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

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x 0 H 2 + x 2 x 1 Fig. 1. Geometry of the 2-hyperboloid. integral vanishes when it is conveniently weighted, that is for p>0. ∫ H 2 + dµ(χ, ϕ) [ ] 1 sinh 2pχ 2 ψ(χ, ϕ) =0, sinh χ Finally we conclude this paper with illustrating examples of hyperbolic wavelets and wavelet transforms and give directions for future work. 2 Geometry of the two-sheeted hyperboloid. Projective structures. We start by recalling basic facts about the upper sheet of the two-sheeted hyperboloid of radius ρ, H 2 +ρ. Letχ, ϕ be a system of polar coordinates for H 2 +ρ . To each point θ =(χ, ϕ) we shall associate the vector x =(x 0,x 1 ,x 2 )of R 3 given by x 0 = ρ cosh χ, x 1 = ρ sinh χ cos ϕ, ρ > 0, χ 0, 0 ≤ ϕ
0 x 0 C 2 + H 2 + r 0 x 2 x 1 Fig. 2. Cross-section of a conic projection called Lobachevskian metric, whereas the measure element on the hyperboloid is dµ = ρ 2 sinh χdχdϕ. (2) In the sequel, we shall put ρ = 1 for convenience and designate the unit hyperboloid H 2 +ρ=1 by H 2 +. Various projections can be used to endow H+ 2 with a local Euclidean structure. One of them is immediate : it suffices to flatten the hyperboloid onto R 2 ≃ C. Another possibility is to project the hyperboloid onto a cone. Let us consider a half null cone C+ 2 ∈ R 3 of equation (x 0 ) 2 − 1 tan ψ 0 ((x 1 ) 2 +(x 2 ) 2 )=0,x 0 0. This cone C+ 2 has Euclidean nature. The cone surface unrolled is a circular sector and all points of H+ 2 will be mapped onto C+ 2 using a specific conic projection. The characteristic parameter of a conic projection is the constant of the cone m =cosψ 0 ,whereψ 0 is the Euclidean angle of inclination of the generatrix of the cone as shown in Figure 2. Considering a radial conic projection, it is more convenient to use a radius r defined by the Euclidean distance between a point on the cone, conic projection of the point (χ, ϕ) ∈ H 2 + ,andthex 0-axis: r = f(χ), dr = f ′ (χ)dχ with dr =1. (3) dχ∣ χ=0 Each suitable projection is determined by a specific choice of f(χ). It is clear that dilation of the cone C 2 + ↦→ aC 2 + = C 2 + entails r ↦→ ar. Consequently, the resulting map χ ↦→ χ a is determined by f(χ a )=af(χ). This is precisely the point at the heart of our approach and we shall discuss this more precisely in Section 4. 5
Page 1 and 2: Continuous <strong
Page 3: and on the sphere, it is natural to
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 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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