  Digital co-addition of astronomical images is a common technique for
increasing signal-to-noise and image depth. A modification of this simple
technique has been applied to the detection of minor bodies in the Solar
System: first stationary objects are removed through the subtraction of a
high-SN template image, then the sky motion of the Solar System bodies of
interest is predicted and compensated for by shifting pixels in software prior
to the co-addition step. This "shift-and-stack" approach has been applied with
great success in directed surveys for minor Solar System bodies. In these
surveys, the shifts have been parameterized in a variety of ways. However,
these parameterizations have not been optimized and in most cases cannot be
effectively applied to data sets with long observation arcs due to objects'
real trajectories diverging from linear tracks on the sky. This paper presents
two novel probabilistic approaches for determining a near-optimum set of
shift-vectors to apply to any image set given a desired region of orbital space
to search. The first method is designed for short observational arcs, and the
second for observational arcs long enough to require non-linear shift-vectors.
Using these techniques and other optimizations, we derive optimized grids for
previous surveys that have used "shift-and-stack" approaches to illustrate the
improvements that can be made with our method, and at the same time derive new
limits on the range of orbital parameters these surveys searched. We conclude
with a simulation of a future applications for this approach with LSST, and
show that combining multiple nights of data from such next-generation
facilities is within the realm of computational feasibility.
