# Tutorial: AstroImageJ Align and Drizzle

## 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 change and correct a lot here in the beginning. But follow it if you find it interesting to see how I develop the function 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.

• 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:

• Transformation matrix, Wiki/
•  Unit matrix 1 0 0 0 1 0 0 0 1

Unit matrix, does nothing to the image.

 Translation matrix 1 0 X 0 1 Y 0 0 1

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

 Scale matrix W 0 0 0 H 0 0 0 1

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

 Rotation matrix Cos(v) Sin(v) 0 -Sin(v) Cos(v) 0 0 0 1

Rotation matrix, rotate the image by angle v.

Altogether this matrixes will correspond to: T1 T2 T3 T4 = Ttot

 Overall matrix, Ttot W*Cos(v) W*Sin(v) X -H*Sin(v) H*Cos(v) Y 0 0 1

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 one pixel coordinate, xij and yij, 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 Ttot will do a affine transformation, all the three transformations above, to that we need three reference stars. It will not correct optical distortion.

 X coordinates (reference star in sub image) x1 x2 x3 y1 y2 y3 1 1 1

 X' coordinates (reference star in reference image) x'1 x'2 x'3 y'1 y'2 y'3 1 1 1

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.

### Inverse matrix:

Now one problem, we need to know how to calculate the matrix T, its internal figures.

X' = T X.

With help of the invers matrix X-1 we can get T alone on the right side, in matrix manipulation we normally can't change order of the matrices. It looks a bit strange if you are not used with it.

X-1 is of the construction that X-1 X give the unit matrix I (I=1), se above.

X' X-1 = T X X-1 = T I = T

T = X' X-1

Now we get one more problem, to calculate X-1.

Later when we have find T values for each sub image we can put in each sub images coordinates one pixel by one to the transformation T matrix and get the new coordinates in the reference image.