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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
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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.
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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.
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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…
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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
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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

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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
2ixy /(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
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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.
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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 !
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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.
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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
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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.
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