The Media Lab 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 1

Motivation Conventional camera PSF measurement: Conventional PSF Blur misfocus Circular Aperture I(z 0 )I(z 1 )I(z 2 ) z0z0 z1z1 z2z2 f Pt. Source 2

Motivation Design apertures for specific imaging tasks Engineer mask Aperture mask: amplitude/phase misfocus Desired set of PSFs I(z 0 ), I(z 1 ), I(z 2 ) z0z0 z1z1 z2z2 3

Examples Misfocus ApplicationEDOFDetect DepthApplication-specific Example:Cubic PhaseRotating PSF1, 2, 4 points Mask: TBD Other examples: Log-Asphere, Sinusoidal, Annular aperture Levin et al.e.g: Deblurring, Tracking, Super-res., Object ID PSFs:Depth-InvariantRotatingArbitrary 4,

Problem Statement Inverse problem: Find A(x)e i ϕ (x) from I(x,z 1 ), I(x,z 2 ), I(x,z 3 )… Examples: Gerchberg-Saxton, Fienup, TIE, etc. I(z 0 ), Multiple PSFs (OTFs) Inputs I(z 1 ),I(z 2 ),… Linked by Propagation U(x,z=0)=A(x)e i ϕ (x) Output Not exact Amp. + Phase at aperture Attempt problem in phase space 5

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

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) PropagationF-TransformIntensityInversion Shear 90 ̊ rotation Integration (u)Unique (const phase) 7 1D -> 2D introduces redundancies [1] M.J. Bastiaans, JOSA 69(12), 1979

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 8 [1] M.J. Bastiaans, JOSA 69(12), 1979 PropagationF-TransformIntensityInversion Shear 90 ̊ rotation Integration (u)Unique (const phase) 1D -> 2D introduces redundancies

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

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

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

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

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

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

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

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

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

1 st Approach: Full Search Example peaks, but limited Performance ExperimentalModeled (normed) Intensity -50 μ 50 μ z=fz=f+.1mmz=f+.3mm Intensity

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

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

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

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

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

2 nd 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 24 [1] Ozaktas et al., JOSA 19 (8), 2002

2 nd 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 For any MI guess, 1 st SVD = good rank-1 estimate AF interpolationMI “guess” F x -1 rotate 45 ̊ Physically realistic MI e.g., open aperture, 3 slices: 1 st SVD 25 [1] Ozaktas et al., JOSA 19 (8), 2002

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 1 st SVD constraint mask 3 OTFs

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 1 st SVD constraint MI: RecoveredOriginal AF: 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 1 st SVD constraint Inversion not exact: -Used only 3 constraints MI: RecoveredOriginal AF: mask 3 OTFs 28

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

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

Conclusion and Future Work In the future: -1 st SVD is not optimal solution: find unique solution -Application-specific PSFs -Holography and 3D display -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 31 3 inputs mask AF