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Abstract
The formalism of Wiener filtering is developed here for the purpose of
reconstructing the largescale structure of the universe from noisy,
sparse, and incomplete data. The method is based on a linear minimum
variance solution, given data and an assumed prior model which specifies
the covariance matrix of the field to be reconstructed. While earlier
applications of the Wiener filer have focused on estimation, namely
suppressing the noise in the measured quantities, we extend the method
here to perform both prediction and dynamical reconstruction. The Wiener
filter is used to predict the values of unmeasured quantities, such as
the density field in unsampled regions of space, or to deconvolve
blurred data. The method is developed, within the context of linear
gravitational instability theory, to perform dynamical reconstruction of
one field which is dynamically related to some other observed field.
This is the case, for example, in the reconstruction of the real space
galaxy distribution from its redshift distribution or the prediction of
the radial velocity field from the observed density field.
When the field to be reconstructed is a Gaussian random field, such as
the primordial perturbation field predicted by the canonical model of
cosmology, the Wiener filter can be pushed to its fullest potential. In
such a case the Wiener estimator coincides with the Bayesian estimator
designed to maximize the posterior probability. The Wiener filter can be
also derived by assuming a quadratic regularization function, in analogy
with the "maximum entropy" method. The mean field obtained by the
minimal variance solution can be supplemented with constrained
realizations of the Gaussian field to create random realizations of the
residual from the mean.
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