Continuous Wavelet Transform on the Hyperboloid - Université de ...

a=0.5, χ=0, φ=0
a=1, χ=0, φ=0
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−0.1
1
0.5
1
0.5
1
0
0.5
0
0.5
−0.5
−0.5
0
−0.5
−0.5
0
−1
−1
−1
−1
a=0.5, χ=1, φ=π/2
a=0.5, χ=1, φ=3π/4
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
1
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
1
0.5
1
0.5
1
0
0.5
0
0.5
−0.5
−0.5
0
−0.5
−0.5
0
−1
−1
−1
−1
a=0.5, χ=0.75, φ=π
a=0.5, χ=2.75, φ=π
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
1
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
1
0.5
1
0.5
1
0
0.5
0
0.5
−0.5
−0.5
0
−0.5
−0.5
0
−1
−1
−1
−1
Fig. 10. The hyperbolic DOG wavelet fψ ϑ ,forϑ = 2 at different scales a and positions
(χ, ϕ), viewed on the unit disk in 3-D perspective.
any ρ [Alonso et al., 2002]:
( ) −
1
E ρ ν,ξ (x) = x0
2
− ˆn⃗x
−iνρ , (83)
ρ
for x ∈ H 2 +ρ , (x2 = ρ 2 ). The Inönü-Wigner contraction limit of the Lorentz
to the Euclidean group SO(2, 1) + → ISO(2) + is the limit at ρ →∞for (83)
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