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Image Refocus in Geometrical Optical Phase Space

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Image Refocus in Geometrical Optical Phase Space 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 Oct27,2010 1 / 18 Introduction Introduction Objectives To provide a convenient mathematical framework for light-field analysis 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 field before imaging; or equivalently, the thin-lens operation in the Fourier domain followed by a slice. Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 2 / 18 Introduction Introduction Objectives To provide a convenient mathematical framework for light-field analysis 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 field before imaging; or equivalently, the thin-lens operation in the Fourier domain followed by a slice. Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 2 / 18 Introduction Introduction Objectives To provide a convenient mathematical framework for light-field analysis 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 field before imaging; or equivalently, the thin-lens operation in the Fourier domain followed by a slice. Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 2 / 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 be discussed. Concluding remarks, merits, limitations and further work. Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 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 be discussed. Concluding remarks, merits, limitations and further work. Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 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 be discussed. Concluding remarks, merits, limitations and further work. Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 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 be discussed. Concluding remarks, merits, limitations and further work. Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 3 / 18 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 Oct27,2010 4 / 18 Theoretical Background Coordinates Coordinates x z α y dx dy Optical direction cosines: p1 = n ds and p2 = n ds . dx p1 = n ds = n sin α ≈ nα and similarly for p2. n is the refractive index. T T T q = (x, y) , p = (p1, p2) and u = (q, p) . Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 5 / 18 Theoretical Background Coordinates Coordinates x z α y dx dy Optical direction cosines: p1 = n ds and p2 = n ds . dx p1 = n ds = n sin α ≈ nα and similarly for p2. n is the refractive index. T T T q = (x, y) , p = (p1, p2) and u = (q, p) . Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 5 / 18 Theoretical Background Hamilton Equations Hamilton Equations The vectors q and p obey the Hamilton equations Hamilton Equations dq ∂H(q,p) dp − ∂H(q,p) dz = ∂p dz = ∂q . The Hamiltonian describes the environment in which the ray is travelling. General Optical Hamiltonian H(n,θ)= −n cos θ Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 6 / 18 Theoretical Background Hamilton Equations Hamilton Equations The vectors q and p obey the Hamilton equations Hamilton Equations dq ∂H(q,p) dp − ∂H(q,p) dz = ∂p dz = ∂q . The Hamiltonian describes the environment in which the ray is travelling. General Optical Hamiltonian H(n,θ)= −n cos θ Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 6 / 18 Theoretical Background Hamilton Equations Hamilton Equations The vectors q and p obey the Hamilton equations Hamilton Equations dq ∂H(q,p) dp − ∂H(q,p) dz = ∂p dz = ∂q . The Hamiltonian describes the environment in which the ray is travelling. General Optical Hamiltonian H(n,θ)= −n cos θ Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 6 / 18 Theoretical Background Hamilton Equations Hamilton Equations The vectors q and p obey the Hamilton equations Hamilton Equations dq ∂H(q,p) dp − ∂H(q,p) dz = ∂p dz = ∂q . The Hamiltonian describes the environment in which the ray is travelling. General Optical Hamiltonian H(n,θ)= −n cos θ Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 6 / 18 Theoretical Background Phase Space Density Hamilton Equations for a Ray and a Bundle of Rays {f g} ∂f ∂g − ∂f ∂g Poisson Bracket , = ∂q ∂p ∂p ∂q Lie Operator LˆH = {·, H}. Hamilton Equations for Hamilton Equations for Single Ray Bundle of Rays du =+Lˆ u ∂ρ −ˆ dz H ∂z = LHρ Solution Solution u(z)= exp[(z − zi)LˆH]u(zi) ρ(z)= exp[−(z − zi)LˆH]ρ(zi) Goto Details on Hamiltonian Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 7 / 18 Theoretical Background Phase Space Density Hamilton Equations for a Ray and a Bundle of Rays {f g} ∂f ∂g − ∂f ∂g Poisson Bracket , = ∂q ∂p ∂p ∂q Lie Operator LˆH = {·, H}. Hamilton Equations for Hamilton Equations for Single Ray Bundle of Rays du =+Lˆ u ∂ρ −ˆ dz H ∂z = LHρ Solution Solution u(z)= exp[(z − zi)LˆH]u(zi) ρ(z)= exp[−(z − zi)LˆH]ρ(zi) Goto Details on Hamiltonian Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 7 / 18 Theoretical Background Phase Space Density Hamilton Equations for a Ray and a Bundle of Rays {f g} ∂f ∂g − ∂f ∂g Poisson Bracket , = ∂q ∂p ∂p ∂q Lie Operator LˆH = {·, H}. Hamilton Equations for Hamilton Equations for Single Ray Bundle of Rays du =+Lˆ u ∂ρ −ˆ dz H ∂z = LHρ Solution Solution u(z)= exp[(z − zi)LˆH]u(zi) ρ(z)= exp[−(z − zi)LˆH]ρ(zi) Goto Details on Hamiltonian Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 7 / 18 Theoretical Background Phase Space Density Hamilton Equations for a Ray and a Bundle of Rays {f g} ∂f ∂g − ∂f ∂g Poisson Bracket , = ∂q ∂p ∂p ∂q Lie Operator LˆH = {·, H}. Hamilton Equations for Hamilton Equations for Single Ray Bundle of Rays du = +Lˆ u ∂ρ −ˆ dz H ∂z = LHρ Solution Solution u(z)= exp[(z − zi)LˆH]u(zi) ρ(z)= exp[−(z − zi)LˆH]ρ(zi) Goto Details on Hamiltonian Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 7 / 18 Theoretical Background Phase Space Density Hamilton Equations for a Ray and a Bundle of Rays {f g} ∂f ∂g − ∂f ∂g Poisson Bracket , = ∂q ∂p ∂p ∂q Lie Operator LˆH = {·, H}. Hamilton Equations for Hamilton Equations for Single Ray Bundle of Rays du =+Lˆ u ∂ρ −ˆ dz H ∂z = LHρ Solution Solution u(z)= exp[(z − zi)LˆH]u(zi) ρ(z)= exp[−(z − zi)LˆH]ρ(zi) Goto Details on Hamiltonian Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 7 / 18 Theoretical Background Phase Space Density Hamilton Equations for a Ray and a Bundle of Rays {f g} ∂f ∂g − ∂f ∂g Poisson Bracket , = ∂q ∂p ∂p ∂q Lie Operator LˆH = {·, H}. Hamilton Equations for Hamilton Equations for Single Ray Bundle of Rays du =+Lˆ u ∂ρ −ˆ dz H ∂z = LHρ Solution Solution u(z)= exp[(z − zi)LˆH]u(zi) ρ(z)= exp[−(z − zi)LˆH]ρ(zi) Goto Details on Hamiltonian Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 7 / 18 Theoretical Background Phase Space Density There are three basic solutions: the free space propagation solution, the thin lens solution, and the magnification solution. 1 2 1 2 ˆ Choosing H = 2 p1 + 2 p2 , then LH = p.∂q, one obtains Free Space Propagation Solution Tˆ (ξ) = exp ξp.∂q h i Solution for Ray Bundle Solution for Ray ρ(q, p, z)= u(z)= Tˆ(z − zi)ui Tˆ[−(z − zi)]ρi(q, p). Goto Table: Optical Lie Group Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 8 / 18 Theoretical Background Phase Space Density There are three basic solutions: the free space propagation solution, the thin lens solution, and the magnification solution. 1 2 1 2 ˆ Choosing H = 2 p1 + 2 p2 , then LH = p.∂q, one obtains Free Space Propagation Solution Tˆ (ξ) = exp ξp.∂q h i Solution for Ray Bundle Solution for Ray ρ(q, p, z)= u(z)= Tˆ(z − zi)ui Tˆ[−(z − zi)]ρi(q, p). Goto Table: Optical Lie Group Chan, Lam (Univ. Hong Kong) Image Refocus Oct27,2010 8 / 18 Theoretical Background Phase Space Density There are three basic solutions: the free space propagation solution, the thin lens solution, and the magnification solution.
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