In this section we give some examples of both nonstationary prewavelets and biorthogonal bases in the case when n = 3. In this case the nonstationary refinable functions ϕ(3,m), m ≥ 0, belong to C2(R), i.e. they have the same smoothness as the cubic B-spline. Interestingly enough, any ϕ(3,m) with m > 0 has the same support as B(3,m), i.e. [0, 4 · 2−m], while ϕ(3,0) is more localized in the scale-time plane having supp ϕ(3,0) = [0, 5/2], a property that appears very useful in applications (see the example below). In order to
Table 1: Numerical values (rounded to the forth digit) of the mask coefficients a(3,m)0 , a(3,m)1 , a(3,m)2 for m = 0, . . . , 8. Here µ = 1.1
m 0 1 2 3 4 5 6 7 8
a(3,m)
0 0.5 0.0313 0.0452 0.0508 0.0537 0.0555 0.0567 0.0576 0.0583 a(3,m)
1 0.5 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 0.2500 a(3,m)
2 0 0.4375 0.4095 0.3984 0.3925 0.3889 0.3865 0.3848 0.3835
0 1 2 3 4
Figure 1: The nonstationary mask coefficients listed in Tab. 1 (left) and ϕ(3,0) (right). The stationary mask of the cubic B-spline and the cubic B-spline itself are also displayed (dashed line)
obtain refinable beses and nonstationary filters significantly different from those ones generated by the cubic B-spline, we choose µ = 1.1 as a value for the tension parameter. The numerical values (rounded to the forth digit) of the mask coefficients are listed in Tab. 1, while their behavior is shown in Fig. 1 (right). The behavior of ϕ(3,0) in comparison with B(3,0) is displayed in Fig. 1 (left). The
−4 −3 −2 −1 0 1 2 3
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2 0.3 0.4
−2 −1 0 1 2 3 4
−0.2
−0.15
−0.1
−0.05 0 0.05 0.1 0.15 0.2
Figure 2: The prewavelet mask {g(3,0)} (left) and ψ(3,0) (right) nonstationary prewavelets ψ(3,m) are given by:
ψ(3,0)= X4 α=−3
(−1)αgα−1(3,0) ϕ(3,1)(· − 2−1α) ,
ψ(3,m)= X4 α=−6
(−1)αgα−1(3,m) ϕ(3,m+1)(· − 2−(m+1)α) , m > 0 ,
where the prewavelet coefficients {gα(3,n)} can be evaluated by the algorithm in Th. 5.3. From (5.9) it follows that supp ψ(3,0)= [−3/2, 4], while for m > 0 supp ψ(3,m) = [−2−m3, 2−m4]. We notice that ψ(3,0) is more localized in the scale-time plane than both ψ(3,m), m > 0, and the B-spline prewavelet.
The prewavelet mask coefficients, rounded to the forth digit, are g−4(3,0) =
−g(3,0)3 = −0.0015, g−3(3,0) = −g(3,0)2 = 0.0259, g−2(3,0) = −g1(3,0) = −0.1479, g−1(3,0)= −g0(3,0) = 0.3244.
In Fig. 2 the behavior of {gα(3,0)} and ψ(3,0) are displayed. Finally, we give the explicit expression of the biorthogonal masks ea(3,m), m ≥ 0. It is well known that the biorthogonal mask of a(3,0) is ea(3,0) = {ea(3,0)0 , ea(3,0)1 } =1
2, −12 . In order to fulfill conditions ensuring the existence of the biorthogonal re-finable function eϕ(3,m), for m > 0 we construct the biorthogonal maskea(3,m) with support [0, 14]. Its explicit expression is given by
ea(3,m)0 = ea(3,m)14 = 8−3−h
−4 + 2h(128 + 26+h+ 5 · 41+h+ 5 · 8h) ea(3,m)1 = ea(3,m)13 = − 4−5−h
−4 + 2h(128 + 26+h+ 5 · 41+h+ 5 · 8h) ea(3,m)2 = ea(3,m)12 = − 8−3−h
−4 + 2h(640 + 7 · 26+h+ 33 · 41+h+ 29 · 8h− 5 · 16h) ea(3,m)3 = ea(3,m)11 = 2−9−2·h
−4 + 2h(128 + 3 · 26+h+ 17 · 41+h+ 17 · 8h) ea(3,m)4 = ea(3,m)10 = 8−3−h
−4 + 2h(1152 + 15 · 26+h+ 133 · 41+h+ 89 · 8h− 39 · 16h) ea(3,m)5 = ea(3,m)9 = 4−5−h
−4 + 2h(128 + 26+h− 123 · 41+h− 123 · 8h) ea(3,m)6 = ea(3,m)8 = − 8−3−h
−4 + 2h(640 + 9 · 26+h− 81 · 21+4·h+ 105 · 41+h+ 577 · 8h) ea7 = − 4−4−h
−4 + 2h(128 + 3 · 26+h+ 81 · 41+h− 175 · 8h)
where h = 3 + m−µ. The biorthogonal mask coefficients {ea(3,m)α } (rounded to the forth digit) are listed in Tab. 2. In Fig. 3 the behavior of {ea(3,m)α } and
e
ϕ(3,0) are displayed. The biorthogonal wavelet mask coefficients q(3,m) and e
q(3,m) can be obtained by qα(3,m) = (−1)αea(3,m)−α+1, eq(3,m)α = −(−1)αa(3,m)−α+1. In Fig. 4 the behavior of ψ(3,0) and eψ(3,0) is displayed.
Just to show how the properties of the constructed nonstationary biorthog-onal filters can affect the analysis of a given signal, we evaluate the coefficients {λmα} and {ζαm}, obtained after three steps of the decomposition algorithm (6.11), when the starting sequence is a spike-like signal (see Fig. 5 (left)).
The coefficients are plotted in Fig. 5 (right) in comparison with the coeffi-cients obtained when using the stationary cubic spline biorthogonal filters.
The figure shows that the nonstationary decomposition algorithm has higher compression properties: actually the number of nonzero coefficients are 26 in the nonstationary case, while they are 39 in the B-spline case.
Table 2: Numerical values (rounded to the forth digit) of the mask coefficients ea(3,m)α , for α = 0, . . . , 7, and m = 0, . . . , 8
m 0 1 2 3 4 5 6 7 8
ea(3,m)0 0.5 0.0011 0.0021 0.0026 0.0030 0.0064 0.0034 0.0035 0.0036 ea(3,m)1 0.5 -0.0085 -0.0114 -0.0129 -0.0138 -0.0288 -0.0148 -0.0151 -0.0154 ea(3,m)2 0 0.0066 0.0028 0.0005 -0.0010 -0.0039 -0.0027 -0.0032 -0.0036 ea(3,m)3 0 0.0574 0.0760 0.0857 0.0914 0.1905 0.0979 0.0999 0.1014 ea(3,m)4 0 -0.0810 -0.0790 -0.0768 -0.0752 -0.1480 -0.0732 -0.0725 -0.0720 ea(3,m)5 0 -0.1998 -0.5108 -0.2834 -0.2998 -0.6211 -0.3180 -0.3236 -0.3278 ea(3,m)6 0 0.3233 0.3241 0.3237 0.3232 0.6456 0.3225 0.3222 0.3220 ea(3,m)7 0 0.8019 0.8816 0.9212 0.9443 1.9187 0.9698 0.9776 0.9835
0 2 4 6 8 10 12 14
Figure 3: The first 8 nonstationary biorthogonal masksea(3,m) (left) and eϕ(3,0) (right)
0 200 400 600 800 1000 1200 1400 1600
0 200 400 600 800 1000 1200 1400 1600 1800
−0.2
0 200 400 600 800 1000 1200 1400 1600 1800
−0.2 0 0.2 0.4 0.6
Figure 5: The spike-like initial sequence (left) and the decomposition coef-ficients obtained after 3 steps of the nonstationary decomposition algorithm (upper right). The decomposition coefficients obtained by the stationary cubic biorthogonal filters are also shown (bottom right)
8 Conclusion
We studied the properties of a class of refinable ripplets associated with se-quences of nonstationary scaling masks. One of the most interesting property of these functions is in that they have a smaller support than the stationary refinable ripplets with the same smoothness. This localization property is crucial in several applications, from geometric modeling to signal processing.
After proving some approximation properties, such as Strang–Fix conditions, polynomial reproduction and approximation order, we proved also that any refinable function in the family is bell-shaped, so that they can efficiently approximate a Gaussian. Moreover, since these refinable functions gener-ate nonstationary multiresolution analyses, we constructed the minimally supported nonstationary prewavelets and proved that their 2m-shifts form L2-stable bases. We note that this construction can be generalized to other class of nonstationary refinable functions, like exponential splines. Moreover, we constructed nonstationary biorthogonal bases and filters to be used in ef-ficient decomposition and reconstruction algorithms.
The localization property of the refinable ripplets we studied implies that the corresponding nonstationary wavelets have a small support too, a property which is very desirable in the case when the relevant information of a func-tion to be approximated or of a signal to be analyzed are focused in small regions of the scale-time plane. The preliminary test in Section 6 shows the good performances of the constructed nonstationary wavelets in a simple compression test. More tests will be the subject of a forthcoming paper.
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