Continuous Wavelet Transform on the Hyperboloid - Université de ...
Continuous Wavelet Transform on the Hyperboloid - Université de ... Continuous Wavelet Transform on the Hyperboloid - Université de ...
x 0 H 2 + N P P P P 0 x 1 C x 2 Fig. 3. Conic projection and flattening. Thus, this conic projection Φ : H+ 2 → C2 + is a bijection given, after flattening, by Π 0 Φ(x) =2sinh χ 2 eiϕ , with x ≡ (χ, ϕ), χ ∈ R + , 0 ≤ ϕ
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
- Page 1 and 2: Continuous <strong
- Page 3 and 4: and on the sphere, it is natural to
- Page 5 and 6: 0 x 0 C 2 + H 2 + r 0 x 2 x 1 Fig.
- Page 7: The action of a motion on a point x
- Page 11 and 12: x 0 H 2 + a N x 1 a S x 2 H 2 - Fig
- Page 13 and 14: E ν,ξ (x)=(ξ · x) − 1 2 −i
- Page 15 and 16: get the more elaborate expression
- Page 17 and 18: We now have all the basic ingredien
- Page 19 and 20: that 0
- Page 21 and 22: By performing the change of variabl
- Page 23 and 24: and so α(a) should behave at least
- Page 25 and 26: Fig. 9. The hyperbolic DOG wavelet
- Page 27 and 28: Fig. 11. The hyperbolic DOG wavelet
- Page 29 and 30: lim ˆψ ρ (ν, ξ)= 1 ∫ ψ(⃗x
- Page 31: April 2004. I. Tosic, I. Bogdanova,
x<br />
0<br />
H 2 +<br />
N<br />
P<br />
P<br />
P<br />
P<br />
0<br />
x<br />
1<br />
C<br />
x<br />
2<br />
Fig. 3. C<strong>on</strong>ic projecti<strong>on</strong> and flattening.<br />
Thus, this c<strong>on</strong>ic projecti<strong>on</strong> Φ : H+ 2 → C2 + is a bijecti<strong>on</strong> given, after flattening,<br />
by<br />
Π 0 Φ(x) =2sinh χ 2 eiϕ ,<br />
with x ≡ (χ, ϕ), χ ∈ R + , 0 ≤ ϕ
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