Prescription Retrieval

Performance of prescription-based PSF generation is limited by the accuracy of the prescription data. Prescription retrieval provides a means of improving estimates of prescription parameters by computer processing of image data (Fig. 2). Prescription retrieval takes as a starting point the

optical design prescription, including the location and orientation of each optical element specified in global coordinates, plus figure (conic constant, radius, and Zernike or other deformation parameters) and index. Higher spatial-frequency figure errors determined by optical testing of the elements are added to the respective element surfaces. Obscurations are placed at the appropriate physical coordinates.

The prescription retrieval process takes image data as input. It uses SCOMP functions to generate images that match the data on a pixel-by-pixel basis. Residual pixel differences are minimized in an iterative optimization loop that varies selected prescription parameters. We have used Levenberg-Marquardt nonlinear least-squares and the NPSOL sequential quadratic programming algorithms with good success. The result, when the process is converged, is improved estimates of those prescription parameters. The best results are obtained using several images simultaneously, preferably images that use different field and focus points. Image diversity improves the observability and separability of individual parameters.

We developed prescription retrieval as part of the Hubble Aberration Recovery Project. The main objective here was to obtain an accurate estimate of the true conic constant of the OTA primary mirror, which is the source of most of the spherical aberration seen in pre-refit HST images. It turned out that other effects were significant also, especially conic constant errors in the WFPC repeater cameras. Alone among several groups analyzing the HST images we were able to determine those effects and separate them from the primary mirror errors (Redding, Dumont, and Yu 1993). The resulting estimate of the primary conic constant was in agreement with the results of workers who examined the HST primary mirror fabrication and test hardware, and is validated by the excellent quality of the images from the refurbished cameras.

Example simulated and actual Hubble/PC6 images are shown in Fig. 3. The simulated image was computed tracing 65,000 rays to update a 512512

diffraction grid. Two images, taken at different focus and field settings, were used in the retrieval process (the second is not shown). A total of about 40 parameters were estimated, including various element alignments and figure parameters. Convergence required many days running on a SPARC 1+ workstation.

The images show good agreement with respect to major features such as the spherical aberration rings and the location and structure of the diffracted obscuration patterns. The match is not perfect and can be improved by the addition of the primary mirror zonal aberrations and other effects.

We also applied prescription retrieval to diagnostic star images that were returned by the Mars Observer Narrow-Angle Camera during its approach to Mars. This camera was equipped with heaters on its primary mirror that, when activated, cause a nearly pure focus change to be imparted to the system. A typical example image showing simulated and actual data is given in Fig. 4.

Retrieval here was complicated by limited dynamic range and noise contributed by a lossy data compression algorithm used in the approach phase of the flight.

Results of prescription retrieval using 7 images taken on both sides of focus and across the camera field indicated that the primary mirror had deformed slightly, with the main error being 45-degree astigmatism. Other possible errors, such as decenter of the secondary mirror, did not accurately reproduce the observed images. Performance at the best focus setting was good, however, with a 1-3 pixel PSF.

There are several potential problems to be considered when applying prescription retrieval. These are:

Noise

Detection noise, background noise, self-noise, stray light and similar effects obscure structure that can be important in the retrieval process.

Local minima

Nonlinear least squares and similar optimization algorithms can be trapped by local minima. The likelihood of this is reduced with a sufficiently diverse data set and a good initial guess.

Inseparable parameters

If two parameters have a proportionally similar effect on the images, the contribution of one cannot be distinguished from that of the other. Separability can be improved by looking at different field angles and focus positions. Some parameters that cannot be separated, need not be separated - their effect on the image is accounted for without knowing precisely the contribution of each parameter individually.

Inadequate model

It is important that major factors in the imaging process be captured by the model. This means getting the physics right, both in the optics and in the detectors. It also includes correctly identifying which factors are common to particular images, and which change between images, so they can be correctly solved for. Unmodeled changes between images, such as incomplete pyramid mirror rotations or thermal deformations of the telescope structure, add confusion and reduce the quality of the solutions.

Meaningless parameters

Solving directly for noise or jitter was found to improve the optimization cost function without improving the accuracy of individual parameter estimates.

Besides its role in image restoration, prescription retrieval also has significant advantages for optical test and calibration, and for active optical control. These advantages include:

• It provides the ability to optically test aspheric elements or systems without null correctors or interferometers, removing a major potential source of testing error.
• It provides end-to-end instrument testing, including the science detector.
• It is a non-invasive testing procedure that does not require instrument disassembly.
• It is applicable over the full operational range of the instrument, rather than at 1 or 2 field points.
• As we have demonstrated, it can be performed remotely on instruments in the field, as well as in a laboratory environment.
• For instruments equipped with low-bandwidth controllable optics, such as deformable or movable mirrors, prescription retrieval can determine corrections based on image data.

Prescription retrieval does require a large amount of computational power. The best results require simultaneous processing of numerous images while exercising an appropriate number of prescription parameters. This requires a large amount of memory as well as high-speed computation.

For most instruments, prescription retrieval will be performed intensively once, to provide the best possible understanding of the basic instrument configuration. This initial retrieval will utilize diagnostic star images taken at multiple focus and field points. This computationally intensive process need not be repeated unless the instrument radically changes.

Subsequent limited prescription retrieval might be optionally performed on a reduced parameter set, without the need for special diagnostic images. Here the objective would be to account for small, localized shifts that are known to occur relatively frequently, such as the HST thermally induced ``breathing'' effects. This process will be much less computationally intensive. Retrieved prescription data will be archived by the application and referenced by date, so that appropriate prescriptions will be available for recovery.