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FOURIER IMAGING Wayne Lawton Department of Mathematics National University of Singapore [email protected] (65)96314907 Hyper-Resolution Imaging Using Digital Phase Retrieval 1 ABSTRACT I first describe the principles of imaging using rays to explain pinhole cameras, waves to explain what a lens does, and Fourier transforms to explain the magic of holograms. Then I describe my own contributions to the analysis and design of computation-based coherent imaging systems at four organizations from Nov 1978 - June 1987 with an emphasis on phase retrieval based systems. Such systems are capable of sub-pico radian angular resolution whereas the space telescope is merely capable of sub-micro radian angular resolution. 2 HISTORY http://itg1.meteor.wisc.edu/wxwise/AckermanKnox/chap5/light_theory.html The Corpuscular Theory of Light Newton proposed this theory that treats light as being composed of tiny particles. We use this theory to describe reflection. While the theory can explain the primary and secondary rainbows, it cannot explain the supernumerary bow, the corona, or an iridescent cloud. The Wave Theory of Light Proposed by Huygens, this theory describes light as waves that spread out from the source that generates the light. Each color is a different wavelength. Supernumerary bows are explained by assuming that light is a wave. 3 PINHOLE CAMERAS http://www.pinhole.org//make/build.cfm a basic camera I usually build my cameras out of black foam-core board. I use white carpenters glue and black masking tape to put it all together… 4 RAY THEORY OF IMAGING da Image Object do Magnification m d i d o Image Resolution Angular Resolution Object Resolution i d If do di a do di di d o then d a i d i d a d i o d o d a d o di d a m 5 COHERENT LIGHT WAVES Electric Fields (single component) Object eal ( Eoo ( x) e Radiated eal ( Er ( y)e Er ( y ) Object Wavenumber e i t ik|| x y|| k 2 / i t ) ) Eoo ( x) || x y || dx 2 Angular Frequency 6 FOURIER TRANSFORM Eo ( x) proj ( Eoo )( x) Er ( y) Pythagorean Theorem & Approximation 1 1 / 2 imply that the radiated field in a plane far away is approx. = Fourier transform of the field in a plane in front of the object (planes orthogonal to line of sight) Er ( y) e ik y y /(2 d ) OP Eo ( x)e ik x x /(2 d ) e 2ixy /(d ) dx ~ˆ ~ e Eo ( y /(d )) where Eˆ o = Fourier Trans. ~ of E ( x) E ( x)eik x x /(2 d ) and is a constant. o o 7 ik y y /(2 d ) PINHOLE CAMERAS If the pinhole aperture is sufficiently small the quadratic phase term in y can be ignored and the aperture field ~ˆ Ea ( y) Eo ( y /(do )) The same reasoning implies that the image field Ei ( z) Eˆ a ( z /(di )) ˆ The Fourier inversion formula hˆ( x) h( x) ~ implies that E ( x) E (mx) i o where the approximation from the truncation of Ea due to the finite aperture size. Note the inversion. 8 LENS-BASED CAMERAS Lens-based cameras use the fact that the difference index of refraction between glass and air/space permits the removal of the aperture quadratic phase term, this decreases (ie improves) angular resolution since / da A typical space based telescope with a 1-meter diameter at the near infrared wavelenth of 1-micron is one microradian. This is costly since the tolerance of the lens is about 0.1-micron. The cost is in the tens of millions of dollars and increases nonlinearly with d a Imagine a budget of tens of billions of dollars - more than the annual R&D budget of Singapore. And imagine free time on hundreds of supercomputers and dozens of programmers. Now design a phantasie Kamera for Dr Strangelove ! 9 HOLOGRAPHIC IMAGING Fact 1. Need to illuminate object with coherent light and capture the radiated field across a large aperture. Fact 2. You lose all phase – can only use intensity = squared magnitude of the aperture field. Fact 3. The Fourier transform of the intensity yields the cross-correlation of the object field with itself. The problem is to do a blind deconvolution where the blur function is the flip = conjugated rotated 180 deg of the object field. Fact 4. Conventional holography does this BUT it uses a reference beam that effectively chooses an object field that is very special. Fact 5. The problem is provably ambiguous because an object field and its flip always have the same aperture field intensities. 10 PHASE RETRIEVAL Fact 1. The Fourier transform of the object field can be extended to five an analytic function of two complex variables F(z,w). Fact 2. The magnitude of F(z,w) for z and w real valued gives the function G(z,w) = F(z,w) F*(z,w) where F* is the Fourier transform of the flip object. Fact 3. Either F or F* can be computed from the magnitude of F if and only if F is irreducible – that means that it can not be factored. Fact 4. Most F are irreducible. Fact 5. Computing F or F* from |F| is HARD Fact 6. 1978 I solve for 10 x 10 array Fact 7. 1979 Russian publish solution for 2 x 2 array Fact 8. 1979 I am allowed to publish results 11 REFERENCE Hurt, Norman E., Phase Retrieval and Zero Crossings - Mathematical Methods in Image Reconstruction, Kluwer, Boston, 1989. Cites and explains most of my papers and the Russians Bruck and Sodin’s 1979 paper. The author was my officemate at MRJ, Inc. 12