How to coadd images? II. A coaddition image that is optimal for any
  purpose in the background dominated noise limit
  Image coaddition is one of the most basic operations that astronomers
perform. In Paper~I, we presented the optimal ways to coadd images in order to
detect faint sources and to perfrom flux measurements under the assumption that
the noise is approximately Gaussian. Here, we build on these results and derive
from first principles a coaddition technique which is optimal for any
hypothesis testing and measurement (e.g., source detection, flux or shape
measurements and star/galaxy separation), in the background-noise-dominated
case. This method has several important properties. The pixels of the resulting
coadd image are uncorrelated. This image preserves all the information (from
the original individual images) on all spatial frequencies. Any hypothesis
testing or measurement that can be done on all the individual images
simultaneously, can be done on the coadded image without any loss of
information. The PSF of this image is typically as narrow, or narrower than the
PSF of the best image in the ensemble. Moreover, this image is practically
indistinguishable from a regular single image, meaning that any code that
measures any property on a regular astronomical image can be applied to it
unchanged. In particular, the optimal source detection statistic derived in
paper~I is reproduced by matched filtering this image with its own PSF. This
coaddition process, which we call proper coaddition, can be understood as a the
maximum signal-to-noise ratio measurement of the Fourier transform of the
image, weighted in such a way that the noise in the entire Fourier domain is of
equal variance. This method has important implications for multi-epoch
seeing-limited deep surveys, weak lensing galaxy shape measurements, and
diffraction-limited imaging via speckle observations. The last topic will be
covered in depth in future papers. We provide an implementation of this
algorithm in MATLAB.
