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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
objects 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

• Humans often qualitatively perceive depth from
  a scenes 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

• 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.

40 20 0 -20
(µm)
3D image of Abraham Lincoln
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 objects 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, 3484-3489
(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)

• 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
RPSFs main lobes
Funding National Science Foundation, CDM Optics
fellowship, CU Technology Transfer Office,
Photonics Technology Access Program, and Honda