Continuous Wavelet Transform on the Hyperboloid - Université de ...
Continuous Wavelet Transform on the Hyperboloid - Université de ... Continuous Wavelet Transform on the Hyperboloid - Université de ...
More precisely, using the hyperbolic function ψ = e − sinh2 χ 2 , we dilate it using the conic projection and obtain we get: D ϑ ψ = 1 ϑ e− 1 ϑ 2 sinh2 χ 2 , (79) f ϑ ψ (χ, ϕ) =e− sinh2 χ 2 − 1 ϑ 2 e− 1 Now, applying a dilation operator on (80) we get ϑ 2 sinh2 χ 2 . (80) D a f ϑ = 1 1 a e− a 2 sinh2 χ 1 2 − 1 aϑ 2 e− a 2 ϑ 2 sinh2 χ 2 . (81) One particular example of hyperbolic DOG wavelet at ϑ =2is: fψ(χ, 2 ϕ) = 1 1 a e− a 2 sinh2 χ 1 2 − 1 4a e− 4a 2 sinh2 χ 2 . The resulting hyperbolic DOG wavelet at different values of the scale a and the position (χ, ϕ) on the hyperboloid is shown on Figures 9 , while Figures 10 and 11 show the same wavelet but viewed on the unit disk. Of course, similar admissible DOG wavelets can be constructed for generic p>0. 6.4 An example of
Fig. 9. The hyperbolic DOG wavelet fψ ϑ ,forϑ = 2 at different scales a and positions (χ, ϕ). and H = L 2 (R 2 , d 2 ⃗x) be the Hilbert space of square integrable functions on the plane. One can easily adapt the Fourier-Helgason transform by updating E ν,ξ (x) for 25
- 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 and 8: The action of a motion on a point x
- Page 9 and 10: 4 p=0.5 4 p=1 3.5 3.5 3 3 2.5 2.5 2
- 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 so α(a) should behave at least
- 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,
Fig. 9. The hyperbolic DOG wavelet fψ ϑ ,forϑ = 2 at different scales a and positi<strong>on</strong>s<br />
(χ, ϕ).<br />
and H = L 2 (R 2 , d 2 ⃗x) be <strong>the</strong> Hilbert space of square integrable functi<strong>on</strong>s <strong>on</strong><br />
<strong>the</strong> plane.<br />
One can easily adapt <strong>the</strong> Fourier-Helgas<strong>on</strong> transform by updating E ν,ξ (x) for<br />
25
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