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.
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