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When the data are consistent with a single ideal PSF, the reconstruction is exact. However, when the data are noisy and inconsistent with the ideal PSF, the reconstructed image is a linear combination of the true PSF and the PSF as modified by the current set of weights. The PSF as modified by the current set of weights is a nonlinear function of the data and this nonlinearity may cause artifacts in the reconstructed image. It is possible that some artifacts will be more pronounced than others and certain artifacts will also depend on the number of data points included in the patch, but the overall quality of the image will be improved.
The OASIS algorithm attempts to solve this problem by taking into account the noise in the data and matching the expected statistical fluctuations in the observed data with the statistical fluctuations of the data expected from the ideal PSF. In the presence of noise, the measured PSF is a noisy version of the ideal PSF and the OASIS algorithm searches for a set of weights that minimizes the difference between the ideal PSF and the PSF as modified by the current set of weights. The algorithm is initialized by setting the weights to the reciprocal of the square root of the data variance. At each step, the weights are adjusted by a small amount such that the mean-square difference between the ideal PSF and the PSF as modified by the current set of weights is reduced. The algorithm is terminated when the weights remain unchanged for a pre-specified number of iterations.
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