Mt department Nazirova E. Sh

Multiresolution Approximations Adapting the signal resolution allows one to process only the relevant details for a particular task. In computer vision, Burt and Adelson introduced a multiresolution pyramid that can be used to process a low-resolution image first and then selectively increase the resolution when necessary. This section formalizes multiresolution approximations, which set the ground for the construction of orthogonal wavelets.

The approximation of a function at resolution is specified by a discrete grid of samples that provides local averages of over neighborhoods of size proportional to Thus, a multiresolution approximation is composed of embedded grids of approximation. More formally,the approximation of a function at a resolution is defined as an orthogonal projection on a space The space regroups all possible approximations at the resolution . The orthogonal projection of is the function f_j ∈V_j that minimizes . The following definition, introduced by Mallat and Meyer, specifies the mathematical properties of multiresolution spaces. To avoid confusion, let us emphasize that a scale parameter is the inverse of the resolution .

Definition 1.3: Multiresolutions. A sequence of closed subspaces of s a multiresolution approximation if the following six properties are satisfied:

Let us give an intuitive explanation of these mathematical properties. Property (7.1) means tha is invariant by any translation proportional to the scale . As we shall see later, this space can be assimilated to a uniform grid with intervals , which characterizes the signal approximation at the resolution . The inclusion (7.2) is a causality property that proves that an approximation at a resolution contains all the necessary information to compute an approximation at a coarser resolution . Dilating functions in by 2 enlarges the details by 2 and (7.3) guarantees that it defines an approximation at a coarser resolution . When the resolution goes to 0 (7.4) implies that we lose all the details of and

On the other hand, when the resolution goes , property (7.5) imposes that

the signal approximation converges to the original signal:

When the resolution increases, the decay rate of the approximation error depends on the regularity of . We relate this error to the uniform Lipschitz regularity of .

The existence of a Riesz basis of he function can be interpreted as a unit resolution cell; the definition of a Riesz basis. It is a family of

linearly independent functions such that there exist , which satisfy

This energy equivalence guarantees that signal expansions over are numerically stable. One may verify that the family s a Riesz

basis of with the same Riesz bounds A and B at all scales Theorem proves that is a Riesz basis if and only if