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Designing Aperture Masks in Phase Space

Designing Aperture Masks in Phase Space. Roarke Horstmeyer 1 , Se Baek Oh 2 , and Ramesh Raskar 1 1 MIT Media Lab 2 MIT Dept. of Mechanical Engineering. Motivation. Conventional camera PSF measurement:. z 0. z 1. z 2. Circular Aperture. Pt. Source. f. misfocus.

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Designing Aperture Masks in Phase Space

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  1. Designing Aperture Masks in Phase Space RoarkeHorstmeyer1, Se Baek Oh2, and Ramesh Raskar1 1MIT Media Lab 2MIT Dept. of Mechanical Engineering

  2. Motivation Conventional camera PSF measurement: z0 z1 z2 Circular Aperture Pt. Source f misfocus Conventional PSF Blur I(z0) I(z1) I(z2)

  3. Motivation Design apertures for specific imaging tasks z0 z1 z2 Aperture mask: amplitude/phase misfocus Engineer mask Desired set of PSFs I(z0), I(z1), I(z2)

  4. Examples Misfocus PSFs: Depth-Invariant Rotating Arbitrary ,

  5. Problem Statement Inputs Output Multiple PSFs (OTFs) Amp. + Phase at aperture I(z0), I(z1), I(z2),… U(x,z=0)=A(x)e iϕ(x) Linked by Propagation Not exact Inverse problem: Find A(x)e iϕ(x) from I(x,z1), I(x,z2), I(x,z3)… Examples: Gerchberg-Saxton, Fienup, TIE, etc. Attempt problem in phase space

  6. What is Phase Space? Joint space-space freq. rep.= Wigner Dist. (WDF)1: Similar to ray space with ray (position, angle) [1] M.J. Bastiaans, JOSA 69(12), 1979

  7. What is Phase Space? Joint space-space freq. rep.= Wigner Dist. (WDF)1: Similar to ray space with ray (position, angle) Fourier-dual space = A(u,x) = Ambiguity Function (AF) 1D -> 2D introduces redundancies [1] M.J. Bastiaans, JOSA 69(12), 1979

  8. What is Phase Space? Joint space-space freq. rep.= Wigner Dist. (WDF)1: Similar to ray space with ray (position, angle) Fourier-dual space = A(u,x) = Ambiguity Function (AF) PSFs OTFs 1D -> 2D introduces redundancies [1] M.J. Bastiaans, JOSA 69(12), 1979

  9. A Simple 1D Example z=0 z x Open aperture U(z=0) = t(x) = rect function: x

  10. A Simple 1D Example W(z=0; x,u) u 1 Open aperture -.4 U(z=0) = t(x) x = rect function: x

  11. A Simple 1D Example Rotate W(z=f; x,u) u x/λf U(z=f) f x/λf Fourier Transform = 90̊ rotation

  12. A Simple 1D Example u Integrate f x/λf I(z=f) Intensity from Integration x/λf

  13. A Simple 1D Example W(z=f+d; x,u) u U(z=f+d) f+d x/λf Propagation = shear shear

  14. A Simple 1D Example u Integrate f+d x/λf I(z=f+d) x/λf

  15. 1st Approach: Full Search Wigner space connects PSFs + t(x): Demonstrated: Rotate, shift, integrate WDF I(z=f) Mask t(x) I(z=f+d)

  16. 1st Approach: Full Search Wigner space connects PSFs + t(x): Unknown aperture: Search I(z=f) Mask t(x) I(z=f+d) Search: Perform same procedure for many t(x), pick t(x) that creates best set of PSFs (lowest MSE match)

  17. 1st Approach: Full Search Example Inputs: Output: WDF 3 PSFs 1 Search all masks Constraints: 1D (2D separable) Binary amplitude Symmetric 40 elements f/4, λ=500nm d=0.1mm, 0.3mm Z=f -.4 Z=f+d1 Mask pattern with lowest MSE Z=f+d2 x=50μ

  18. 1st Approach: Full Search Example Modeled (normed) Experimental Intensity Intensity -50μ 50μ 124 peaks, but limited Performance z=f z=f+.1mm z=f+.3mm

  19. 2nd Approach: Direct Design • Instead of searching, can I design a function directly? • Populate values directly in phase space • Need a constraint: require a valid function

  20. 2nd Approach: Direct Design • Instead of searching, can I design a function directly? • Populate values directly in phase space • Need a constraint: require a valid function Phase-space tomography WDF PSF(2) PSF(1) PSF(0)

  21. 2nd Approach: Direct Design • Instead of searching, can I design a function directly? • Populate values directly in phase space • Need a constraint: require a valid function Phase-space tomography AF: Slices WDF F-slice OTF(1) OTF(0) OTF(2) AF easier than WDF 1 PSF(2) PSF(1) PSF(0) [1] Tu, Tamura, Phys. Rev. E 55 (1997)

  22. 2nd Approach: Direct Design Algorithm: Populate AF with OTF slices (z=f, z=f+d, …) Interpolate: Taylor power series approximation 1 OTF Inputs Populate Interpolate Closest AF OTF0 OTF1 Unique Inversion Apply Constraint Desired Mask [1] Ojeda-Castaneda et al., Applied Optics 27 (4), 1988

  23. 2nd Approach: Direct Design Algorithm: Populate AF with OTF slices (z=f, z=f+d, …) Interpolate: Taylor power series approximation 1 OTF Inputs Populate Interpolate Closest AF OTF0 OTF1 Unique Inversion Apply Constraint ? Desired Mask [1] Ojeda-Castaneda et al., Applied Optics 27 (4), 1988

  24. 2nd Approach: Direct Design • Constraint • Physical restriction: spatially coherent source • Coherent mutual intensity (MI) constrained • - 2D matrix is an outer product (1D) 1 • AF space to MI space: 2 operations [1] Ozaktas et al., JOSA 19 (8), 2002

  25. 2nd Approach: Direct Design • Constraint • Physical restriction: spatially coherent source • Coherent mutual intensity (MI) constrained • - 2D matrix is an outer product (1D) 1 • AF space to MI space: 2 operations Physically realistic MI AF interpolation MI “guess” e.g., open aperture,3 slices: Fx-1 rotate 45̊ 1st SVD For any MI guess, 1st SVD = good rank-1 estimate [1] Ozaktas et al., JOSA 19 (8), 2002

  26. Example: Known Input • -Known Mask: 5 different slits (e.g.) • - Take 3 OTF’s from z=f, f+.1mm, f+.2mm • Iterate algorithm 5x, apply 1st SVD constraint mask 3 OTFs

  27. Example: Known Input • -Known Mask: 5 different slits (e.g.) • - Take 3 OTF’s from z=f, f+.1mm, f+.2mm • Iterate algorithm 5x, apply 1st SVD constraint mask 3 OTFs Original Recovered MI: AF:

  28. Example: Known Input • -Known Mask: 5 different slits (e.g.) • - Take 3 OTF’s from z=f, f+.1mm, f+.2mm • Iterate algorithm 5x, apply 1st SVD constraint mask 3 OTFs Original Recovered MI: Inversion not exact: -Used only 3 constraints AF:

  29. Example: Desired Input • Same desired PSFs of 1,2 and 4 peaks: • Input 3 OTF’s into AF, populate, constrain AF MI Amp. Mask Old Mask • Now have higher spatial res., continuous (A,ϕ)

  30. Example: Desired Input • Same desired PSFs of 1,2 and 4 peaks: • Input 3 OTF’s into AF, populate, constrain AF MI Amp. Mask Old Mask • Now have higher spatial res., continuous (A,ϕ) Experimental results: - Thresheld amplitude mask - Still not optimal performance Intensity -50μ 50μ

  31. Conclusion and Future Work • Mask design method w/few PSF inputs • Populate and constrain • Phase/Ray space benefits: • I(x) at all planes from few inputs • Adapts to additional inputs, scales well mask 3 inputs AF • In the future: • 1st SVD is not optimal solution: find unique solution • Application-specific PSFs • Holography and 3D display

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