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 0 5 10 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 -60 -30 0 30 60 3 2 1 Cramer-Rao Bound Defocus RPSF Standard Phase-only QRPSF mask -20 -10 0 10 20 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, 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) 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