11.3 Denoising

We begin with the simplest example. One wants to recover X from the given data Y, where we assume that Y = X + ? Z. The term ? Z is considered to be noise; typically Z will be standard white noise and ? > 0 is a small parameter. In Donoho's work, the object X is a function f of a real variable t or an image, in which case we will assume that t belongs to the unit square. To develop an algorithm for recovering X, it is necessary to make some mathematical assumptions about the nature of f. These assumptions should reflect our a priori knowledge about the object X. Making assumptions about f based on our knowledge of what Xshould be is called modeling, and this issue will be addressed again in section 11.4. For the moment, we are going to follow Donoho, so our modeling of f says that fshould be smooth or should belong to some ball B in a given function space. We will argue in the next section that images naturally belong to the space of functions of bounded variation in the plane. For convenience of notation, we will use X to denote both the object we wish to recover and the function that models this object.

Our goal is to construct an estimator of X.