Phase Retrieval of Scattered Fields

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Phase Retrieval of Scattered Fields
Greg Hislop and Andrew Hellicar CSIRO, ICT
Centre, Sydney greg.hislop_at_csiro.au
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Introduction
As the frequency of electromagnetic radiation
increases so does the difficulty and financial
cost of measuring phase information. CSIROs
Wireless Laboratorys Imaging Project is
developing an all-electronic terahertz imaging
system. Measuring phase with such a system would
prove extremely difficult so an investigation has
been done into the possibility of reconstructing
phase from amplitude only measurements. This work
considers the retrieval of phase information from
unknown scatterers placed in a known plane wave
field. To retrieve the phase, the fields
amplitude is measured across two parallel planes
and signal processing is applied to reconstruct
the phase. Very little literature exists on this
problem so two techniques were taken from the
related field of phase retrieval for antenna
characterisation and adjusted for the problem at
hand. These two techniques are the well
established method of successive projections and
a more recent conjugate gradient approach.
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Measurement Setup
The scenario of interest consists of a known
field incident upon an unknown scatterer centered
in a hypothetical scatterers plane. The
scatterer may be compact, as depicted
(constrained), or may completely obscure the
incident field (unconstrained). The transmitted
amplitude is measured across two parallel planes
allowing our algorithms to reconstruct the phase.

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Successive Projections

• This is an iterative technique which
  creates and updates an estimate of the amplitude
  and phase of the field for constrained scatterers
  as follows
• Create an estimate of the field and set the
  estimates phase to that which would have been
  measured at the first plane had no scatterer been
  present (ie phase of the incident field).
• Set the estimates amplitude to that measured on
  the first plane.
• Back propagate to the scatterers plane and set
  the estimates field terms, outside the
  scatterers physical extent, equal to the
  incident field.
• Propagate to the second measurement plane and
  change the estimates amplitude to the measured
  amplitude.
• Back propagate to the scatterers plane as per
  point 3 and again change the estimates terms
  outside the scatterers physical extent to the
  incident field.
• Propagate to the first measurement plane and
  repeat steps 2-6 until a suitable cost function
  stabilises.

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Conjugate Gradients

• The method of conjugate gradients operates as
  follows for constrained scatterers
• Start with an initial estimate of the field at
  the scatterers plane (note the field outside the
  scatterers physical extent does not change and
  is set to the incident field).
• Propagate this field to the two measurement
  planes.
• Evaluate a quadratic cost function (and its
  gradient) relating the estimate's power to the
  measured power.
• Use the gradient and the previous search
  direction to determine via the method of
  conjugate gradients a new direction in which to
  step the field estimate.
• Solve for the optimum step distance in the given
  direction by algebraically minimising the cost
  function.
• Update the estimate of the field at the
  scatterers plane using the distance and
  direction determined.
• Continue steps 2-7 until the cost function
  stabilises.

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Unconstrained Scatterers
For unconstrained scatterers (scatterers larger
then the incident fields extent), no restraint
is available at the scatterers plane. This
greatly increases the number of local minima
making the correct solution hard to find. To
cater for unconstrained scatterers our techniques
initially include only small spatial frequency
terms and then progressively include the higher
terms. This allows for false minima avoidance by
increasing the ratio of data to unknowns.
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Testing

• Two test scatterers (described below) were used
  in synthetic experiments, one in a constrained
  scenario (large measurement plane) and the other
  unconstrained (measurement plane same size as
  target).

Unbracketed parameters used in constrained case.
Bracketed parameters are changes made for
unconstrained case.
1?
3?
2?
3?
2?
2?
3?
3?
s 0 S/m er 7 (7.5) µr 1
s 0 (50) S/m er 3 (4) µr 1 (1.5)
s 0 S/m er 5 µr 1
s 0 (50) S/m er 3 (4) µr 1 (1.5)
s 0 S/m er 9 (2) µr 1
s 0 S/m er 7 (7.5) µr 1
s 0 S/m er 5 µr 1
8?
18?
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Example Reconstructions for the Constrained
Scatterer across a Range of Errors
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Histograms of Average Phase Error at Different
SNR for the Constrained Scatterer
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Example Reconstructions for the Unconstrained
Scatterer across a Range of Errors
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Histograms of Average Phase Error at Different
SNR for the Unconstrained Scatterer
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Comparison Between the Two Techniques

• The successive projections performs slightly
  better then the conjugate gradients technique.
• Successive Projections is simpler to implement,
  faster to run and the input parameters are more
  logical.
• Unlike the successive projections cost function,
  that of the conjugate gradients method is
  guaranteed to decrease monotonically.

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Conclusions

• By using two parallel measurement planes, the
  phase of a scattered field may be reconstructed
  using numerical techniques.
• The successive projections technique slightly out
  performs the conjugate gradients technique.
• Phase reconstruction is possible for constrained
  scatterers at significant noise levels.
• For unconstrained scatterers reconstruction is
  possible at more moderate noise levels.