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NASA NTRS2007

Status on Iterative Transform Phase Retrieval Applied to the GBT Data

This slide presentation reviews the use of iterative transform phase retrieval in the analysis of the Green Bank Radio Telescope (GBT) Data. It reviews the NASA projects that have used phase retrieval, and the testbed for the algorithm to be used for the James Webb Space Telescope. It shows the comparison of phase retrieval with an interferometer, and reviews the two approaches used for phase retrieval, iterative transform (ITA) or parametric (non-linear least squares model fitting). The concept of ITA Phase Retrieval is reviewed, and the application to Radio Antennas is reviewed. The presentation also examines the National Radio Astronomy Observatory (NRAO) data from the GBT, and the Fourier model that NRAO uses to analyze the data. The challenge for ITA phase retrieval is reviewed, and the coherent approximation for incoherent data is shown. The validity of the approximation is good for a large tilt. There is a review of the proof of concept of the Phase Review simulation using the input wavefront, and the initial sampling parameters estimate from the focused GBT data.

Dean, Bruce↗

NASA NTRS2007

Iterative-Transform Phase Retrieval Using Adaptive Diversity

A phase-diverse iterative-transform phase-retrieval algorithm enables high spatial-frequency, high-dynamic-range, image-based wavefront sensing. [The terms phase-diverse, phase retrieval, image-based, and wavefront sensing are defined in the first of the two immediately preceding articles, Broadband Phase Retrieval for Image-Based Wavefront Sensing (GSC-14899-1).] As described below, no prior phase-retrieval algorithm has offered both high dynamic range and the capability to recover high spatial-frequency components. Each of the previously developed image-based phase-retrieval techniques can be classified into one of two categories: iterative transform or parametric. Among the modifications of the original iterative-transform approach has been the introduction of a defocus diversity function (also defined in the cited companion article). Modifications of the original parametric approach have included minimizing alternative objective functions as well as implementing a variety of nonlinear optimization methods. The iterative-transform approach offers the advantage of ability to recover low, middle, and high spatial frequencies, but has disadvantage of having a limited dynamic range to one wavelength or less. In contrast, parametric phase retrieval offers the advantage of high dynamic range, but is poorly suited for recovering higher spatial frequency aberrations. The present phase-diverse iterative transform phase-retrieval algorithm offers both the high-spatial-frequency capability of the iterative-transform approach and the high dynamic range of parametric phase-recovery techniques. In implementation, this is a focus-diverse iterative-transform phaseretrieval algorithm that incorporates an adaptive diversity function, which makes it possible to avoid phase unwrapping while preserving high-spatial-frequency recovery. The algorithm includes an inner and an outer loop (see figure). An initial estimate of phase is used to start the algorithm on the inner loop, wherein multiple intensity images are processed, each using a different defocus value. The processing is done by an iterative-transform method, yielding individual phase estimates corresponding to each image of the defocus-diversity data set. These individual phase estimates are combined in a weighted average to form a new phase estimate, which serves as the initial phase estimate for either the next iteration of the iterative-transform method or, if the maximum number of iterations has been reached, for the next several steps, which constitute the outerloop portion of the algorithm. The details of the next several steps must be omitted here for the sake of brevity. The overall effect of these steps is to adaptively update the diversity defocus values according to recovery of global defocus in the phase estimate. Aberration recovery varies with differing amounts as the amount of diversity defocus is updated in each image; thus, feedback is incorporated into the recovery process. This process is iterated until the global defocus error is driven to zero during the recovery process. The amplitude of aberration may far exceed one wavelength after completion of the inner-loop portion of the algorithm, and the classical iterative transform method does not, by itself, enable recovery of multi-wavelength aberrations. Hence, in the absence of a means of off-loading the multi-wavelength portion of the aberration, the algorithm would produce a wrapped phase map. However, a special aberration-fitting procedure can be applied to the wrapped phase data to transfer at least some portion of the multi-wavelength aberration to the diversity function, wherein the data are treated as known phase values. In this way, a multiwavelength aberration can be recovered incrementally by successively applying the aberration-fitting procedure to intermediate wrapped phase maps. During recovery, as more of the aberration is transferred to the diversity function following successive iterations around the ter loop, the estimated phase ceases to wrap in places where the aberration values become incorporated as part of the diversity function. As a result, as the aberration content is transferred to the diversity function, the phase estimate resembles that of a reference flat.

Dean, Bruce H.↗

NASA NTRS1991

Phase retrieval for the Hubble Space Telescope using iterative propagation algorithms

Phase retrieval algorithms, including the iterative transform algorithm and gradient search algorithms, were generalized to include the effects of propagation through a complicated optical system and to discount the effects of bad CCD pixels. For the gradient search algorithms, analytic gradients were derived that greatly speed up the computation over finite difference methods. For the Hubble Space Telescope (HST), the aperture function was reconstructed and the phase errors were retrieved. This information is useful to design correction optics for the telescope and for the deconvolution of blurred images from the HST.

Fienup, J. R.↗

NASA NTRS2009

Discrete Fourier Transform Analysis in a Complex Vector Space

Alternative computational strategies for the Discrete Fourier Transform (DFT) have been developed using analysis of geometric manifolds. This approach provides a general framework for performing DFT calculations, and suggests a more efficient implementation of the DFT for applications using iterative transform methods, particularly phase retrieval. The DFT can thus be implemented using fewer operations when compared to the usual DFT counterpart. The software decreases the run time of the DFT in certain applications such as phase retrieval that iteratively call the DFT function. The algorithm exploits a special computational approach based on analysis of the DFT as a transformation in a complex vector space. As such, this approach has the potential to realize a DFT computation that approaches N operations versus Nlog(N) operations for the equivalent Fast Fourier Transform (FFT) calculation.

Dean, Bruce H.↗

NASA NTRS2011

Phase Retrieval for Radio Telescope and Antenna Control

Phase-retrieval is a general term used in optics to describe the estimation of optical imperfections or "aberrations." The purpose of this innovation is to develop the application of phase retrieval to radio telescope and antenna control in the millimeter wave band. Earlier techniques do not approximate the incoherent subtraction process as a coherent propagation. This approximation reduces the noise in the data and allows a straightforward application of conventional phase retrieval techniques for radio telescope and antenna control. The application of iterative-transform phase retrieval to radio telescope and antenna control is made by approximating the incoherent subtraction process as a coherent propagation. Thus, for systems utilizing both positive and negative polarity feeds, this approximation allows both surface and alignment errors to be assessed without the use of additional hardware or laser metrology. Knowledge of the antenna surface profile allows errors to be corrected at a given surface temperature and observing angle. In addition to imperfections of the antenna surface figure, the misalignment of multiple antennas operating in unison can reduce or degrade the signal-to-noise ratio of the received or broadcast signals. This technique also has application to the alignment of antenna array configurations.

Dean, Bruce↗

NASA NTRS2007

NASA Tech Briefs, April 2007

Topics include: Wearable Environmental and Physiological Sensing Unit; Broadband Phase Retrieval for Image-Based Wavefront Sensing; Filter Function for Wavefront Sensing Over a Field of View; Iterative-Transform Phase Retrieval Using Adaptive Diversity; Wavefront Sensing With Switched Lenses for Defocus Diversity; Smooth Phase Interpolated Keying; Maintaining Stability During a Conducted-Ripple EMC Test; Photodiode Preamplifier for Laser Ranging With Weak Signals; Advanced High-Definition Video Cameras; Circuit for Full Charging of Series Lithium-Ion Cells; Analog Nonvolatile Computer Memory Circuits; JavaGenes Molecular Evolution; World Wind 3D Earth Viewing; Lithium Dinitramide as an Additive in Lithium Power Cells; Accounting for Uncertainties in Strengths of SiC MEMS Parts; Ion-Conducting Organic/Inorganic Polymers; MoO3 Cathodes for High-Temperature Lithium Thin-Film Cells; Counterrotating-Shoulder Mechanism for Friction Stir Welding; Strain Gauges Indicate Differential-CTE-Induced Failures; Antibodies Against Three Forms of Urokinase; Understanding and Counteracting Fatigue in Flight Crews; Active Correction of Aberrations of Low-Quality Telescope Optics; Dual-Beam Atom Laser Driven by Spinor Dynamics; Rugged, Tunable Extended-Cavity Diode Laser; Balloon for Long-Duration, High-Altitude Flight at Venus; and Wide-Temperature-Range Integrated Operational Amplifier.

Source record↗

NASA NTRS1993

Artificial neural network for the determination of Hubble Space Telescope aberration from stellar images

An artificial-neural-network method, first developed for the measurement and control of atmospheric phase distortion, using stellar images, was used to estimate the optical aberration of the Hubble Space Telescope. A total of 26 estimates of distortion was obtained from 23 stellar images acquired at several secondary-mirror axial positions. The results were expressed as coefficients of eight orthogonal Zernike polynomials: focus through third-order spherical. For all modes other than spherical the measured aberration was small. The average spherical aberration of the estimates was -0.299 micron rms, which is in good agreement with predictions obtained when iterative phase-retrieval algorithms were used.

Barrett, Todd K.↗

NASA NTRS2013

Implementation of a Wavefront-Sensing Algorithm

A computer program has been written as a unique implementation of an image-based wavefront-sensing algorithm reported in "Iterative-Transform Phase Retrieval Using Adaptive Diversity" (GSC-14879-1), NASA Tech Briefs, Vol. 31, No. 4 (April 2007), page 32. This software was originally intended for application to the James Webb Space Telescope, but is also applicable to other segmented-mirror telescopes. The software is capable of determining optical-wavefront information using, as input, a variable number of irradiance measurements collected in defocus planes about the best focal position. The software also uses input of the geometrical definition of the telescope exit pupil (otherwise denoted the pupil mask) to identify the locations of the segments of the primary telescope mirror. From the irradiance data and mask information, the software calculates an estimate of the optical wavefront (a measure of performance) of the telescope generally and across each primary mirror segment specifically. The software is capable of generating irradiance data, wavefront estimates, and basis functions for the full telescope and for each primary-mirror segment. Optionally, each of these pieces of information can be measured or computed outside of the software and incorporated during execution of the software.

Smith, Jeffrey S.↗

NASA NTRS2009

Hybrid diversity method utilizing adaptive diversity function for recovering unknown aberrations in an optical system

A method of recovering unknown aberrations in an optical system includes collecting intensity data produced by the optical system, generating an initial estimate of a phase of the optical system, iteratively performing a phase retrieval on the intensity data to generate a phase estimate using an initial diversity function corresponding to the intensity data, generating a phase map from the phase retrieval phase estimate, decomposing the phase map to generate a decomposition vector, generating an updated diversity function by combining the initial diversity function with the decomposition vector, generating an updated estimate of the phase of the optical system by removing the initial diversity function from the phase map. The method may further include repeating the process beginning with iteratively performing a phase retrieval on the intensity data using the updated estimate of the phase of the optical system in place of the initial estimate of the phase of the optical system, and using the updated diversity function in place of the initial diversity function, until a predetermined convergence is achieved.

Dean, Bruce H.↗

NASA NTRS1993

Phase-retrieval algorithms for a complicated optical system

Phase-retrieval algorithms have been developed that handle a complicated optical system that requires multiple Fresnellike transforms to propagate from one end of the system to the other including the absorption by apertures in more than one plane and allowance for bad detector pixels. Gradient-search algorithms and generalizations of the iterative-transform phase-retrieval algorithms are derived. Analytic expressions for the gradient of an error metric, with respect to polynomial coefficients and with respect to point-by-point phase descriptions, are given. The entire gradient can be computed with the number of transforms required to propagate a wave front from one end of the optical system to the other and back again, independent of the number of coefficients or phase points. This greatly speeds the computation. The reconstruction of pupil amplitude is also given. A convergence proof of the generalized iterative transform algorithm is given. These improved algorithms permit a more accurate characterization of complicated optical systems from their point spread functions.

Fienup, J. R.↗

NASA NTRS2012

Variable Sampling Mapping

The performance of an optical system (for example, a telescope) is limited by the misalignments and manufacturing imperfections of the optical elements in the system. The impact of these misalignments and imperfections can be quantified by the phase variations imparted on light traveling through the system. Phase retrieval is a methodology for determining these variations. Phase retrieval uses images taken with the optical system and using a light source of known shape and characteristics. Unlike interferometric methods, which require an optical reference for comparison, and unlike Shack-Hartmann wavefront sensors that require special optical hardware at the optical system's exit pupil, phase retrieval is an in situ, image-based method for determining the phase variations of light at the system s exit pupil. Phase retrieval can be used both as an optical metrology tool (during fabrication of optical surfaces and assembly of optical systems) and as a sensor used in active, closed-loop control of an optical system, to optimize performance. One class of phase-retrieval algorithms is the iterative transform algorithm (ITA). ITAs estimate the phase variations by iteratively enforcing known constraints in the exit pupil and at the detector, determined from modeled or measured data. The Variable Sampling Mapping (VSM) technique is a new method for enforcing these constraints in ITAs. VSM is an open framework for addressing a wide range of issues that have previously been considered detrimental to high-accuracy phase retrieval, including undersampled images, broadband illumination, images taken at or near best focus, chromatic aberrations, jitter or vibration of the optical system or detector, and dead or noisy detector pixels. The VSM is a model-to-data mapping procedure. In VSM, fully sampled electric fields at multiple wavelengths are modeled inside the phase-retrieval algorithm, and then these fields are mapped to intensities on the light detector, using the properties of the detector and optical system, for comparison with measured data. Ultimately, this model-to-data mapping procedure enables a more robust and accurate way of incorporating the exit-pupil and image detector constraints, which are fundamental to the general class of ITA phase retrieval algorithms.

Smith, Jeffrey, S.↗

NASA NTRS2005

Actuated Hybrid Mirror Telescope

The figure depicts the planned Actuated Hybrid Mirror Telescope (AHMT), which is intended to demonstrate a new approach to the design and construction of wide-aperture spaceborne telescopes for astronomy and Earth science. This technology is also appropriate for Earth-based telescopes. The new approach can be broadly summarized as using advanced lightweight mirrors that can be manufactured rapidly at relatively low cost. More specifically, it is planned to use precise replicated metallic nanolaminate mirrors to obtain the required high-quality optical finishes. Lightweight, dimensionally stable silicon carbide (SiC) structures will support the nanolaminate mirrors in the required surface figures. To enable diffraction- limited telescope performance, errors in surface figures will be corrected by use of mirror-shape-control actuators that will be energized, as needed, by a wave-front-sensing and control system. The concepts of nanolaminate materials and mirrors made from nanolaminate materials were discussed in several previous NASA Tech Briefs articles. Nanolaminates constitute a relatively new class of materials that can approach theoretical limits of stiffness and strength. Nanolaminate mirrors are synthesized by magnetron sputter deposition of metallic alloys and/or compounds on optically precise master surfaces to obtain optical-quality reflector surfaces backed by thin shell structures. As an integral part of the deposition process, a layer of gold that will constitute the reflective surface layer is deposited first, eliminating the need for a subsequent and separate reflective-coating process. The crystallographic textures of the nanolaminate will be controlled to optimize the performance of the mirror. The entire deposition process for making a nanolaminate mirror takes less than 100 hours, regardless of the mirror diameter. Each nanolaminate mirror will be bonded to its lightweight SiC supporting structure. The lightweight nanolaminate mirrors and SiC supporting structures will be fabricated from reusable master molds. The mirror-shape-control actuators will be low-power, high-capacitance lead magnesium niobate electrostrictive actuators that will be embedded in the SiC structures. The mode of operation of these actuators will be such that once power was applied, they will change in length and once power was removed, they will maintain dimensional stability to nanometer precision. This mode of operation will enable the use of low-power, minimally complex electronic control circuitry. The wave-front-sensing and control system will be designed and built according to a two-stage architecture. The first stage will be implemented by a Shack- Hartmann (SH) sensor subsystem, which will provide a large capture range. The second, higher-performance stage will be implemented by an image-based wave-front-sensing subsystem that will include a phase-retrieval camera (PRC), and will utilize phase retrieval and other techniques to measure wavefront error directly. Phase retrieval is a process in which multiple images of an unresolved object are iterated to estimate the phase of the optical system that acquired the images. The combination of SH and phase-retrieval sensors will afford the virtues of both a dynamic range of 105 and an accuracy of <10 nm.

Hickey, Gregory↗

NASA NTRS2011

Autonomous Phase Retrieval Calibration

The Palomar Adaptive Optics System actively corrects for changing aberrations in light due to atmospheric turbulence. However, the underlying internal static error is unknown and uncorrected by this process. The dedicated wavefront sensor device necessarily lies along a different path than the science camera, and, therefore, doesn't measure the true errors along the path leading to the final detected imagery. This is a standard problem in adaptive optics (AO) called "non-common path error." The Autonomous Phase Retrieval Calibration (APRC) software suite performs automated sensing and correction iterations to calibrate the Palomar AO system to levels that were previously unreachable.

Estlin, Tara A.↗

NASA NTRS2004

Above-Cloud Precipitable Water Retrievals using the MODIS 0.94 micron Band with Applications for Multi-Layer Cloud Detection

In order to perform satellite retrievals of cloud properties, it is important to account for the effect of the above-cloud atmosphere on the observations. The solar bands used in the operational MODIS Terra and Aqua cloud optical and microphysical algorithms (visible, NIR, and SWIR spectral windows) are primarily affected by water vapor, and to a lesser extent by well-mixed gases. For water vapor, the above-cloud column amount, or precipitable water, provides adequate information for an atmospheric correction; details of the vertical vapor distribution are not typically necessary for the level of correction required. Cloud-top pressure has a secondary effect due to pressure broadening influences. For well- mixed gases, cloud-top pressure is also required for estimates of above-cloud abundances. We present a method for obtaining above-cloud precipitable water over dark Ocean surfaces using the MODIS 0.94 pm vapor absorption band. The retrieval includes an iterative procedure for establishing cloud-top temperature and pressure, and is useful for both single layer water and ice clouds. Knowledge of cloud thermodynamic phase is fundamental in retrieving cloud optical and microphysical properties. However, in cases of optically thin cirrus overlapping lower water clouds, the concept of a single unique phase is ill- defined and depends, at least, on the spectral region of interest. We will present a method for multi-layer and multi-phase cloud detection which uses above-cloud precipitable water retrievals along with several existing MODIS operational cloud products (cloud-top pressure derived from a C02 slicing algorithm, IR and SWIR phase retrievals). Results are catagorized by whether the radiative signature in the MODIS solar bands is primarily that of a water cloud with ice cloud contamination, or visa-versa. Examples in polar and mid-latitude regions will be shown.

Platnick, S.↗

NASA NTRS2015

Discrete Fourier Transform in a Complex Vector Space

An image-based phase retrieval technique has been developed that can be used on board a space based iterative transformation system. Image-based wavefront sensing is computationally demanding due to the floating-point nature of the process. The discrete Fourier transform (DFT) calculation is presented in "diagonal" form. By diagonal we mean that a transformation of basis is introduced by an application of the similarity transform of linear algebra. The current method exploits the diagonal structure of the DFT in a special way, particularly when parts of the calculation do not have to be repeated at each iteration to converge to an acceptable solution in order to focus an image.

Dean, Bruce H.↗

NASA NTRS2016

Phase Retrieval Implementation for the WFIRST Coronagraph Development Testbed

The Wide-Field Infrared Survey Telescope (WFIRST) is a 2.4m diameter space telescope NASA program. The payload will include two instruments, including a coronagraph instrument (CGI). The CGI will be the first active, high-contrast stellar coronagraph in space intended for imaging, discovery, and spectral characterization of exoplanets and debris discs. The CGI designs under development use deformable mirrors (DM) to create a point spread function (PSF) with a dark region around the obscured star object. Each DM consists of an array of actuators, and each actuator causes a local displacement to a thin, mirror-coated, glass face sheet. The actuator displacements are designed to create the PSF. Typically, phase retrieval is used to set the DM actuator displacements to their designed pattern, and then the actuators are further modified by an iterative control loop to correct for system non-idealities. To observe an exoplanet near the star object, the contrast in the annular region must be better than 10^8 (10^9 after post processing) relative to the unobstructed image of the star [1,2]. The coronagraph testbed at Jet Propulsion Laboratory is used to test coronagraph design and control algorithms. The optical system (Fig. 1) is maintained in a vacuum tank so that the PSF is stable for long integration times. Phase retrieval is an important tool for the testbed and serves several purposes, in addition to setting initial DM actuator displacements. Other uses include evaluation of systematic wavefront error introduced by system components, and alignment of the DMs and pupil masks. For example, the location of specific DM actuators relative to the Lyot mask can be determined by displacing the actuator, then measuring the location of the phase disturbance in the pupil plane with phase retrieval. In the next section we further describe the optical train of the testbed and the collection of source and pupil images available as inputs for phase retrieval, as well as one phase retrieval algorithm employed. Section 3 shows example phase retrieval results.

Marx, David↗

NASA NTRS2001

Columnar Aerosol Single-Scattering Albedo and Phase Function Retrieved from Sky Radiance Over the Ocean: Measurements of African Dust

The single-scattering albedo and phase function of African mineral dust are retrieved from ground-based measurements of sky radiance collected in the Florida Keys. The retrieval algorithm employs the radiative transfer equation to solve by iteration for these two properties which best reproduce the observed sky radiance using an assumed aerosol vertical structure and measured aerosol optical depth. Thus, no assumptions regarding particle size, shape, or composition are required. The single-scattering albedo, presented at fourteen wavelengths between 380 and 870 nm, displays a spectral shape expected of iron-bearing minerals but is much higher than current dust models allow. This indicates the absorption of light by mineral dust is significantly overestimated in climate studies. Uncertainty in the retrieved albedo is less than 0.02 due to the small uncertainty in the solar-reflectance-based calibration (12.2%) method employed. The phase function retrieved at 860 nm is very robust under simulations of expected experimental errors, indicating retrieved phase functions at this wavelength may be confidently used to describe aerosol scattering characteristics. The phase function retrieved at 443 nm is very sensitive to expected experimental errors and should not be used to describe aerosol scattering. Radiative forcing by aerosol is the greatest source of uncertainty in current climate models. These results will help reduce uncertainty in the absorption of light by mineral dust. Assessment of the radiative impact of aerosol species is a key component to NASA's Earth System Enterprise.

Cattrall, Christopher↗

NASA NTRS1989

Physical retrieval of precipitation cell parameters using passive 118-GHz observations

Physical retrieval of precipitation parameters from 118-GHz spectra is investigated using an iterative planar-stratified numerical radiative-transfer model. Liquid and frozen hydrometeors are modeled as spherical Marshall-Palmer and Sekhon-Srivastava (SS) distributed Mie-scattering polydispersions, respectively, with Henyey-Greenstein phase functions. A comparison of 118-GHz rain-cell spectral perturbations observed during Cohmex with model calculations based on coincident radar data yields + or - 10 percent agreement over the convective core region, although a mean ice size 50 percent larger than that given by the SS distribution is required for agreement over the anvil region. A rain-cell model parameterized by top-altitude and total water density suggests that the dominant 118-GHz spectral modes can be used to retrieve top-layer altitude with rms errors of about 1-1.5 km, consistent with statistical retrieval results. Model calculations also suggest that cell density is not observable using 118-GHz spectra alone.

Gasiewski, A. J.↗