**What is Wavelet?**

Wavelets means ¡°small waves¡±. They are waves which oscillates and satisfies ,where
is the wavelet function. Wavelets provide convenient sets of
basis functions for function spaces such as L^{2}(R). The fact that the wavelets are localized
in time and frequency enables them to represent functions that are localized
both in time and frequency. Wavelets allows us to represent functions with sharp spikes
or edges, as for example in two-dimensional images, with fewer terms. Such
property results in wavelets being more advantageous in many applications; for
example, in data compression and transmission.

Obviously, unlike the case of Fourier transforms (Fourier basis consists of sines and cosines, and it does not have rapid decay
property), depending on the choice of mother wavelet, there are varieties of
wavelet families. However, we are more interested in wavelets with desirable
properties such as orthogonality, rapid decay, wavelets
with compact support, and smoothness.

Now, how do we construct wavelets? One starts from the basic
wavelet called the "mother wavelet" and
generates the basis by dilation and translation .
In my examples I used discrete wavelets where the parameters by which one
translates, b, and dilates, a, are restricted to
a discrete set, in most of the time a = 2^{j}, b = k, where j, k are in
Z.

So where does ¥÷(t) come from?

To answer this question, I must
talk about ¥õ(t) the scaling function which satisfies
heuristic conditions of multiresolution
analysis.

**Then what is Multiresolution Analysis(MRA)? **

A multiresolution analysis of is
a nested sequence of subspaces such that

(i)

(ii)

(iii)

(iv)

(v) There exists a function ,
called the scaling function(¡°father function¡±), such that is
an orthonormal basis of .

A function (or signal) can be
represented as composition of a smooth background and fluctuations or details. The
smooth part and the details part are distinguished by the resolution, meaning
that by the scale below which the details of a signal cannot be discerned. This indeed can be described more
precisely as follows. We label the resolution level by j in Z. The scale
associated with the level j = 0 is set to, say, unity and that with the level is .
Consider a function f(t). At resolution level j it is approximated by .
At the next level of resolution j + 1, the details at that level denoted by are
included and we have the approximation to f(t) at the
new resolution level, .
The original function f(t) is recovered when we let
the resolution go to infinity

above equation represents one way of decomposing the
function into
a smooth part plus details. Similarly, we can view the space of functions that
are square integrable, as composed of a sequence of
subspaces and
,
such that the approximation of at
resolution ,
,
is in and
the details are
in .

Consider
L^{2}(R). We define a sequence of
resolutions labeled by the integers such that all details of the signal on
scales smaller than are
suppressed at resolution .
We denote the subspace of functions that contain signal information down to
scale by .
The multiresolution analysis involves a decomposition
of the function space into a sequence of subspaces .
The first requirement is that the subspace be
contained in all the higher subspaces. If we denote the approximation to at
level by
,
then .
Since information at resolution level is
necessarily included in the information at a higher resolution, must
be contained in ,
that is, for
all .

The
additional information about details at scale is .
So

We
then can decompose our subspaces accordingly, such as

where is
the detail space at resolution level and
is orthogonal to .

So to
summarize the above we have:

**¥õ(****t) and ¥÷(t)**

,
so can
be expanded in terms :

¥õ(t) = ¢²_{k} h_{k}¥õ_{1,k}(t)
= ¡î2¢²_{k} h_{k}¥õ (2t - k) (*)

are orthonormal, the coefficients are
obtained by computing the inner product (assuming is ¥õ real):

h_{k}_{ } = <¥õ_{ 1,k}, ¥õ> = ¡î2¡ò_{R}
¥õ(t) ¥õ (2t - k)dt

We can derive some properties of
the coefficients immediately
from the dilation equation. Integrating both sides of (*), we obtain (note that
normalization gives )

On the other hand, if we multiply
both sides of (*) by and
integrate, we get

By the orthogonality
of we
have or

Now, let¡¯s take a look at the spaces ,
the detail spaces. They are orthogonal to each other. From above result we can
derive

and letting ,
gives us

we can decompose into
orthogonal subspaces, each containing information about details at a given
resolution. So for a multiresolution analysis, the
detail space has
an orthonormal basis .
So has
an orthonormal basis called
the wavelet basis. Each wavelet is
generated from a single function by
translation and dilation. Since the function that
generates all the basis functions of the space
we call it mother wavelet.

Since is
in and
,
can
be written as a superposition of the basis functions for ,
.
In particular, we have

¥÷(t) = ¢²_{k} g_{k}
¥õ_{ 1,k}(t) = ¡î2¢²_{k} g_{k} ¥õ
(2t - k)

This is called the *wavelet equation.Note*
the similarity between this equation and the dilation equation. This is an
equation relating the mother wavelet to the scaling function at the next finer
scale. Again, using the fact that are
orthonormal, the coefficients can
be obtained by computing the inner product:

g_{k}_{ } = <¥õ_{ 1,k}, ¥÷> = ¡î2¡ò_{R}
¥÷ (t) ¥õ (2t - k)dt

From
the requirement that ,
we get .

So
why orthogonal wavelets?

Well, I was told(and
I read) that orthogonality provides conveniences in
computations in our applications. I
don¡¯t exactly know what they mean by conveniences in computations in our
applications. I decided to compute
some coefficients that results in orthogonal wavelets for certain scaling
functions, and see what I could observe from the graphs:

Graphs of Scaling Functions and Wavelet Functions of Wavelet Functions for Box Function

Graphs of Scaling Functions and Wavelet Functions of Wavelet Functions for Shannon Function

Graphs of Scaling Functions and Wavelet Functions of Wavelet Functions for Spline Function

Source:

Wavelet Explorer Documentation-Wolfram Research