Abstract
As emphasized in many places above, RHESSI and STIX are “Fourier imagers”—the native form of the data is a set of spatial Fourier components, deduced from analysis of either the temporal modulation of the detector count rates as the spacecraft rotates (in the case of RHESSI) or the sets of Moiré patterns (in the case of STIX). In this Chapter we formalize this concept through the definition of “visibilities”—components of the spatial Fourier transform of the source image at spatial frequencies sampled by the instrument. We then proceed to discuss a variety of image reconstruction algorithms that are optimized to a dataset that consists of a sparse number of measured visibilities, and we compare their strengths and limitations. Finally, we discuss an ingenious method that inverts the order of the spatial and spectral inversion processes in proceeding from count-based visibilities to images of the mean source electron spectrum, first by spectrally inverting the count-based visibilities to obtain the visibility values associated with the electron flux spectrum, and then performing the spatial Fourier-based inversion to obtain images of the mean source electron spectrum. By virtue of the manner in which they are constructed, such images vary sufficiently smoothly with electron energy E to permit application to flare studies, discussed in the next chapter.
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Notes
- 1.
A copy of the poster presented with the abstract in [97] is available at https://hesperia.gsfc.nasa.gov/rhessi3/pics/rhessi_visibility_poster_SPD_2005.pdf.
- 2.
See further discussion of this issue in Sect. 3.3.
- 3.
The default numbers of visibilities used in the IDL analysis software are as high as 64 for the detectors behind the finer grids, decreasing to as low as 6 for detector #9, depending on the flare location with respect to the spin axis.
- 4.
An analogous approach can be realized when the measurements are the recorded counts.
- 5.
Of course, other choices are possible, depending on the a priori information one has at one’s disposal.
- 6.
It should be noted that the phase of the visibilities is determined by the distance of the (arbitrarily chosen) phase center (x o, y o) from the center (x 1, y 1) of the Gaussian. Because of the factors multiplying the Fourier-space coordinates u and
in this expression, a large distance between the phase center and the image centroid produces a visibility function 
that varies very rapidly, making construction of a useful set of discrete “observed” visibilities very challenging, if not impossible. Success of the method therefore depends on choosing the phase center at a point relatively close to the Gaussian centroid of the image. - 7.
- 8.
The form of the quantity Q(𝜖, E) depends on the emission process being considered. In principle, this could include a host of emission processes, such as gyrosynchrotron emission, inverse Compton emission, free-bound emission, and bremsstrahlung. We here take the form of Q(𝜖, E) as that corresponding to bremsstrahlung, and we use the isotropic form of the cross-section in [105].
in this expression, a large distance between the phase center and the image centroid produces a visibility function 
that varies very rapidly, making construction of a useful set of discrete “observed” visibilities very challenging, if not impossible. Success of the method therefore depends on choosing the phase center at a point relatively close to the Gaussian centroid of the image.References
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Piana, M., Emslie, A.G., Massone, A.M., Dennis, B.R. (2022). Visibility-Based Imaging Methods. In: Hard X-Ray Imaging of Solar Flares. Springer, Cham. https://doi.org/10.1007/978-3-030-87277-9_6
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