For i = 1 the original o-pattern can be seen.

For i = 2^n-1 we see again the starting points with different gray values corresponding to the arrangement in the the original o-pattern.

For i = 2^n steps (where n = 1, 2, ...) the o-pattern made of o-patterns made of o-patterns ... can be seen.

FItIn - algorithm in 2 dimensional space:

P1: starting image pattern, P2: fractal filter image pattern, P1' : resulting image pattern

repeat {

for all points (x1, y1) in P1 {
P1'(x2, y2)=P1(x2, y2)

for all points (x2, y2) in P2 {

P1'(x2, y2)= P1'(x2, y2) XOR (P1(x1-x2, y1-y2) AND P2(x2, y2))
}
}

Application:

P2 can be a key for encryption. Any mapping (x2, y2) => (x2', y2') dependent on (x1, y1) can provide for additional encryption strength. The strength increases with the number of iterations. The encryption can be applied partially in the image.

P1: starting image pattern, P2: fractal filter image pattern, P1' : resulting image pattern

repeat {

for all points (x11, x21, x31 , ... , xn1) in P1 {
P1'(x11, x21, x31 , ...) = P1(x11, x21, x31 , ...)

for all points (x12, x22, x32 , ... , xn2) in P2 {

P1'(x11, x21, x31 , ...) = P1'(x11, x21, x31 , ...) XOR (P1(x11-x12, x21-x22, x31-x32, ...) AND P2(x12, x22, x32 , ...))
}
}