The idea for this text emerged over a number of years as the authors participated in various research projects related to analysis and interpretation of data from NASA's RHESSI Small Explorer mission. The data produced over the (unexpectedly long; February 2002 -- April 2018) operational lifetime of this mission inspired a large number of investigations related to the overarching science questions that RHESSI was designed to address: the when, where, and how of electron acceleration during solar flares in the stressed magnetic environment of the active Sun.A vital key to unlocking this science problem is the ability to produce high-quality images of the so-called ``hard'' X-rays produced by bremsstrahlung radiation from electrons that are accelerated over the course of a solar flare. To adequately address the science objectives, these images must cover the energy range from less than 10~keV up to more than 100~keV with ~1 keV energy resolution, spatial resolution in the arcsecond range, and a time cadence of order 1 second. The only practical way to do this within the technological and budgetary limitations of the RHESSI era was to eschew imaging via some form of focusing optics in favor of indirect imaging techniques. Such modality involves the use of multiple ``rotating modulation collimators,'' through which imaging information is encoded not in the usual ``pixel-by-pixel'' fashion of direct imaging devices, but rather as a set of modulation ``patterns'' produced by rotating an instrument made up of multiple bi-grid X-ray collimators, with each one placed in front of a detector that has no inherent spatial resolution. It can be shown that each such pattern corresponds to a different two-dimensional spatial Fourier component of the source and can be identified through the appearance of a particular signature in the time profile of the measured count rate in the corresponding detector. The image is ``reconstructed'' from this information using one of many computational techniques that have been developed, in many cases expressly for this purpose.It is fair to say that this way of thinking about imaging was not one with which the solar physics community was entirely comfortable, at least initially. Radio astronomers, working with large interferometer arrays, had employed Fourier imaging techniques for many years, primarily for observations of astronomical sources other than the Sun. They typically were able to utilize many hundreds of Fourier components because of the many different baselines that can be formed with an array of radio telescope dishes. They also benefited from a strong signal-to-noise ratio, a consequence both of the very large collecting area of their Earth-based telescopes and of the very small energy per radio photon. By contrast, X-ray images produced by RHESSI had to be constructed from a much more limited number (typically 100) of sparsely distributed Fourier components. Furthermore, the limited collection area available on a low-cost space mission, combined with the relatively high energy per X-ray photon, meant that, despite the relative proximity of the Sun, there were many fewer photons available with which to construct each image; the associated Poisson statistics for such low count rates meant that observers had to deal with much noisier data. Combined, these essential differences meant that the extensive analysis software that already existed for radio astronomy provided only a starting point for developing new techniques optimized for this new context. To make things even more interesting, construction of images from sparse, noisy, Fourier data is hardly intuitive, and extensive testing and validation of the methods was necessary to ensure that they produced images with sufficient accuracy and fidelity for solar scientists to employ them in addressing compelling science problems.This book summarizes the results of this development of image reconstruction techniques specifically designed for this form of data. It covers a set of published works that span over two decades, during which, it is fair to say, there was very little in the way of a guiding ``script.'' Over this extended period of time, as the various image reconstruction techniques were introduced, developed, validated, and applied to observations, it became more and more apparent to the authors that it would be a good idea to put together a compendium of these methods and their applications. Hence the book you are now reading. The order in which the various image reconstruction methods are presented reflects not so much the chronological order of their development but rather the similarities and differences among them, and the degree to which they may (or may not) be useful in addressing the science problems for which they were created.weiterlesen