0.2 A multiresolution formulation of wavelet systems  (Page 4/8)

or at even $j=-\infty$ where [link] becomes

eliminating the scaling space altogether and allowing an expansion of the form in [link] .

Another way to describe the relation of $V_0$ to the wavelet spaces is noting

which again shows that the scale of the scaling space can be chosen arbitrarily. In practice, it is usually chosen to represent the coarsestdetail of interest in a signal.

Since these wavelets reside in the space spanned by the next narrower scaling function, $W_0\subset V_1$ , they can be represented by a weighted sum of shifted scaling function $\phi \left(2t\right)$ defined in [link] by

for some set of coefficients $h_1\left(n\right)$ . From the requirement that the wavelets span the “difference" or orthogonal complement spaces, and theorthogonality of integer translates of the wavelet (or scaling function), it is shown in the Appendix in [link] that the wavelet coefficients (modulo translations by integer multiples of two) are required byorthogonality to be related to the scaling function coefficients by

One example for a finite even length- $N$ $h\left(n\right)$ could be

The function generated by [link] gives the prototype or mother wavelet $\psi \left(t\right)$ for a class of expansion functions of the form

where $2^j$ is the scaling of $t$ ( $j$ is the ${log}_2$ of the scale), $2^{-j}k$ is the translation in $t$ , and $2^{j/2}$ maintains the (perhaps unity) $L^2$ norm of the wavelet at different scales.

The Haar and triangle wavelets that are associated with the scaling functions in [link] are shown in [link] . For the Haar wavelet, the coefficients in [link] are $h_1\left(0\right)=1/\sqrt{2},h_1\left(1\right)=-1/\sqrt{2}$ which satisfy [link] . The Daubechies wavelets associated with the scaling functions in Figure: Daubechies Scaling Functions are shown in Figure: Daubechies Wavelets with corresponding coefficients given later in the book in  Table: Scaling Function and Wavelet Coefficients plus their Discrete Moments for Daubechies-8 and Table: Daubechies Scaling Function and Wavelet Coefficients plus their Moments .

We have now constructed a set of functions ${\phi }_k\left(t\right)$ and ${\psi }_{j,k}\left(t\right)$ that could span all of $L^2\left(\mathbf{R}\right)$ . According to [link] , any function $g\left(t\right)\in L^2\left(\mathbf{R}\right)$ could be written

as a series expansion in terms of the scaling function and wavelets.

In this expansion, the first summation in [link] gives a function that is a low resolution or coarse approximation of $g\left(t\right)$ . For each increasing index $j$ in the second summation, a higher or finer resolution function is added, which adds increasing detail. This is somewhat analogous to a Fourierseries where the higher frequency terms contain the detail of the signal.

Later in this book, we will develop the property of having these expansion functions form an orthonormal basis or a tight frame, whichallows the coefficients to be calculated by inner products as

and

The coefficient $d(j,k)$ is sometimes written as $d_j\left(k\right)$ to emphasize the difference between the time translation index $k$ and the scale parameter $j$ . The coefficient $c\left(k\right)$ is also sometimes written as $c_j\left(k\right)$ or $c(j,k)$ if a more general “starting scale" other than $j=0$ for the lower limit on the sum in [link] is used.