Given these transformations, we know how to construct a number of specifically placed and oriented smaller versions of the original image. If we overlay all of the available smaller versions (one for each transformation), then the image formed should look exactly like the original image too. This gives rise to the two simple rules constraining the IFS transformations:

Rule 1:

All transformations must be contractive. This means that the image produced by each transformation must be smaller than the area of the original whole image. It is this property that makes the overlayed transformed copies of an approximation a better approximation. This idea is played upon by the death construction algorithm detailed below.

Rule 2:

The transformed copies must cover the original image properly when overlayed. This is demonstrated below. This is required when re-constructing the image so that no part of the image is missed out, or over-written by a badly fitting copy. Note that we need at least two transformations in all cases.

For a more compact coding of the transformations, we can represent all the scalings, rotations and skews by four numbers representing a single 2x2 matrix. The translation vector is simply stored as two numbers for the distance along each axis. For consistency in this document, we always apply the matrix before translation. We label these values a-f.

            (x') = (a b)(x) + (e)
            (y')   (c d)(y)   (f)
            

Note that values a-f control scaling and rotation (and possibly skewing in more advanced transformations). The values e and f are the absolute translations.

A matrix that simply scales everything by a factor s relative to the origin looks like this:

            (s 0)
            (0 s)
            

A matrix that rotates everything t degress/radians anti-clockwise around the origin looks like this:

            (cos t  -sin t)
            (sin t   cos t)
            

To form the final matrix used in the IFS transform above, we multiply these two matrices. Due to the simple nature of the above matrices, the order does not matter in this case. This is not generally true, however:

            (cos t  -sin t) (s 0)   (s.cos t  -s.sin t)
            (sin t   cos t) (0 s) = (s.sin t   s.cos t)
            

But what use is that?

Now, lets say we want to recreate the image from just these IFS transformatation values. On the face of it this looks impossible, since we start from nothing and hence all we can do is make smaller copies of the nothing we have!!!

However, if we start with some approximation to the image, then we will get a number of smaller approximations to the image that can be fitted together to form another approximation to the image.

The cunning thing is, when we overlay all the approximations we have (i.e. one from each transformation), we actually obtain a better approximation to the original image! This idea is fundamentally behind all the IFS reconstruction methods explained in the following documents.