2.3 Wavelets and multiresolution analysis
2.3.3 Multiresolution analysis
Multiresolution techniques exploit the idea of analysing a given signal at different resolutions or scales. This can be compared with measuring an ocean coast line with rigid rulers of different sizes (e.g., powers of two). The representation of the coast line will look different depending on what ruler size had been used and the resulting length would be different as well. However, in multiresolution analy- sis as defined by S. Mallat (e.g.,Mallat,1989) for image processing the so-called
scaling function takes the part of the ruler and extracts a series of images at res- olutions differing by a factor of two, which include only the low frequency part of the image at each level. In one direction, moving towards higher resolutions each successive level contains all previous ones and eventually approximates the origi- nal image. In the other direction increasingly less information is represented. The difference between two successive levels can be encoded using a wavelet function. We will now describe multiresolution analysis more formally following
Raghuveer and Bopardikar (1998) and Daubechies (1992), and return to a one- dimensional signal for the time being, but again the reader must be warned that the account will be very superficial and he or she should consult the already mentioned wavelet literature andVaidyanathan(1993).
A multiresolution analysis consists of a sequence of nested linear vector spaces . . . ⊂ V1 ⊂ V0 ⊂ V−1. . .. The closed subspaces must satisfy the following five
conditions:
i. Every subspace is entirely included in the next one, forming a sequence of successive approximation spaces:
V
k⊂V
k−1 for allk∈Z
(2.32)[
k∈Z
V
k=L
2(R)
(2.33)where the over-line denotes theset closure (for an explanation of the term ’dense’ and ’set closure’ see texts on mathematical analysis or Weisstein,
1999).
iii. All subspaces only have the set containing the all-zero function or zero vector in common.
\
k∈Z
V
k={0}
(2.34)The following last two conditions make the nested vector space sequence an actual multi-resolution analysis:
v. Elements in a space Vk are simply scaled versions of the elements in sub-
spaceVk−1
f(t)∈V
k⇔
f(2t)∈V
k−1 (2.35)A function in a certain subspace dilated by factor 2 yields a function in the next coarser subspace. The value 2 of the factor is not a necessity, actually it is required only to be a power of two, but we confine ourselves in this presentation to dyadic relationships.
vi. There exist a so-calledscaling function φ(t)such that
{φ(t−n);
n∈Z}
is a basis inV
0 (2.36)According toRaghuveer and Bopardikar(1998)
... the final property requires that there be a scaling function φ(t)such that the set {φ(t−n) :ninteger}is linearly independent,
and any functionf0(t)∈V0is expressible as
f0(t) = ∞
X n=−∞
a(0, n)φ(t−n)
for a sequence of scalarsa(0, n)wheren=0,±1,±2, and so on.
We can now formulate the recursive so-called Multiresolution Analysis (MRA) equation (also calledrefinement ordilationequation).
φ(t) =
∞
X
n=−∞
c(n)√2 φ(2t−n)
(2.37) that allows us to write φ as a weighted sum of its translates at a resolution twice as fine. The coefficientsc(n)are called the scaling function coefficientsandthe factor√2ensures a constant norm over all scales.
Two properties of an MRA are now of special interest for us. Firstly, as shown inDaubechies (1992) for every sequence of closed subspaces fulfilling conditions (2.32)-(2.36) with the additional requirement in (2.36) that {φ(t−n); n∈Z}must
be anorthonormalbasis,6 there exists an orthonormal wavelet basis{ψ
s,l;s, l∈Z}
ofL2(R),ψ
s,l(t) =2−s/2ψ(2−st−l)such that, for allf∈L2(R)
P
s−1f=P
sf+
X
l∈Z
hf, ψ
s,liψ
s,l (2.38)where Psf is the orthogonal projection of f on Vs. This means that the dif-
ference between each MRA level is covered by a certain wavelet function on the corresponding level and all its translates:
V
s−1=V
s⊕W
s (2.39)whereWs is the subspace generated by the set{ψ(2−st−l);s, l∈Z}}, and⊕is
the orthogonal sum (sinceVs−1⊥Ws).
Secondly, a MRA can be expressed and realized as a cascade filter bank of pairwise low and high pass filters. The filters must be half-band filters and more- over satisfy certain criteria which led to the nameQuadratureMirrorFilters(QMV, see the already cited wavelet literature, in particular Strang and Nguyen,1997). Their coefficients can be calculated from the scaling function and vice versa.
Therefore the wavelet transformation and in particular its discrete time version DTWT (Discrete-timeWaveletTransform) can be realized as filter bank, too. The resulting output signals at each level of the DWT/DTWT will be bandlimited with theoretically a bandwidth of exactly one octave. In practise, however, no filter is perfect and the results will vary slightly with the steepness of the filter slope.
Note that we will use the term multiresolution analysis in a less formal way in chapter3to characterise analogies between the motion tracking coarse-to-fine strategy per se and its included image processing part.