To detect objects that vary in brightness or spatial coordinates over time, C. Alard and R. H. Lupton in 1998 proposed an “optimal image subtraction” (OIS) method that constructs a convolution kernel from a set of matching stars distributed across the two images to be subtracted. Using multivariable least squares, the kernel is derived and can be designed to vary by pixel coordinates across the convolved image. Local effects in the optics, including aberrations or other spatially sensitive perturbations to a perfect image, can be mitigated. This paper presents the specific systems of equations that originate from the OIS method. Also included is a complete description of the Gaussian components basis vectors used by Alard & Lupton to construct the convolution kernel. An alternative set of basis vectors, called the delta function basis, is also described. Important issues are addressed, including the selection of the matching stars, differential background correction, constant photometric flux, contaminated pixel masking, and alignment at the subpixel level. Computer algorithms for the OIS method were developed, written using the Interactive Data Language (IDL), and applications demonstrating these algorithms are presented.