1. Passive 3D imaging with rotating point spread functions
Sri Rama Prasanna Pavani, Adam Greengard, and Rafael Piestun
Department of Electrical and Computer Engineering, University of Colorado at Boulder
The Problem: Passive 3D imaging
To obtain an object’s three dimensional information without imposing constraints on illumination
3D cues in 2D images
RPSF fundamentals
RPSF is obtained from a linear superposition of Gauss-Laguerre (GL) modes lying along a straight line in the GL modal plane.
3D computational optical imaging
Two images of an object are obtained; one with the RPSF mask (Irot) and the other without the mask (Iref).
The depth of a particular region of the object is estimated from the angle of rotation of the RPSF in the corresponding region of Irot. The RPSF of a region of Irot is estimated from the following deconvolution procedure: 10 5 m n
GL modal plane
4Zo
Spiral trajectory of a point
(1,1)
(1, 1) + (3, 5) + (5, 9) +
(7, 13) + (9, 17)
Superposition
Mode 1
0.5  -
Intensity
Phase
Human’s often qualitatively perceive depth from a scene’s context
No quantitative 3D information
Parallax (stereo) estimates depth from two images of an object
obtained from two different angles
Suffers from occlusion and correspondence problems
Object
Digital decoding
Detector
hrot(zo)
hstd
Irot
Istd
Context
Parallax
Focus / Defocus
Near
Far
Experimentally estimated 3D image of Abraham Lincoln in the backside of a US one cent coin is shown below.
Defocus estimators using RPSFs have an order of magnitude lower estimator variance (Cramer-Rao bound) than those using standard PSFs. RPSFs offer a 10 fold increase in Fisher Information over standard PSFs (“axial super-resolution”)
Since RPSFs are eigen Fourier transforms, the transfer function of a RPSF system is a scaled version of the RPSF itself.
By simultaneously optimizing RPSFs in the GL modal plane, Fourier domain, and spatial domain, efficient phase-only transfer functions of quasi RPSFs (QRPSFs) can be obtained. QRPSFs present rotating features within a 3D domain of interest and they form a cloud around a straight line in the GL modal plane.
RPSF image
Every 2D image has 3D information in the form of defocus.
Two prominent methods are depth from focus (DFF) and depth from defocus (DFD). While DFF estimates depth by continuously refocusing until a focused image is obtained, DFD uses a defocused image and an image with extended depth.
Both DFF and DFD are largely based on geometric optics models that do not optimize optics and post processing together.
Object 40 20
-20
(µm)
3D image of
Abraham Lincoln 3 2 1
Cramer-Rao Bound
Defocus
RPSF
Standard
Phase-only QRPSF mask
80400
-40
-80
Z (mm)
Rotation angle (degrees)
500nm
550nm
600nm
Standard image
Rotating point spread functions (RPSFs)
Unlike standard point spread functions (PSFs), RPSFs have circularly asymmetric transverse profiles that rotate continuously with defocus.
Conclusion
Passive 3D imaging can be achieved with high depth accuracy (“axial super-resolution”) using RPSFs. RPSF systems are hybrid computational optical imaging systems that engineer the PSF of an imaging system to optimally encode an object’s 3D information.
References
[1] A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Optics Letters 31, 183 (2006)
[2] S.R.P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Optics Express 16, (2008)
[3] R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A 17, 294 (2000)
Lens f
Mask
RPSF slices
RPSF
Standard PSF
In-focus
Positive defocus
Negative defocus
A QRPSF mask designed for a particular wavelength exhibits different rotation rates for other neighboring wavelengths. This phenomenon can be used for simultaneous 3D measurements with a broad band source.
QRPSF masks can be fabricated either as continuous phase masks or more easily as masks with quantized phase levels (with minimal quantization effects). Alternatively, they can also be implemented as computer generated holograms (CGHs).
Depth is estimated from the angle of rotation of RPSF’s main lobes
Funding: National Science Foundation, CDM Optics fellowship, CU Technology Transfer Office, Photonics Technology Access Program, and Honda