Graphics Reference
In-Depth Information
Step 2.
We repeat this process with the two averages 3 and 5. Their average is 4 and
to get the actual values back we have to add +1 and -1, respectively. In other words,
()
+
()
=
()
-
()
3
ux ux u x w x
,
5
4
1
.
(21.25)
10
11
,
00
,
00
,
See Figure 21.12(d). Substituting (21.25) into (21.24) finally gives
()
=
()
-
()
-
()
+
()
gx
4
f
x
1
y
x
2
y
x
2
y
x
.
(21.26)
00
,
00
,
10
,
11
,
This is the wavelet representation of g(x) that we were looking for.
Example 21.9.1 really describes how one can find the wavelet approximation to
sampled functions in general by successively taking averages of adjacent values and
specifying the differences between the actual and average values. This is the basis for
a compression algorithm since the differences, called
detail coefficients
, are smaller
numbers than the actual ones and so it takes fewer bits to represent the sampled values
of the function. For example, the sequence 4, -1, -2, +2 completely represents the
samples 1, 5, 7, 3 if one understands it: From 4 and -1 one forms 3 and 5, then from
the sequence 3, 5, -2, +2, one gets 1, 5, 7, 3.
Now in the definitions of f
j,i
and Y
j,i
above we had some constraints on the range
of i and j, but that was only because we were temporarily interested in functions
defined on [0,1]. The definitions actually make sense for all integers i and j and we
need to allow this if we want to analyze functions defined on
R
. The following holds:
The functions 2
j/2
21.9.2
Theorem.
Y
j,i
, i, j Œ
Z
, form an orthonormal basis for
L
2
(
R
).
Proof.
See [GomV98].
One extremely nice feature of wavelets is that they allow a multi-resolution analy-
sis of functions, which means that one can specify the degree of detail that one wants
to capture. This should be clear from the above. To get higher resolutions, we simply
increase the j above so that we are approximating with smaller-width step functions.
See [GomV98] or [Glas95] for a definition of what is meant by a multi-resolution
representation. For multi-resolution analysis on arbitrary surfaces, not just
R
2
, see
[LoDW97].
The Haar wavelet basis and the associated spaces V
j
are only one possible choice.
We can use smoother functions. For a continuous basis one can use hat functions or,
more generally, one can use higher-order B-splines for more smoothness. See
[Four95]. They would play the role of the f
j,i
above. Aside from the particular wavelet
basis, the basic constructions would stay the same. One does lose orthogonality,
however. Once one has a basis, then the analysis of a function in L
2
(
R
) is simply to
take the orthogonal projection of the function onto the space spanned by the wavelet
basis.
Definition.
Given an orthonormal wavelet basis Y
j,i
of functions in L
2
(
R
), the
discrete wavelet transform
WT is defined on a function f in L
2
(
R
) by
