Department of Mathematics. University of Washington, Seattle. Seismic Imaging Summer School. University of Washington, 2
The Curvelet Frame Applications of Curvelets
An Introduction to Curvelets Hart F. Smith Department of Mathematics University of Washington, Seattle
Seismic Imaging Summer School University of Washington, 2009
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Curvelets
A curvelet frame {ϕγ } is a wave packet frame on L2 (R2 ) based on second dyadic decomposition.
f (x) =
X
cγ ϕγ (x)
γ
Z cγ
=
Hart F. Smith
f (x) ϕγ (x) dx
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Dyadic Decomposition Frequency shells: 2k < |ξ| < 2k +1
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Second Dyadic Decomposition Angular Sectors: ∠(ω, ξ) ≤ 2−k /2
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Second Dyadic Decomposition Parabolic scaling: ∆ξ2 ∼
√
ξ1
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Second Dyadic Decomposition Associated partition of unity: k /2
1 = ψˆ0 (ξ) + 2
2 ∞ X X
ψˆω,k (ξ)2
k =0 ω=1
supp(ψˆω,k ) ⊂ ξ : |ξ| ≈ 2k , ω −
ξ |ξ|
. 2−k /2
Second dyadic decomposition of f : k /2
1 = ψˆ0 (ξ)2 ˆf (ξ) +
∞ X 2 X
ψˆω,k (ξ)2 ˆf (ξ)
k =0 ω=1
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Second Dyadic Decomposition Associated partition of unity: k /2
1 = ψˆ0 (ξ) + 2
2 ∞ X X
ψˆω,k (ξ)2
k =0 ω=1
supp(ψˆω,k ) ⊂ ξ : |ξ| ≈ 2k , ω −
ξ |ξ|
. 2−k /2
Second dyadic decomposition of f : k /2
1 = ψˆ0 (ξ)2 ˆf (ξ) +
∞ X 2 X
ψˆω,k (ξ)2 ˆf (ξ)
k =0 ω=1
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Final step: expand ψˆω,k (ξ) ˆf (ξ) in Fourier series If supp(g(ξ)) ⊂ L1 × L2 rectangle: g(ξ) = (L1 L2 )−1/2
X
cp,q e−i p ξ1 −i q ξ2
p,q −1/2
cp,q = (L1 L2 )
Z
g(ξ)e i p ξ1 +i q ξ2
Points (p, q) belong to dilated lattice: p, q ∈
2π 2π Z× Z L1 L2
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
ψˆω,k (ξ)ˆf (ξ) supported in rotated 2k × 2k /2 rectangle !^",k(#) 2k/2 2k
Points (p, q) belong to rotated 2−k × 2−k /2 lattice Ξω,k Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Let (p, q) = x ∈ Ξω,k
cx,ω,k = 2
−3k /4
Z
e ihx,ξi ψˆω,k (ξ)ˆf (ξ) dξ
ψˆω,k (ξ) ˆf (ξ) = 2−3k /4
X
cx,ω,k e−ihx,ξi
x∈Ξω,k
Reconstruction is periodic, so localize: X cx,ω,k e−ihx,ξi ψˆω,k (ξ) ψˆω,k (ξ)2 ˆf (ξ) = 2−3k /4 x∈Ξω,k
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Let (p, q) = x ∈ Ξω,k
cx,ω,k = 2
−3k /4
Z
e ihx,ξi ψˆω,k (ξ)ˆf (ξ) dξ
ψˆω,k (ξ) ˆf (ξ) = 2−3k /4
X
cx,ω,k e−ihx,ξi
x∈Ξω,k
Reconstruction is periodic, so localize: X ψˆω,k (ξ)2 ˆf (ξ) = 2−3k /4 cx,ω,k e−ihx,ξi ψˆω,k (ξ) x∈Ξω,k
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Sum over ω then k to recover f ˆf (ξ) =
X X X
cx,ω,k 2−3k /4 e−ihx,ξi ψˆω,k (ξ)
ω x∈Ξω,k
k
Z
2−3k /4 e ihx,ξi ψˆω,k (ξ)ˆf (ξ) dξ
cx,ω,k =
Take inverse Fourier transform: X X X f (y ) = cx,ω,k 2−3k /4 ψω,k (y − x) k
ω x∈Ξω,k
Z cx,ω,k =
2−3k /4 ψω,k (y − x) f (y ) dy
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Sum over ω then k to recover f ˆf (ξ) =
X X X
cx,ω,k 2−3k /4 e−ihx,ξi ψˆω,k (ξ)
ω x∈Ξω,k
k
Z
2−3k /4 e ihx,ξi ψˆω,k (ξ)ˆf (ξ) dξ
cx,ω,k =
Take inverse Fourier transform: X X X f (y ) = cx,ω,k 2−3k /4 ψω,k (y − x) k
ω x∈Ξω,k
Z cx,ω,k =
2−3k /4 ψω,k (y − x) f (y ) dy
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Curvelets: ϕγ (y ) = 2−3k /4 ψω,k (y − x) ,
Frequency support:
γ = (x, ω, k )
Spatial support: !^",k(#) 2k/2 2k 2!k/2 !",k(y!x)
Hart F. Smith
2!k
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Plancherel identity
X
2
|cx,ω,k | =
Z
ψˆω,k (ξ)2 |ˆf (ξ)|2 dξ
x∈Ξω,k
Sum over ω and k : X
2
Z
|cγ | =
|ˆf (ξ)|2 dξ =
Z
|f (y )|2 dy
γ
Frame, not basis: ψω,k (y − x) not orthogonal, but almost.
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Plancherel identity
X
2
|cx,ω,k | =
Z
ψˆω,k (ξ)2 |ˆf (ξ)|2 dξ
x∈Ξω,k
Sum over ω and k : X
2
Z
|cγ | =
|ˆf (ξ)|2 dξ =
Z
|f (y )|2 dy
γ
Frame, not basis: ψω,k (y − x) not orthogonal, but almost.
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Curvelets and the Second Dyadic Decomposition
Plancherel identity
X
2
|cx,ω,k | =
Z
ψˆω,k (ξ)2 |ˆf (ξ)|2 dξ
x∈Ξω,k
Sum over ω and k : X
2
Z
|cγ | =
|ˆf (ξ)|2 dξ =
Z
|f (y )|2 dy
γ
Frame, not basis: ψω,k (y − x) not orthogonal, but almost.
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Motivation for SDD: homogeneous phase functions Consider Φ(ξ) smooth for ξ 6= 0, homogeneous degree 1: Φ(sξ) = sΦ(ξ) ,
s>0
Example: Φ(ξ) = |ξ| Claim: on supp(ψˆω,k ),
Φ(ξ) = ∇Φ(ω) · ξ + r (ξ) ,
where |r (ξ)| . 1 k
n −km− 2 n |∇m ω ∇ω ⊥ r (ξ)| . 2
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Motivation for SDD: homogeneous phase functions Consider Φ(ξ) smooth for ξ 6= 0, homogeneous degree 1: Φ(sξ) = sΦ(ξ) ,
s>0
Example: Φ(ξ) = |ξ| Claim: on supp(ψˆω,k ),
Φ(ξ) = ∇Φ(ω) · ξ + r (ξ) ,
where |r (ξ)| . 1 k
n −km− 2 n |∇m ω ∇ω ⊥ r (ξ)| . 2
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Example: |ξ| = ω · ξ + r (ξ) Solution to half-wave Cauchy problem: p ∂t + i −∆y u(t, y ) = 0 , u(0, y ) = ϕγ (y ) Z
e ihy ,ξi−it|ξ| ϕˆγ (ξ) dξ
Z
e ihy −tω,ξi e−itr (ξ) ϕˆγ (ξ) dξ
u(t, y ) = =
≈ ϕγ (y − tω) Initial data: u(0, y ) =
P
γ
cγ ϕγ (y ), then
u(t, y ) ≈
X
ϕγ (y − tω)
γ
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Example: |ξ| = ω · ξ + r (ξ) Solution to half-wave Cauchy problem: p ∂t + i −∆y u(t, y ) = 0 , u(0, y ) = ϕγ (y ) Z
e ihy ,ξi−it|ξ| ϕˆγ (ξ) dξ
Z
e ihy −tω,ξi e−itr (ξ) ϕˆγ (ξ) dξ
u(t, y ) = =
≈ ϕγ (y − tω) Initial data: u(0, y ) =
P
γ
cγ ϕγ (y ), then
u(t, y ) ≈
X
ϕγ (y − tω)
γ
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Evolution of waves A wavefront consisting of a few curvelets:
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Evolution of waves First approximation to wave flow:
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Córdoba-Fefferman Decomposition: 2k /2 × 2k /2 cubes Alternative phase-space decomposition being explored to compute wave propagators: (Demanét-Ying)
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
C. Fefferman [1973] Decompose support function of unit disc:
f(x)=1
Hart F. Smith
f(x)=0
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
C. Fefferman [1973]
Frequency sectors:
Spatial decomposition:
f^(!) 2k/2
2!k/2 2!k
2k
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Candés-Donoho (2003) Image with sharp jump along smooth curve:
f(x)=1
Hart F. Smith
f(x)=0
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Candés-Donoho (2003) 2k /2 dominant terms in curvelet expansion at frequency 2k :
f(x)=1
Hart F. Smith
f(x)=0
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Candés-Donoho (2003)
Approximation rate is optimal: Choose n largest coefficients cγ in f =
P
γ
cγ ϕγ
kf − fn k2L2 . n−2 log(n)3 No frame can do better for jumps along C 2 curves. Wavelet expansion: kf − fn k2L2 . n−1
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Candés-Donoho (2003)
Approximation rate is optimal: Choose n largest coefficients cγ in f =
P
γ
cγ ϕγ
kf − fn k2L2 . n−2 log(n)3 No frame can do better for jumps along C 2 curves. Wavelet expansion: kf − fn k2L2 . n−1
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Candés-Donoho (2003)
Approximation rate is optimal: Choose n largest coefficients cγ in f =
P
γ
cγ ϕγ
kf − fn k2L2 . n−2 log(n)3 No frame can do better for jumps along C 2 curves. Wavelet expansion: kf − fn k2L2 . n−1
Hart F. Smith
An Introduction to Curvelets
The Curvelet Frame Applications of Curvelets
Application to wave flow Images with jumps along curves
Denoising Images with Curvelets
Hart F. Smith
An Introduction to Curvelets
![Introduction to Programming Using Java - Department of Mathematics ... [PDF]](https://m.moam.info/img/260x300/introduction-to-programming-using-java-department-_649d773a098a9e163c8b4646.jpg)
