Linking Approximation Theory and Statistical Estimation in Wavelet Image Processing

Decomposition in a basis: (b j) j∈J basis of L 2 ([0, 1]). ∀f ∈ L 2 ([0, 1], f = j∈J f, b j b j with f, b j = 1 0 f (t)b j (t)dt. The Parseval-Bessel norm energy conservation equality yields f 2 = j∈J |f, b j | 2 1.2 Linear and non linear approximation For any I ∈ P(J), one defines f |I by f |I = j∈I f, b j b j. The question is now how to select " a good subset " of a given size M. A first possibility is to select the N first coefficients. This assume that there is a natural order on the basis elements and thus J ≡ N. The resulting function f L M is thus f L M = N −1 j=0 f, b j b j where the L in the notation is here to stress that it is a linear approximation. A second possibility is to try to minimize the error between f and f |I. Indeed by the Parseval-Bessel equality, we know that f − f |I 2 = j ∈I |f, b j | 2 = f 2 − j∈I |f, b j | 2 .