2 Basic equations

Here we briefly summarize the basics of image subtraction (Alard & Lupton 1998). The essence of the method is to find a convolution kernel (K), that will transform a reference image (R) to fit a given image I.

In terms of least-squares this is equivalent to solving for a kernel that will minimize the sum:

$$\sum_i \left ([R \otimes K](x_i,y_i) - I(x_i,y_i) \right )^2.$$ (1)

Provided the kernel can be decomposed onto basis functions, the above equations become a simple linear least-squares problem. For the kernel decomposition we take:

$$K(u,v) = \sum_n a_n \ K_n(u,v)$$

with:

$$K_n(u,v) = {\rm e}^{-(u^2+v^2)/{2 {\sigma_k}^2}} \ u^i \ v^j.$$

and the generalized index n = (i,j,k).

The kernel solution can be calculated by solving the following linear system:

$$M \bf a = \bf B$$ (2)

with:

$$M_{ij} = \int [R \otimes K_i](x,y) \ \frac{[R \otimes K_j](x,y)}{\sigma(x,y)^2} \ {\rm d}x{\rm d}y.$$

and:

$$B_i = \int I(x,y) \ \frac{[R \otimes K_i](x,y)}{\sigma(x,y)^2} \ {\rm d}x{\rm d}y.$$