  A simple, yet general, formalism for the optimized linear combination of
astrophysical images is constructed and demonstrated. The formalism allows the
user to combine multiple undersampled images to provide oversampled output at
high precision. The proposed method is general and may be used for any
configuration of input pixels and point spread function; it also provides the
noise covariance in the output image along with a powerful metric for
describing undesired distortion to the image convolution kernel. The method
explicitly provides knowledge and control of the inevitable compromise between
noise and fidelity in the output image. We present a first prototype
implementation of the method, outlining steps taken to generate an efficient
algorithm. This implementation is then put to practical use in reconstructing
fully-sampled output images using simulated, undersampled input exposures that
are designed to mimic the proposed \emph{Wide Field InfraRed Survey Telescope}
(\emph{WFIRST}). We examine results using randomly rotated and dithered input
images, while also assessing better-known "ideal" dither patterns: comparing
results we illustrate the use of the method as a survey design tool. Finally,
we use the method to test the robustness of linear image combination when
subject to practical realities such as missing input pixels and focal plane
plate scale variations.
