functions considered here.…”

“…This approach consists of determining the maxiset of a thresholding procedure that is the maximal functional space for which the quadratic risk of the procedure reaches a given rate of convergence. As previously discussed in Cohen, De Vore, Kerkyacharian, and Picard (2001b), Picard (2000, 2002), Autin (2004Autin ( , 2008a, Autin, Le Pennec, Loubes, and Rivoirard (2010), Autin, Freyermuth, and von Sachs (2011a) and Autin, Freyermuth, and von Sachs (2011b), this approach can be successful at differentiating between minimax-equivalent procedures whenever their maxisets are nested. Without such embeddings, the comparison would be impossible.…”

“…As examples, we cite estimators that rely on vertical-block thresholding rules (see Cohen et al 2001b;Autin 2008a,b;Autin et al 2011a) or horizontal-block thresholding rules (see, among others, Cai 1997Cai , 1999Cai , 2008Hall, Kerkycharian, and Picard 1998a,b;Cai and Silverman 2001;Cai and Zhou 2009;Autin et al 2011b). When looking at these procedures, their maxisets contain those of procedures based on elitist rules, including Hard and Soft thresholding estimators, and also many Bayes procedures (see Autin, Picard, and Rivoirard 2006).…”

“…The (μ T , m) thresholding estimator was already studied by Autin (2008a) and Autin et al (2011a). It relies on a rule which keeps empirical wavelet coefficients with level strictly less than j mt that are larger in absolute value than the threshold mt and keeps their ancestors in the dyadic tree rooted a ( j 0 , k 0 ) := (0, 0) too.…”

“…the empirical wavelet coefficients that have been kept for the signal reconstruction satisfy the hereditary constraint of Engel 1994). They can be viewed as both a hybrid wavelet version of Lepski's kernel method (see Autin 2008a) and a vertical-block thresholding method (see Autin et al 2011a).…”