Fractal Coding

Dr. Qi TianCS 4763 Spring 2007

Fractals and Image ProcessingThe word “Fractal” less than 20 years by one of the history’s most creative mathematician Benoit MandelbrotOther contributors: Cantor, Hausdorff, Julia, Koch, Peano, Poincare, Sierpinski, Weierstrass, Richardson

“Fractal” from Latin verb frangere, meaning to break or fragment

Basically, a fractal is any pattern that reveals greater complexity as it is enlarged, “worlds within worlds”

Examples

Examples

Examples

Examples

Fractal Transform

Typical transform:Discrete Cosine Transform (DCT)Wavelet Transform (WT)

Fractal Transform:rather than describing the image data directly, describe a system that can generate the image.

Fractal image compression offers enormous compression ratios.

Fractals and Image Processing

ReferenceBenoit B. Mandelbrot, Michael F. Barnsley, Amaud E. Jacquin

Fractals for the Classroom, Heinz-Otto Peitgen, Hartmut Jürgens, DietmarSaupe, Springer Verlag, New York, 1992.

Fractal Image Compression: Theory and Application to Digital Images,Yuval Fisher (Ed.), Springer Verlag, New York, 1995 is a collection of articles on Fractal Image Encoding.

Fractal Image Encoding and Analysis: A NATO ASI Series Book, Yuval Fisher (Ed.), Springer Verlag, New York, 1996

What is a fractal?

A fractal is a geometric shape which

1. is self-similar and2. has fractional (fractal) dimension

Courtesy Mary Ann Connors, University of Massachusetts

An Introduction

Fractal geometry and chaos theory are providing us with a new perspective to view the world.

For centuries we've used the line as a basic building block to understand the objects around us. Chaos science uses a different geometry called fractal geometry.

Fractal geometry is a new language used to describe, model and analyze complex forms found in nature.

An Introduction

Traditionally Euclidean pattern appear simpler as they are magnified. For example, your home in one area, the shape looks more and more like a straight line.

But fractals like bumps of broccoli, are not differentiable: the more closer you come, the more detail you see.

More Examples

An IntroductionA few things that fractals can model are:

plants weather fluid flow geologic activity planetary orbits human body rhythms animal group behavior socioeconomic patterns music and more …

An IntroductionSome ideas: Broccoflower

This is how nature creates a magnificent tree from a seed the size of a pea ... or broccoflower

An IntroductionFerns

An IntroductionOthers:

TreesBushesRoots and Shoots of Plants Mountains Coastlines Clouds Galaxies The Human Brain Human Circulatory System

Geometry

Euclidean Traditional (>2000 yr)Based on characteristic size of scaleSuits man-made objectsDescribe by formulaTopological dimension

FractalsModern monster (~10-20 yr)

No specific size or scaling

Appropriate for natural shapes

(recursive) algorithms

Fractal dimension

Coastline of An Island

DT=1

L=?

ε·N(ε)=? Depends on ε

ε: yardstick

Yardstick(ε) ---> 0

L= ε·N(ε) ---> ∞

Coastline of An IslandBecause the roughness of the coastline does not vanish in smaller scales, i.e., it does not become smoother.

This is contrast to the traditional geometrical sets

For these curves

εε 1)( ⋅≠ kN for small ε

Spain vs. Portugal Netherlands vs. Belgium

987 km 1214 km 380 449 km

over 20% difference

Coastline of An IslandRichardson showed that

depends on roughnessDkNε

ε 1)( ⋅≅

kND =→

)(lim0

εεε

Higher D -> 0

Right D -> k

Lower D -> ∞

∞

k

D

Fractal Dimension

The D for which

∞≠≠

=→

0)(lim

0kND εε

ε

is called the fractal dimension of the curve

Fractal DimensionFractal dimension can measure the texture and complexity of everything from coastline to mountains to storm clouds. We can now use fractals to store photographic quality images in a tiny fraction of the space ordinarily needed.

Fractals provide a different way of observing and modeling complex phenomena than Euclidean Geometry or the Calculus developed by Leibnitz and Newton.

An arising cross disciplinary science of complexity coupled withthe power of desktop computers brings new tools and techniques for studying real world systems.

Self-SimilarityA fractal

Looks the same

Over all the range

Self-Similarityof scale

Self-SimilarityExact Self-Similarity

Statistical Self-Similarity

What is a dimension?What is dimension? How do we assign dimension to an object? In what dimension does each move?

a train moving along railroad tracks ?

a boat sailing on a lake?

a plane in the sky

Try a more difficult one

Crumple it up into a ball

What is the dimension of

the ball?

When you carefully reopen the ball of foil, what dimension has it become?

1. Train: 1

2. Ship: 2

3. Plane: 3

4. Aluminum a) 2

b) 3

c) somewhere between 2 and 3

Answers

1. Notice that the line segment is self-similar. It can be separated into 4 = 4^1 "miniature" pieces. Each is 1/4 the size of the original. Each looks exactly like the original figure when magnified by a factor of 4 (magnification or scaling factor).

Mathematical Interpretation

2. The square can be separated in to miniature pieces with each side = 1/4 the size of the original square. However, we need 16 = 4^2 pieces to make up the original square figure

Mathematical Interpretation

2. The cube can be separated into 64 = 4^3 pieces with each edge 1/4 the size of the original cube

Mathematical Interpretation

In these simple cases the exponent gives the dimension:

4 = 4^1pieces

16 = 4^2pieces

64 = 4^3pieces

Mathematical Interpretation

Therefore, N (the number of miniature pieces in the final figure) is equal to S (the scaling factor) raised to the power D (dimension).

N = S^D

In the previous cases it is easy to find the dimension by simply reading the exponent.

Dimension of a Fractal

However it's not always so easy. Consider theSierpinski Triangle - an example of a fractal.

Let's look at how it is generated: Begin with a triangle

Draw the lines connecting the midpoints of the sides and cut out the center triangle

Dimension of a Fractal

Note that we have in our new triangle 3 “miniature” triangles. Each side = 1/2 the length of a side of the original triangle. Each “miniature” triangle looks exactly like the original triangle when magnified by a factor of 2 (magnification or scaling factor).

Dimension of a Fractal

Take the result and repeat (iterate).

Dimension of a Fractaland again and forever

Notice that the lower left portion of the triangle is exactly the same as the entire triangle when magnified by a factor of two. It is self-similar.

Now we compute the dimension of the SierpinskiTriangle: Notice the second triangle is composed of 3 miniature triangles exactly like the original.

The length of any side of one of the miniature triangles could be multiplied by 2 to produce the entire triangle (S = 2).

The resulting figures consists of 3 separate identical miniature pieces. (N = 3).

What is D?

In general,

Fractal Dimension

This method of finding fractal dimension can be used for only exact self-similar fractals.

Other ways of computing fractal dimension include: mass, box, compass, etc.

Generating Fractals

Three transformations are enough

Generating Fractals

What is a fractal?Fractals are self-similar geometrical objects.

How to construct a fractal?

The fractal can be constructed by iterating the same process.

Two properties:

1. The further the process goes, the more detail is added;

2. A different initial image can be used to create the same fractal. Only the process is important.

Transformation

Affine Transform

Fractal CodingConsider a image like the square

2. Do the same thing again, but using (b) as the original to be transformed

3. Keep iteratively, and repeat the whole process four more times.

(b)

(a)

(c)

This is the famous fractal called Sierpinski triangle

Transformations:

1. Take the origin as the bottom left corner, shrink the square by half; the second is shrunk and moved up; the third is shrunk and moved up and right

So the original square

+

three simple transformations

this very complex detailed image

Compression ratio is very high!

The final image is independent of the original image, it is uniquely determined by the transforms alone.

Take a very different initial picture

Apply exactly the same transformation five times.

But if I take a different set of transformation…

Affine TransformAffine transformations include the basic transformations of rotation, translation, reflection, scaling, and shear.

A shear transform in X-direction

Affine TransformAffine Transform

TAXXW +=)( ⎥⎦

⎤⎢⎣

⎡=

dcba

A ⎥⎦

⎤⎢⎣

⎡=

fe

T

ebyaxx ++='fdycxy ++='

Or

⎥⎦

⎤⎢⎣

⎡ −=⎥

⎦

⎤⎢⎣

⎡

2211

2211

cossinsincosθθθθ

rrrr

dcba

The matrix A can always be written in the form

Affine Transform

A single affine transformation W=[A, T] is defined by just 6 real numbers.

The Sierpinski triangle was defined by three affine transformations, or just 18 real numbers.

Fantastic compression ratio with fractal compression.

Generating Exactly Self-Similar Fractal

Fractal Dimension

Fractal Dimension

A contraction mapping doing its work, drawing all of a compact metric space X towards the fixed point.

Iterated Function System (IFS)

Iterated Function System (IFS)

Iterated Function System (IFS)

Iterated Function System (IFS)

Iterated Function System (IFS)

Iterated Function System (IFS)