Image Refocus in Geometrical Optical Phase Spacecwachan/FiO2010.pdf · Chan, Lam (Univ. Hong Kong)...
Transcript of Image Refocus in Geometrical Optical Phase Spacecwachan/FiO2010.pdf · Chan, Lam (Univ. Hong Kong)...
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Image Refocus in Geometrical Optical Phase Space
Aaron C. W. Chan Edmund Y. Lam
Imaging Systems Laboratory,Department of Electrical and Electronic Engineering,
University of Hong Kong.
http://www.eee.hku.hk/isl
2010 OSA Frontiers in Optics
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 1 / 18
http://www.eee.hku.hk/isl
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Introduction
Introduction
Objectives
To provide a convenient mathematical framework for light-fieldanalysis based on Hamiltonian Optics.
To demonstrate some mathematical symmetries that arise —
useful for computation and algorithm design.
To explain the refocus process in this formulation — Ren Ng,
Stanford (2006). Simply the free space operation on the light fieldbefore imaging; or equivalently, the thin-lens operation in theFourier domain followed by a slice.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 2 / 18
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Introduction
Introduction
Objectives
To provide a convenient mathematical framework for light-fieldanalysis based on Hamiltonian Optics.
To demonstrate some mathematical symmetries that arise —
useful for computation and algorithm design.
To explain the refocus process in this formulation — Ren Ng,
Stanford (2006). Simply the free space operation on the light fieldbefore imaging; or equivalently, the thin-lens operation in theFourier domain followed by a slice.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 2 / 18
-
Introduction
Introduction
Objectives
To provide a convenient mathematical framework for light-fieldanalysis based on Hamiltonian Optics.
To demonstrate some mathematical symmetries that arise —
useful for computation and algorithm design.
To explain the refocus process in this formulation — Ren Ng,
Stanford (2006). Simply the free space operation on the light fieldbefore imaging; or equivalently, the thin-lens operation in theFourier domain followed by a slice.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 2 / 18
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Introduction
Presentation Outline
Theoretical Background: Hamilton Equations; Lie Operator
formulation of ray equation.
Analysis of imaging system in spatial and spatial frequency
domains. The symmetries will be highlighted.
Refocus process derived. The computational costs will briefly bediscussed.
Concluding remarks, merits, limitations and further work.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 3 / 18
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Introduction
Presentation Outline
Theoretical Background: Hamilton Equations; Lie Operator
formulation of ray equation.
Analysis of imaging system in spatial and spatial frequency
domains. The symmetries will be highlighted.
Refocus process derived. The computational costs will briefly bediscussed.
Concluding remarks, merits, limitations and further work.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 3 / 18
-
Introduction
Presentation Outline
Theoretical Background: Hamilton Equations; Lie Operator
formulation of ray equation.
Analysis of imaging system in spatial and spatial frequency
domains. The symmetries will be highlighted.
Refocus process derived. The computational costs will briefly bediscussed.
Concluding remarks, merits, limitations and further work.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 3 / 18
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Introduction
Presentation Outline
Theoretical Background: Hamilton Equations; Lie Operator
formulation of ray equation.
Analysis of imaging system in spatial and spatial frequency
domains. The symmetries will be highlighted.
Refocus process derived. The computational costs will briefly bediscussed.
Concluding remarks, merits, limitations and further work.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 3 / 18
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Theoretical Background
Theoretical Background
This analysis is based on the geometrical optics (paraxial)approximation — light is incoherent, non-monochromatic...
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 4 / 18
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Theoretical Background Coordinates
Coordinates
x
y
z
α
Optical direction cosines: p1 = n dxds and p2 = ndyds .
p1 = n dxds = n sinα ≈ nα and similarly for p2.
n is the refractive index.
q = (x, y)T , p = (p1, p2)T and u = (q,p)T .
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 5 / 18
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Theoretical Background Coordinates
Coordinates
x
y
z
α
Optical direction cosines: p1 = n dxds and p2 = ndyds .
p1 = n dxds = n sinα ≈ nα and similarly for p2.
n is the refractive index.
q = (x, y)T , p = (p1, p2)T and u = (q,p)T .
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 5 / 18
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Theoretical Background Hamilton Equations
Hamilton Equations
The vectors q and p obey the Hamilton equations
Hamilton Equations
dqdz =
∂H(q,p)∂p
dpdz = −
∂H(q,p)∂q .
The Hamiltonian describes the environment in which the ray istravelling.
General Optical Hamiltonian
H(n, θ) = −n cosθ
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 6 / 18
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Theoretical Background Hamilton Equations
Hamilton Equations
The vectors q and p obey the Hamilton equations
Hamilton Equations
dqdz =
∂H(q,p)∂p
dpdz = −
∂H(q,p)∂q .
The Hamiltonian describes the environment in which the ray istravelling.
General Optical Hamiltonian
H(n, θ) = −n cosθ
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 6 / 18
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Theoretical Background Hamilton Equations
Hamilton Equations
The vectors q and p obey the Hamilton equations
Hamilton Equations
dqdz =
∂H(q,p)∂p
dpdz = −
∂H(q,p)∂q .
The Hamiltonian describes the environment in which the ray istravelling.
General Optical Hamiltonian
H(n, θ) = −n cosθ
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 6 / 18
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Theoretical Background Hamilton Equations
Hamilton Equations
The vectors q and p obey the Hamilton equations
Hamilton Equations
dqdz =
∂H(q,p)∂p
dpdz = −
∂H(q,p)∂q .
The Hamiltonian describes the environment in which the ray istravelling.
General Optical Hamiltonian
H(n, θ) = −n cosθ
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 6 / 18
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Theoretical Background Phase Space Density
Hamilton Equations for a Ray and a Bundle of Rays
Poisson Bracket {f , g} = ∂f∂q∂g∂p −
∂f∂p∂g∂q
Lie Operator L̂H = {·,H}.
Hamilton Equations forSingle Ray
dudz = +L̂Hu
Hamilton Equations forBundle of Rays
∂ρ∂z = −L̂Hρ
Solution
u(z) = exp[(z − zi)L̂H]u(zi)
Solution
ρ(z) = exp[−(z − zi)L̂H]ρ(zi)
Goto Details on Hamiltonian
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 7 / 18
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Theoretical Background Phase Space Density
Hamilton Equations for a Ray and a Bundle of Rays
Poisson Bracket {f , g} = ∂f∂q∂g∂p −
∂f∂p∂g∂q
Lie Operator L̂H = {·,H}.
Hamilton Equations forSingle Ray
dudz = +L̂Hu
Hamilton Equations forBundle of Rays
∂ρ∂z = −L̂Hρ
Solution
u(z) = exp[(z − zi)L̂H]u(zi)
Solution
ρ(z) = exp[−(z − zi)L̂H]ρ(zi)
Goto Details on Hamiltonian
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 7 / 18
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Theoretical Background Phase Space Density
Hamilton Equations for a Ray and a Bundle of Rays
Poisson Bracket {f , g} = ∂f∂q∂g∂p −
∂f∂p∂g∂q
Lie Operator L̂H = {·,H}.
Hamilton Equations forSingle Ray
dudz = +L̂Hu
Hamilton Equations forBundle of Rays
∂ρ∂z = −L̂Hρ
Solution
u(z) = exp[(z − zi)L̂H]u(zi)
Solution
ρ(z) = exp[−(z − zi)L̂H]ρ(zi)
Goto Details on Hamiltonian
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 7 / 18
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Theoretical Background Phase Space Density
Hamilton Equations for a Ray and a Bundle of Rays
Poisson Bracket {f , g} = ∂f∂q∂g∂p −
∂f∂p∂g∂q
Lie Operator L̂H = {·,H}.
Hamilton Equations forSingle Ray
dudz = +L̂Hu
Hamilton Equations forBundle of Rays
∂ρ∂z = −L̂Hρ
Solution
u(z) = exp[(z − zi)L̂H]u(zi)
Solution
ρ(z) = exp[−(z − zi)L̂H]ρ(zi)
Goto Details on Hamiltonian
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 7 / 18
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Theoretical Background Phase Space Density
Hamilton Equations for a Ray and a Bundle of Rays
Poisson Bracket {f , g} = ∂f∂q∂g∂p −
∂f∂p∂g∂q
Lie Operator L̂H = {·,H}.
Hamilton Equations forSingle Ray
dudz = +L̂Hu
Hamilton Equations forBundle of Rays
∂ρ∂z = −L̂Hρ
Solution
u(z) = exp[(z − zi)L̂H]u(zi)
Solution
ρ(z) = exp[−(z − zi)L̂H]ρ(zi)
Goto Details on Hamiltonian
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 7 / 18
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Theoretical Background Phase Space Density
Hamilton Equations for a Ray and a Bundle of Rays
Poisson Bracket {f , g} = ∂f∂q∂g∂p −
∂f∂p∂g∂q
Lie Operator L̂H = {·,H}.
Hamilton Equations forSingle Ray
dudz = +L̂Hu
Hamilton Equations forBundle of Rays
∂ρ∂z = −L̂Hρ
Solution
u(z) = exp[(z − zi)L̂H]u(zi)
Solution
ρ(z) = exp[−(z − zi)L̂H]ρ(zi)
Goto Details on Hamiltonian
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 7 / 18
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Theoretical Background Phase Space Density
There are three basic solutions: the free space propagationsolution, the thin lens solution, and the magnification solution.
Choosing H = 12p21 +
12p
22 , then L̂H = p.∂q, one obtains
Free Space Propagation Solution
T̂ (ξ) = exp[
ξp.∂q]
Solution for Ray
u(z) = T̂(z − zi)ui
Solution for Ray Bundle
ρ(q,p, z) =T̂ [−(z − zi)]ρi(q,p).
Goto Table: Optical Lie Group
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 8 / 18
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Theoretical Background Phase Space Density
There are three basic solutions: the free space propagationsolution, the thin lens solution, and the magnification solution.
Choosing H = 12p21 +
12p
22 , then L̂H = p.∂q, one obtains
Free Space Propagation Solution
T̂ (ξ) = exp[
ξp.∂q]
Solution for Ray
u(z) = T̂(z − zi)ui
Solution for Ray Bundle
ρ(q,p, z) =T̂ [−(z − zi)]ρi(q,p).
Goto Table: Optical Lie Group
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 8 / 18
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Theoretical Background Phase Space Density
There are three basic solutions: the free space propagationsolution, the thin lens solution, and the magnification solution.
Choosing H = 12p21 +
12p
22 , then L̂H = p.∂q, one obtains
Free Space Propagation Solution
T̂ (ξ) = exp[
ξp.∂q]
Solution for Ray
u(z) = T̂(z − zi)ui
Solution for Ray Bundle
ρ(q,p, z) =T̂ [−(z − zi)]ρi(q,p).
Goto Table: Optical Lie Group
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 8 / 18
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Theoretical Background Phase Space Density
There are three basic solutions: the free space propagationsolution, the thin lens solution, and the magnification solution.
Choosing H = 12p21 +
12p
22 , then L̂H = p.∂q, one obtains
Free Space Propagation Solution
T̂ (ξ) = exp[
ξp.∂q]
Solution for Ray
u(z) = T̂(z − zi)ui
Solution for Ray Bundle
ρ(q,p, z) =T̂ [−(z − zi)]ρi(q,p).
Goto Table: Optical Lie Group
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 8 / 18
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Theoretical Background Phase Space Density
qq
p p
q-shear
T̂(ξ) and performs q-shears in phase space, that is q → q + ξp orρ (q, p)→ ρ (q − ξp, p).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 9 / 18
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Theoretical Background Phase Space Density
q
z
Imaging Integral
I(q) =∫
Ωpρ (q,p)dp
Imaging is the integration of all the rays from all directions pimpinging onto a single point with position q = (x, y)T on theimage plane.
The integral projection of ρ(q,p) onto the q-plane.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 10 / 18
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Theoretical Background Phase Space Density
q
z
Imaging Integral
I(q) =∫
Ωpρ (q,p)dp
Imaging is the integration of all the rays from all directions pimpinging onto a single point with position q = (x, y)T on theimage plane.
The integral projection of ρ(q,p) onto the q-plane.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 10 / 18
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Theoretical Background Phase Space Density
q
z
Imaging Integral
I(q) =∫
Ωpρ (q,p)dp
Imaging is the integration of all the rays from all directions pimpinging onto a single point with position q = (x, y)T on theimage plane.
The integral projection of ρ(q,p) onto the q-plane.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 10 / 18
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Analysis of Imaging System
Analysis of Imaging System
f
u v
object image
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 11 / 18
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Analysis of Imaging System Single Lens Imaging System
Single Lens Optical System
If we consider a simple biconvex lens optical system, that acts onthe phase space position vector, it is evident from the thin lensequation 1f =
1u +
1v that if one wants to refocus to a further
distance v, u must be shortened.
fu v
object image
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 12 / 18
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Analysis of Imaging System Refocus
Refocus – In Spatial Domain
unewuold
vnewvold ξ
object image
Imaging of Refocused Light Field
I′(q) =∫
ΩpT̂(−ξ)ρ(q,p)dp =
∫
Ωpρ(q − ξp,p)dp.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 13 / 18
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Analysis of Imaging System Refocus
Refocus – In Spatial Domain
unewuold
vnewvold ξ
object image
Imaging of Refocused Light Field
I′(q) =∫
ΩpT̂(−ξ)ρ(q,p)dp =
∫
Ωpρ(q − ξp,p)dp.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 13 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Definitions
Fq =1
2π
∫ ∞
−∞
∫ ∞
−∞dq exp(−ikq.q), similarly for p.
kq = (kx , ky)T as the spatial frequency of q, similarly for p.
ρ̂(
kq, kp)
= FpFqρ(q,p).
Realistically, max(q)� max(p), and the allowable angles are such that|p| < n(q) under the paraxial approximation, so it is better to define
F(L)p =
12π
∫ L
−L
∫ L
−Ldp exp(−ikp.p).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 14 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Definitions
Fq =1
2π
∫ ∞
−∞
∫ ∞
−∞dq exp(−ikq.q), similarly for p.
kq = (kx , ky)T as the spatial frequency of q, similarly for p.
ρ̂(
kq, kp)
= FpFqρ(q,p).
Realistically, max(q)� max(p), and the allowable angles are such that|p| < n(q) under the paraxial approximation, so it is better to define
F(L)p =
12π
∫ L
−L
∫ L
−Ldp exp(−ikp.p).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 14 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Definitions
Fq =1
2π
∫ ∞
−∞
∫ ∞
−∞dq exp(−ikq.q), similarly for p.
kq = (kx , ky)T as the spatial frequency of q, similarly for p.
ρ̂(
kq, kp)
= FpFqρ(q,p).
Realistically, max(q)� max(p), and the allowable angles are such that|p| < n(q) under the paraxial approximation, so it is better to define
F(L)p =
12π
∫ L
−L
∫ L
−Ldp exp(−ikp.p).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 14 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Definitions
Fq =1
2π
∫ ∞
−∞
∫ ∞
−∞dq exp(−ikq.q), similarly for p.
kq = (kx , ky)T as the spatial frequency of q, similarly for p.
ρ̂(
kq, kp)
= FpFqρ(q,p).
Realistically, max(q)� max(p), and the allowable angles are such that|p| < n(q) under the paraxial approximation, so it is better to define
F(L)p =
12π
∫ L
−L
∫ L
−Ldp exp(−ikp.p).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 14 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Definitions
Fq =1
2π
∫ ∞
−∞
∫ ∞
−∞dq exp(−ikq.q), similarly for p.
kq = (kx , ky)T as the spatial frequency of q, similarly for p.
ρ̂(
kq, kp)
= FpFqρ(q,p).
Realistically, max(q)� max(p), and the allowable angles are such that|p| < n(q) under the paraxial approximation, so it is better to define
F(L)p =
12π
∫ L
−L
∫ L
−Ldp exp(−ikp.p).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 14 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
1 FpFqT̂ (−ξ)ρ(q,p)2 = FpFq exp(−ξp.∂q)ρ(q,p)
3 = exp(
ξkq.∂kp)
ρ̂(
kq, kp)
4 = ρ̂(kq, kp + ξkq).
Free space operator.
Thin lens operator L̂(ξ) ink-space.p-shear in k-space.
F(L)p FqT̂ (−ξ)ρ(q,p) = ρ̂(kq, kp + ξkq) ∗
[
sin(kp1 L)kp1
sin(kp2 L)kp2
]
if p is defined over a finite interval.Alternative Derivation Table of Operators
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 15 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
1 FpFqT̂ (−ξ)ρ(q,p)2 = FpFq exp(−ξp.∂q)ρ(q,p)
3 = exp(
ξkq.∂kp)
ρ̂(
kq, kp)
4 = ρ̂(kq, kp + ξkq).
Free space operator.
Thin lens operator L̂(ξ) ink-space.p-shear in k-space.
F(L)p FqT̂ (−ξ)ρ(q,p) = ρ̂(kq, kp + ξkq) ∗
[
sin(kp1 L)kp1
sin(kp2 L)kp2
]
if p is defined over a finite interval.Alternative Derivation Table of Operators
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 15 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
1 FpFqT̂ (−ξ)ρ(q,p)2 = FpFq exp(−ξp.∂q)ρ(q,p)
3 = exp(
ξkq.∂kp)
ρ̂(
kq, kp)
4 = ρ̂(kq, kp + ξkq).
Free space operator.
Thin lens operator L̂(ξ) ink-space.p-shear in k-space.
F(L)p FqT̂ (−ξ)ρ(q,p) = ρ̂(kq, kp + ξkq) ∗
[
sin(kp1 L)kp1
sin(kp2 L)kp2
]
if p is defined over a finite interval.Alternative Derivation Table of Operators
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 15 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
1 FpFqT̂ (−ξ)ρ(q,p)2 = FpFq exp(−ξp.∂q)ρ(q,p)
3 = exp(
ξkq.∂kp)
ρ̂(
kq, kp)
4 = ρ̂(kq, kp + ξkq).
Free space operator.
Thin lens operator L̂(ξ) ink-space.p-shear in k-space.
F(L)p FqT̂ (−ξ)ρ(q,p) = ρ̂(kq, kp + ξkq) ∗
[
sin(kp1 L)kp1
sin(kp2 L)kp2
]
if p is defined over a finite interval.Alternative Derivation Table of Operators
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 15 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
1 FpFqT̂ (−ξ)ρ(q,p)2 = FpFq exp(−ξp.∂q)ρ(q,p)
3 = exp(
ξkq.∂kp)
ρ̂(
kq, kp)
4 = ρ̂(kq, kp + ξkq).
Free space operator.
Thin lens operator L̂(ξ) ink-space.p-shear in k-space.
F(L)p FqT̂ (−ξ)ρ(q,p) = ρ̂(kq, kp + ξkq) ∗
[
sin(kp1 L)kp1
sin(kp2 L)kp2
]
if p is defined over a finite interval.Alternative Derivation Table of Operators
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 15 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
1 FpFqT̂ (−ξ)ρ(q,p)2 = FpFq exp(−ξp.∂q)ρ(q,p)
3 = exp(
ξkq.∂kp)
ρ̂(
kq, kp)
4 = ρ̂(kq, kp + ξkq).
Free space operator.
Thin lens operator L̂(ξ) ink-space.p-shear in k-space.
F(L)p FqT̂ (−ξ)ρ(q,p) = ρ̂(kq, kp + ξkq) ∗
[
sin(kp1 L)kp1
sin(kp2 L)kp2
]
if p is defined over a finite interval.Alternative Derivation Table of Operators
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 15 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
1 FpFqT̂ (−ξ)ρ(q,p)2 = FpFq exp(−ξp.∂q)ρ(q,p)
3 = exp(
ξkq.∂kp)
ρ̂(
kq, kp)
4 = ρ̂(kq, kp + ξkq).
Free space operator.
Thin lens operator L̂(ξ) ink-space.p-shear in k-space.
F(L)p FqT̂ (−ξ)ρ(q,p) = ρ̂(kq, kp + ξkq) ∗
[
sin(kp1 L)kp1
sin(kp2 L)kp2
]
if p is defined over a finite interval.Alternative Derivation Table of Operators
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 15 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Defining the projection operator Pkp as the operation that sets kp = 0,
1 FqI′(q) = Fq∫
ΩpT̂(−ξ)ρ (q,p)dp
2 = 2πPkpFpFqT̂(−ξ)ρ(q,p)
3 = 2πPkpL̂(ξ)ρ̂(kq, kp)4 = 2πρ̂(kq, ξkq)⇒
Fourier Slice
I′(q) = 2πF −1q ρ̂(kq, ξkq).
Fourier Slice – For finite p
I′(q) = 2πF −1q ρ̂(kq,U) ∗[
sin(U1L)U1
sin(U2L)U2
]
.U = ξkq.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 16 / 18
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Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Defining the projection operator Pkp as the operation that sets kp = 0,
1 FqI′(q) = Fq∫
ΩpT̂(−ξ)ρ (q,p)dp
2 = 2πPkpFpFqT̂(−ξ)ρ(q,p)
3 = 2πPkpL̂(ξ)ρ̂(kq, kp)4 = 2πρ̂(kq, ξkq)⇒
Fourier Slice
I′(q) = 2πF −1q ρ̂(kq, ξkq).
Fourier Slice – For finite p
I′(q) = 2πF −1q ρ̂(kq,U) ∗[
sin(U1L)U1
sin(U2L)U2
]
.U = ξkq.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 16 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Defining the projection operator Pkp as the operation that sets kp = 0,
1 FqI′(q) = Fq∫
ΩpT̂(−ξ)ρ (q,p)dp
2 = 2πPkpFpFqT̂(−ξ)ρ(q,p)
3 = 2πPkpL̂(ξ)ρ̂(kq, kp)4 = 2πρ̂(kq, ξkq)⇒
Fourier Slice
I′(q) = 2πF −1q ρ̂(kq, ξkq).
Fourier Slice – For finite p
I′(q) = 2πF −1q ρ̂(kq,U) ∗[
sin(U1L)U1
sin(U2L)U2
]
.U = ξkq.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 16 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Defining the projection operator Pkp as the operation that sets kp = 0,
1 FqI′(q) = Fq∫
ΩpT̂(−ξ)ρ (q,p)dp
2 = 2πPkpFpFqT̂(−ξ)ρ(q,p)
3 = 2πPkpL̂(ξ)ρ̂(kq, kp)4 = 2πρ̂(kq, ξkq)⇒
Fourier Slice
I′(q) = 2πF −1q ρ̂(kq, ξkq).
Fourier Slice – For finite p
I′(q) = 2πF −1q ρ̂(kq,U) ∗[
sin(U1L)U1
sin(U2L)U2
]
.U = ξkq.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 16 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Defining the projection operator Pkp as the operation that sets kp = 0,
1 FqI′(q) = Fq∫
ΩpT̂(−ξ)ρ (q,p)dp
2 = 2πPkpFpFqT̂(−ξ)ρ(q,p)
3 = 2πPkpL̂(ξ)ρ̂(kq, kp)4 = 2πρ̂(kq, ξkq)⇒
Fourier Slice
I′(q) = 2πF −1q ρ̂(kq, ξkq).
Fourier Slice – For finite p
I′(q) = 2πF −1q ρ̂(kq,U) ∗[
sin(U1L)U1
sin(U2L)U2
]
.U = ξkq.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 16 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Defining the projection operator Pkp as the operation that sets kp = 0,
1 FqI′(q) = Fq∫
ΩpT̂(−ξ)ρ (q,p)dp
2 = 2πPkpFpFqT̂(−ξ)ρ(q,p)
3 = 2πPkpL̂(ξ)ρ̂(kq, kp)4 = 2πρ̂(kq, ξkq)⇒
Fourier Slice
I′(q) = 2πF −1q ρ̂(kq, ξkq).
Fourier Slice – For finite p
I′(q) = 2πF −1q ρ̂(kq,U) ∗[
sin(U1L)U1
sin(U2L)U2
]
.U = ξkq.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 16 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Defining the projection operator Pkp as the operation that sets kp = 0,
1 FqI′(q) = Fq∫
ΩpT̂(−ξ)ρ (q,p)dp
2 = 2πPkpFpFqT̂(−ξ)ρ(q,p)
3 = 2πPkpL̂(ξ)ρ̂(kq, kp)4 = 2πρ̂(kq, ξkq)⇒
Fourier Slice
I′(q) = 2πF −1q ρ̂(kq, ξkq).
Fourier Slice – For finite p
I′(q) = 2πF −1q ρ̂(kq,U) ∗[
sin(U1L)U1
sin(U2L)U2
]
.U = ξkq.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 16 / 18
-
Analysis of Imaging System Spectral Analysis
Imaging and Refocus in Spatial Frequency Domain
Defining the projection operator Pkp as the operation that sets kp = 0,
1 FqI′(q) = Fq∫
ΩpT̂(−ξ)ρ (q,p)dp
2 = 2πPkpFpFqT̂(−ξ)ρ(q,p)
3 = 2πPkpL̂(ξ)ρ̂(kq, kp)4 = 2πρ̂(kq, ξkq)⇒
Fourier Slice
I′(q) = 2πF −1q ρ̂(kq, ξkq).
Fourier Slice – For finite p
I′(q) = 2πF −1q ρ̂(kq,U) ∗[
sin(U1L)U1
sin(U2L)U2
]
.U = ξkq.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 16 / 18
-
Conclusion
Conclusion
Conclusion:
Hamiltonian Optics and Operator methods provide a powerfulway of analysing incoherent optical systems.
This can be extended into k-space.
There is a symmetry between the operators in physical space andk-space. This means in computation, one only needs to optimizefor three or fewer operations.
However, computation in 4 dimensions still expensive with theFFT. Still, it is often desirable to work in Fourier Domain in ImageProcessing.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 17 / 18
-
Conclusion
Conclusion – Further Work
Further Work
Provide optimized image processing algorithms for light fieldscaptured in 4D.
Compressed sensing/sampling of phase space in ordinary camerasand reconstruction algorithms.
Provide analysis to incorporate stochastic elements/noise/rayscattering (Mie Scattering and Rayleigh Scattering).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 18 / 18
-
References
References
1 J. N. Mait, Optics and Photonics News 17, 22 (2006).
2 J. Mait, R. Athale, and J. van der Gracht, Optics Express 11, 2093(2003).
3 T. Mirani, D. Rajan, M. P. Christensen, S. C. Douglas, and S. L.Wood, Applied Optics 47, B86 (2006).
4 E. R. Dowski andW. T. Cathey, Applied Optics 34, 1859 (1995).
5 A. Accardi and G. Wornell, J. Opt. Soc. Am. A 26, 2055 (2009).
6 A. Torre, Linear Ray and Wave Optics in Phase Space (Elsevier,2004).
7 K. B.Wolf, Geometric Optics on Phase Space (Springer, 2004).
8 R. Ng, ACM Transactions on Graphics pp. 19 (2005).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 19 / 18
-
References
References
9 M. Levoy and P. Hanrahan, in ACM SIGGRAPH (1996), pp. 3142.
10 R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P.Hanrahan, Light field photography with a hand-held plenopiccamera, Tech. rep., Stanford University (2005).
11 A. L. Rivera, S. M. Chumakov, and K. B.Wolf, J. Opt. Soc. Am. A12, 1380 (1995).
12 K. B.Wolf, J. Opt. Soc. Am. A 8, 1389 (1991).
13 E. H. Adelson and J. R. Bergen, The Plenoptic Function and theElements of Early Vision (1991), pp. 320.
14 E. H. Adelson and J. Y. Wang, IEEE Trans. Pattern Anal. Mach.Intell. 14, 99 (1992).
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 20 / 18
-
Acknowledgment
Acknowledgement
Research Grants Council of the Hong Kong Special AdministrativeRegion, China under Projects HKU 713906 and 713408.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 21 / 18
-
Acknowledgment
Acknowledgement
Research Grants Council of the Hong Kong Special AdministrativeRegion, China under Projects HKU 713906 and 713408.
Group Website
http://www.eee.hku.hk/isl/
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 21 / 18
http://www.eee.hku.hk/isl/
-
Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem
The total derivative of ρ is given bydρdz =
∂ρ∂z +
∂ρ∂q
dqdz +
∂ρ∂p
dpdz .
Using the Hamilton Equations, this can be written asdρdz =
∂ρ∂z +
∂ρ∂q∂H∂p −
∂ρ∂p∂H∂q =
∂ρ∂z + {ρ,H}.
The rate of change ofNV(z) with respect to z of this volume isequal to the rate of flow out of the 3D surface.
Consider a 4D volume in phase spaceV ∈ R4. The number of rayscontained in this volume isNV(z) =
∫
Vρ(q,p, z)dV .
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 22 / 18
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Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem
The total derivative of ρ is given bydρdz =
∂ρ∂z +
∂ρ∂q
dqdz +
∂ρ∂p
dpdz .
Using the Hamilton Equations, this can be written asdρdz =
∂ρ∂z +
∂ρ∂q∂H∂p −
∂ρ∂p∂H∂q =
∂ρ∂z + {ρ,H}.
The rate of change ofNV(z) with respect to z of this volume isequal to the rate of flow out of the 3D surface.
Consider a 4D volume in phase spaceV ∈ R4. The number of rayscontained in this volume isNV(z) =
∫
Vρ(q,p, z)dV .
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 22 / 18
-
Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem
The total derivative of ρ is given bydρdz =
∂ρ∂z +
∂ρ∂q
dqdz +
∂ρ∂p
dpdz .
Using the Hamilton Equations, this can be written asdρdz =
∂ρ∂z +
∂ρ∂q∂H∂p −
∂ρ∂p∂H∂q =
∂ρ∂z + {ρ,H}.
The rate of change ofNV(z) with respect to z of this volume isequal to the rate of flow out of the 3D surface.
Consider a 4D volume in phase spaceV ∈ R4. The number of rayscontained in this volume isNV(z) =
∫
Vρ(q,p, z)dV .
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 22 / 18
-
Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem
The total derivative of ρ is given bydρdz =
∂ρ∂z +
∂ρ∂q
dqdz +
∂ρ∂p
dpdz .
Using the Hamilton Equations, this can be written asdρdz =
∂ρ∂z +
∂ρ∂q∂H∂p −
∂ρ∂p∂H∂q =
∂ρ∂z + {ρ,H}.
The rate of change ofNV(z) with respect to z of this volume isequal to the rate of flow out of the 3D surface.
Consider a 4D volume in phase spaceV ∈ R4. The number of rayscontained in this volume isNV(z) =
∫
Vρ(q,p, z)dV .
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 22 / 18
-
Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem 2
The rate of change ofNV(z) with respect to z of this volume isequal to the rate of flow out of the 3D hyper-surface S∂∂zNV(z) = −
∫
Sρ(q,p, z)dudz .n̂dS .
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 23 / 18
-
Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem 2
The rate of change ofNV(z) with respect to z of this volume isequal to the rate of flow out of the 3D hyper-surface S∂∂zNV(z) = −
∫
Sρ(q,p, z)dudz .n̂dS .
From the generalized divergence theorem∂∂zNV(z) = −
∫
V∇.
(
ρ(q,p, z)dudz)
dV .
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 23 / 18
-
Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem 2
The rate of change ofNV(z) with respect to z of this volume isequal to the rate of flow out of the 3D hyper-surface S∂∂zNV(z) = −
∫
Sρ(q,p, z)dudz .n̂dS .
From the generalized divergence theorem∂∂zNV(z) = −
∫
V∇.
(
ρ(q,p, z)dudz)
dV .
Taking all arguments to the left hand side and moving the partialderivative to inside the integral gives∫
V
(
∂ρ∂z + ∇.(ρ
dudz )
)
dV = 0.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 23 / 18
-
Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem 3
Using the result ∇.(ρ udz ) =∂ρ∂q
dqdz +
∂ρ∂p
dpdz =
∂ρ∂q∂H∂p −
∂ρ∂p∂H∂q = {ρ,H},
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 24 / 18
-
Appendix Proof of Liouville’s Theorem
Detour - Proof of Liouville’s Theorem 3
Using the result ∇.(ρ udz ) =∂ρ∂q
dqdz +
∂ρ∂p
dpdz =
∂ρ∂q∂H∂p −
∂ρ∂p∂H∂q = {ρ,H},
and the fact that the volume integral is true for anyV,dρdz =
∂ρ∂z + {ρ,H} = 0.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 24 / 18
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Appendix Details of Hamiltonian
Detour - More on the Hamiltonian
Return to Presentation
Explicitly, the optical Hamiltonian derived from Fermat’sprinciple is
H (x, y, p1, p2, z) = −√
n2(x, y, z) − p21 − p22 ≈
12n(x,y,z)
(
p2x + p2y
)
− n(x, y, z).Terms up to the second order give the paraxial approximation.
Note p(n, θ) =√
p2x + p2y = n sinθ and H(n, θ) = −n cosθ, where
θ is the angle the ray makes with the optic axis.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 25 / 18
-
Appendix Details of Hamiltonian
Detour - More on the Hamiltonian
Return to Presentation
Explicitly, the optical Hamiltonian derived from Fermat’sprinciple is
H (x, y, p1, p2, z) = −√
n2(x, y, z) − p21 − p22 ≈
12n(x,y,z)
(
p2x + p2y
)
− n(x, y, z).Terms up to the second order give the paraxial approximation.
Note p(n, θ) =√
p2x + p2y = n sinθ and H(n, θ) = −n cosθ, where
θ is the angle the ray makes with the optic axis.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 25 / 18
-
Appendix Details of Hamiltonian
Detour - More on the Hamiltonian 2
Return to Presentation
Taylor expanding the refractive index from the optic axis, andassume that there is no z variation, one obtainsn(x, y) ≈ n(0, 0) − ∂n∂x x −
∂n∂y y −
12
(
∂2n∂x2 x
2 + ∂2n∂y2 y
2 + ∂2n∂x∂y xy
)
.
Ignoring the first order terms, which relate to shifts and tilts of theoptic axis,
H(x, y, p1, p2) ≈ 12n(0,0)p21 +
∂2x n|(0,0)2 x
2 + 12n(0,0)p22 +
∂2y n|(0,0)2 y
2.
In general, in the paraxial approximation, we make use of thequadratic monomials q2, p2 and qp.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 26 / 18
-
Appendix Details of Hamiltonian
Detour - More on the Hamiltonian 2
Return to Presentation
Taylor expanding the refractive index from the optic axis, andassume that there is no z variation, one obtainsn(x, y) ≈ n(0, 0) − ∂n∂x x −
∂n∂y y −
12
(
∂2n∂x2 x
2 + ∂2n∂y2 y
2 + ∂2n∂x∂y xy
)
.
Ignoring the first order terms, which relate to shifts and tilts of theoptic axis,
H(x, y, p1, p2) ≈ 12n(0,0)p21 +
∂2x n|(0,0)2 x
2 + 12n(0,0)p22 +
∂2y n|(0,0)2 y
2.
In general, in the paraxial approximation, we make use of thequadratic monomials q2, p2 and qp.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 26 / 18
-
Appendix Details of Hamiltonian
Detour - More on the Hamiltonian 2
Return to Presentation
Taylor expanding the refractive index from the optic axis, andassume that there is no z variation, one obtainsn(x, y) ≈ n(0, 0) − ∂n∂x x −
∂n∂y y −
12
(
∂2n∂x2 x
2 + ∂2n∂y2 y
2 + ∂2n∂x∂y xy
)
.
Ignoring the first order terms, which relate to shifts and tilts of theoptic axis,
H(x, y, p1, p2) ≈ 12n(0,0)p21 +
∂2x n|(0,0)2 x
2 + 12n(0,0)p22 +
∂2y n|(0,0)2 y
2.
In general, in the paraxial approximation, we make use of thequadratic monomials q2, p2 and qp.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 26 / 18
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Appendix Solutions to Optical Hamilton Equations
Solutions to Optical Hamilton Equations
Return to Presentation
The solutions to these equations take a simple form. H is a linearcombination of q2, p2 and qp under the paraxial approximation.
If we take H = q2, H = p2 and H = qp separately, then we obtain aSp(2,R) Lie group of solutions.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 27 / 18
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Appendix Solutions to Optical Hamilton Equations
Solutions to Optical Hamilton Equations
Return to Presentation
The solutions to these equations take a simple form. H is a linearcombination of q2, p2 and qp under the paraxial approximation.
If we take H = q2, H = p2 and H = qp separately, then we obtain aSp(2,R) Lie group of solutions.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 27 / 18
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Appendix Solutions to Optical Hamilton Equations
sp(2,R) Optical Lie Algebra
Return to Presentation Return to Spectral Analysis
Table: This table shows the sp(2,R) optical Lie algebras. They are the generators foroptical operations as shown in the next slide
H L̂H K matrix
12p
2 L̂− = p ∂∂q K− =(
0 10 0
)
12q
2 L̂+ = −q ∂∂p K+ =(
0 0−1 0
)
qp L̂3 = p ∂∂p − q∂∂q K3 =
(
1 00 −1
)
.
This operator is of interest.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 28 / 18
-
Appendix Solutions to Optical Hamilton Equations
sp(2,R) Optical Lie Algebra
Return to Presentation Return to Spectral Analysis
Table: This table shows the sp(2,R) optical Lie algebras. They are the generators foroptical operations as shown in the next slide
H L̂H K matrix
12p
2 L̂− = p ∂∂q K− =(
0 10 0
)
12q
2 L̂+ = −q ∂∂p K+ =(
0 0−1 0
)
qp L̂3 = p ∂∂p − q∂∂q K3 =
(
1 00 −1
)
.
This operator is of interest.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 28 / 18
-
Appendix Solutions to Optical Hamilton Equations
sp(2,R) Optical Lie Algebra
Return to Presentation Return to Spectral Analysis
Table: This table shows the sp(2,R) optical Lie algebras. They are the generators foroptical operations as shown in the next slide
H L̂H K matrix
12p
2 L̂− = p ∂∂q K− =(
0 10 0
)
12q
2 L̂+ = −q ∂∂p K+ =(
0 0−1 0
)
qp L̂3 = p ∂∂p − q∂∂q K3 =
(
1 00 −1
)
.
This operator is of interest.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 28 / 18
-
Appendix Solutions to Optical Hamilton Equations
Sp(2,R) Optical Lie Group
Return to Presentation Return to Spectral Analysis Return to Phase Space Discussion
Table: This table shows the Sp(2,R) Lie groups of optical operations. They aregenerated by the elements in the previous table through exponentiation.
exp(ζL̂H) exp(ζK)
Free Space T̂(ξ) = eξL̂− T(ξ) = eξK− =(
1 ξ0 1
)
Thin Lens L̂(f) = e1f L̂+ L(f) = e
1f K+ =
(
1 0−1f 1
)
Magnifier M̂(s) = esL̂3 M(s) = esK3 =(
es 00 e−s
)
Anti-homomorphism. Ray Transfer Matrices.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 29 / 18
-
Appendix Solutions to Optical Hamilton Equations
Sp(2,R) Optical Lie Group
Return to Presentation Return to Spectral Analysis Return to Phase Space Discussion
Table: This table shows the Sp(2,R) Lie groups of optical operations. They aregenerated by the elements in the previous table through exponentiation.
exp(ζL̂H) exp(ζK)
Free Space T̂(ξ) = eξL̂− T(ξ) = eξK− =(
1 ξ0 1
)
Thin Lens L̂(f) = e1f L̂+ L(f) = e
1f K+ =
(
1 0−1f 1
)
Magnifier M̂(s) = esL̂3 M(s) = esK3 =(
es 00 e−s
)
Anti-homomorphism. Ray Transfer Matrices.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 29 / 18
-
Appendix Solutions to Optical Hamilton Equations
Sp(2,R) Optical Lie Group
Return to Presentation Return to Spectral Analysis Return to Phase Space Discussion
Table: This table shows the Sp(2,R) Lie groups of optical operations. They aregenerated by the elements in the previous table through exponentiation.
exp(ζL̂H) exp(ζK)
Free Space T̂(ξ) = eξL̂− T(ξ) = eξK− =(
1 ξ0 1
)
Thin Lens L̂(f) = e1f L̂+ L(f) = e
1f K+ =
(
1 0−1f 1
)
Magnifier M̂(s) = esL̂3 M(s) = esK3 =(
es 00 e−s
)
Anti-homomorphism. Ray Transfer Matrices.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 29 / 18
-
Appendix Solutions to Optical Hamilton Equations
Sp(2,R) Optical Lie Group
Return to Presentation Return to Spectral Analysis Return to Phase Space Discussion
Table: This table shows the Sp(2,R) Lie groups of optical operations. They aregenerated by the elements in the previous table through exponentiation.
exp(ζL̂H) exp(ζK)
Free Space T̂(ξ) = eξL̂− T(ξ) = eξK− =(
1 ξ0 1
)
Thin Lens L̂(f) = e1f L̂+ L(f) = e
1f K+ =
(
1 0−1f 1
)
Magnifier M̂(s) = esL̂3 M(s) = esK3 =(
es 00 e−s
)
Anti-homomorphism. Ray Transfer Matrices.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 29 / 18
-
Appendix Solutions to Optical Hamilton Equations
Sp(2,R) Optical Lie Group
Return to Presentation Return to Spectral Analysis Return to Phase Space Discussion
Table: This table shows the Sp(2,R) Lie groups of optical operations. They aregenerated by the elements in the previous table through exponentiation.
exp(ζL̂H) exp(ζK)
Free Space T̂(ξ) = eξL̂− T(ξ) = eξK− =(
1 ξ0 1
)
Thin Lens L̂(f) = e1f L̂+ L(f) = e
1f K+ =
(
1 0−1f 1
)
Magnifier M̂(s) = esL̂3 M(s) = esK3 =(
es 00 e−s
)
Anti-homomorphism. Ray Transfer Matrices.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 29 / 18
-
Appendix Algebra
Composition Property
exp(
ζL̂H)
ρ(q,p) = ρ(
exp(
ζL̂H)
q, exp(
ζL̂H)
p)
.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 30 / 18
-
Appendix Spectral Analysis
Sheared phase space density in k-space
Return to Spectral Analysis
F(L)p Fqρ(q − ξp,p) = F
(L)p exp(−iξkq.p)Fq (ρ(q,p))
= 12π
∫ L
−L
∫ L
−Ldp exp(−i(kp + ξkq).p)Fq (ρ(q,p))
= ρ̂(kq, kp + ξkq) ∗[
sin(kp1L)kp1
sin(kp2 L)kp2
]
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 31 / 18
-
Appendix Microlens Array
Off-axis lens
The Operator for shifting the z-axis is: Ŝ(s) = exp(
s∂q)
. Note that
Ŝ−1(s) = Ŝ(−s). Ŝ has no matrix counterpart.The action on the phase space density of an off-axis lens distance sfrom the optical axis is ρ̃(q, p) = Ŝ(−s)L̂(−f)Ŝ(s)ρ(q, p).Using the result exp(−t )B exp(−tÂ) = exp(−t[ , ·])B , it can beshown thatρ̃(q, p) = exp
(
q−sf∂∂p
)
ρ(q, p) = ρ(
q, p + q−sf)
.
Chan, Lam (Univ. Hong Kong) Image Refocus Oct 27, 2010 32 / 18
IntroductionTheoretical BackgroundCoordinatesHamilton EquationsPhase Space Density
Analysis of Imaging SystemSingle Lens Imaging SystemRefocusSpectral Analysis
ConclusionReferencesAcknowledgmentAppendixAppendixProof of Liouville's TheoremDetails of HamiltonianSolutions to Optical Hamilton EquationsAlgebraSpectral AnalysisMicrolens Array