UBI 516
Advanced Computer Graphics
Fractal Geometry Methods
Aydın Öztürk
ozturk@ube.ege.edu.tr
http://www.ube.ege.edu.tr/~ozturk
Fractals


Many objects in
nature aren't formed
of squares or
triangles, but of more
complicated
geometric figures.
Many natural objects
- ferns, coastlines,
etc. - are shaped like
fractals.
Fractals(cont)


All the object representations we have considered
so far used Euclidean geometry methods that is
object shapes were described with equations.
Natural objects can be realistically described with
fractal geometry methods, where procedures rather
than equations are used.
Fractals(cont.)

A fractal has two basic charecteristics
- Infinite detail at every point
- Self similarity between the object parts.
Self Similarity

Geometric figures are similar if they
have the same shape.

The two squares are similar.

The two rectangles are not similar.

But the two rectangles are similar.
Self Similarity

Many figures that are not
fractals are self-similar.

Notice the figure to the
right. The outline of the
figure is a trapezoid. All the
trapezoids inside make up
the larger trapezoid.
Self Similarity: Examples
Construction of Self Similar Fractals


To construct a deterministic self-similar fractal,
we start with a geometric shape called initiator.
Subparts of of initiator are then replaced with a
pattern called generator.
Initiator
Generator
Construction of Self Similar Fractals


To construct a deterministic self-similar fractal,
we start with a geometric shape called initiator.
Subparts of of initiator are then replaced with a
pattern called generator.
Construction of Self Similar Fractals
The Sierpinski Triangle
The Sierpinski Triangle(cont.)
Step One
Draw an equilateral
triangle with sides of
2 triangle lengths
each.
Connect the
midpoints of each
side.
Shade out the triangle
in the center
The Sierpinski Triangle(cont.)
Step Two
Draw another equilateral
triangle with sides of 2
triangle lengths each.
Connect the midpoints of the
sides and shade the triangle
in the center as before.
Notice the three small
triangles that also need to be
shaded out in each of the
three triangles on each
corner - three more holes.
The Sierpinski Triangle(cont.)
Step Three
Draw an equilateral triangle
with sides of 4 triangle lengths
each.
The Sierpinski Triangle(cont.)
Step Four
We follow the above
pattern and complete
the Sierpinski
Triangle.
Example-1
Example-2
Example-3
Example-4
Example-5
Example-6
Example-7
Example-8
Example-9
Example-10
Example-11
Example-12
Example-13
Example-14
Example-15
Example-16
Example-17
Example-18
Example-19
Example-20
Classification of Fractals



Self-similar fractals have parts that are
scaled-down versions of the entire object.
Self-affine fractals have parts that are
formed with different scaling parameters sx ,
sy , sz in different coordinate directions.
Invariant fractal sets are formed with
nonlinear transformations.
Fractal Dimension


The detail variation in a fractal object can be
described with a number D, called fractal
dimension, which is a measure of roughness,
or fragmentation of the object.
More jagged-looking objects have larger
fractal dimensions.
Fractal Dimension(cont.)
A point has no dimensions –
no length, no width, no height.
A line has one dimension length. It has no width and no
height, but infinite
A plane has two dimensions length and width, no depth.
Fractal Dimension(cont.)


A cube has three
dimensions, length,
width, and depth,
extending to infinity in
all three directions
Fractals can have
fractional (or fractal)
dimension. A fractal
might have dimension
of 1.6 or 2.4.
Fractal Dimension(cont.)

A unit straight-line segment is
divided into two equal length subparts. This gives two scaled
copies of the original segment.

A unit square is divided into four
sub-squares. Scaling the length
and width by 2 gives four copies of
the original square.

A unit cube is divided into eight
sub-cubes. Scaling the length,
width, and height by 2 gives eight
copies of the original cube.
Fractal Dimension(cont.)
Relationship between the number of copies,
scaling factor and the dimension.
Figure
Dimension
No. of Copies
Line
1
2 = 21
Square
2
4 = 22
Cube
3
8 = 23
Similarity with
scaling factor s=1/2
d
n = 2d
Fractal Dimension(cont.)
Sierpinski triangle. Scale the length of the sides of an equilateral
triangle by 2.
Scaling the sides gives us three copies, so 3 = 2d, where d is the
dimension. Solving this equation we obtain the corresponding fractal
dimension d=1.584.
Figure
Line
Sierpinski's Triangle
Square
Cube
Doubling Similarity
Dimension
1
1.584
2
3
d
No. of Copies
2 = 21
3 = 2d
4 = 22
8 = 23
n = 2d
Fractal Dimension(cont.)
Let s be the scaling factor.
Relationship between the number of copies, the
scaling factor and the dimension can be
written as
ns
d
1
Solwing this expression for d, we have
d 
ln n
ln( 1 / s )
Fractal Dimension: Examples
Generator
Segment length Fractal dimension
1/√7
d=ln 3/ln(√7)=1.129
1/4
d=ln 8/ln 4=1.500
1/6
d=ln 18/ln 6=1.613
Creating an Image by Means of Iterated
Function Systems




Suppose that we take an initial image I0 , and put it
through a special photocopier that produces a new
image I1.
I1 is not a simple copy of I0 ; rather, it is a
superposition of several reduced versions of I0.
We then take I1 and feed it back into the copier
again, to produce image I2 ...etc.
We investigate whether these images converge to
some image.
Multiple Reduction Copying Machine
Multiple Reduction Copying Machine(cont.)




There are multiple lens arrangements to create
multiple copies of the original.
Each lens arrangement reduces the size of the
original.
The copier operates in a feedback loop, with the
output of one stage the input to the next.
The initial input may be any image.
Multiple Reduction Copying Machine(cont.)
Sierpinski’s Triangle is a typical example
http://www.arcytech.org/java/fractals/sierpinski.shtml
Multiple Reduction Copying Machine:
Sierpinski’s Triangle



The Sierpinski’s triangle is a most well known and
simplest example of Iterated Function Systems(IFS).
It is comprised of three component functions(lenses),
each of which shrinks the input image by one half
and translates it to a new position.
This contractive property guarantees convergence of
the iterative process.
Multiple Reduction Copying Machine:
Sierpinski’s Triangle(cont.)


Mathematically, each reducing lense is represented as a
contractive affine transformation that acts to scale, rotate, shear
and translate a copy of the input image.
The three transformations that produce Sierpinski’s Triangle
are:
1
2

W1   0

 0
0
1
2
0
0

0

1

1
2

W 2  0

 0
0
1
2
0
1

2

0

1

1
2

W3  0

 0
0
1
2
0
1

4

1

2

1

Multiple Reduction Copying Machine:
Sierpinski’s Triangle(cont.)


Mathematically, each reducing lense is represented as a
contractive affine transformation that acts to scale, rotate, shear
and translate a copy of the input image.
The three transformations that produce Sierpinski’s Triangle
are:
1
2

W1   0

 0
0 0

1
0
2

0 1

1
2

W 2  0

 0
0

1
0
2

0 1

1
2
1
2

W 3  0

 0
0

1 1

2 2

0 1

1
4
Fractal Tree
The coefficiens of four transformation matrices
Transform
W1
W2
W3
W4
a
0.53
-0.31
-0.25
0.29
b
-0.08
-0.42
-0.05
0.54
c
d
0.08
-0.44
-0.07
-0.04
0.53
0.33
0.29
0.29
e
f
-0.88
-15.19
1.48
18.74
33.44
19.43
11.73
9.87
Multiple Reduction Copying Machine:Example
Partitioned Iterated Function System
The basic idea:
Find self similarity between larger parts and
smaller parts of the image.
 Partitioning the original image into blocks is a
natural choice.
 Large partitions are called domain blocks,
and the small partitions are range blocks.

Partitioned Iterated Function System
Fractal Image Compression
Resim İçindeki Benzer Parçalar


Range : Küçük bloklar
Domain: Büyük bloklar
Resmi Parçalarına Ayırmak
Orjinal resim
4x4 Range Parçaları
Orjinal 8x8 Domain parçaları ve
bunların 4x4’e indirgenmiş halleri
Parçaların Benzerliği
ri  a  d i  e i
ri : i' th range
d i : i' th domain
Parçaların Benzerliği-3





Benzerini arayacağımız
Range üsttedir.
Domainler’in orijinal halleri
ilk kolonda gösterilmiştir.
Regresyon modeline göre
düzeltilmiş domainler ise
ikinci sütunda gösterilmiştir.
Yandaki rakamlar residual
mean square(rms)
değerlerini göstermektedir.
Rms değeri en küçük olan
domain eldeki range ile
eşleştirilir.
Rangeleri Domainler Cinsinden
İfade Etmek






Benzer Range-Domainler bulunduktan sonra bunlar bir dosyada
saklanarak resim bilgisini oluştururlar.
Örnek resmimiz 320x200x8 = 512,000 bit (64,000 byte) içeriyor.
Domain numaraları 0-999 arasında olduğu için 10 bitle ifade edilebilir.
Parlaklık ve konrast tamsayıya çevrilip, 0-15 ve 0-31 aralığında
kuantize edilirse, 4 ve 5 bitle ifade edilebilir.
Dolayısıyla, 4000 Range’e karşılık gelen Domain bilgilerini dosyaya
yazarsak :
4000*(Domain Numarası+Parlaklık+Kontrast)
= 4000*(10+4+5) = 76,000 bit (9,500 byte) olmaktadır.
512/76 = 6.73:1, (512-76)/512 = %85 sıkıştırmak demektir, üstelik
%85 sıkıştıran diğer algoritmalardan çok daha iyi bir kalitede !
Görüntü Kalitesi

Decompression, siyah bir ekrandan başlanarak
8-10 iterasyonla gerçekleştirilir.

Resmin kalitesi sonucun orjinale ne kadar yakın
olduğuna yani herbir Range-Domain çiftinin ne
kadar benzer olduk-larına bağlıdır.
Decompression: Iteration-1
Decompression: Iteration-2
Decompression: Iteration-3
Decompression: Iteration-10
Aynı Resimden Daha Fazla Domain Elde
Etmek : Domain Transformasyonları




Range’e uygun Domaini ararken
Domainleri olduğu gibi bırakmayıp
transformasyon işlemlerine
sokabiliriz.
Orjinal, 90o, 180o ve 270o derece
döndürülür.
X eksenine göre yansımaları
alınır.
Elde edilen 8 ayrı Domainin
negatifi alınarak, bir Domain-den
hareketle 16 farklı Domain elde
edilir.