\setcounter{equation}{0}\section{Connection to Wavelet Theory}

In this section we discuss
how the well-known concepts 
in wavelet theory are related to our $W$-matrix transforms.
Suppose that $ J $ levels of multiresolution analysis have been applied
to a signal \bdx\ of sufficiently long length $ N $.  
For compression purposes, the ideal situation is 
such that we can throw away most of the components of 
each $ \yt $ at each level.  The compressed 
data of the signal thus
consists mainly of the last-level $ \yo $, supplemented by a
few components from the $ \yt $ of earlier levels.
The approximate signal restored from the compressed data
is then the $ J $ times inverse transform of the last-level $ \yo $,
plus some detail adjustments using the additional $ \yt $ data.

By linearity, the $ J $-times-inverse transform of $ \yo $ is the
sum of the $ J $ times inverse transform of each of the components
of $ \yo $.  By the shift invariance character of the $W$-matrix
transform, the inverse transform of each component is simply a 
multiple of the translated inverse transform of another component
(except for the boundary components).  The signal restored from
$ \yo $ is thus a linear combination of translates of some basic
signal, which is the $ J $ times inverse transform of the ($\yo$) vector
$ [0,1,0]' $--- by this, we mean that we take 
the $ J $-th level $ \yo $ vector to be $ [0,1,0]' $, and $ \yt $
at all levels to be zero, and we compute the original signal that
gives this decomposition.  We normalize this signal by multiplying
it with a constant so that the maximum of the signal is 1.
This basic signal is called the $ J $-th scaling signal.

What happens when we let $ J\rightarrow \infty  $?   The length of the scaling
signal increases as $ J $ increases, but we can consider it as
the sampling of a continuous signal defined on a fixed interval, say
$ [0,a] $.  If there exists a continuous signal on $ [0,a] $ such that
the $ J $-th scaling signals converge to as $ J\rightarrow \infty  $, this
continuous signal is defined to be the scaling function of the multiresolution analysis.

In a similar way, the $ J $-times-inverse transform of the ($\yt$)
vector $ [0,1,0]' $, after normalizing to have maximum 1,
is the $ J $-th wavelet signal.  As $ J\rightarrow \infty  $, the 
$ J $-th wavelet signal may converge to a continuous wavelet.

A signal restored from a $ J $-level multiresolution analysis is then a linear combination of
translates of the $ J $-th scaling signal and translates
of wavelet signals of various levels (lower than or equal to $ J) $.
In practice, the continuous scaling function and continuous wavelet will never be used,
but the discrete scaling signals and wavelet signals, for $ J $ large,
closely approximate their continuous relatives.  For instance,
the $ D_4 $ scaling functions and wavelets are not smooth.  Therefore, the
high-level $ D_4 $ scaling and wavelet signals are also not smooth.
This phenomenon explains why, when a signal is compressed using several
levels of the $ D_4 $  transform and a majority of the $ \yt $ components are
discarded, the restored signal has numerous sharp cusps.
On the other hand, the scaling function of the QS transform is $ C^1 $ smooth,
so that the corresponding discrete scaling signals are smooth
and the restored signal will look smooth (except where adjustments
are made with the $ \yt $ components retained in the compressed
data).

We can see that
the scaling function we define here coincides with the classical scaling
function defined as in \cite{ma} by the following theorem.

\begin{THEOREM}
Let $ \phi  $ be the scaling function, if it exists, corresponding to a $W$-matrix.
Then $ \phi  $ satisfies the dilation equation
\begin{equation}    \phi (x) = \frac{2}{\sum \overline g_i}  \sum \overline g_i \phi (2x - i +1), \end{equation}
where $ \bg=[\overline g_1,\cdots ,\overline g_n] $ is the first basic vector of 
the inverse $W$-matrix.
\end{THEOREM}

In particular, the QS scaling function satisfies
\begin{equation}    \phi (x) = \frac{1}{4} (\phi (2x)+3\phi (2x-1)+3\phi (2x-2)+\phi (2x-3)) . \end{equation}
It is well known (see, for example, \cite{ch}) that $ \phi  $ is the classical
quadratic spline.

The wavelet defined here coincides with the classical wavelet in 
the case where the
multiresolution analysis is orthogonal.  In the contrary case, we can consider our
wavelet as a generalization of the classical wavelet.  Our wavelet has the following property:

\begin{THEOREM}
\begin{equation}    \psi (x) = C \sum \overline h_i \phi (2x - i +1), \end{equation}
where $ \bh=[\overline h_1,\cdots ,\overline h_n] $ is the second basic vector of 
of the inverse $W$-matrix and $ C $ is some scaling constant.
\end{THEOREM}

Figures 6 and 7 show the QS scaling function and wavelet,
respectively.

\psfull
\begin{center}
%%
\mbox{\psfig{file=../sf.eps,width=4in,height=2in}} 
\\
Figure 6. The QS scaling function
\end{center}
\begin{center}
%%
\mbox{\psfig{file=../wf.eps,width=4in,height=2in}}
\\
Figure 7. The QS wavelet
\end{center}

\setcounter{equation}{0}\section{Conclusion}

We have introduced the $W$-matrices, shown how they are
constructed, and discussed how each $W$-matrix 
leads to a transform and a multiresolution analysis
for signals of arbitrary length. 
We have demonstrated that the new QS (quadratic spline)
transform is suitable for compression purposes
and that, for reasonable signals and images, 
it performs better than the $ D_4 $  transform.

The generality of our $W$-matrices opens up some possibilities that
are worth further investigation.  By varying the parameters
(the $ h $- and $ g $-vectors that define the $W$-matrix), one can
choose an optimal transform according to the input signal.
Using generalized $W$-matrices (that are generated by more than one
set of basic vectors), one can transform different parts of the
signal differently to enhance the compression performance.
The optimal $W$-matrix for a subregion of an image may also be
used to characterize the texture.
One can also employ different transforms at different levels
of the multiresolution analysis.

That the QS transform does not work as well as the 
$ D_4 $  transform in compressing signals that are wildly oscillating
or that are degraded by noise really means that the QS
transform does not retain much of the high-frequency
part of the signal in the $ \yo $ vector.
This fact can be turned into an advantage for the QS
transform if it is used for denoising purposes.  Further study
is planned for this topic.
