Fractal Image Compression: Understanding the Concepts and Techniques, Study notes of Electrical and Electronics Engineering

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Fractal Image Compression

Dennis Lin

ECE 547

18 September 2008

Iterated Function System (IFS)

A class of fractals Consists of: A complete metric space (X , d) – Usually R^2 with the Euclidean metric A set of contractive mappings wi defined on X – Usually linear like

wi

([ x y

])

[ ai bi ci di

] [ x y

]

[ ei fi

]

Can shrink (never enlarge), shift, rotate (and skew) Definitions of the terms and other rigorous details later – For now, think of a copy machine...

If we keep going...

open in external viewer

What if we started differently?

... with, say, a single dot

open in external viewer

Recall...

An IFS consists of: A complete metric space (X , d) – Usually R^2 with the Euclidean metric A set of contractive mappings wi defined on X – Usually linear like

wi

([

x y

])

[

ai bi ci di

] [

x y

]

[

ei fi

]

Metric Space

A set of points X and a distance function d The d(·) function must have some nice properties, such as satisfying the triangle inequality We won’t worry about the details here A common example is R^2 with the Euclidean distance - i.e. the “normal” 2D plane

A Bigger Space

The fern doesn’t live in R^2 – it lives in (H(R^2 ), h) H(R^2 ) is the space of all non-empty (compact) subsets of R^2 A single point in our space is an entire black & white image An IFS W defined on (X , d) is actually a mapping H(X ) → H(X ). Let B ∈ H(X ), then

W (B) =

i

wi (B)

wi (B) =

b∈B

wi (b)

h(·) is Hausdorff distance (next slide)

Hausdorff Distance

Allows us to decide how similar two sets (images) are Let A, B ∈ H(X ) Then we define h(A, B) as:

h(A, B) = max(d(A, B), d(B, A)) d(A, B) = max a∈A

min b∈B d(a, b)

where d(a, b) is the normal distance in X (Euclidean distance of R^2 in our case). Intuitively, it’s the length of the biggest “gap” between A and B “Clearly^1 ” if all the wi ∈ W are contractive under d with contractivity factor si , then W itself is contractive under h with contractivity factor maxi si (^1) We will state without proof...

Proof Sketch – A∗^ exists

Construct a sequence:

x 0 , x 1 = W (x 0 ), x 2 = W (x 1 ),... , xi = W (xi− 1 ),...

The distance between two successive elements

h(xi , xi+ 1 ) = h(W (xi− 1 ), W (xi )) ≤ sh(xi− 1 , xi )

By induction, we get

h(xi , xi+ 1 ) ≤ si^ h(x 0 , x 1 )

which means that the sequence is Cauchy Since X is complete, the limit A∗^ = limi→∞ xi must exist and is a fixed point

Proof Sketch – A∗^ is unique

Proof by contradiction Assume that the fixed point is not unique Let A∗^ and B∗^ be two distinct fixed points

W (A∗) = A∗, W (B∗) = B∗^ =⇒ h(W (A∗), W (B∗)) = h(A∗, B∗)

However, the definition of contractive says for some s < 1

h(W (A∗), W (B∗)) ≤ sh(A∗, B∗) < h(A∗, B∗)

Which gives us the contradiction

h(A∗, B∗) < h(A∗, B∗)

Inverse Problem

Given an image (compact set) C, can we find an IFS W such that the fixed point of W is C? Very hard – still an unsolved problem Collage Theorem (Barnsley 1986)

The Collage Theorem

For a set C and a contractive transform W with fixed point A∗, there exists s ∈ [ 0 , 1 ) such that

h(C, A∗) ≤ h(C, W (C)) 1 − s Thus, to ensure that we have an A∗^ that is close to the C that we are given, all we have to do is make sure that W (C) comes close to mapping C back to itself W(C) is called the collage of C Even with this theorem, there is no good way to automatically come up with W – I think that the best known solution is the “Graduate Student Algorithm”

Sierpinski – Solution

(^00) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.

w 1 :

[

] [

x y

]

[

]

w 2 :

[

] [

x y

]

[

]

w 3 :

[

] [

x y

]

[

]

(we did a bunch on Tuesday)

Local Iterated Function System (LIFS)

Arnaud E. Jacquin (1992) A.k.a Fractal Transform Instead of applying the transform to the whole image, cut and paste little pieces of it We can now exploit local self-similarity