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

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3 Affine transformations on the 2-hyperboloid We recall that our purpose is to build a total family of functions in L 2 (H 2 +, dµ) by picking a wavelet or probe ψ(χ) with suitable localization properties and applying on it hyperbolic motions, belonging to the group SO 0 (1, 2), supplemented by appropriate dilations ψ(x) → λ(a, x)ψ(d 1/a g −1 x) ≡ ψ a,g (x), g ∈ SO 0 (1, 2), a∈ R + ∗ . (4) Dilations d a will be studied below. Hyperbolic rotations and motions, g ∈ SO 0 (1, 2), act on x in the following way. Amotiong ∈ SO 0 (1, 2) can be factorized as g = k 1 hk 2 ,wherek 1 ,k 2 ∈ SO(2), h∈ SO 0 (1, 1), and the respective action of k and h are the following ⎛ ⎞ ⎛ ⎞ 1 0 0 cosh χ k(ϕ 0 ).x(χ, ϕ)= ⎜ 0cosϕ 0 − sin ϕ 0 ⎟ ⎜ sinh χ cos ϕ ⎟ ⎝ ⎠ ⎝ ⎠ 0sinϕ 0 cos ϕ 0 sinh χ sin ϕ = x(χ, ϕ + ϕ 0 ), (6) ⎛ ⎞ ⎛ ⎞ cosh χ 0 sinh χ 0 0 cosh χ h(χ 0 ).x(χ, ϕ)= ⎜ sinh χ 0 cosh χ 0 0 ⎟ ⎜ sinh χ cos ϕ (7) ⎟ ⎝ ⎠ ⎝ ⎠ 0 0 1 sinh χ sin ϕ = x(χ + χ 0 ,ϕ) . (8) (5) On the other hand, the dilation is a homeomorphism d a : H 2 + → H 2 + and we require that d a fulfills the two conditions: (i) it monotonically dilates the azimuthal distance between two points on H 2 +: where dist(x, x ′ ) is defined by dist(d a (x), d a (x ′ )), (9) dist(x, x ′ )=cosh −1 (x · x ′ ), (10) and the dot product is the Minkowski product in R 3 ;notethatdist(x, x ′ ) reduces to |χ − χ ′ | when ϕ = ϕ ′ ; (ii) it is homomorphic to the group R + ∗ ; R + ∗ ∋ a → d a , d ab = d a d b , d a −1 = d −1 a , d 1 = I d . 6
The action of a motion on a point x ∈ H+ 2 is trivial: it displaces (rotates) by a hyperbolic angle χ ∈ R + (respectively by an angle ϕ). It has to be noted that, as opposed to the case of the sphere, attempting to use the conformal group SO 0 (1, 3) for describing dilation, our requirements are not satisfied. In particular, conformal dilations do not preserve the upper sheet H+ 2 of the hyperboloid. In this paper we adopt an alternative procedure that describes different maps for dilating the hyperboloid. 4 Dilations on hyperboloid Considering the half null-cone of equation { } C+ 2 = ξ ∈ R 3 : ξ · ξ = ξ0 2 − ξ1 2 − ξ2 2 =0, ξ 0 > 0 , (11) there exist the SO 0 (1, 2)-motions and the obvious Euclidean dilations ξ ∈ C 2 + → aξ ∈ C2 + ≡ dC2 + a (ξ), (12) which form a multiplicative one-parameter group isomorphic to R + ∗ . In order to lift dilation (12) back to H+ 2 , it is natural to use possible conic projections of H+ 2 onto C2 + , as defined in Section 2. H 2 + ∋ x → Φ(x) ∈ C2 + → Π 0Φ(x) ∈ R 2 ≃ C, (13) where Π 0 stands for flattening, defined by Π 0 Φ(x) :x(r, ϕ) ∈ C 2 + ↦→ reiϕ ∈ C. (14) Flattening reveals the Euclidean nature of the conic projection and the full action of (14) is depicted on Figure 3. Then, we might wish to find a form of Π 0 Φ such that, expressed in polar coordinates, the measure is dµ = rdrdϕ. (15) In this case, dilating r will quadratically dilate the measure dµ as well. By expressing the measure (15) with the radius defined in (3) we obtain f(χ)f ′ (χ) =sinhχ =⇒ f(χ) =2sinh χ 2 . (16) Consequently, the radius of this particular conic projection is r =2sinh χ 2 . 7
Page 1 and 2: Continuous <strong
Page 3 and 4: and on the sphere, it is natural to
Page 5: 0 x 0 C 2 + H 2 + r 0 x 2 x 1 Fig.
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,

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