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Partially Coherent Ambiguity Functions for Depth-Variant Point Spread Function Design

Partially Coherent Ambiguity Functions for Depth-Variant Point Spread Function Design. Roarke Horstmeyer 1 Se Baek Oh 2 Otkrist Gupta 1 Ramesh Raskar 1. 1 MIT Media Lab 2 MIT Department of Mechanical Engineering. PIERS Marrakesh, 3/20/11. Problem Statement.

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Partially Coherent Ambiguity Functions for Depth-Variant Point Spread Function Design

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  1. Partially Coherent Ambiguity Functions for Depth-Variant Point Spread Function Design Roarke Horstmeyer1 Se Baek Oh2 Otkrist Gupta1 Ramesh Raskar1 1MIT Media Lab 2MIT Department of Mechanical Engineering PIERS Marrakesh, 3/20/11

  2. Problem Statement Conventional camera PSF measurement: z0 z1 z2 Circular Aperture Pt. Source f defocus Conventional PSF Blur I(z0) I(z1) I(z2)

  3. Problem Statement Design apertures for specific imaging tasks z0 z1 z2 Aperture mask: amplitude/phase defocus Determine mask Desired set of PSFs I(z0), I(z1), I(z2)

  4. Examples Defocus PSFs: Depth-Invariant Rotating Arbitrary Gauss-Laguerre modes Cubic phase Iterative Design ?

  5. PSF Design Similar to Phase Recovery z0 z1 z2 Aperture Element f (A,ϕ) I(z0), I(z1), I(z2) I(z0), I(z1), I(z2) (A,ϕ) Similar to Measurement for phase recovery 3D PSF Design z0 z1 z2 Diffractive Element General Goal: Find (A,ϕ) from multiple intensities Models: Fresnel propagation, k-space, light fields, phase space d

  6. Overview of 3D Design Techniques … U(A,ϕ) I(x0,z0) I(x1,z1) U(A,ϕ) U(A,ϕ) I(x0,z0) I(x0,z0) I(x1,z1) I(x1,z1) U(A,ϕ) I(x0,z0) I(xn,zn) iterative iterative iterative simultaneous simultaneous simultaneous (d) Mode Selective (proposed) (c) Phase Space Tomography (a) Phase Retrieval (b) Transport of Intensity

  7. Overview of 3D Design Techniques … U(A,ϕ) I(x0,z0) I(x1,z1) U(A,ϕ) U(A,ϕ) I(x0,z0) I(x0,z0) I(x1,z1) I(x1,z1) U(A,ϕ) I(x0,z0) I(xn,zn) Phase space extends nicely to partially coherent design (b) Transport of Intensity iterative iterative iterative simultaneous simultaneous simultaneous (c) Phase Space Tomography (a) Phase Retrieval (d) Mode Selective (proposed)

  8. Phase Space Functions Wigner Distribution (WDF) Ambiguity Function (AF) AF Slices: OTFs WDF Projections: PSFs x' u F-slice OTF(z1) OTF(z0) OTF(-z1) x u PSF(z1) AF “easier” than WDF Tu, Tamura, Phys. Rev. E 55, 1997 PSF(-z1) PSF(z0)

  9. z0 z1 Phase Space Camera Model Aperture mask r U(x) Δz f x z1 z0 I I xʹ xʹ OTFs

  10. z0 z1 Phase Space Camera Model Aperture mask r U(x) Why AF is useful: 1. Polar display of the OTF Δz f x u z1 z0 tan(θ1)=(W20k/π) I I z0 tan(θ0)=0 z1 xʹ xʹ xʹ OTFs

  11. Aperture mask z0 z1 Phase Space Camera Model r U(x) Why AF is useful: 1. Polar display of the OTF Δz f x u z1 z0 tan(θ1)=(W20k/π) I I z0 tan(θ0)=0 z1 xʹ xʹ xʹ OTFs 2. Convert AF to Mutual Intensity: inverse FT, 45° rotation, scale by 2 3. Recovery of U(x) from AF (up to constant Δϕ)

  12. (2) One-time AF Interpolation PSF OTF Inputs OTF(z0) u (1) AF Population (3) Mutual Intensity J OTF(z1) 1 u x2 θn xʹ OTF(z2) 0 xʹ x1 Iterate & xʹ Rank Constraint Error Check (5) Optimized AF (4) Optimized J Output: Desired Aperture Mask, 1D 1 1 x2 u 0 0 x1 xʹ

  13. Rank-constraint on Mutual Intensity, J Represent J with coherent mode decomposition1 Coherent, orthogonal modes from singular value decomposition J = UΛVT = Σ λi Ui(x1)Ui*(x2) x2 x2 n Assume: J symmetric, nxn λi = Singluar Values Uiorthogonal to Ujfor all i≠j = λ1 + λ2 + λ3 +… i=1 J x1 x1 Imperfect J guess: Many coherent modes 1 E. Wolf, JOSA 72 (3), 1982

  14. Rank-constraint on Mutual Intensity, J Represent J with coherent mode decomposition1 Coherent, orthogonal modes from singular value decomposition x2 x2 PSF = response to a point source: restricted to 1 mode = λ1 + λ2 + λ3 +… Jest = λ1U1(x1)U1*(x2) J x1 x1 1st Mode: Coherent 1 E. Wolf, JOSA 72 (3), 1982

  15. Reconstruction Example: Cubic Phase Mask Example aperture mask function: exp(jαx3), α=40, 20 iterations Ground Truth and Reconstructed OTFs W20=0 W20=λ/2 W20=λ xʹ xʹ xʹ Computed Phase Mask Ground Truth AF Reconstructed AF u u +π -π xʹ xʹ

  16. Simple Example: Arbitrary Input Input I1(x) I2(x) I3(x) z1=50mm z3=50.2mm z2=50.1mm Simulation Amplitude One (fixed) mask 50μ Experiment Rank-1 constraint 25μ – resolution, 1cm2 binary mask in 50mm f/1.8 Nikkor, 200μ pinhole @ z=4m

  17. Constrained Decompositions In Experiment: Amplitude-only or Phase-only required Phase-only constraint Amplitude-only constraint No Constraints on (A,ϕ) +π Phase (rad.) Amplitude (AU) -π x (cm) MSE vs. # iterations Aperture mask constraints: MSE • Varied performance • Algorithm still converges # of iterations

  18. Keeping More than One Mode • More accurate estimate found with n> 1 modes Several Modes: Partially Coherent x2 J x2 = λ1 + λ2 + λ3 +… x1 x1 Eckert-Young Thm.: 1stn-modes of SVD(J) = optimal rank-n estimate 3 n= 3: (J - Σ λiUi(x1)Ui(x2))2 = global minimum i=1 SVD = Optimal estimate (L2 norm, no prior knowledge)

  19. Simulating Partial Coherence • More accurate estimate found with n> 1 modes Several Modes: Partially Coherent x2 J x2 = λ1 + λ2 + λ3 +… x1 x1 Multiple modes can be multiplexed over time1,2 Spatial Light Modulator: Vary over time CPU 1 P. De Santis, JOSA 3 (8), 1986, 2 Z. Zhang, private communication, 2011

  20. Example: Benefit of Several Modes Input I1(x) I2(x) I3(x) z1=50mm z3=50.2mm z2=50.1mm Display 3 Optimal masks Simulation Rank-3 constraint 1 2 50μ 3 MSE improvement ~100x (modes contain both A and ϕ) “Weights”: 1 - 1 2 - .61 3 - .38 Experimentally: A,ϕ over time = hard 1cm2 masks

  21. Adding a Constraint to Several Modes SVD & constrain in separate operations: No convergence Operation On(J) = find closest nrank-1 outer-products, constrained On(J) x2 x2 W1=? W2=? W3=? = μ1 + μ2 +μ3 x1 x1 General solution: convex optimization e.g.: amplitude-only, phase-only, spatial1 and coherence constraints n min || J – ΣμiWiWi*|| 2 subject to constraints on W, given n i=1 1 Flewett et al., Optics Letters 34 (14) 2010

  22. Example Constraint: Amplitude-only Problem: n optimal coherent modes that are real, positive min || J – WWT|| 2W ≥ 0, real (J = kxk, W=kxn)

  23. Example Constraint: Amplitude-only Problem: n optimal coherent modes that are real, positive min || J – WWT|| 2W ≥ 0, real (J = kxk, W=kxn) • Solution: Non-negative matrix factorization1 • e.g. Netflix challenge: low-rank rep. of 0-5 star movie scores • Symmetric NMF: add to update rules (solve for W &H, W≈H) 1. H=WT 2. W=W.*(HTJ)./((HHT)H+δ) 3. H=H.*(WTJ)T./H(WW)+δ) Update Rules: Add line 1 δ=tiny value, error ~2-5% Note: Optimal “Coherent modes” are no longer orthogonal 1Lee and Seung, Nature 401, 2001

  24. A Simple Example: Multiple Amplitude Modes Input I1(x) I2(x) I3(x) z1=50mm z3=50.2mm z2=50.1mm Amplitude-only, 3 masks Simulation Amp-only masks: Sym. NMF 1 2 3 50μ “Weights”: 1 - 1 2 - .78 3 - .08 MSE improvement ~7x (vs. 1 Amp. mode) Buildup of a baseline bias… 1cm2 masks

  25. Conclusion & Future Work • Phase space functions = intuitive window into 3D PSF design • Multiple modes (partially coherent) = increased flexibility • Constrained searches can be achieved w/ convex methods • Amplitude-only: Symmetric NMF • Other constraints: Phase-only (another convex implementation), coherence length (weighted SVD) • Subtracting modes: J=U1U1* ± U2U2*±… (take 2 images) • Current: Initial Experimental tests using an SLM • Next Step: Find a nice application Thanks! Questions?

  26. Partially CoherentReconstruction: 3 Modes 3 Coherent Modes Ground Truth Reconstructed 1cm Mask u u AFpc(x',u) -5e4 x'(cm-1) 5e4 -5e4 x'(cm-1) 5e4 x2 x2 Jpc(x1,x2) x1(cm) .5 -.5 x1(cm) .5 x(cm) .5 -.5 -.5

  27. Constraining Several Modes Apply Constraint: Amplitude-only, Phase-only, prior knowledge, etc. SVD(J) x2 x2 = λ1 + λ2 + λ3 +… x1 x1 individual constraint individual constraint individual constraint Example: Amplitude-only mask amplitude-only amplitude-only amplitude-only Sum=No longer optimal (localized constraints will not converge) x2 λ1 + λ2 + λ3 x1

  28. A Simple Example: Prior Knowledge 3 modes hitting unknown structure A B x SVD(J) = 2 orthogonal modes SVD(J) U1U1*ε (0,.9) U2U2*ε (-.1,.4) C x2 = + (A=.4, B=.5, C=.7) x1 Negative values = phase

  29. A Simple Example: Prior Knowledge 3 modes hitting unknown structure A B x SVD(J) = 2 orthogonal modes + C Negative (A=.4, B=.5, C=.7) Symmetric NMF: Assume no phase change - 3 modes>0, more info about structure U2U2*ε (0,.6) U3U3*ε (0,.6) U1U1*ε (0,.47) SNMF(J) x2 + + = x1

  30. Keeping More than One Mode • More accurate estimate with n>1 mutual intensity modes • Jest= Σ Ji <-> AFest = Σ AFi= Σ L(Ji)1,2 (L=linear transformation) • If Jest more accurate, then AFestmore accurate x2 x2 J = λ1 + λ2 + λ3 +… x1 x1 Several Modes: Partially Coherent Multiple Modes can be: a. Multiplexed over time (Desantis, Zheng) b. Could also multiplex over space and/or angle 1M. Bastiaans, JOSA 3(8) 1986, 2Lohmann and Rhodes, Appl. Opt. 17, 1978

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