Astronomy, Cosmology and Control Systems: Astronomy, Cosmology and Control Systems: Instrumentation for Understanding the Instrumentation for Understanding the

Universe through Adaptive OpticsUniverse through Adaptive Optics

By Donald M. Wiberg,Emeritus Professor of Electrical Engineering,

University of California, Santa Cruz,Researcher, University of California Lick Observatories,

Center for Adaptive Optics and Gordon and Betty Moore Laboratory for Adaptive Optics

30 meter telescope

Outline of the TalkOutline of the Talk• Background: Dark matter & energy• Why new instruments? >95% of the universe consists

of stuff we do not experience on earth• Large telescopes: 30 Meter Telescope = TMT• Explanation of adaptive optics: MCAO & MOAO• Exact stochastic optimal AO: A Hilbert space

approach, Kalman filtering, & why this is not needed• Predictive AO & extensions to tomography

30 meter telescope

The Hubble Space TelescopeThe Hubble Space Telescope

Hubble Floats Free Credit: STS-82 Crew, STScI, NASA

Hubble Ultra Deep FieldHubble Ultra Deep Field

Credit: S. Beckwith & the HUDF Working Group (STScI), HST, ESA, NASA

GalaxiesGalaxies

Milky Way Galaxy Looks Like AndromedaMilky Way Galaxy Looks Like Andromeda

Dark MatterDark Matter• About 200±40 billion stars in the Milky Way• (Many more galaxies in the universe than this)• Milky Way rotates as if its mass is about 200 billion

solar masses, except outer stars move faster• But observations of how galaxies and clusters of

galaxies move indicate that there is 10 times as much mass between them.

• Remaining mass must not be detectible by any kind of telescope, i. e. not luminous

• This is called dark matter

Not enough dust to be dark matter Not enough dust to be dark matter

M16: Pillars of Creation Credit: J. Hester, P. Scowen (ASU), HST, NASA

Could dark matter be all Black Holes?Could dark matter be all Black Holes?• Fine tuning the Keck 10 meter telescope’s adaptive optics recently

(2005) permitted Prof. Andrea Ghez of UCLA to verify that a supermassive black hole of 3.7 ± 0.2 million solar masses is at the center of the Milky Way

• There are many black holes of less than 10 solar masses scattered through the Milky Way, but none much bigger than 10 solar masses

• Normal galaxies all seem to have supermassive black holes at their center

• The mass of these supermassive black holes correlates well with ½of one percent of the mass of all material in any galaxy

• Not enough mass in all black holes to account for dark matter• We do not know what dark matter is now.• Could be new type of particle not observed on earth

Milky Way Galactic CenterMilky Way Galactic Center

Credit: Andrea Ghez, UCLAThe orbits of stars within the central 1.0 X 1.0 arcseconds of our Galaxy. In the background, the central portion of a diffraction-limited image taken in 2004 is displayed. While every star in this image has been seen to move over the past 9 years, estimates of orbital parameters are only possible for the seven stars that have had significant curvature detected. The annual average positions for these seven stars are plotted as colored dots, which have increasing color saturation with time. Also plotted are the best fitting simultaneous orbital solutions. These orbits provide the best evidence yet for a supermassive black hole, which has a mass of 3.7 million times the mass of the Sun

Abell 2218: A Galaxy Cluster Lens

Explanation: Gravity can bend light, allowing huge clusters of galaxies to act as telescopes. Almost all of the bright objects in this released Hubble Space Telescope image are galaxies in the cluster known as Abell 2218. The cluster is so massive and so compact that its gravity bends and focuses the light from galaxies that lie behind it. As a result, multiple images of these background galaxies are distorted into long faint arcs - a simple lensing effect analogous to viewing distant street lamps through a glass of wine. The cluster of galaxies Abell2218 is itself about three billion light-years away in the northern constellation Draco. The power of this massive cluster telescope has recently allowed astronomers to detect a galaxy at redshift 5.58, the most distant galaxyyet measured. This young, still-maturing galaxy is faintly visible to the lower right of the cluster core.

Credit: Andrew Fruchter (STScI) et al., WFPC2, HST, NASA

Gravitational lensing:Galaxy cluster Abell 2218bends light from distant galaxies behind Abell2218 to form arcs whose radius is proportional to the mass of Abell 2218.

Abell 2218

Dark Matter Dark Matter MapMap

Credit: J.-P. Kneib (Observatoire Midi-Pyrenees, Caltech) et al., ESA, NASAExplanation: The total mass within giant galaxy cluster CL0025+1654, about 4.5 billion light-years away, produces a cosmic gravitational lens -- bending light as predicted by Einstein's theory of gravity and forming detectable images of even more distant background galaxies. Of course, the total cluster mass is the sum of the galaxies themselves, seen as ordinary luminous matter, plus the cluster's invisible dark matter whose natureremains unknown. But by analyzing the distribution of luminous matter and the properties of the gravitational lensing due to total cluster mass, researchers have solved the problem of tracing the dark matter layout. Their resulting map shows the otherwise invisible dark matter in blue, and the positions of the cluster galaxies in yellow. The work, based on extensive Hubble Space Telescope observations, reveals that the cluster's dark matter is not evenly distributed but follows the clumps of luminous matter closely

Dark matter between galaxies that is detectible by the galaxies’ motion can be mapped using gravitational lensing

Blue shading in picture is computer generated supposition about where dark matter is

Finding Dark Matter Finding Dark Matter using a computer using a computer

generated simulationgenerated simulationdark matter in gray and galaxies ascolored circles.

Dark matter was born with the big bang, not created later.

Credit & Copyright: A. J. Benson (Caltech) et al., U. Durham, PPARCExplanation: Where is dark matter? Galaxies rotate and move in clusters as if a tremendous amount of unseen matter is present. But does dark matter exist in the greater universe too -- and if so, where? The answer can be found by comparing the distribution of galaxies observed with numerical simulations. This comparison became much more accurate recently when over 100,000 galaxy observations from the 2-Degree Field Galactic RedshiftSurvey were used. In the above frame from a computer simulation of our universe, a 300 million light-year slice shows dark matter in gray and galaxies as colored circles. The red box indicates the location of a rich cluster of galaxies, while the green box shows a more typical cross-section of our universe. Analyses indicate that the immense gravity of the pervasive dark matter pulls normal matter to it, so that light matter and dark matter actually cluster together.

Conclusions about dark matterConclusions about dark matter• Dark matter is not luminous or reflective.• Dark matter is more than 80% of the mass in the universe• Dark matter was created with the big bang• Dark matter is spread like foam, as if an airy sponge with

huge voids in space, joining galactic clusters• MACHOs, Massive Compact Halo Objects, such as black

holes, neutron stars, dwarfs stars, etc., that are normal baryonic material emitting almost no radiation, have been ruled out in the range 0.00000001 to 100 solar masses

• WIMPs, Weakly Interacting Massive Particles, are possible.• WIMPs are hypothetical, not baryonic, have huge mass, are

predicted by supersymmetry theory, but have not been observed experimentally, and do not come from the standard model of particle physics (neutralinos?)

• We really do not know what dark matter is.

Dark EnergyDark Energy

• Big bang creation of the universe • What happens at the end of the universe?• Will there be a big crunch or merely a

slower expansion due to gravity?

Dark EnergyDark Energy

• Big bang creation of the universe • What happens at the end of the universe?• Will there be a big crunch or merely a slower

expansion due to gravity?• Answer: Neither!• Galactic clusters are accelerating away from

each other!• Acceleration implies a force due to dark energy

counteracting the gravity force that pulls galactic clusters towards one another

Dark Energy, continuedDark Energy, continued• Initial findings of galactic acceleration by Riess, et al. 1998 and

Perlmutter, et al. 1999 who made observations at high red shift that were impossible to make previously

• Since confirmed by other observations including the 2005 Supernova Legacy Survey

• Dark energy overcame the decelerating effects of gravity some 5 – 7 billion years ago, about half the age of the universe

• Dark energy is a weak repulsive force spread uniformly in space• Dark energy could be explained mathematically by:• 1. Einstein’s cosmological constant: the vacuum energy (negative)

2. A vacuum field called “quintessance”: perfect gas law3. An exception to general relativity: Einstein must be revised

confirmationconfirmation

• The recent WMAP survey confirmed that all known matter + dark matter + dark energy = just about correct to account for the measured curvature of space (“flat”)

• WMAP is a map of the background radiation remaining from the big bang, black body radiation cooled to about 3.7 degrees Kelvin

WMAP anisotropy surveyWMAP anisotropy survey

Credit: WMAP Science Team, NASAExplanation: Analyses of a new high-resolution map of microwave light emitted only 380,000 years after the Big Bang appear to define our universe more precisely than ever before. The eagerly awaited results announced last year from the orbiting Wilkinson Microwave Anisotropy Probe resolve several long-standing disagreements in cosmology rooted in less precise data. Specifically, present analysesof above WMAP all-sky image indicate that the universe is 13.7 billion years old (accurate to 1 percent), composed of 73 percent dark energy, 23 percent cold dark matter, and only 4 percent atoms, is currently expanding at the rate of 71 km/sec/Mpc (accurate to 5 percent), underwent episodes of rapid expansion called inflation, and will expand forever. Astronomers will likely research the foundations and implications of these results for years to come.

Implications of dark matter & energyImplications of dark matter & energy

• I believe discovery of dark matter and energy is comparable to the discovery of Jupiter’s moons by Galileo and to the discovery of radium by the Curies.

• Galileo’s and the Curies’ discoveries were made possible by instrumentation: the telescope and the photographic plate

• To find out about dark matter and energy, we need better instrumentation

Astronomical observatories with Astronomical observatories with AO on 3AO on 3--5 m telescopes5 m telescopes

• ESO 3.6 m telescope, Chile

• University of Hawaii

• Canada France Hawaii

• Mt. Wilson, CA

• Lick Observatory, CA

• Calar Alto, Spain

• Nordic Optical Telescope (2.5m), Canaries

The new generation: The new generation: adaptive optics on 8adaptive optics on 8--10 m 10 m

telescopestelescopesSummit of Mauna Kea volcano in Hawaii:

Subaru (8.2m)

2 Kecks (10m)

Gemini North(8m)

others: Mt. Palomar (6.5m), MMT (6.5m), 4 VLTs (8m), 2 LBTs (8.4m), Gemini South (8m)

Telescopes in SpaceTelescopes in Space• 1990: Hubble – visible light, 2.6 m diameter• 1991-2000: Compton – gamma ray, ~1 m diam• 1996: Solar and Heliospheric – study sun• 1999: Chandra X-Ray• 2000: ESO XMM-Newton – X rays and optical• 2001: Microwave Anisotropy Probe (WMAP)• 2003: Spitzer – infra red, 0.85 m dia, solar orbit• 2003: Galaxy Evolution Explorer (GALEX)

Ultraviolet light, 0.5 m diameter• 2008: Herschel – far infrared, 3.5 m diam• 2009: Kepler – search for earth sized planets, 1.4 m• 2013: James Webb – infrared, 6.5 meter diameter

30 meter telescope30 meter telescope

Cut away drawing of TMT showing the primary mirror with hexagonal segments, secondary mirror and tertiary mirror

TMT cut away with elevation and azimuth axis gimbals

16 m

GROUND LEVEL

23 m

EL AXIS

46.59 m

M2 VERTEX

50.56 m

17.5 R 27.6 R

1.8 m

Boeing 747

Hale (Palomar)5m (1948)

TMT 30m (2016?)

Evolution of ScaleEvolution of ScaleKeck TMT

Segments 36 492

Actuators 108 1476

Sensors 168 2772

Keck 10m (1992)

Image credit Doug MacMynowski

Possible Possible StrehlStrehl improvement with TMTimprovement with TMT

Primary mirror M1 ControlPrimary mirror M1 Control

Two steps:• Estimation of segment

position from edge sensors

• Use pseudo-inverse because A is rectangular matrix

• SISO control of segment motion using estimated position

Control:Mirror cell introduces structural coupling that limits bandwidth

yAx 1ˆ −≈

TMT Control LoopsTMT Control Loops

M1: 492 segments1476 actuators2772 sensors

Compensate forgravity, thermal, wind;~10 nm rms residual1-2 Hz bandwidth

Mount control Pointing, guiding(El, Az)

M1 actuatorservo loops

AO (NFIRAOS):7500+ actuatorsCompensate for atmosphericturbulence, ~kHz sample rates

M2 5dof position+ figure

Image credit: Doug MacMynowski

Instruments on TMTInstruments on TMT• NFIRAOS: Narrow field infrared adaptive optics system, multi-

conjugate AO (MCAO) with conjugates at 0 and 12 km• IRIS: Infrared imaging spectrometer (with AO)• IRMS: Infrared multi-slit spectrometer (with AO)• WFOS: Wide field optical spectrometer (no AO)• MIRES: Mid-infrared Eschelle spectrometer (with special AO

optimized for infrared - MIRAO)• NIRES: Mid-infrared Eschelle spectrometer (with AO)• IRMOS: Infrared multi-object spectrometer (with multi-object

adaptive optics - MOAO) 10 -20 objects of 1 - 5 arc sec distributed through a 5 arc min field

• HROS: High spatial resolution optical spectrometer (no AO)• PFI: Planet formation instrument (EXAO) special high

resolution AO with coronagraph

Motivation for Adaptive Optics (AO)Motivation for Adaptive Optics (AO)Turbulence in the Earth’s atmosphere distorts light from distant objects, limiting the resolution and contrast that can be achieved by ground-based telescopes.

Flat wavefronts come from point light sources located far away, and the atmosphere causes them to become aberrated irregularly.

Adaptive optics attempts to correct for these optical aberrations by actively sensing the localized slopes of the incoming wavefront and using a deformable mirror in order to “flatten” the wavefront before it is sensed by the science camera. Image credit: Claire E. Max, UCSC

If thereIf there’’s no closes no close--by by ““realreal”” star, create star, create one with a laserone with a laser

• Bright enough “real” stars cover only 4% of the sky.

• Use a laser beam to create artificial “star” at altitude of 100 km in atmosphere, where 5 -10 km layer of sodium exists.

• 20 watt laser excites ~0.2 watts of light, enough for AO wavefront sensor.

Laser is operating at Lick Laser is operating at Lick Observatory and at KeckObservatory and at Keck

Lick Observatory

Keck Observatory

Adaptive optics on 8m ground based Adaptive optics on 8m ground based telescope can give better image than Hubbletelescope can give better image than Hubble

(depends on exposure time)

NAOS AO on VLTAn 8 meter diameter λ = 2.3 microns

Hubble Space TelescopeAn 8 foot diameter λ = 800 nm

Neptune in infraNeptune in infra--red light (1.65 red light (1.65 microns)microns)

With Keck adaptive opticsWithout adaptive optics

2.3

arc

sec

May 24, 1999 June 27, 1999

Adaptive optics at UCSCAdaptive optics at UCSC• Center for Adaptive Optics (CfAO)

– AO Building at UCSC is CfAO headquarters– NSF Science and Technology Center (10 yrs, $40M)– AO for astronomy and for looking into the living human eye– 12 U. S. universities are members, as well as JPL and

LLNL national laboratories• Laboratory for Adaptive Optics

– Funded 2005 by the Gordon and Betty Moore Foundation– 6 years, $9M– Two labs in Thiemann Building at UCSC– Experiments on “Extreme AO” to search for planets, and on

AO for Extremely Large Telescopes

Typical AO SystemTypical AO System

f

Image credit: Jason Porter, University of Rochester

wavefrontsensing(WFS)

wavefront correction(DM)

laserbeacon

imaging

LGS

control system

Pupil plane Image plane

Image credit: Claire E. Max, UCSC

Feedback controller calculates next wavefront correction based

on the error signal from the current correction

Shack-Hartmannwavefront sensor

Deformable mirrors come in many sizesDeformable mirrors come in many sizesRange from 13 to > 900 actuators (degrees of freedom)

XineticsA couple of inches

About 12”

Piezoelectric deformable mirrors

MEMS is the next generation of deformable mirrors

AO Control SystemAO Control SystemIncoming wavefront

Residual error

),( yxr Tr K Λr

-

Gain MatrixWavefront estimate Spatial weight vector

),,(0 tyxφ ),,(ˆ),,(),,( 0 tyxtyxtyx φφ −=e

)(),(),,(ˆ

tcyxrtyx

T rr=φ

Integrator

delay

Sensor reading

)1( +tcr)(tcr )(tsr

)()(),,(),()(

)(

)()(

)(

)()(

)()()1(

11

tetsdxdxytyxeyxwts

signalsensorts

tstssignalcontrol

tc

tctc

tsKtctc

ii

mn

Λ==

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=

+=+

∫rr

Mr

rM

r

r

rvr

+

Algorithm inside computer

AO control system algorithmsAO control system algorithms

• The computer uses an integration algorithm as diagrammed on the previous slide.

• Algorithm input is measurement m-vector s(t)• Algorithm output is command n-vector c(t)• Poke matrix P is measured by commanding a unit n-

vector that determines a column of P by the resulting measured m-vector.

• Usual AO algorithm is the pseudo-inverse K ~ P¯ , except modified somewhat for practical reasons.

• Optimal algorithms seek to maximize the average Strehlby a different choice of K.

¹

Optimal AO AlgorithmsOptimal AO Algorithms• Wallner (1983) proposed choosing K to minimize the squared

error variance over the aperture.• Wallner could only consider open loop.• Gardner et al. (1990), Ellerbroek (1994), Wild (1996), Le Roux

et al. (2002) minimized closed loop variance in different ways assuming Komolgorov turbulence

• Wiberg (2004 CDC) proved an exact mathematical closed loop minimum solution on the Hilbert space of stochastic piston and tip/tilt removed ₤2 functions over the aperture assuming no wind, but any turbulence, not necessarily Komolgorov

• This solution had an extra computational step called a “Kalman filter”, that reduces to the Wallner solution in very low noise

ConclusionsConclusions• The null-seeker (pseudo-inverse), the Wallner (open Loop),

and the exact stochastic optimal algorithm (Wiberg) control all have about the same performance in practice, due to the low measurement noise of the CCD sensors as verified by simulation and by use on 5 experiments on the Lick AO

• Therefore, for adaptive optics on large telescopes with nowind, there is no point to using a Kalman filter approach.

• But, with wind, there is a definite improvement in Strehl to move the image to a predicted position after one control cycle. (Gavel, D. T., & Wiberg, D. M. 2003, Proc. SPIE 4839, pp. 890-901)

• This is called “predictive control”• Before presenting predictive control, there are 2 major

error sources in adaptive optics to consider

Atmospheric Cone EffectAtmospheric Cone Effect(not taking all the atmosphere into account)(not taking all the atmosphere into account)

• With only one laser guide star and one WFS, not all of the atmosphere can be sensed

Image credit: Don Gavel, LAO

IsoplanaticIsoplanatic angleangle(Looking at the science object, not the guide star)(Looking at the science object, not the guide star)

• Since light from off-center objects takes a different optical path than the guide star light, the un-sensed turbulence limits the effectiveness of the AO correction

θ0h

r0

Image credit: Don Gavel, LAO

• Multi-Conjugate AO– Multiple DM placed

conjugate to different layers in the atmosphere

• Multi-Object AO– Each object in the

field of view gets its own DM

AO systems are growing in complexity, AO systems are growing in complexity, size, demands on performancesize, demands on performance

–MCAO• 2-3 conjugate DMs• 5-7 LGS• 3 TTS

–MOAO• Up to 20 IFUs each with a DM• 8-9 LGS• 3-5 TTS

Atmospheric TomographyAtmospheric Tomography

• Break up atmosphere above telescope into 3D voxels.

• Use tomographicreconstruction methods to estimate 3D structure of atmosphere

Image credit: Marc Reinig, LAO

• Each wavefront sensor measures the integral of index variation along the ray lines• The line integral along z determines the kz=0 Fourier spatial frequency component• Projections at several angles sample the kx,ky,kz volume

kZ

( ) ( )∫∞

−∆=0

, dzzzxnx θφ ( ) ( )θxxx kkNk −∆=Φ ,

Fourier slice theorem in tomography(Kak, Computer Aided Tomography, 1988)

How tomography worksHow tomography works

zkk z

x ∆=

∆<

θθ1

kX

Motivation for Predictive ControlMotivation for Predictive ControlTemporal Error – caused by time delay needed to compute and apply DM

correction 35

0

2 4.28 ⎟⎟⎠

⎞⎜⎜⎝

⎛=

ττσ td

v0

0r

=τ = time it takes the wind to blow frozen turbulence over a subaperture of size r0

Typical values can range from 10 – 1000 msec.

9.0Strehl035.00

≤⇒≈ττ

Currently, the only way to reduce temporal error is by increasing the speed of the controller. This is costly and increases errors due to photon noise.

Predictive control attempts to reduce temporal error by incorporPredictive control attempts to reduce temporal error by incorporating knowledge of atmospheric ating knowledge of atmospheric dynamics into the control loop.dynamics into the control loop.

Predictive Control SystemPredictive Control System-

Incoming wavefront Residual error

KP

)(tφ )(te

)1(ˆ −+ tφ

+

delay

Predictor

F

)(ˆ t+φ

Wavefront estimate

)(ˆ t−φ

Internal wind model

delaydelay

)1(ˆ)(ˆ)()(ˆ)(ˆ

),...,1,0(ˆ)(ˆ

)1,...,1,0(ˆ)(ˆ

−=

+=

=

−=

+−

−+

+

−

tt

ttt

tt

tt

φφ

φφ

φφ

φφ

F

PKe

e

e

Taylor frozen flowTaylor frozen flowAssumes that turbulence features do not change significantly betAssumes that turbulence features do not change significantly between time steps:ween time steps:

Turbulence is effectively a fixed sheet that is blown across theTurbulence is effectively a fixed sheet that is blown across the aperture aperture of the telescope with constant velocityof the telescope with constant velocity

Image credit: Tokovinin A., CTIO http://www.ctio.noao.edu/~atokovin/tutorial/

This means the evolution of the turbulence between time steps isThis means the evolution of the turbulence between time steps is simple simple translation along the vector translation along the vector vv

Characteristic evolution time for atmospheric boiling effects isCharacteristic evolution time for atmospheric boiling effects is generally generally on the order of >100 on the order of >100 msecmsec so temporal evolution at a rate >100 Hz is so temporal evolution at a rate >100 Hz is

primarily due to frozen flow wind layers. At these speeds, boilprimarily due to frozen flow wind layers. At these speeds, boiling effects ing effects can be neglected. can be neglected.

SingleSingle--Wind Predictive ControlWind Predictive Control

-

Incoming wavefront Residual error

KP

)(tφ )(te

)1(ˆ+ tφ

+

delay

Predictor

F

)(ˆ t+φ

Wavefront estimate

)(ˆ t−φ

Predictor uses bilinear interpolation to shift by)1(ˆ −+ tφ dtvRegularized least-squares reconstructor

Internal wind model

Time delay

Problem: some values along the edge of the aperture will be unknownSolution: replace unknown values with prior values

dtv

'AAAB ∩−=

)1,(ˆ),(ˆ −= +− tt BB φφ

A A’

Single wind estimationSingle wind estimation

)1,(),( −−= tt vxx φφ

If we assume that all phase aberrations are caused by a single wind layer in frozen flow conditions then the phase aberrations between two consecutive timesteps can be described by:

If we view the measurements from the WFS at each time as images with the intensity of each “pixel” equal to the phase aberration in the corresponding subaperture then this is a direct analogue to the Brightness Constancy Equation, commonly used in optical flow estimation. There is a wealth of techniques available in image processing to estimate the shift vector, v.

In order for our predictive controller to be successful, we need to have a reasonably accurate estimate of the wind velocity.

Image Motion EstimationImage Motion Estimation

• Fourier Domain– Utilizes shift property of

Fourier Transform– 3D Fourier Transform can

detect multiple layers simultaneously

– Must compute FFT at each timestep

– Edge effects can be difficult to deal with

• Image Domain– Uses Brightness Constancy

Assumption to set up a cost function

– Gradient-based methods used to minimize cost function

– Sub-Optimal search algorithms can reduce computations

– Radon Transform can also be used to reduce computations

• Applications– Motion estimation– Image registration– Video compression

• Methods

• ReferencesAlliney (1986), Barron (1994), Bergen (1992), Brown (1992), Bruton (1985), Burt (1991), Cain (2001), Heeger (1987), Irani (1994), Liu (1996), Milanfar (1999), Robinson (2003), Stiller (1999), Tham (1998)

Wind Estimation MethodsWind Estimation Methods2

)1,(ˆ),(ˆ −−−= ++ ttJ vxx φφ

Let }{minarg* Jvv =

Gauss-Newton Method 2D Binary Search Radon Transform Method

- Uses gradient descent to estimate v*

- Very accurate

- Relatively slow

- Intelligently searches a subspace of possible values of v* and returns the best one.

- Slightly less accuratethan Gauss-Newton

- Faster than Gauss-Newton

- Reduces the 2D probleminto two 1D problems using the Radon Transform

- Least accurate method

- Fastest method

Comparison of EstimatorsComparison of EstimatorsSimulation settings:Aperture size: 10 m

r0: 0.2 mWFS: 41 x 41

td : 1 millisecondSNR: 30

wind velocity: (15,0) m/s

Results after 20 simulations:

method x Std dev y Std devGauss-Newton 14.22 1.790 -0.140 0.666

2D Binary Search

15.28 2.062 -0.106 0.897

Radon Tform 12.08 3.245 0.0414 1.574

High order wind estimationHigh order wind estimationSimulation settings:Aperture size: 10 m

r0: 0.2 mWFS: 60 x 60

td : 1 millisecondSNR: 30

wind velocity: (15,0) m/sWindow size: 15 x 15

Results after 20 simulations:

method x Std dev y Std dev2D

Binary Search

17.17 1.006 2.5e-4 .9668

Radon Tform 11.49 3.080 0.0795 1.520

Wind model refinementWind model refinementIncoming wavefront Residual error

),( txφ ),( txe

)1,(ˆ −+ txφ ),(ˆ tx+φ

Wavefront estimate

),(ˆ tx−φ

)1,(ˆ −+ td xφ)1*,ˆ(ˆ −−+ td vxφ

)1,(ˆ −+ ts xφ

Internal wind model

The wavefront splitter calculates the static wavefront by averaging a user-defined number of past wavefront estimates. The number of frames averaged is a important parameter that depends on the strength of the turbulence and the rate at which the wind and the quasi-static errors are changing.

Data from PALAOData from PALAOTwo open-loop WFS data sets from two different nights at Palomar Observatory

Wind estimation done with 2D binary search method

Single wind layer detected in both data sets

Agrees with wind measurements taken by L. Poyneer at the same location using different methods

When closed-loop DM is simulated, the predictive controller shows a 1.1% decrease in error residuals compared to the non-predictive case. This is consistent with simulations

Relative magnitude of dynamic layer ranges from 10-40% and wind speeds are somewhat low. Gains would be much higher if the dynamic layer was stronger or wind was faster.

Page 2645

Future workFuture work• Incorporate multiple winds into internal model

– Single conjugate AO• Predictive Fourier Control (Poyneer 2007) already exists

– Unable to measure wind at every time step– Only updates wind estimates every few minutes– Estimates wind differences at each frequency, not individual velocities

• Multi-layer methods for image domain exist, however they are expensive• Is there an efficient way to do this in the image domain using suboptimal parameter search

methods?• If not, is there a better Fourier Domain estimator (Milanfar 1999?) that can be used as a

stand-alone wind estimator for our suboptimal wind predictor?

– Atmospheric tomography• Multi-Conjugate and Multi-Object AO systems• Divides atmosphere into discrete layers so current model applies• Necessary component of future >30m telescopes due to cone effect• Easily parallelized – FPGA design completed for TMT• MCAO bench in LAO ready to test wind estimator/predictor

ReferencesReferences• Alliney, S. & Morandi, C. 1986, ITPAM, 8, pp. 222-223 • Barron, J. L., Fleet, D. J.,& Beauchemin, S. S. 1994, JCV, 12:1, pp. 43-77 • Bergen, J. R., Anandan, P., Hanna, K. J., & Hingorani, R. 1992, Proc. of the European Conf. on Computer Vision, pp. 237-252 • Bergen, J. R., Burt, P. J., Hingorani, R., & Peleg, S. 1992b, PAMI, 14, 9, pp. 886-895 • Brown, J. G. 1992, ACM Computing Surveys, 24, 4, pp. 325-376 • Bruton, L. T. & Bartley, N. R. 1985, TCAS, 32, 7 • Burt, P. J., Hingorant, R., & Kolczynski, R. J. 1991, Proc. of the IEEE Workshop on Visual Motion, pp.187-193 • Cain, S. C., Hayat, M. M., & Armstrong, E. E. 2001, ITIP, 10, 12, pp. 1860-1872 • Deans, S. R. 1993, The Radon Transform and Some of its Applications (Malabar: Krieger) • Gavel, D. T., & Wiberg, D. M. 2003, Proc. SPIE 4839, pp. 890-901 • Gavel, D. T. 2003, Proc. SPIE 4839, pp. 972-980 • Gavel, D. T. 2004, SPIE Astronomical Telescopes and Instrumentation, 5490, pp. • Heeger, D. J. 1987 J. Opt. Soc. Am. A, 4, 8 • Hinnen, K., & Verhaegen, M. 2007, J. Opt. Soc. Am. A, 24, 6 • Irani, M., Rousso, B., & Peleg, S. 1994, IJCV, 12, 5-16 • Jain, J. R. & Jain, A. K. 1981, T-COM, 29, 12, pp. 1799-1808 • Kim, J-S. & Park, R-K. 1992, IJSAC, 10, 5, pp. 968-971 • Le Roux, B., Conan, J-M., Kulc‘sar, C., Raynaud, H-F., Mugnier, L. M., & Fucsco, T. 2004, J. Opt. Soc. Am. A, 21, 7 • Liu, H., Hong, T-H., Herman, M., & Chellappa, R. 1996, CVIU, 72, 3, pp. 271-286 • Liu, Y. & Oraintara, S. 2004, ICASSP, pp. 341-344 • Milanfar, P. 1996, J. Opt. Soc. Am. A, 13, 11 • Milanfar, P. 1999, T-IP, 8, 3 • Milanfar, P. 1999, T-IP, 8, 9 • Park, S. & Schowengerdt, R. 1983, CVGIP, 23, 256 • Poyneer, L. A., Macintosh, B. A., & V‘eran, J.-P. 2007, J. Opt. Soc. Am. A, 24, 9 • Poyneer, L. A. 2008, to be submitted • Robinson, D. & Milanfar, P. 2003, JMIV, 18, pp. 35-54 • Rifman, S. S. & McKinnon, D. M. 1974, , Report 20634-6003-TU-00, TRW Systems • Stiller, C. & Konrad, J. 1999, ISPM, 16, pp. 70-91 • Tham, J. Y., Ranganath, S., Ranganath, M., & Kassim, A. A. 1998, T-CSVT, 8, 4 • Tu, C., Tran, T., Prince, J., & Topiwala, P. 2000, Proc. SPIE Applications of Digital Image Processing XXIII, pp.374-384 • Turaga, D. & Alkanhal, M. Mid-Term project • Wiberg, D. M., Johnson, L. C., & Gavel, D. T. 2006, Proc. SPIE, 6272 62722X

Thanks to:Thanks to:

• Everyone in the UCSC Center for Adaptive Optics and Lab for Adaptive Optics!

• Especially Don Gavel, Luke Johnson, Claire Max, Sandy Faber, Jerry Nelson, and Doug MacMynowski

• And thanks for the funding from the National Science Foundation and the Gordon and Betty Moore Foundation

Present Present ““nullnull--seekingseeking”” controllercontroller• Sensor number = m, actuator number = n• desired local wavefront slopes as seen by

sensors, an m-vector• sensor signals after deformable mirror, m-

vector• local wavefront slopes produced by

deformable mirror, m-vector• n-vector of actuator commands• Adaptive optics model: where P is mxn

poke matrix

=0φr

( ) =ter

( ) =tφr

( ) =tcr

( ) ( )tcPt rr=φ

∫ === njandmifordxdyyxryxwP jiij ,...,2,1,...,2,1),(),(

Inte

grat

ed e

rror

squa

red

Gain =0.3, leak =0.99 Gain =0.25, leak =0.7

Comparison of Kalman filter controller on right with Wallner instantaneous open loop optimal controller on left.

Thanks to Zuobing Xu for the simulation of Lick AO

Experimental resultsExperimental results

• Intensity Wallner Null-seeker Photons Strehl Strehl

• 100 0.2 0.14• 100 0.18 0.24• 50 0.12 0.30• 50 0.19 0.20• 25 0.05 0.07

Analysis of AO feedback with integratorAnalysis of AO feedback with integrator

r φ (t) = P

r c ( t)

r e ( t) =

r φ 0 −

r φ (t)

r c ( t) =

r c (t − T ) + K

r e ( t − T )

Math model:Error definition:Integrator:

c is n vector, P is mxn poke matrixe and φ are m vectors

K is nxm matrix

KintegratorP+_

r φ 0

r φ (t)

[ ] [ ])()()()()()( 0 tPKtteKtcPTtcPTt φφφφrrrrrrr

−+=+=+=+

For t = 0, T, 2T,…

Solution:( ) )0()( /)( φφφ

rrr TTto PKITt +−+=+

If all | (I-PK)| < 1, system is stable and (t) →r φ 0

If we set K= then goes to regardless of the value of or

r φ 0

r φ 0 GOOD AO SYSTEM!

r e (t)

r c (t)

λi r φ

P−1

I is unit matrix, λ is eigenvalue

in one time step, r φ (0).

r φ (t)

r φ (t + T) = (I − PK )

r φ (t) + PK

r φ 0

Check solution:

r φ (t) =

r φ o + I − PK( )t /T r

φ (0)0)()( φφrr

PKtPKI +−=

Explanation of poke matrix P, cont.

φ1

M

φm

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=p11 L p1n

M M

pm1 L pmn

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

c1

M

cn

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

r φ = P

r c

Command , measure the m local wavefrontslopes produced by the deformable mirror deflectinglaser light put into the telescope.

c1 =1, c2 = c3 =Lcn = 0

φ1,φ 2,L ,φ m

φ1

M

φm

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=p11 L p1n

M M

pm1 L pmn

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

1M

0

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=p11

M

pm1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Then command , etc., in turn to to get

c1 =0,c2 =1,c3 =L=cn =0 c1 = c2 =L = cn−1 = 0, cn =1

p12

M

pm 2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ,L,

p1n

M

pmn

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ to fill out the poke matrix P.

Other ideas on how to find K Other ideas on how to find K ““optimallyoptimally””(some use the Kalman filter)

[1] E. .P. [1] E. .P. WallnerWallner, , ““Optimal waveOptimal wave--front correction using slope measurementsfront correction using slope measurements””, J. , J. Opt. Soc. Am. 73,1771Opt. Soc. Am. 73,1771--76(1983) 76(1983)

[2] C. S. Gardner, B. M. Welsh , L.A. Thompson, [2] C. S. Gardner, B. M. Welsh , L.A. Thompson, ““ Design and performance analysis Design and performance analysis of adaptive optical telescopes using lasing guide stars.of adaptive optical telescopes using lasing guide stars.”” , Proc. IEEE 78, 1721, Proc. IEEE 78, 1721--1743 1743 (1990) (1990)

[3] W. J. Wild, [3] W. J. Wild, ““Predictive optimal estimation for adaptive optics systemsPredictive optimal estimation for adaptive optics systems””, Optics , Optics Letters 21, (18); 1433Letters 21, (18); 1433--1435 (1996) 1435 (1996)

[4] B. L. [4] B. L. EllerbroekEllerbroek, , ““First order performance evaluation of adaptive optics systems First order performance evaluation of adaptive optics systems for atmospheric turbulence compensation in extendedfor atmospheric turbulence compensation in extended--fieldfield--ofof--view astronomical view astronomical telescopestelescopes””, J. Optical Soc. Am. A11(2):783, J. Optical Soc. Am. A11(2):783--805(1994). 805(1994).

[5] B. Le Roux, J.M. Conan, C. [5] B. Le Roux, J.M. Conan, C. KulcsarKulcsar, H.F. Reynaud, L. M. , H.F. Reynaud, L. M. MugnierMugnier and T. Fusco, and T. Fusco, ““Optimal control law for Optimal control law for multiconjugatemulticonjugate adaptive opticsadaptive optics”” Proc. SPIE Conf. Telescopes Proc. SPIE Conf. Telescopes and Inst., Paper 4839and Inst., Paper 4839--106,33, Waikoloa Hawaii (2002). 106,33, Waikoloa Hawaii (2002).

““A Geometric View of Adaptive OpticsA Geometric View of Adaptive Optics””, D. M. Wiberg, C. E. Max, and D. T. Gavel, , D. M. Wiberg, C. E. Max, and D. T. Gavel, SPIE Astronomical Telescopes and Instrumentation Conf., June 21 SPIE Astronomical Telescopes and Instrumentation Conf., June 21 --25, 2004, 25, 2004, Glasgow, Scotland, paper 5490Glasgow, Scotland, paper 5490--9393

Stochastic optimal control: LQG

Atmospheric modelAtmospheric model( ) ( )tyxtyxtyx ,,),,(1,, ϕαϕϕ ∆+=+

Aperture average removed phase = ϕ(x,y,t)Spatial dimensions on aperture domain D = (x,y)Discrete time steps 0,1,2,… = tAtmospheric “boiling”constant = αTemporally white 0 mean random field =CovarianceUsually spatially Komolgorov distributed

∆2φ x,y,t( )

( )tyx ,,φ∆determined from data

Phenomenological model produces slowly changing phase

ControllabilityControllability

∫

++=

==

=

D

T

nn

n

Tn

dxdyyxryxr

nxnilitycontrollabDefinetuyxrtuyxrtuyx

yxryxryx

tututu

),(),( = R

by Rmatrix )(),(....)(),()(),(r

ismirror deformable by the induced change Phasefcn. influence spatialactuator )),( .... ),((),(r

signalsactuator =))( .... )(()(

11T

1T

1

Controllability cont.Controllability cont.

ϕ c(x, y, t) = r T(x,y) R−1 r( ′ x D∫ , ′ y , t)ϕ( ′ x , ′ y , t)d ′ x d ′ y

Controllable part of phase ϕ is called ϕ c where

Uncontrollable part of ϕ is

← K K c(t) K K →We call c(t) the “generator” of ϕ c

ϕ c = ϕ − ϕ c

ϕ c ∈ range space r T ,ϕ c ∈ null space r

at time t

r(x , y)ϕ c

D∫ (x , y, t)dxdy = 0

ϕ c ⊥ ϕ c ⇔ ϕ c(x, y)ϕ c

D∫ (x, y)dxdy = 0

Geometrically, is the projection of ϕ on to r(x,y).ϕ c

ObservabilityObservabilityMeasurements from m sensors at time t are

s(t) = (s1(t),s2(t ),K, sm (t))T

Define linear operator Λ, Λϕ = w(x, y)ϕ(x,y,t)dxdyD∫

where w(x,y) is m vector of sensor spatial influence functions.The sensor signal is modeled as

s(t) = Λ ϕ x, y,t( )− rT x,y( )u t( )[ ]+ v(t)where v(t) is zero mean,independent, white electronic and photonic noise m vector at time t, with covariance matrix V.The unobservable part of is defined byϕ o x, y,t( )

Λϕo = 0

ϕ x, y,t( )

Note this is an m vector equation

ObservabilityObservability cont.cont.Then the observable part of ϕ is ϕ o = ϕ − ϕ o

Fact: ϕ o⊥ ϕ o in that < ϕo ϕo > = 0ϕ c asDefine “generator” of observable part of

c(t) = (Λϕ c)( t)Physically, (t) is the m vector signal produced by the m sensors due to the uncontrollable part of ϕ in the absence of noise. The mxn poke matrix P = Λr is the signal observed by commanding the actuators one at a time.

T

cr

s t( ) = Λϕ( ) t( ) − Pu t( ) + v t( )

Exact finite dimensional modelExact finite dimensional model

To develop an exact finite dimensional model, recall the atmospheric model

ϕ x, y,t +1( )= αϕ(x,y,t) + ∆ϕ x, y,t( )

We use this as an example model, but it can be any set of linear difference equations in ϕ.

Finite model cont.Finite model cont.c(t)An equation like that of c(t) can be found similarly for

c t +1( ) = αc t( ) + ∆ t( )− P∆ t( )where ∆ t( ) = Λ∆ ϕ t( ) and ∆ t( ) = R −1 r x, y( )

D∫ ∆ ϕ x , y, t( )dxdy

( ) ( ) ( )tyxthroughcorrelatedspatiallyaretandtNote ,,ϕ∆∆∆Recall the sensor signal m vector s(t) obeys

s t( ) = Λ ϕ c + ϕ c( )t( ) − Pu t( ) + v t( )ϕ c x, y, t( ) = r T x, y( )c t( )and P = ΛrT soRecall

s t( ) = Pc t( ) + c t( ) − Pu t( ) + v t( )

Finite model cont.Finite model cont.ϕ x, y,t +1( )= αϕ(x,y,t) + ∆ϕ x, y,t( )Multiply by r(x,y), integrate over D, and multiply by R inverse.

R−1 r(x ,y)D∫ ϕ x ,y , t + 1( )dxdy =

R−1α r(x , y)D∫ ϕ(x, y, t)dxdy + R−1 r (x , y)

D∫ ∆ϕ x ,y , t( )dxdy

c(t+1) = c(t) + ∆(t)R−1 r(x,y)∆ϕ

D

αDefinition of c(t) gives

∫ (x, y, t)dxdywhere ∆(t) =

Finite model cont.Finite model cont.In summary, the finite dimensional state space model is

c t +1( ) = αc t( )+ ∆ t( ) c t( ) is n vector

c t +1( ) = αc t( ) + ∆ t( )− P∆ t( ) c t( ) is m vector

s t( ) = Pc t( )+ c t( ) − Pu t( )+ v t( ) s t( ) is m vector

Note s t( ) is made up of1( ) Pc t( ) = signal due to controllable part of ϕ

2( ) c t( ) = signal due to uncontrollable part of ϕ

3( ) Pu t( ) = deformable mirror actions4( )v t( )= photonic and instrument noise

Note this model is exact, there is no modal truncation.

Change state variablesChange state variablesInstead of c(t) and s(t), use

z t( ) = c t( )+ Pc t( ) = Λϕ c t( ) + Λϕ t( ) = Λϕ t( )h t( ) = s t( )+ Pu t( )

Then the state space model becomesc t +1( ) = αc t( ) + ∆ t( )z t +1( ) = α z t( )+ ∆ t( )h t( ) = z t( )+ v t( )

Note only z(t) is seen in the measurements h(t), and the unob-servable part of (x,y,t) contained in c(t) is not seen in h(t).ϕ c

Optimal Control LawOptimal Control LawThe controller u(t) can do nothing about (x,y,t) and so the best u(t) must optimally predict c(t) in based on all past measurements s(0),s(1),…s(t-1).Therefore the minimum variance (maximum Strehl) feedback control law is

ϕc

ϕ c x,y,t( )= rT x, y( )c t( )

uopt t( )= ˆ c t t−1 = c t( ) s 0( ), s 1( )Ks t −1( )

The Kalman filter is used to recursively compute the observable conditional mean which is then algebraically related to the optimal control See “A Geometric View of Adaptive Optics”, D. M. Wiberg, C. E. Max, and D. T. Gavel, SPIE Astronomical Telescopes and Instrumentation Conf.June 21 -25, 2004, Glasgow, Scotland, paper 5490-93, also available on the CfAO website

1ˆ −ttz( )tuc opt

tt =−1/ˆ