Tutorial:
AstroImageJ Align and Drizzle

Content:

1. Introduction matrix
2. Align points test
3. Size of output matrix
4. Inverse matrix
5. Determinant
6. Matlab and Octave
7. Parallel Super-Resolution Plug in
8. Parallel Super-Resolution Plug in, test 2
9. Temporary solution with new workflow

Note:
I take no responsibility or liability for what are written here, you use the information at your own risk!

1, Introduction:

This is an attempt from me to create a macro to AstroImageJ to align and stack sub images with a Drizzle function.

Expect me to edit and correct a lot here in the beginning. But follow it if you find it interesting to see how I develop the functions and see if I reach the goal, a working Drizzle function!

Drizzle:

Drizzle is a way to increase the resolution of under sampled images, i.e. when you have big size pixels on a high resolution telescope.

More information about Drizzle:

• Drizzle image, Wiki/

Matrix:

First we, especially me have to dig deeper in the mathematical word about matrices and transformation. Uses of matrices in the calculations can make it a lot easier to do it.

Here you find deeper information about matrices:

• Transformation matrix, Wiki/

The base tranformation matrices:

Unit matrix, does nothing to the image.

Translation matrix, shifts the image in x and y axis.

Scale the x (W=width) and y (H=height) axis, many times W could be set equal to H, symmetry.

Rotation matrix, rotate the image by angle v.

Altogether these matrices correspond to: T₁ T₂ T₃ T₄ = T_tot

Translation, Scale and rotation matrices (functions) put together in one matrix.

How to use T:

X' = T X, T transform the coordinates of X to X'.

X consist of the pixel coordinates, x_ij and y_ij, i=row and j=column. X' is the new coordinates. First we must find out what T is. To that we use our reference star as we get when we align our sub images. To X we put in our reference stars coordinates from our sub images. X' is the reference stars coordinates reference image (one of the sub images normally). One T matrix for each of your sub images will transform them to the reference image coordinates.

The T_tot will do an Affine transformation, all the three transformations above, to that we need three reference stars. Note: It will not correct optical distortion.

Each column are the coordinates of the center of a reference star, x₁, y₁ and x'₁, y'₁ and so on should align on each other after the transformation (align) process. That is what matrix T_tot take care of.

Warping:

My first plan was to limit it my translation to an Affine translation, I don't want it be to complex to solve. But after some talk in the Swedish forum Astronet about optical distortion I feel I must also have correction of distortion from the beginning.

This is a more complex translation and it often is called warping. With that I need a higher degree of polynomial in the transfer function.

Here is a very useful explanation about what I aim for:

• Image Warping by Mikkel Stegmann

Higher degree polynomial:

The general form of a high degree polynomial looks like this:

x' = Sum_i Sum_j a_ijxⁱy^j
y' = Sum_i Sum_j b_ijxⁱy^j

Remember from school: x⁰ = 1 and x¹ = x, always!

If we limit it to a second degree polynomial it will look like this:

x' = a₀ + a₁x + a₂y + a₃x² + a₄y² + a₅xy
y' = b₀ + b₁x + b₂y + b₃x² + b₄y² + b₅xy

Or in matrix form:

X' = Zt

X' is a two row column vector.

Z is a matrix of two rows and 12 columns.

t is a vector of 12 rows and one column.

Twelve unknown parameters that we have to solve. Each reference star give us two equations, x' and y'.

Now I have some understanding and overview of what I have to do.

There is also a couple of Warp plugins to ImageJ that I maybe could built my own function around, however, they are more specialized to work with microscope images.