Efficient rotating PSFs for 3D imaging

1
Efficient rotating PSFs for 3D imaging
Sri Rama Prasanna Pavani and Rafael Piestun Micro
Optical - Imaging Systems Laboratory, Dept. of
Electrical and Computer Engineering, University
of Colorado at Boulder http//moisl.colorado.edu
Frontiers in Optics - 9/16/2007
2
Overview

• Depth from diffracted rotation
• Rotating point spread functions (PSF)
• Quasi-rotating PSFs
• Optimal quasi-rotating PSFs
• Conclusion

3
Depth estimation

• Obtain 3D information from 2D image(s)

Parallax
Focus/Defocus
Context
Back
Front

• Defocus is inherently related to depth
• Defocus parameter (y) depends on
• wavelength (l)
• aperture size (r)
• best focus distance (zobj)
• object distance (zobj)

4
Depth from diffracted rotation
Rotating PSF
Standard PSF
f
Slices
Mask
Lens
3 2 1 0

• Use rotating PSF to estimate defocus
• Axial superresolution
• Cramer Rao Bound
• Lower bound of estimator variance
• One order of magnitude lower
• Constant through a wide range

CRB / Estimator variance
-60 -30 0 30
60
(y)
Defocus
5
Experimental Results
Rotating PSF Image
Standard PSF Image
Rotating PSF Image
Standard PSF Image
Optically estimated depth map
Optically estimated depth map
40 20 0 -20
80 40 0 -20
A. Greengard, Y. Y. Schechner, and R. Piestun,
Depth from diffracted rotation, Opt. Lett. 31,
181-183 2006 A. Greengard, Y. Y. Schechner, and
R. Piestun, to be published
6
Rotating PSFs
GL modal plane
10 5 0

• Gauss Laguerre (GL) basis
• All paraxial fields are superposition of GL
  modes
• Rotating beams
• A GL mode superposition along a straight line
  rotates upon propagation

n
-10 -5 0 5
10
m
Intensity
Phase
Intensity
Phase
R. Piestun et al, Propagation-invariant wave
fields with finite energy , J. Opt. Soc. Am. A
17, 294-303 2000
7
Low efficiency problems

• 1. Transfer function is absorbing

Transfer function intensity
Transfer function phase
1.09

• 2. Low diffraction efficiency of
  diffractive optical element

4 2 0
Continuous amplitude CGH
Binary phase CGH
8
Quasi rotating PSFs - a solution to the transfer
function efficiency problem

• Definition Rotates approximately while keeping
  the main features of the PSF intact
• In our case, the two prominent peaks in the PSF
  rotate
• Advantage High transfer function efficiency
• Design approach
• For rotation, phase of a GL mode or GL
  superposition is more important than amplitude
• Option 1
• Option 2
• Option 3 Optimize with one of the above designs
  as the starting point

9
Importance of GL phase
Mask amplitude and phase
Fourier Transform intensity and phase
GL mode (1,1)
Phase of GL (1,1) with Gaussian apodization
GL superposition
Phase of GL superposition with Gaussian
apodization
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GL modal plane comparison
Exact transfer function
High efficiency design 1
High efficiency design 2
modes
modes
Exact superposition
Exact rotating PSF
High efficiency rotating PSF 1
High efficiency rotating PSF 2
1.09
2.25 to 15.08
11
Optimal quasi-rotating PSFs

• Features
• Phase only transfer function
• Main lobes rotate with defocus
• No side lobes
• Starting guess
• Quasi-rotating PSFs
• Method
• Iterative Optimization

12
Iterative Optimization
Quasi Rotating
1
2
N
Starting guess
Constraints
Phase only
Rotating
No side lobes
Good enough ?
Nope!
Oh yeah!
Optimal
13
Constraint 1
After an iteration
Enforce phaseonlyness
Starting guess quasi rotating

• Phase only transfer function
• How to enforce?
• Just remove the amplitude!

Constraint 2
Phase
Intensity

• Main lobes must rotate with defocus
• In general, a very hard constraint to impose
• Easy to design DOE for any field in one axial
  plane
• Fields at all other planes are dictated by
  diffraction

Slices
Mask
maxima line
Lens
Weight function
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Constraints 3,4,5,,11

• No side lobes in the PSF
• How to enforce?
• Boost the main lobes of the PSF
• Leave alone the phase!
• Apply this constraint for 9 rotated PSF planes

Weight function
Results
Transfer function evolution
Initial guess
Optimal

• Phase becomes smoother
• Locations of the singularities are the
• same before and after optimization

Phase
Intensity
15
Comparison
Quasi-rotating PSFs
Exact rotating PSF
Optimal quasi-rotating PSFs
1.84
42.07
57.01
Standard Exact Optimized
CRB / Estimator variance
Phase only super-resolving mask
Defocus
Pavani and Piestun, Quasi-rotating point spread
functions, Optics Letters (in preparation)
16
Conclusion

• Rotating point spread functions improve accuracy
  of depth estimation - Axial super-resolution
• Quasi-rotating PSFs have about 30 times higher
  efficiency
• For the first time, we now know the recipe of
  Optimal quasi-rotating PSFs

17
Acknowledgements

• Prof. Carol Cogswell
• Dr. Adam Greengard
• Prof. Gregory Beylkin
• Prof. Yoav Y. Schechner

CDMOptics PhD fellowship National Science
Foundation under Grant No. ECS-225533 Technology
Transfer Office, University of Colorado Photonics
Technology Access Program
18
Optimization Results
Tranfer function evolution
15 7.5 0
transparency
0 25 50
Iterations
GL modal plane

• Optimized transfer function forms
• a cloud in the GL plane
• Phase becomes smoother
• Locations of the singularities are the
• same before and after optimization

10 5 0
n
-10 -5 0 5
10
m
19
3D imaging simulation
f
f
f
f

• Parabola focussed a little farther from the
  apex
• Efficient rotating transfer function in the
  Fourier plane

Detector
Textured Parabolic surface
Lens (f)
Lens (f)
Mask
Reconstructed depth map
Traditional system
Rotating PSF system
0.2 0 -0.2
1 0.5
0 0.5 1
(cm)
(cm)
Z Depth Z0 Rayleigh range
20
Experiment SLM implementation

• Problems
• Orders due to sampling
• Off-axis imaging

• Device used - Holoeye SLM
• 1920 x 1080 pixels
• 8 bit addressing
• Phase Only Modulation

21
Experimental results - Lee and Blazed CGHs
Binary phase only with Gaussian apodization
Binary exact GL
Binary phase only
Blazed Phase only
PSF
PSF
PSF
PSF
22
Experimental efficiencies