  Detection of moving sources over complicated background is important for
several reasons. First is measuring the astrophysical motion of the source.
Second is that such motion resulting from atmospheric scintillation, color
refraction, or astrophysical reasons is a major source of false alarms for
image subtraction methods. We extend the Zackay, Ofek, and Gal-Yam image
subtraction formalism to deal with moving sources. The new method, named
translient (translational transient) detector, applies hypothesis testing
between the hypothesis that the source is stationary and that the source is
moving. It can be used to detect source motion or to distinguish between
stellar variability and motion. For moving source detection, we show the
superiority of translient over the proper image subtraction, using the
improvement in the receiver-operating characteristic curve. We show that in the
small translation limit, Translient is an optimal detector of point source
motion in any direction. Furthermore, it is numerically stable, fast to
calculate, and presented in a closed form. Efficient transient detection
requires both the proper image subtraction statistics and the translient
statistics: when the translient statistic is higher, then the subtraction
artifact is likely due to motion. We test our algorithm both on simulated data
and on real images obtained by the Large Array Survey Telescope (LAST). We
demonstrate the ability of translient to distinguish between motion and
variability, which has the potential to reduce the number of false alarms in
transients detection. We provide the translient implementation in Python and
MATLAB.
