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Lecture 13: Basic Concepts of Wavefront
Reconstruction
Astro 289
Claire MaxFebruary 25, 2016
Based on slides by Marcos van Dam and Lisa Poyneer
CfAO Summer School
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• System matrix, H: from actuators to centroids
• Reconstructor, R: from centroids to actuators
• Hardware
Outline
Wavefront sensor Wavefront (phase)
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• System matrix describes how a signal applied to the actuators, a, affects the WFS centroids, s.
• Can be calculated theoretically or, preferably, measured experimentally
System matrix generation
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• Poke one actuator at a time in the positive and negative directions and record the WFS centroids
• Set WFS centroid values from subapertures far away from the actuators to 0
Experimental system matrix
System matrix
Noise
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• We have the system matrix• We need a reconstructor matrix to
convert from centroids to actuator voltages
Inverting the system matrix
Least-squares reconstructor
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• Least squares reconstructor is • Minimizes• But is not invertible because some
modes are invisible!• Two invisible modes are piston and waffle
Least-squares reconstructor
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• The SVD reconstructor is found by rejecting small singular values of H.
• Write
• The pseudo inverse is
Singular value decomposition (SVD)
are the eigenvalues of
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• The pseudo inverse is
• Replace all the with 0 for small values of
Singular value decomposition
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• Example: Keck Observatory
Singular value decomposition
Set to zero
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• Suppose we only have centroid noise in the system with variance
• Variance of actuator commands is:
Noise propagation
0
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• For well-conditioned H matrices, we can penalize piston, p, and waffle, w:
• Minimizes
Least-squares reconstructor
Choose the actuator voltages that best cancel the measured centroids
Invertible
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• For well-conditioned H matrices, we can penalize piston, p, and waffle, w:
• Minimizes
Least-squares reconstructor
Choose the actuator voltages such that there is no piston
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• For well-conditioned H matrices, we can penalize piston, p, and waffle, w:
• Minimizes
Least-squares reconstructor
Choose the actuator voltages such that there is no waffle
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• Penalize waffle in the inversion: 1. Inverse covariance matrix of Kolmogorov
turbulence or2. Waffle penalization matrix
Least-squares reconstructor
1 2
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Slaved actuators
• Some actuators are located outside the pupil and do not directly affect the wavefront
• They are often “slaved” to the average value of its neighbors
Slaved to average value of its neighbors
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Modal reconstructors
• Can choose to only reconstruct certain modes
• Avoids reconstructing unwanted modes (e.g., waffle)
Zernike modes
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Modal reconstructors
Zernike modes
Centroids measured by applying Zernike modes to the DM
Zernike reconstructor
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Hardware approaches
• Systems until recently could use fast CPUs• For advanced AO systems, need to use more
processors and be able to split the problem into parallel blocks
• GPU – Graphics Processing Unit• DSP – Digital Signal Processor • FPGA – Field Programmable Gate Array (lots
and lots of logic gates)