Since the pioneering contributions of Labeyrie, researchers have made tremendous strides in developing techniques for imaging through turbulent media such as the atmosphere. Imaging through turbid (scattering) media is a more challenging problem. Historically, researchers assumed that scattered light is so chaotic that it carries little or no information. Their approach was to retrieve the weak ballistic (unscattered) signal in the presence of the dominant and confounding scattered signal. In more recent years, researchers have demonstrated focusing light and imaging through thin scattering volumes (diffusers) by actually utilizing the scattered light, illustrating that scattered light also carries information. However, these demonstrations require access to the object plane for inserting detectors, beacons, or a fluorescing agent. We introduce a novel method for imaging through an unknown diffuser with a strictly one-sided observation, wherein the observer has no access to the object plane (for illumination or diffuser characterization) nor does the observer need to label the object with a fluorescing agent. The method requires laser illumination, a digital-holographic data collection, and the use of an image-sharpness criterion to jointly estimate the specific diffuser response and the image that would be obtained by removing the diffuser. Estimation is accomplished with off-line processing after the data are acquired. We demonstrate the method in simulation using a thin diffuser. We also suggest a framework under which the method can be generalized for use with thick diffusers. Our approach shows promise for imaging into human tissue, clouds, fog, smoke, suspended particulates, tree canopy, or other scattering media.
There are many approaches to incoherent imaging through the atmosphere that involve joint estimation of multiple turbulence-induced wavefront-aberration realizations and an object that is common across realizations. These approaches, all of which use short-exposure or “speckle” data, include Multi-Frame Blind Deconvolution (MFBD), Phase-Diverse Speckle (PDS), and Wavelength-Diverse Speckle (WDS). We enumerate fundamental estimation ambiguities that arise within each of these modalities and identify strategies to eliminate some of the ambiguities.
A reflective microelectromechanical mirror array was used to control the intensity distribution of a coherent beam that
was propagated through a strongly scattering medium. The controller modulated phase spatially in a plane upstream of
the scattering medium and monitored intensity spatially in a plane downstream of the medium. Optimization techniques
were used to maximize the intensity at a single point in the downstream plane. Intensity enhancement by factors of
several hundred were achieved within a few thousand iterations using a MEMS segmented deformable mirror (e.g. a
spatial light modulator) with 1020 independent segments. Experimental results are reported for alternate optimization
approaches and for optimization through dynamically translating scattering media.
Previously, we demonstrated useful and novel features of the General Dynamics QuickStar adaptive-optics testbed
utilizing Phase Diversity (PD) as the wavefront sensor operating on a point object. Point objects are relatively easy to
produce in the laboratory and simplify the calibration procedure. However, for some applications, natural or artificial
beacons may not be readily available and a wavefront sensor that operates on extended scenes is required. Accordingly,
the QuickStar testbed has been augmented to allow PD to operate on natural three-dimensional solar-illuminated scenes
external to the QuickStar laboratory. In addition, a computationally efficient chip-selection strategy has been developed
that allows PD to operate on chips with favorable scene content. Finally, a covariance matrix has been developed that
provides an accuracy estimate for PD wavefront-parameter estimates. The covariance can be used by the controls
algorithm to properly weight the correction applied according to the accuracy of the estimates. These advances suggest
that PD is a sufficiently mature technology for use in adaptive optics systems that require operation with extended
scenes.
KEYWORDS: Wavefronts, Process control, Calibration, Imaging systems, Image processing, Fermium, Frequency modulation, Beam splitters, Cameras, Control systems
A proof-of-concept phase diversity (PD) wavefront sensing and control (WFS&C) testbed has been developed that
displays 5/1,000 wave RMS accuracy, operates at a sample rate of 100Hz, uses the extended scene of interest in lieu of a
guide star, and is comprised of all low-cost commercial-off-the-shelf (COTS) parts - including the PD processor. This testbed allows closed-loop
operation via a dual deformable-mirror (DM) concept where two DMs are optically conjugate to the exit pupil: one
acting as an independent disturbance and the other reacting to PD WFS&C commands in order to correct the system
wavefront. The use of low-cost, COTS components demonstrated the flexibility of a PD-only
WFS&C approach, and additionally allowed for this system to be conceived, designed, assembled and brought to
operation in approximately nine months. Automatic calibration efforts begun on this testbed have allowed for the quick
discrimination of prominent PD forward-model parameters and a more rapid verification and validation (V&V) process.
Also aiding the V&V process is a novel spatial-heterodyning optical interferometer that collects all information in a
single snapshot and may be made synchronous with the fast PD sample rate. This demonstration proves a PD-only
WFS&C subsystem capability suitable for use on a wide variety of adaptive-optics imaging systems.
Phase Diversity (PD) is a wavefront-sensing technology that offers certain advantages in an Adaptive-Optics
(AO) system. Historically, PD has not been considered for use in AO applications because computations have
been prohibitive. However, algorithmic and computational-hardware advances have recently allowed use of PD
in AO applications. PD is an attractive candidate for AO applications for a variety of reasons. The optical
hardware required is simple to implement and eliminates non-common path errors. In addition, PD has also
been shown to work well with extended scenes that are encountered, for example, when imaging low-contrast solar
granulation. PD can estimate high-order continuous aberrations as well as wavefront discontinuities characteristic
of segmented-aperture or sparse-aperture telescope designs. Furthermore, the fundamental information content
in a PD data set is shown to be greater than that of the correlation Shack-Hartmann wavefront sensor for the
limiting case of unresolved objects. These advantages coupled with recent laboratory results (extended-scene
closed-loop AO with PD sampling at 100 Hz) highlight the maturation of not only the PD concept and algorithm
but the technology as an emerging and viable wavefront sensor for use in AO applications.
This paper will report on efforts to automatically calibrate in situ a phase-diversity (PD) wavefront sensing and control
(WFS&C) system, the results of which are demonstrated on the General Dynamics Advanced Information System's
(GDAIS') QuickStar testbed1, a dual deformable mirror (DM) system which operates at 100Hz sampling rate. The
iterative automatic calibration (AutoCal) process includes both coarse and fine calibration modes, initial closed-loop
flattening of the commercial-off-the-shelf (COTS) DMs, estimation of the system's static wavefront - including DM
print-through, determination of PD-derived actuator influence functions, formulating the resulting system matrix and the
resulting forward-model parameters. Analyses of the system after the calibration routines shows low-order WFS
accuracy of ~0.005λ RMS and closed-loop residual wavefront measurement of ~0.002λ. All of these results were
accomplished with a software package that takes on the order of one hour to operate.
Space-variant blur occurs when imaging through volume turbulence over sufficiently large fields of view. Space- variant effects are particularly severe in horizontal-path imaging, slant-path (air-to-ground or ground-to-air) geometries, and ground-based imaging of low-elevation satellites or astronomical objects. In these geometries, the isoplanatic angle can be comparable to or even smaller than the diffraction-limited resolution angle. Clearly, space-invariant methods used in conjunction with mosaicing will fail in this regime. Our approach to this problem has been to generalize the method of Phase-diverse Speckle (PDS) by using a physically motivated distributed phase-screen model to accomplish both pre- and post-detection correction. Previously reported simulation results have demonstrated the reconstruction of near diffraction-limited imagery using imagery which was severely degraded by space-variant blur. In this paper, we present a novel adaptation of the space- variant PDS scheme for use as a beacon-less wavefront sensor in a multi-conjugate AO system when imaging extended scenes. We then present results of simulation experiments demonstrating that this multi-conjugate AO-compensation scheme is very effective in improving the quality and resolution of collected imagery.
There is currently much interest in deploying large space- based telescopes for various applications including fine- resolution astronomical imaging and earth observing. Often a large primary mirror is synthesized by the precise alignment of several smaller mirror segments. Misalignment or misfigure of these segments results in phase error which degrade the resolution of collected imagery. Phase diversity (PD) is a technique used to infer unknown phase aberrations form image data. It requires the collection of two or more images of the same object, each incorporating a known phase perturbation in addition to the unknown aberrations. Statistical estimation techniques are employed to identify a combination of object and aberrations that is consistent with all of the collected images. The wavefront- sensing performance of PD is evaluated through simulation for a variety of signal and aberration strengths. The aberrations are parameterizes by piston and tilt misalignment of each segment. An unknown extended scene is imaged, complicating the estimation procedure. Since wavefront correction is often an iterative process, moderate estimation errors can be corrected by subsequent estimates. The interpretation of iterative wavefront adjustments as creating new phase-diversity channels suggests a more sophisticated processing approach, called Actuated Phase Diversity. This technique is shown to significantly improve PD wavefront-sensing performance.
The Lockheed Martin phased-array telescope developed at the Palo Alto Research Laboratory is a 0.75-m imaging system consisting of 9 separate 90-mm telescopes. One of the technology drivers behind this design is the ability to maintain the phasing of the individual telescopes to sub- wavelength tolerances. We demonstrate here the use of the focal-plane method of phase diversity for maintaining the phased-array alignment. The telescope is designed to operate with white light, so the phase diversity concept is extended to accommodate a broad optical bandwidth. A simulation of white-light phase-diverse wavefront sensing is presented as a demonstration of the robustness of the method with respect to sparse pupil and wavelength sampling. The simulation is validated with laboratory experiments using a point source. Finally, a closed-loop experiment is conducted that demonstrates the ability of phase diversity to sense piston error and maintain the alignment of the phased-array system.
Imaging through volume turbulence gives rise to anisoplanatism (space-variant blur). The effects of volume turbulence on imaging are often modeled through the use of a sequence of phase screens distributed along the optical path. Wallner recently derived a prescription for the optimal functional form and location of multiple phase screens for use in simulating the effects of volume turbulence in infinite-range imaging geometries. We generalized Wallner's method to accommodate the finite range case and to have a more optimal functional form for the phase screens. These methods can also be used for designing a multi-conjugate AO system. Examples of optimal solutions are given for horizontal-path finite-range imaging cases.
Space-variant blur occurs when imaging through volume turbulence over sufficiently large fields of view. This condition arises in a variety of imaging geometries, including astronomical imaging, horizontal-path imaging, and slant-path (e.g. air-to-ground) imaging. Space-variant effects are particularly severe when much of the optical path is immersed in turbulent media. We present a novel post-processing algorithm based on the technique of phase- diverse speckle (PDS) and a physical model for the space- variant blur. PDS imaging is a combination of phase diversity and speckle imaging which has proven to be an effective post-processing technique for cases with space- invariant blur. We present the details of the algorithm modified to accommodate space-variance and demonstrate its performance with results from both simulation experiments and real-data experiments. The results show that the space- variant PDS algorithm is very effective in cases involving severe space-variant blur, which cause correction techniques based on space-invariant models to fail.
Phase-Diverse Speckle (PDS) is a short-exposure data- collection and processing technique that blends phase- diversity and speckle-imaging concepts. PDS has been successfully used for solar astronomy to achieve near diffraction-limited resolution in ground-based imaging of solar granulation. Variants of PDS that involve narrow-band, spectroscopic, and polarimetric data provide more information observations. We present results from processing data collected with the 76-cm Richard B. Dunn Solar Telescope (DST) on Sacramento Peak, NM. Three-channel data sets consisting of a pair of phase-diverse images of the solar continuum and a narrow-band image were collected over spans of 15 - 20 minutes. Point-spread functions that are estimated from the PDS data are used in a multi-frame deconvolution algorithm to correct the narrow-band imagery. The data were processed into a number of time series. A rare, short-lived continuum bright point with a peak intensity at a factor of 2.1 above the mean intensity in the continuum was observed in one such sequence. The field of view spans multiple isoplanatic patches, and strategies for processing these large fields were developed. We will discuss these methods along with other techniques that were explored for accelerating the processing. Finally, we show the first PDS reconstruction of adaptive-optics (AO) compensated solar granulation taken at the DST. As expected, we find that these data are less aberrated and, thus, the use of AO in future experiments is planned.
By illuminating an object with a laser and collecting far- field speckle intensity patterns, at a regularly spaced sequence of wavelengths, one obtains the squared magnitude of the 3D Fourier transform of the object. Performing 3D phase retrieval to reconstruct a 3D image (consisting of complex-valued voxels) is relatively difficult unless one has a tight support constraint. An alternative is to perform averaging of the autocovariance of the far-field speckle intensities, over an ensemble of speckle realizations, to estimate the square magnitude of the Fourier transform of the underlying (incoherent) reflectivity of the object, by the correlography method. This also gives us an incoherent- image-autocorrelation estimate, from which we can derive an initial support constraint. Since the image, being incoherent, is real-valued and nonnegative, performing phase retrieval on this data is easier and more robust. Unfortunately the resolution for correlography is only moderate since the SNR is low at the higher spatial frequencies. However, one can then use a thresholded version of that reconstructed incoherent image as a tight support constraint for performing phase retrieval on the original speckle intensity patterns to reconstruct a fine-resolution, coherent image. The fact that the objects are opaque plays an important role in the robustness of this approach. We will show successful reconstruction results from real data collected in the laboratory as part of the PROCLAIM (Phase Retrieval with an Opacity Constraint for LAser IMaging) effort.
Multi-frame blind deconvolution (MFBD) has been shown to be useful for overcoming the blurring effects of turbulence- and instrument-induced aberrations in ground-based imaging of satellites. In this scenario, the object has a finite extent that is often entirely contained within the sensor field-of-view. We report on the generalization of MFBD to accommodate objects that extend beyond the field of view, as would be encountered, for example, in solar and planetary astronomy or in down-looking scenarios. We simulate both down-looking and up-looking scenarios, and vary parameters such as the level of scene illumination and the number of data realizations included. In the simulations, MFBD performance is evaluated by comparing results to the true scenes as well as to reconstructions using more established Phase-Diverse Speckle techniques. Using real data, MFBD reconstructions of solar-granulation scenes are validated by comparison with well-accepted PDS results.
We present preliminary results from a comparison of image estimation and recovery algorithms developed for use with advanced telescope instrumentation and adaptive optics systems. Our study will quantitatively compare the potential of these techniques to boost the resolution of imagery obtained with undersampled or low-bandwidth adaptive optics; example applications are optical observations with IR- optimized AO, AO observations in server turbulence, and AO observations with dim guidestars. We will compare the algorithms in terms of morphological and relative radiometric accuracy as well as computational efficiency. Here, we present qualitative comments on image results for two levels each of seeing, object brightness, and AO compensation/wavefront sensing.
We compare phase diversity and curvature wavefront sensing. Besides having completely different reconstruction algorithms, the two methods measure data in different domains: phase diversity very near to the focal plane, and curvature wavefront sensing far from the focal plane in quasi-pupil planes, which enable real-time computation of the wavefront using analog techniques. By using information- theoretic lower bounds, we show that the price of measuring far from the focal lane is an increased error in estimating the phase. Curvature wavefront sensing is currently operating in the field, but phase diversity should produce superior estimates as real-time computing develops.
We are developing a technique to measure segment misalignment of large telescopes based on wavefront estimation using phase-diverse images. We report the current results of an experiment to measure piston errors on the Keck II primary segmented mirror, through atmospheric turbulence, using phase-diverse phase retrieval. The segment piston errors are separated from the random turbulence by averaging phase estimates from many frames. Phase estimates from real data collected with segments intentionally moved in piston reproduce the observed speckle patterns well. However, average phase maps do not reveal the segment piston errors. Simulations show that the observed data were collected in a regime of turbulence where the current algorithm often fails, but would be expected to work very well when the adaptive optics system is operating. There is reason to believe that we can eventually make the algorithm work with these or similar data if apparent mismatches between the data and our current imaging model are removed.
Space-object identification from ground-based telescopes is challenging because of the degradation in resolution arising from atmospheric turbulence. Phase-diverse speckle is a novel post-detection correction method that can be used to overcome turbulence-induced aberrations for telescopes with or without adaptive optics. We present a simulation study of phase-diverse speckle satellite reconstructions for the Air Force Maui Optical station 1.6-meter telescope. For a given turbulence strength, satellite reconstruction fidelity is evaluated as a function of quality and quantity of data. The credibility of this study is enhanced by reconstructions from actual compensated data collected wit the 1.5-meter telescope at the Starfire Optical Range. Consistent details observed across a time series of reconstructions from a portion of a satellite pass enhance the authenticity of these features. We conclude that phase-diverse speckle can restore fine-resolution features not apparent in the raw aberrated images of space objects.
Generating a 3D image can be accomplished by gathering 3D far-field heterodyne array data with multiple laser wavelengths and performing a 3D Fourier transform. However, since heterodyne detection is difficult at optical frequencies, the collection system can be greatly simplified if direct detection is performed instead. Then to reconstruct an image one would need a phase-retrieval algorithm. To assist the reconstruction algorithm, we place bounds on the support of the illuminated object, derived from the support of the autocorrelation function, which can be computed from the Fourier intensity data. We have developed 3D locator sets for getting tight bounds on the object support. These new locator sets are more powerful than those for 2D imaging, and some of them make explicit use of the fact that the illuminated opaque object is effectively a 2D surface embedded in 3D space. For those cases in which it is tight enough, the locator set itself may be all that we need to give an accurate height profile of the object.
3D imaging provides important profile information not available with conventional 2D image products. Profile information can be extremely valuable for industrial- inspection and remote target-characterization applications. In this paper, we discuss a novel imaging modality, called PROCLAIM, that utilizes the powerful constraint that opaque objects can be described by a 2D surface embedded in 3D space. Far-field Fourier intensity measurements are collected by flood-illuminating an object with a frequency- tunable laser and direct detecting the backscattered signal with a lensless sensor. This technique allows for precise, non-contract surface measurements, without the stringent coherence and mechanical stability requirements of related interferometric techniques. We present reconstruction results form simulated data and from laboratory measurements.
The effect of focus anisoplanatism upon the performance of an astronomical laser guide star (LGS) adaptive optics (AO) system can in principle be reduced if the lowest order wavefront aberrations are sensed and corrected using a natural guide star (NGS). For this approach to be useful, the noise performance of the wavefront sensor (WFS) used for the NGS measurements must be optimized to enable operation with the dimmest possible source. Two candidate sensors for this application are the Shack-Hartmann sensor and “phase-diverse phase retrieval,” a comparatively novel approach in which the phase distortion is estimated from two or more well-sampled, full-aperture images of the NGS measured with known adjustments applied to the phase profile. We present analysis and simulation results on the noise-limited performance of these two methods for a sample LGS AO observing scenario. The common parameters for this comparison are the NGS signal level, the sensing wavelength, the second-order statistics of the phase distortion, and the RMS detector read noise. Free parameters for the two approaches are the Shack-Hartmann subaperture geometry, the focus biases used for the phase-diversity measurements, and the algorithms used to estimate the wavefront. We find that phase-diverse phase retrieval provides consistently superior wavefront estimation accuracy when the NGS signal level is high. For lower NGS signal levels on the order of 103 photodetection events, the Shack-Hartmann (phase diversity) approach is preferred at a RMS detector read noise level of 5 (0) electrons/pixel.
Many processes of interest in the solar atmosphere have spatial scales of much less than one second of arc. If the processes are related to magnetic fields, the relevant scales are even smaller. Noticeable evolutions of solar features occur on time-scales of less than a minute if a spatial resolution of better than one second of arc is reached. It is, therefore, of great interest to recover time-series imagery with near diffraction-limited spatial resolution and good temporal resolution on a consistent basis and over extended periods of time using ground-based techniques. Phase diversity is a post-collection technique for restoring fine-resolution detail when imaging in the presence of phase aberrations such as atmospheric turbulence. Incident energy is split into two channels: one is collected at the conventional focal plane, the other is intentionally defocussed a known amount and collected by a second detector array. Phase-diverse speckle is an extension of phase diversity whereby a time sequence of short-exposure image pairs is collected. The maximum-likelihood estimate of a common object and a set of phase aberrations is performed jointly using all images. A phase-diverse speckle set of images of a plage region was collected over a span of 13.5 minutes using the 76-cm Vacuum Tower Telescope at the National Solar Observatory on Sacramento Peak. A phase- diverse pair of broad-band images at 6563 angstroms was collected along a third, narrow-band image in the wing of H- (alpha) . A set of restorations was made into a movie depicting the highly dynamic photosphere at scales below 0.3 arcsec. We conclude that the combination of fine spatial and temporal resolution achieved with phase-diverse speckle opens a new window to the study of the dynamics of the solar atmosphere from ground-based observatories.
In phase diverse speckle imaging, one collects a time series of phase-diversity image sets. From these data it is possible to jointly estimate the object and each realization of the aberrations. Current approaches model the total aberration phase screen in some deterministic, parametric fashion. For a typical scenario, however, one has more information than this. Specifically, the total aberration phase screen is caused by fixed aberrations combined with dynamic (time-varying), turbulence-induced aberrations for which we have some knowledge about the stochastic behavior. One important example is where the dynamic aberrations derive from Kolmogorov turbulence. In this context, utilizing this extra information has the potential for being a powerful aid in the joint aberration/object estimation. In addition, such a framework would provide a relatively simple method for calibrating fixed aberrations in an imaging system. The natural framework for utilizing the stochastic nature of the wavefronts is that of Bayesian statistical inference, where one imposes an a priori probability distribution on the turbulence-induced wavefronts. In this paper, we present the general Bayesian approach for this joint-estimation problem of the fixed aberrations, the dynamic aberrations, and the object from phase-diverse speckle data. We then discuss issues related to theoretical performance, numerical implementation, and applications. Finally we provide simulation results which demonstrate improvement in PDS image reconstructions resulting from the Bayesian estimation approach.
The method of phase diversity has been used in the context of incoherent imaging to estimate jointly an object that is being imaged and phase aberrations induced by atmospheric turbulence. The method requires a parametric model for the phase-aberration function. Typically, the parameters are coefficients to a finite set of basis functions. Care must be taken in selecting a parameterization that properly balances accuracy in the representation of the phase- aberration function with stability in the estimates. It is well known that over parameterization can result in unstable estimates. Thus a certain amount of model mismatch is often desirable. We derive expressions that quantify the bias and variance in object and aberration estimates as a function of parameter dimension.
Phase-diverse speckle imaging is a novel imaging modality that makes use of both speckle-imaging and phase-diversity concepts. A phase- diverse speckle data set consists of one conventional image and at least one additional image with known phase diversity for each of multiple atmospheric phase realizations. We demonstrate the use of a phase-diverse speckle data set collected at the Swedish Vacuum Solar Telescope on La Palma to overcome the effects of atmospheric turbulence and to restore a fine-resolution image of solar granulation. We present preliminary results of simultaneously reconstructing an object and a sequence of atmospheric phase aberrations from these data using a maximum-likelihood parameter- estimation framework. The consistency of the reconstructions is demonstrated using subsets of the sequence of images pairs. The use of different phase-aberration parameterization schemes and their affect on parameter estimates are discussed. Insight into the desired number of atmospheric realizations is provided.
We discuss a 3D imaging modality called pulsed heterodyne-array imaging (PHI). The relationship between PHI and stepped-frequency methods for 3D coherent image formation is derived. For both cases, we consider flood-illumination of the object and detection with a 2D array of coherent receivers located in the pupil plane. It is shown that PHI can recover the same coherent 3D image as a stepped-frequency method such as holographic laser radar.
Three-dimensional imaging provides profile information not available with conventional 2D imaging. Many 3D objects of interest are opaque to the illuminating radiation, meaning that the object exhibits surface, as opposed to volume, scattering. We investigate the use of an opacity constraint to perform 3D phase retrieval. The use of an opacity constraint in conjunction with frequency-diverse pupil-plane speckle measurements to reconstruct a 3D object constitutes a novel unconventional-imaging concept. This imaging modality avoids the difficulties associated with making phase measurements at a cost of increased computations.
A variety of clever pre- and post-detection methods have been developed to image through atmospheric turbulence. These methods have been developed and exercised primarily for use in the isoplanatic (space-invariant) imaging case. Multi-conjugate methods have been investigated for the accommodation of space-variance with compensated imaging, however multiple guidestars, wavefront sensors, and deformable mirrors are required making a practical implementation uncertain. Post-detection correction methods can be extended to the space-variant case with mosaicing methods, but such approaches provide suboptimal estimates.
Phase-diverse speckle imaging is a novel post-detection correction technique for use in imaging through atmospheric turbulence. When the wavelength dependence is introduced, the data-collection model and problem statement are seen to accommodate a variety of existing and proposed data-collection schemes, including speckle imaging, multi-frame blind deconvolution, deconvolution with wavefront sensing, and hybrid adaptive optics.
In order for a phased-array telescope to achieve its resolution potential, the individual telescopes and beamcombining optics must be precisely aligned. Misalignments could be measured directly with laser interferometers. We present an alternative, based on Gonsalves’ phase-diversity concept, in which misalignments are inferred from the collected imagery. Once the misalignments have been estimated they can be used to actively correct the system or to construct a Wiener-Helstrom filter to deblur the collected imagery. The total process is referred to as multiple-plane measurement for aberration correction (MMAC). In this paper we present simulations that demonstrate the use of MMAC in estimating both piston and tilt misalignments in the presence of noise. Measurements of the sensitivity of MMAC to certain systematic errors are presented and a subframing technique is demonstrated.
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