Continuous Wavelet Transform on the Hyperboloid - UniversitÃ© de ...

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2 1.8 1.6 p=0.1 p=0.5 p=1 1.4 dist(x N , x a ) 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 dilation a Fig. 6. Analysis of the distance (9) as a function of dilation a, withx N being the North Pole and using conic projection for different parameter p. in the equatorial plane: x = x(χ, ϕ) → Φ(x) =tanh χ 2 eiϕ . (23) In the case p = 1 2 by using (17) and basic trigonometric relations, we obtain tanh χ a 2 = √ √√√ a 2 tanh 2 χ 2 1+(a 2 − 1) tanh 2 χ 2 ≡ ζ. (24) In this case, the dilation leaves invariant both ζ =0andζ = 1, the center and the border of the disc, respectively. Figure 7 depicts the action of this transformation on a point x ∈ H+ 2 . A dilation from the North Pole (D N)is considered as a dilation in the unit disc in equatorial plane and lifted back to H+ 2 by inverse stereographic projection from the South Pole. A dilation from any other point x ∈ H+ 2 is obtained by moving x to the North Pole by a rotation g ∈ SO 0 (1, 2), performing dilation D N and going back by inverse rotation: D x = g −1 D N g. The visualization of the dilation on the hyperboloid H+ 2 ,withp = 0.5, is provided in Figure 8. There, each circle represents points on the hyperboloid at constant χ and is dilated by the scale factor a =0.75. 10
x 0 H 2 + a N x 1 a S x 2 H 2 - Fig. 7. Action of a dilation a on the hyperboloid H 2 + through a stereographic projection (case p = 1 2 ). Fig. 8. Visualization of the dilation on the hyperboloid H 2 + (case p = 1 2 ). 5 Harmonic analysis on the 2-hyperboloid 5.1 Fourier-Helgason <strong>Transform</strong> This integral transform is the precise analog of the Fourier-Plancherel transform on R n . It consists of an isometry between two Hilbert spaces FH : L 2 (H 2 +, dµ) −→ L 2 (L, dη), (25) 11
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: 4 p=0.5 4 p=1 3.5 3.5 3 3 2.5 2.5 2
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,

Continuous Wavelet Transform on the Hyperboloid - UniversitÃ© de ...

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