Methods

Alard & Lupton

Alard and Lupton [alard1997] introduced a method to simultaneously fit a background and a convolution kernel that will minimize the difference between a reference image and an another science image.

This method assumes that the convolution kernel can be approximated by a linear combination of fixed Gaussians, where the linear coefficients are left free to vary. The centre, width and orientation of the Gaussians are fixed beforehand as well as the number of Gaussians to use.

In addition, each Gaussian is modulated with a polynomial of an arbitrary degree.

In OIS this is specified with a list of dictionaries, one for each gaussian we want to use. Below is an example:

gausslist = [{center: (5, 5),
              sx: 2.
              sy: 2.
              modPolyDeg: 3},
             {sx: 1.0
              sy: 2.5
              modPolyDeg: 1},
             {sx: 3.0
              sy: 1.0},
            ]

In this example ois will return the kernel with the optimal linear combination of the three Gaussians provided.

Not all the keys need to be present. The dictionary keys are as follows:

center: the pixel x, y coordinates of the center relative to the kernel origin (column=0, row=0). For a 11 rows by 13 columns kernel, the center of the kernel would be at x, y = (6, 5) If not provided, ois will assume the kernel center calculated from its shape.

sx: sigma in the x direction

sy: sigma in the y direction

modPolyDeg: The degree of the modulating polynomial. If not provided, it will default to 2.

Bramich

If no list of Gaussians is provided, ois will default to use the method for image subtraction developed by

Bramich [bramich2008] modifies Alard-Lupton making each pixel of the kernel an independent parameter to fit. This is equivalent as having a vector basis consisting of Delta kernels. It can also simultaneously fit a polynomial background.

This method does not make assumptions on the kernel shape and can thus model completely arbitrary kernels. It can also correct for small translations between the images.

While more accurate, this method is computationally more expensive than Alard & Lupton’s.

Warning

Since each pixel is treated independently, a 11 by 11 kernel will have 121 free parameters just for the kernel. It grows quadratically with the kernel side. This needs to be taken in consideration for large kernels.

Adaptive Bramich

Like Bramich, this method also treats each pixel independently, but it will also multiply each pixel by a polynomial on the coordinates of the image.

This requires a special type of convolution where the kernel varies point to point in the image.

It is especially suited for situations where the PSF varies significatively across the image.

The method is described in more detail in [miller2008].

Warning

Just like Bramich, the number of free parameters scales quadratically with the kernel side. Furthermore, the degree of the polynomial multiplies the number of parameters by (deg + 1) * (deg + 2) / 2. This needs to be taken in consideration for large kernels.

[alard1997]

“A method for optimal image subtraction” - C. Alard, R. H. Lupton, 1997.

[bramich2008]

“A New Algorithm For Difference Image Analysis” - D.M. Bramich, 2008.

[miller2008]

“Optimal Image Subtraction Method: Summary Derivations, Applications, and Publicly Shared Application Using IDL” - J. PATRICK MILLER et al., 2008.