An Efficient Shift-and-Stack Algorithm Applied to Detection Catalogs

The boundary of solar system object discovery lies in detecting its faintest members. However, their discovery in detection catalogs from imaging surveys is fundamentally limited by the practice of thresholding detections at signal-to-noise (SNR) $\geq 5$ to maintain catalog purity. Faint moving objects can be recovered from survey images using the shift-and-stack algorithm, which coadds pixels from multi-epoch images along a candidate trajectory. Trajectories matching real objects accumulate signal coherently, enabling high-confidence detections of very faint moving objects. Applying shift-and-stack comes with high computational cost, which scales with target object velocity, typically limiting its use to searches for slow-moving objects in the outer solar system. This work introduces a modified shift-and-stack algorithm that trades sensitivity for speedup. Our algorithm stacks low SNR detection catalogs instead of pixels, the sparsity of which enables approximations that reduce the number of stacks required. Our algorithm achieves real-world speedups of $10$--$10^3 \times$ over image-based shift-and-stack while retaining the ability to find faint objects. We validate its performance by recovering synthetic inner and outer solar system objects injected into images from the DECam Ecliptic Exploration Project (DEEP). Exploring the sensitivity--compute time trade-off of this algorithm, we find that our method achieves a speedup of $\sim30\times$ with $88\%$ of the memory usage while sacrificing $0.25$ mag in depth compared to image-based shift-and-stack. These speedups enable the broad application of shift-and-stack to large-scale imaging surveys and searches for faint inner solar system objects. We provide a reference implementation via the find-asteroids Python package and this URL: https://github.com/stevenstetzler/find-asteroids.