Doctoral Dissertation 1996,
Stockholm Observatory,
Stockholm University,
S-106 91 Stockholm
This thesis is based on the following papers, which will be referred to in the text by their roman numerals:
[Note: Paper III was never published in the form included in the dissertation. The material was expanded and split into three papers:
An implementation of a phase-diverse speckle imaging (PDS) technique for reducing the effects of aberrations in solar images is described. Such aberrations usually occur in the Earth's atmosphere and in telescopes. PDS is a post-processing technique for measuring such aberrations and for deblurring the images.
The code has been extensively tested. Realistic simulations indicate that the systematic errors are small. The wavefront and object estimates calculated from real solar data, collected with the Swedish Vacuum Solar Telescope (SVST), are spatially and temporally consistent with expectations for anisoplanatism and the assumption of no evolution of the solar features on the time scale of a few seconds. Implementation invariance is demonstrated by comparison of the output with that of a separately developed implementation. External reference for the wavefront determination is provided by comparison with theoretical predictions of temporal variation of the telescopic aberrations at the SVST. High-quality image restorations can be made with much less data than is needed by the more established method of speckle interferometry.
The usefulness of the technique for astronomical purposes is demonstrated by the successful restoration and analysis of a 29"x29" 70-minute time sequence of solar granulation and bright points. The resolution in the restored data is sufficient to allow the evolution and motion of bright points to be followed in detail.
Stars are heated by nuclear processes in the center and cooled by radiation from the surface. In order for a star to maintain equilibrium, energy has to be transported from the hot interior to the surface. In the outer parts of the Sun and similar stars, the energy is transported by convection, which is manifested in the photosphere as granulation, a pattern of bright cells where the hot gas flows up and darker intergranular lanes of descending cooler gas.
The Sun displays many phenomena that are not explained by this simple picture. The magnetic field plays a dominant role in the layers above the photosphere such as in the chromosphere and in the corona. In localized regions, it also strongly disturbs the photosphere. In such regions, the magnetic field can be very strong. This gives rise to observable phenomena, such as sunspots, pores, and faculae. The photon mean free path is about 70 km in the deep photosphere which corresponds to ~0.1" at solar center, so fine structure on this scale or smaller must be expected. Many of the magnetic phenomena are indeed associated with very small-scale fine structures, such as filaments in sunspot penumbrae, umbral dots, filigree, etc. In order to study them, very-high-resolution observations are needed. The angular dimension of the smallest observed magnetic elements is on the order of 0.1"-0.3", which is at or below the diffraction-limited resolution of most present solar telescopes.
Normally, the theoretical resolution of a telescope cannot be reached, since the resolution is in practice limited by seeing and by imperfections in the telescope optical systems. Seeing refers to the time-variable blurring that is caused by fluctuations in the refractive index of the turbulent air mass in front of the telescope. Determination of these aberrations is usually referred to as wavefront sensing.
Seeing varies significantly on a time scale of milliseconds. Frame-selection techniques, developed at the Swedish Vacuum Solar Telescope (SVST) on La Palma, Canary Islands, have made it possible to obtain images during the best moments of seeing, thereby providing resolution close to the diffraction limit. The long sequences of high-resolution observations needed for the study of time-evolution of the magnetic fine structures can, however, rarely be obtained without further reducing the effects of seeing.
This thesis is a collection of three papers, which describe the implementation, testing, and scientific application of one particular technique for eliminating the effects of seeing from solar images. This technique is referred to as phase-diverse speckle imaging, and combines wavefront sensing with compensation for the blurring of the images in post-processing.
There are in principle three possible ways to eliminate or reduce the effects of seeing. The telescope can be put in space, adaptive optics can correct the distorted wavefront in real-time, and deblurring of the images can be performed in post-processing. Each of these strategies have their advantages and disadvantages.
Space-born instruments have access to wavelengths which are absorbed by the Earth's atmosphere and permit diffraction-limited imaging but are extremely expensive.
Adaptive optics is the most attractive solution for many applications, but the requirements of fast processing for the determination of the wavefront and rapid adjustment of the optics causes severe practical problems and the technique is still very expensive.
By comparison, post-processing is simpler and much less expensive. Without the severe constraint of real-time processing, higher-order aberrations can be derived and/or sophisticated models of image formation implemented. The main drawback compared to adaptive optics is that post-processing cannot improve the signal-to-noise ratio. This is a problem particularly at high spatial frequencies, where fine-structure is usually lost in noise. On the other hand, post-processing methods have the advantage over adaptive optics that seeing effects can be compensated over a much larger field-of-view (FOV). The reason for this is anisoplanatism. Light from different parts of an extended object are aberrated by penetration through different parts of the Earth's atmosphere, causing the point spread function (PSF) to vary with position in the image. This restricts the use of a single wavefront determination to a small area referred to as the isoplanatic patch, usually with a linear scale of a few arcseconds. In post-processing, sub-fields corresponding to different isoplanatic patches can be deconvolved with separately determined PSFs and combined as mosaics in the restored image.
A number of different methods have been conceived to find the best estimate of the real object from blurred images. Overviews of post-processing methods used in astronomical imaging are given by e.g Paxman (1993) and Dai (1995).
There are methods that do not require wavefront information and instead use a priori assumptions about the object or the PSF, such as non-negativity and spatial extent. Within this class of methods we find the Richardson-Lucy algorithm, Maximum Entropy methods and Blind Deconvolution.
Speckle methods rely on statistical models of turbulence to predict the average atmospheric point-spread function over a large number of realizations. Speckle interferometry was developed for night-time observations in the beginning of the seventies (Labeyrie 1970; Knox and Thompson 1974), and has later been adapted to solar imaging (von der Lühe 1984, 1993, 1994; Keller and von der Lühe 1992; de Boer et al. 1992; de Boer and Kneer 1992). The technique has provided excellent solar images, but in order to provide one reconstructed image, on the order of 100 images need to be collected in a short interval to justify the statistical approach. In order to make a movie or time-series, a vast amount of data has to be collected.
In wavefront-supported post-processing techniques, images are deconvolved with wavefronts measured with an auxiliary instrument, such as a Hartman-Shack wavefront sensor.
Phase-diversity techniques do not rely on statistical models, nor is any auxiliary wavefront sensor needed. Instead, they take advantage of the fact that information about the aberrations is encoded in the image and seek to use a combination of images which allows this information to be extracted. An appealing property of phase-diversity techniques is that the same images which are used to derive the wavefront are also used to restore the object.
The idea behind phase diversity (PD) is the following: An image is the combination of two unknown quantities, the object and the aberrations. With only one measurement, the unknowns cannot easily be separated. With two cleverly arranged measurements, however, they can. In PD, the clever arrangement is that the aberrations are perturbed with a known difference in phase (hence the name of the method) between two simultaneously collected images, for example by recording one of the images slightly out of focus. Given this extra information, an exact solution for both the unknown aberrations and the unknown object, to within the resolving power of the telescope, can be found if there is no noise in the data. All real data, however, contain noise. Existing PD methods therefore contain a noise model and attempts to provide an optimum estimate of the object with respect to this noise.
The ability of doing wavefront sensing and image deconvolution based on only two frames greatly reduces the amount of data that needs to be collected, as compared to e.g speckle techniques. This is particularly important when large detector arrays are used, since the read-out time of current CCD:s restricts the number of images which can be read out within a time interval which is short compared to the characteristic life time of the solar feature being observed. Using a small number of images also increases the probability for frame selection to pick out images with sufficient signal to noise ratio (SNR) at high spatial frequencies to allow good restorations.
The first PD technique was proposed by Gonsalves (1979, 1982), but went unnoticed within the astronomical community for some time. Independently, Högbom (1988a,b) at the Stockholm Observatory, discussed the possibility of a focal-volume method. He proposed that the information in two simultaneously recorded images, one in focus and the other out of focus, could be combined to find and correct for small wavefront aberrations. He also pointed out that adequate sampling of the focal volume provides all the relevant information for recovering the wavefront also when its amplitude is very large. Paxman et al. (1992) at the Environmental Research Institute of Michigan (ERIM) extended the method of Gonsalves to use several diversity channels and several realizations of the atmospheric turbulence to determine a single common object. As this extends the data set to essentially two data sets of the type used in speckle interferometry, they named this technique phase-diverse speckle imaging (PDS).
The simulations made by Gonsalves, Högbom, and Paxman and co-workers indicated that PD is a realistic concept. None of them had tried the method on real data. Restaino (1992) developed a technique based on a formulation by Roddier and Roddier (1991) to restore solar images from the Sacramento Peak telescope. The quality of the restored images, however, was far below the standard of that or other existing solar telescopes. At this point, no well tested implementation of PD techniques was available to the solar community.
The Swedish Vacuum Solar Telescope (SVST) (Scharmer et al. 1985) is well-known for its high optical quality and good location in La Palma, making it one of the world's best telescopes for high-resolution solar observations. Developing techniques to further improve this ability has been a priority at the SVST for ten years.
The frame-selection technique developed by Scharmer and co-workers consists in measuring the sharpness of recorded images in real-time and to keep only the best frames in any given time interval. This has been very successful and a new generation of this instrumentation has been used to collect the longest solar image sequences ever recorded with consistent high resolution (Simon et al. 1994).
Still, these recorded images are not free from aberrations. A wavefront-sensing and image deblurring system is needed to increase the rate of high quality images that can be collected with the SVST. Guided by Jan Högbom, who demonstrated his theoretical work and his simulations, I started looking at the possibilities of writing a program that could use simultaneously recorded focused and defocused images to produce nearly diffraction limited solar images. Göran Scharmer recorded some phase-diverse data in 1990, but it was not clear what was the best way to extract the information in the data. At the 13th Sacramento Peak Summer Workshop in 1992 (Radick 1994), I was made aware of the work of Gonsalves, Paxman, and Restaino. In particular, the work of Gonsalves provided the proper quantity to optimize. Since no implementation had yet demonstrated high-resolution restorations of real solar data, I decided to continue with my own implementation.
In 1992, Göran Scharmer started working with me on this project. Our guiding strategy with the implementation of PD techniques for solar observations came to be expressed as several partly overlapping goals: First, the main purpose was to develop a practical and useful method for improving the quality of images recorded with the SVST. Second, we wanted to have, and others to have, confidence in the restored images to enable quantitative measurements from the restored images to be made. Since this was a new technique, we decided that it was crucial to test the method and implementation very carefully and demonstrate as clearly as possible that the technique actually works on real data. We also planned to make the code available to colleagues; this required well-structured and transparent programming as well as simple handling of input and output data. Finally, the code had to be efficient to allow the processing of large data sets.
The core of the program was written during the fall and winter of 1992. Many important problems had to be solved before the program worked as expected. In the spring of 1993, we had a working program and a progress report was submitted (Löfdahl and Scharmer 1994a). The details of the implementation are given in Sects. I.2 and II.3.2. Here I will discuss some of the design choices we made.
Much of the solar photosphere is a low contrast object, with its ~10-15% RMS contrast granulation pattern. In such data, Poisson distributed noise can be well approximated with additive white (Gaussian) noise. Such a noise model leads to a considerable simplification in the PD formulation as compared to the more general model (Paxman et al. 1992). This simplification results in a much faster code, and it was therefore a natural choice for our implementation.
Finding the aberrations and the common object in a PD data set involves the nonlinear minimization of an error metric. Previous experience with nonlinear equations in the radiative-transfer code MULTI (Scharmer and Carlsson 1985) guided us to choosing a linearization approach. Methods of this type solve the problem iteratively, such that each iteration is computationally expensive because the solution to the non-linear equations is sought in a few large (accurate) corrections instead of by means of many small (inaccurate) corrections.
In each iteration we solve a linear system of equations. When this is to be done automatically, taking into account both the noise in the data as well as the limited information available from small sub-fields, the program must be able to handle singular or nearly singular matrices gracefully, which is not done by ordinary Gaussian elimination. The method of singular value decomposition (SVD) solves this problem in an optimal fashion by restricting the rank of the solution parameter space (Press et al. 1986). One reason that near-singular matrices may appear is over-parameterization, which is a well known problem in modeling of data. Thelen and Paxman (1994) discuss this for phase diversity. We have found that the SVD method handles this problem well also with severe over-parameterization.
The combination of linearization and SVD makes our algorithm similar to the Levenberg-Marquardt method (Lawson and Hanson 1974).
Anisoplanatism is a well-known problem in seeing compensation, both for adaptive optics and for post-processing. The easiest solution is to work around the problem by applying the wavefront sensing to small subfields. This, however leads to two problems. With small subfields, the information content is smaller, which makes the wavefronts less well determined. A choice of subfield size has to be made. The other problem is that the object as well as the influence of the point-spread function extends beyond the field of view (FOV). Because digital Fourier transforms assume periodic objects, this leads to a mismatch at the borders of the FOV, which becomes comparatively more severe when the FOV is small. To circumvent this problem, we perform all Fourier transforms and convolutions on a FOV which is larger than the assumed isoplanatic patch and then perform the optimization in the image plane, using only those pixels which are unaffected by artifacts or uncertainties at the boundaries of the image.
The work of Gonsalves provides an expression for the best estimate of the object in the presence of additive noise, but contains no mechanism by which to optimize the signal/noise in the restored object. In fact, the expression of Gonsalves leads to unlimited noise amplification at the highest spatial frequencies transmitted by the telescope. To remedy this problem, we have introduced an optimum filter into the formulation of the error metric, see Sects. I.2.3 and II.3.2.3.
For any single realization of an atmospheric wavefront, the optical transfer function will have zeroes in the Fourier plane at some spatial frequencies. At these frequencies, information about the object is lost. During the course of our testing we found that this resulted in artifacts in the estimated objects, but by combining separate inversions corresponding to two or more realizations of atmospheric wavefronts, which normally correspond to zeroes at different spatial frequencies, information about the object can be recovered from all spatial frequencies within the bandwidth limited by noise. We now refer to this approach as partitioned phase-diverse speckle imaging (PPDS), to distinguish it from the joint phase-diverse speckle imaging (JPDS) technique of Paxman et al. (1992).
The decision to use the interactive data processing language ANA (Shine et al. 1988) was more or less guided by momentum: Most of the SVST data analysis code is written in ANA. In retrospect, I believe this was a good choice, although a stand-alone code would probably be faster. I have sometimes missed the powerful data structures, higher-order parameters and recursion available in more general languages like Pascal or C, but working in an interactive environment makes testing easy and gives immediate access to image display and plotting of all variables. I would certainly not have wanted to write this code in FORTRAN.
Guided by the experiences with the 1990 data, we collected a much more extensive set of PD data for testing purposes in April 1993. Less than a year later, we had thoroughly examined the properties of the code both in simulations and with consistency tests using real solar data (Löfdahl and Scharmer 1994b (Paper I); shorter version in Löfdahl and Scharmer 1994c). Simulations are good for testing known error sources, but there is always the possibility that the model is too simple, or that all error sources are not included, or that the simulated data are affected by the same errors as the inversions. Only tests on real data can show that the program can handle real data. With real data, the consistency of the estimated quantities to expected statistical behavior and external references can be examined. We further tested our implementation and that of another group in a joint project (Paxman et al. 1996 (Paper II)).
Our ambition with these tests has been twofold: We wanted to gain confidence in our implementation and to ensure ourselves that our implementation performed well in comparison with the code of Paxman et al. We also wanted to establish suitable procedures for testing the implementation in the future. Here, I discuss the tests we performed.
Simulations (Sect. I.3) demonstrate what can be expected from systematic error sources. Our conclusions from these simulations are the following: The restored images are quite acceptable in good seeing conditions, where the RMS of the wavefront is on the order of 1/6-1/4 wave, with inversions using the 15 first Zernike polynomials and a FOV of ~5" squared. Realistic errors in the PD focus difference, inter-channel registration errors at the level expected after pre-processing, the calibration of the intensities of the two CCDs, as well as small differences in the image scale primarily give (small) systematic errors in the derived wavefronts but negligible effects on the restored images.
Image quality is the most important consideration for our implementation. We demonstrate temporal consistency of image restorations in Sect. I.4.3.2. 100 images of the solar granulation collected within 6 seconds should be almost identical, were it not for seeing. We demonstrate that 50 two-realization PPDS image estimates are very similar, and much more similar than the corresponding raw images. There is also spatial consistency, see Sect. I.4.3.2; inversions made separately in neighboring subfields match almost seamlessly when mosaiced to form a large FOV. In the example given in Fig. I.10 no smoothing or overlap is used. An external reference is provided by the comparison with a speckle restoration in Sect. II.5.4.
Although simulations indicate that errors are more likely in the restored wavefronts than in the image estimates, it is still interesting to examine the consistency of the wavefronts. Wavefronts estimated for the same atmospheric realizations, but at different positions are expected to be different because of anisoplanatism. They should show some degree of correlation, however, because telescope aberrations must be nearly the same across the FOV and, at least in the day time, much of the seeing disturbances occur near ground, which gives a large isoplanatic patch. Such correlations are demonstrated in Sect. I.4.3.1. Anisoplanatic effects are also investigated in Sect. II.5.5. Long-term temporal consistency of estimated wavefronts is demonstrated in Sect. I.4.2, where long-exposure aberrations are shown to evolve according to predictions from telescope tracking. This also provides an external reference.
In Sect. II.5.1 we demonstrate that PD is a robust technique in the sense that two (very) different implementations give virtually identical output from common input. We compare our PDS techniques in Sect. II.5.2, and in Sect. II.5.3, we investigate how the image quality depends on the number of atmospheric realizations.
The solar group of the Lockheed-Martin Palo Alto Research Laboratory have used the SVST for many years to collect high-resolution images of the solar photosphere. One of their main interests is the interaction between solar convective flows and magnetic fields. In order to provide data for a collaborative project, I spent three weeks in August 1995 at the SVST in sub-average seeing conditions, and managed to collect an almost two-hour long time-sequence of an active region, including a sunspot, some pores, and an area of G band bright points.
Tom Berger and Dick Shine of the LMPARL group collected better data in October, however. The raw data did not permit successful tracking of the bright points associated with strong magnetic fields but with the restored images provided by my program, it was possible to follow in detail the evolution of bright points, thereby enabling my co-authors to analyze their motions in detail (Berger et al. 1996).
This work demonstrates that our implementation of PDS is very useful for scientific data, and that PDS can realistically be used to restore long time sequences of data, with a FOV large enough to track dynamic phenomena in active regions on the Sun. I can only regret that it was not possible to record this data with an external shutter, since I feel strongly that the time delay of approximately 4 ms between the focused and defocused images gives noticeable problems with the inversions of this data set.
In this paper we present our implementation of phase diversity. We demonstrate solutions to several practical implementation problems. Using realistic simulations, we investigate the effects of various experimental errors. We present a number of consistency tests, to show that the technique works on real data. We show theoretically and demonstrate practically that aberration parameters derived from averaged images correspond to telescope aberrations. Inter-channel alignment parameters derived the same way are good estimates, that can be used as fixed parameters during the inversions of the individual frames. We demonstrate that using more than one phase-diverse image pair visibly improves the quality and reliability of the restored images.
As far as we know, the results presented here provide the first conclusive evidence to show that phase-diversity techniques allow near diffraction-limited imaging with a small solar telescope.
The implementation and testing of the PDS implementation was done in collaboration with my supervisor, Prof. Göran Scharmer. It is often difficult to reconstruct who did what during such a project, but there are some things that can be said. We discussed all the details of the implementation together. Scharmer contributed significantly to the theoretical formulations, in particular in the beginning of the project. I wrote all of the code, both the PDS programs and the various programs for testing the implementation and simulating aberrated images. We collected the data and designed the tests in Paper I together. I analyzed all the results of the tests.
We demonstrate that, for imaging of solar granulation, the Gaussian noise model is entirely adequate and object estimates are virtually identical using JPDS and PPDS.
The idea of comparing restorations made with different methods came to me early. Already before I had written the code, I suggested to Christoph Keller, who was using von der Lühe's speckle interferometry code for his observations, that we should compare PD restorations to those made with speckle. When I met Rick Paxman at the Sacramento Peak workshop, it felt natural to extend the collaboration to include also his group. With time, the project became more of a comparison between different implementations of PD techniques, using speckle interferometry as an external reference. The scope of the paper materialized when we met at conferences, through several email discussions and while I was visiting ERIM for half a week in October 1994. I performed the fidelity metric analysis in Sect. II.5.2 and the anisoplanatism analysis in Sect. II.5.5 and I did part of the comparison with speckle.
Here we use PDS for the first time to restore solar images to be used in a scientific investigation. We demonstrate that phase-diversity image restoration of solar photospheric images can produce the quality of time-series data necessary for following the evolution of solar fine structure down to nearly the resolution limit of the telescope. We discuss briefly a possible extension of the PDS concept, by using data in two different wavelengths in the estimation of the common wavefronts.
This paper represents the collaborative work of two graduate students with different time constraints. It will very likely evolve somewhat before it is submitted for publication, but even in its present form it provides a good illustration of the usefulness of PDS. The analysis results in conclusions about motions of magnetic elements and granulation in active network regions.
My contribution to Paper III is the PPDS processing of the data. Apart from providing restored images, this involved writing new software to efficiently handle large amounts of data and making repeated tests to ensure that the reconstructed images in two different wavelengths are as consistent as possible.
Phase-diverse speckle imaging, as implemented by us and by others, has been shown to work and to have advantages over other post-processing methods. Our code is developed for solar images, where the low contrast validates the assumption of additive Gaussian noise. In our experience, also higher contrast objects, such as pores and sunspots can be well, maybe even better, restored with our program. Our conjecture is that for such objects, the increased high-frequency information content makes up for the noise model mismatch.
The ERIM implementation has been used to compensate residual aberrations in images of a binary star collected with an adaptive optics system (Seldin et al. 1995). I think the combination of an adaptive mirror to improve the high-frequency content in the images with PDS techniques for providing the final corrected image is a promising approach for the future. A particularly attractive way to implement a combined approach is to use the phase-diversity data collected for post-processing also for slow wavefront sensing to control the adaptive mirror. In my experience, wavefront differences between separate light paths can change significantly during an hour of observing time, particularly when the beam splitter is introduced early in the beam. This effect would cause suboptimal wavefront correction using a separate Hartman-Shack wavefront sensor. By using a deformable mirror suitably located in the telescope and a focus control, it would be possible to ensure that images are always recorded with the highest possible Strehl ratio, thus giving data with the best possible SNR at high spatial frequencies.
Recent experiments (Kendrick et al. 1996) have shown that our linearized formulation of the phase-diversity problem is a good choice for PD mirror control. The estimated aberration parameters from the first iteration (which uses precalculated derivatives of the transfer functions) are good enough to reduce the aberrations in the next frame, also when the mirror is far from optimum compensation of the wavefront. The result is that the wavefront is iteratively corrected. The bandwidth has to be improved significantly, however, to make this useful in practice. The possibility of installing a "small" adaptive mirror, which corrects only for focus and astigmatism in real time has been studied and appears quite feasible with the SVST.
Acton et al. (1995) implemented the ``Löfdahl-Scharmer algorithm'' in IDL, and applied it to small subfields in order to measure the size of the isoplanatic patch of the Vacuum Tower Telescope in Tenerife for adaptive optics applications. Their results, together with our own measurements of the telescope aberrations of the SVST, suggest that one of the future applications of PD techniques may be for the testing of optics.
JPDS has the advantage over PPDS, that using several realizations to derive a single object along with the individual wavefronts makes the estimated quantities better determined by the data. Actually, we found from our collaboration with the ERIM group, that our code does not suffer significantly from this problem, even with inversions of low-contrast targets such as granulation. However, we suspect that in less favorable SNR conditions, JPDS should perform better than PPDS, which effectively determines the object separately for each of the realizations. Although not a high priority at the moment, it seems likely that I will modify my code in the future to make single object estimates from multiple realizations possible. On the other hand, the geometrical distortions caused by the seeing may violate the assumption of a common object, making PPDS a better choice. The most satisfying solution is of course to include anisoplanatism in the model of image formation, and phase-diversity is indeed capable of performing inversions of such models, as shown by Gonsalves (1994) and Paxman et al. (1994a,b). However, because in a strictly anisoplanatic code, the efficient PSF convolution model of incoherent image formation cannot be retained, the inversions become extremely time consuming. Clever approximations must be developed to speed up the computations, for this to be a serious alternative to the successful techniques of subfielding and mosaicing. This is one of many exciting challenges to researchers in this field.
I am deeply indebted to my supervisor, Prof. Göran Scharmer. He has spent a lot of time and effort on guiding me this far. Even more importantly, he didn't let me quit.
The staff at the Royal Swedish Academy of Sciences, Research Station for Astrophysics - in Stockholm and in La Palma - is thanked for assistance and for being good company. Particularly, I want to thank Wang Wei for his invaluable technical support. Dan Kiselman is thanked for many entertaining discussions about science - not necessarily astronomy or even physics. Birgitta Larsson has helped me with a lot of administrative problems.
The Stockholm Observatory has given me an opportunity to see how science is done in practice, and I have learned a lot from this. I am grateful to Jan Högbom for introducing me to the exciting field of image correction and for initiating my project. I have had a lot of fun together with my fellow graduate students.
It has always been a pleasure to collaborate with Rick Paxman and John Seldin. John is particularly thanked for the hospitality of his home. Many thanks to Dick Shine for writing ANA. Special thanks to Tom Berger for disrupting his own schedule in order to rush the completion of paper III.
I don't know how I would have survived without my floor-ball buddies and my friends in Fysikalen. I believe playing floor-ball kept me sane by providing a weekly outlet for all kinds of tensions - Fysikalen let me put any residual insanity to use on stage (Löfdahl 1992).
I owe a lot to Jonna, to my mother, and to my friends, for being there and for putting up with the far too little time I have been giving them from time to time.
My salary as a graduate student was paid by the Swedish Science Research Council (NFR), which is gratefully acknowledged. I also received financial support from the Royal Swedish Academy of Sciences (KVA) and from the Stockholm Observatory.