# Morph Maths

Boredom results in learning!
So I was messing about on this site: http://www.morphthing.com/ when I thought of an old simple crypto program I wrote, it used matricies to encrypt/decrypt sentences. It used a matrix to encode and the inverse matrix to decode: $ABA^{-1} = BI$
For example, $B$ is the original text, $A$ is the cypher text and that makes $A^{-1}$ the key.

That other letter, $I$, is the Identity matrix and it is the equivalent of 1, $BI = B$ like $1a = a$ so it’s nothing here.

Here is an old screenshot:
Anyway, this relates to the site I was on because I wondered is it possible to extract an image from a morph? Well, my idea was to get one of the images used in a morph and “invert” it, then just morph it again. I decided that i’d use the inverted colours as the inverse. It seemed… logical.

The experiment:
Let $A$ be an image of Avril Lavigne and $E$ an image of Emma Watson.

My hypothesis:
$AE * A^{-1} = E$

 $AE$ $A^{-1}$ $AE * A^{-1}$

It seems that $AA^{-1}$ does not equate the Identity matrix $I$, but instead what I will call grey-noise $G$. This makes sense as the colours are probably being overlayed in translucent layers, as we inverted the colour it would neutralise the picture. It roughly worked but there is ofcourse differences where the morph has moved the original image.

Result:
$AE * A^{-1} \approx EG$