Comments on AO Tomography Algorithms

1
Comments on AO Tomography Algorithms

• Can prove that all least squares and minimum
  variance reconstructor algorithms use inverse
  tomography back-projections
• In the zero-measurement noise case
• Slope-to-phase, phase-to-volume, volume-to-DM
  steps are all separable for design, analysis, and
  optimization
• A wide class of algorithms is proven closed-loop
  stable
• The algorithms massively parallelize on every
  ray!

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Stable reconstructor algorithms
Measurement
x volume of delta-OPD y GS phase at WFS A
forward projection AT back-projection
underdetermined
Reconstructor
P, S are any positive-definite matrices S
pre-filter (or pre-conditioner) determines
convergence rate P post-filter determines
solution type PI least squares Pltx0x0Tgt min
variance
3
This is a major insight into AO tomography
architecture

• Back projection is an integral feature
• Meaning of the Filters
• S filters data
• P filters volume estimates
• Can deal with missing modes (piston, tip, tilt,
  focus)
• Can deal with auxilliary data (tip/tilt/focus
  stars)
• It just back-projects those modes back towards
  the auxilliary guidestars
• Hartmann sensors

overdetermined
Chain rule
Always retaining Min.Variance and Iterative and
Closed Loop Stability
4
But with measurement noise,it has warts
Unfortunately, for the non-zero measurement noise
case, the conditional mean of the phase, y, given
phase gradients, s, is not independent of the
statistics of the delta optical paths in the
volume as it is for the zero measurement noise
case. This forces any iterative algorithm to
simultaneously solve for y and x. A convergent
algorithm for y is followed by which
can be implemented by a series of divergence,
back projection, forward projection, gradient,
and pre and post filtering operations.
Ggradient GTdivergence Aforward
projection ATback projection