Continuous Wavelet Transform on the Hyperboloid - Université de ...

4
p=0.5
4
p=1
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
−4 −3 −2 −1 0 1 2 3 4
0
−4 −3 −2 −1 0 1 2 3 4
Fig. 4. Cross-section of conic projections for different values of parameter p.
x
0
H 2 +
a
P
P a
N
P
P
x
1
C
0
P
P
Pa
P
0
x
2
Fig. 5. Action of a dilation a on the hyperboloid H 2 + by conic projection with
parameter p =1.
The action of dilation by conic projection is given by
sinh pχ a = a sinh pχ (21)
The particular case p = 1 is depicted in Figure 5. The dilated point x a ∈ H+
2
is
x a =(coshχ a , sinh χ a cos ϕ, sinh χ a sin ϕ), (22)
with polar coordinates θ a =(χ a ,ϕ). The behaviour of dist(x N , x a ), with x N
being the North Pole, is shown in Figure 6 in the case p =0.1, p =0.5 and
p = 1. We can see that this is an increasing function with respect to the
dilation a.
It is also interesting to compute the action of dilations in the bounded version
of H+ 2 . The latter is obtained by applying the stereographic projection from
the South Pole of H 2 and it maps the upper sheet H+ 2 onto the open unit disc
9