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 L2(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 = 2j, 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 L2(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 .

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) = ¢²k hk¥õ1,k(t) = ¡î2¢²k hk¥õ (2t - k)                                                                                                                     (*)

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

hk  =  <¥õ 1,k, ¥õ> = ¡î2¡òR ¥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) = ¢²k gk ¥õ 1,k(t) = ¡î2¢²k gk ¥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:

gk  =  <¥õ 1,k, ¥÷> = ¡î2¡òR ¥÷ (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

Source:

Wavelet Explorer Documentation-Wolfram Research