ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-CelestialObject O O O O O O B-Telescope B-Telescope O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: quantifies in Fourier space the sum of the square of the difference between the data and the restored object, degraded with the current estimated individual transfer functions . In the usual case that the image data has a good signal to noise ratio (S/N), the regularization constant may be set to zero. Equation ( 1 )makes use of the assumption of additive, Gaussian noise, in which case the common estimate of the Fourier transform of object , , for a given set of transfer functions is given by (see e.g., Paxman et al. 1996 ) in which the object is obtained by deconvolution of the average of a purposely degraded dataset, with the average over the autocorrelation function of the individual transfer functions. Since the data have already been degraded by the , the additional degradation of the data with the complex conjugate of the results in a dataset that has been degraded with a PSF that is centered on the origin, and which can therefore be added to other such data, without further degrading the result. We note that it is implicitly assumed that the transfer function is independent of the spatial coordinates, so that the convolution operator can be conveniently expressed as a multiplication in Fourier transformed space. While this is in reality not quite true, the assumption leads to a considerable reduction in computational effort, and for a 1 m-class solar telescope it is normally a good assumption over an area smaller than about 5 5. In practice, an image is therefore divided in tiles no larger than approximately 5 5, restored on a tile-by-tile basis, and then re-assembled to a single restored image, although this may become more problematic with the large 4 m-class solar telescopes currently planned or under construction (DKIST, EST), where this area is expected to reduce to only 2 2. The initial assumption made by the MFBD process, that the PSF is the result of a pure phase error of the wavefront entering the telescope pupil, is clearly only true if the image is recorded significantly faster than the time scale over which the wavefront changes. This so-called seeing freezing time scale is variable, dependent on the telescope diameter, and has been determined in the past to typically lie at or below 10 ms ( Tarbell Smithson 1981 ) for a 1 m-class telescope. If this time is exceeded significantly, the assumption of the PSF being the result of a pure phase error is no longer valid, and the fitted wavefront cannot be unambiguously interpreted as a wavefront anymore, as it describes a PSF that has been produced by integration over time of several wavefronts, which is inconsistent with the formation model. Due to the non-linearity of the mapping from wavefront to PSF, the correspondence of PSF to wavefront is generally non-linear and not unique. The latter, however, can be addressed by providing additional constraints, such as phase diversity, that are effortlessly included in the formalism used here, and have been used to reduce or even remove