0.2 A multiresolution formulation of wavelet systems  (Page 2/8)

In order to develop the wavelet expansion described in [link] , we will need the idea of an expansion set or a basis set. If we start withthe vector space of signals, $S$ , then if any $f\left(t\right)\in S$ can be expressed as $f\left(t\right)={\sum }_k{a}_k{\phi }_k\left(t\right)$ , the set of functions ${\phi }_k\left(t\right)$ are called an expansion set for the space $S$ . If the representation is unique, the set is a basis. Alternatively, one could start with the expansion set or basis set and define the space $S$ as the set of all functions that can be expressed by $f\left(t\right)={\sum }_k{a}_k{\phi }_k\left(t\right)$ . This is called the span of the basis set. In several cases, the signal spaces that we will need are actuallythe closure of the space spanned by the basis set. That means the space contains not only all signals that can be expressed by alinear combination of the basis functions, but also the signals which are the limit of these infinite expansions. The closure of a space is usuallydenoted by an over-line.

The scaling function

In order to use the idea of multiresolution, we will start by defining the scaling function and then define the wavelet in terms of it. As describedfor the wavelet in the previous chapter, we define a set of scaling functions in terms of integer translatesof the basic scaling function by

The subspace of $L^2\left(\mathbf{R}\right)$ spanned by these functions is defined as

for all integers $k$ from minus infinity to infinity. The over-bar denotes closure. This means that

One can generally increase the size of the subspace spanned by changing the time scale of the scaling functions. A two-dimensional family offunctions is generated from the basic scaling function by scaling and translation by

whose span over $k$ is

for all integers $k\in \mathbf{Z}$ . This means that if $f\left(t\right)\in V_j$ , then it can be expressed as

For $j>0$ , the span can be larger since ${\phi }_{j,k}\left(t\right)$ is narrower and is translated in smaller steps. It, therefore, can represent finerdetail. For $j<0$ , ${\phi }_{j,k}\left(t\right)$ is wider and is translated in larger steps. So these wider scaling functions can represent only coarseinformation, and the space they span is smaller. Another way to think about the effects of a change of scale is in terms of resolution. If onetalks about photographic or optical resolution, then this idea of scale is the same as resolving power.

Multiresolution analysis

In order to agree with our intuitive ideas of scale or resolution, we formulate the basic requirement of multiresolution analysis (MRA) [link] by requiring a nesting of the spanned spaces as

or

with

The space that contains high resolution signals will contain those of lower resolution also.

Because of the definition of $V_j$ , the spaces have to satisfy a natural scaling condition

which insures elements in a space are simply scaled versions of the elements in the next space. The relationship of the spanned spacesis illustrated in [link] .

The nesting of the spans of $\phi (2^{j}t-k)$ , denoted by $V_j$ and shown in [link] and [link] and graphically illustrated in [link] , is achieved by requiring that $\phi \left(t\right)\in V_1$ , which means that if $\phi \left(t\right)$ is in $V_0$ , it is also in $V_1$ , the space spanned by $\phi \left(2t\right)$ . This means $\phi \left(t\right)$ can be expressed in terms of a weighted sum of shifted $\phi \left(2t\right)$ as