creation of images by iterated functions

1.                  Experimental copier

—                 Take an initial image I₀

—     Produce new image I₁ by superimposing several reduced versions of I₀

—     Feed I₁ back to the copier to generate I₂

—     Repeat the process to obtain a sequence of images I_0, I_1, I_2,… called orbit of I₀

2.S-copier

-                      Superimposes three smaller versions of whatever image is fed onto it and repeated iterations may result in Sierpinski triangle

Generating images

—     The 3 lenses present in the copier reduces the i/p image to one half of its size and moves it to a new position

—     The reduced and shifted images are superimposed on printed o/p

—     Each lens performs its own affine transformation

IFS – it is a collection of ‘N’ affine transformations T_i  for I = 1,2,…N         

Theory of copying process

Method:

-                      Using the lenses present in the copier draw the o/p image by transforming the points present in the i/p image

Working:

 Let I be the i/p image to the copier, then the i-th lens builds the new set of points denoted by  T_i (I) and adds them to the image being produced at the current iteration.

The o/p image is obtained by superimposing the three transformed images created by the three lenses

                        o/p image = T₁(I) U  T₂(I) U T₃(I)

Overall mapping from i/p to o/p is denoted by W(.)

                        W(.) = T₁(.) UT₂(.) UT₃(.)

Drawing the  k-th iterate

Choices of Initial images ( I₀ )

1.                  Polyline

2.                  Single point

Chaos Game ( Random Iteration Algorithm)

•                produces the picture of an attractor

•                produces pictures in a non recursive way

Ex. Sierpinski gasket