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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”
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.