\setcounter{equation}{0}\section{Introduction}

Wavelet theory 
and discrete wavelet transforms have had great impact on
the field of signal and image processing.  
For references to the extensive literature on the
subject, we refer to the bibliographies found in 
[1,2].
In this paper we propose a new class of discrete transforms.
It ``includes'' the classical Haar
and Daubechies transforms.  
Our transforms treat the endpoints of a signal in a different 
manner from that of conventional techniques.
This new approach allows us efficiently
to handle signals of any
length; thus, one is not restricted
to work with signal or image sizes that are multiples of
a power of 2.  A different method for dealing with signals
of arbitrary length was given earlier by Taswell and
McGill \cite{ta}.  Our method does not lengthen the output signal
and does not require an additional bookkeeping vector.

An exciting result is the uncovering of a new and simple transform
(presented in Section~3)
that performs very well for compression purposes.
It has compact support of length 4, and so is its inverse.
The coefficients are symmetrical, and the 
associated scaling
function is fairly smooth (it is the quadratic 
B-spline).  The associated dual
wavelet has vanishing moments up to order 2.
Numerical results comparing the performance of our transform 
with that of
the Daubechies $ D_4 $  transform are given in Section~4.
The multiresolution decomposition,
however, is not orthogonal.  We will see 
why this apparent defect is not a real problem in practice.
Furthermore, we will give in Section~5 a method to compute 
an orthogonal
compensation that gives us the best approximation
possible with the given scaling space.

Our transform can be described completely within
the context of matrix theory and linear algebra.
Thus, even without prior knowledge
of wavelet theory, one can easily grasp the concrete
algorithm and apply it to specific problems within a very short time,
without having to master complex
functional analysis.  At the end of the paper, we shall
make the connection to wavelet theory.

The experiments mentioned in this article were carried out
in Matlab; most of the formulas, especially those
involving matrix inverses, were derived by using Maple.
We acknowledge the usefulness of these
excellent packages.  A description of our Matlab implementation
can be found in \cite{k}.  The paper and the Matlab toolbox can
be obtained through ftp at
\verb+info.mcs.anl.gov+ under the directory
\verb+/pub/W-transform+.
