Fractal Geometry Course

Fractals and Noise – Creation and Fractals and Noise – Creation and ApplicationApplicationShort TutorialShort Tutorial Fractal Geometry Properties and Fractal Geometry Properties and Exploitation: 30+ years of ImagesExploitation: 30+ years of Images

Martin J. Turner

Research Computing, University of Manchester

[email protected]

mailto:[email protected]

Research Computing

The University of Manchester

Web: http://www.rcs.manchester.ac.uk/

•Resources at:

•http://wiki.rcs.manchester.ac.uk/community/Fractal_Resources_Tutorial

http://www.rcs.manchester.ac.uk/

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Contents of Presentation

Introduction to Fractal Geometry.

Properties – Deterministic Fractals.

Noise – Creation and Images.

Language for Fractals.

Extra bits:– Fractal Compression

– Fractal Segmentation

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Introduction to Fractals and Chaos

Algorithms used to generate fractals and chaotic fields depend on:

understanding a physical (e.g. non-linear) system;

clear and concise (mathematical) definition(s) of the field properties.

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Example Applications

Signal Processing:

Time Series Analysis, Speech Recognition

Image Processing:

Fractal Compression, Fractal Dimension Segmentation

Simulation:

Terrain Modelling, Image Synthesis, Music, Stochastic Fields

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Financial:

Fractal Market Analysis, Futures Markets

Medicine:

Histology, Monitoring, Epidemiology

Military:

Visual Camouflage, Covert Digital Communications

Example Applications

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Brief History of Fractals/Chaos

1815-1897: K W T Weierstrass Nowhere Differentiable Functions

1854-1912: J H Poincare

Non-Deterministic (Chaotic) Dynamics

1900s: Julia, Koch & others

Julia sets (simple fractals) - arise in connection with the iteration of a function of a complex variable

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1886-1971: P Levy

Fractal Random Walks (Random Fractals)

1960s: E Lorenz

Nonlinear Systems Dynamics and Chaos

1970s: B Mandelbrot

Mandelbrot sets and the development of a general theory on the ‘Fractal Geometry of Nature’ (1975)


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1815-1897:

K W T Weierstrass Nowhere Differentiable Functions

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1815-1897:

K W T Weierstrass

Matlab – speed.

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1815-1897:

K W T Weierstrass Nowhere Differentiable Functions

Dreadful Plague,Hermite

Yesterday, if a new functionwas invented it was to serve some practical end; today they arespecially invented only to show to the arguments of our fathers, andthey will never have any other use

Poincare

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Aside: Fourier Series

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Hilbert – one of the first useful fractals

Hilbert Space Filling Curve

Suddenly we have uses;

Dimension reduction

Dithering style

Compression codecs

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Hilbert – one of the first useful fractals

Hilbert Space Filling Curve

Properties: Function f,

is a one-to-one mapping,

is onto, and

continuous function

There is no continuous inverse mapping.

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Robert Brown – Non-Differentiation in nature

varies in the wildest way in magnitude and direction, and does not tend to a limit as the time taken for an observation decrease

nature contains suggestions of non-differentiable as well as differentiable processes

Jean Perrin

Step: 32, Length: 6392

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Robert Brown – non-differentiation in nature



Jean Perrin


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Jean Perrin


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Jean Perrin


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Texture and Fractals

Texture is an elusive notion which mathematicians and scientists find hard to grasp

Much of Fractal Geometry can be considered to be an intrinsic study of texture

Benoit Mandelbrot

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The Fractal Geometry of Nature: Clouds

Photo from Manchester, UK

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Islamic Art: Self-Repeating Patterns

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Self-Similarity by M C Escher

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Self-Similarity by K Hokusai -Japanese Art from the 1800s

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Self-Similarity by SnowPhotos by Patricia Rasmussen

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Texture by Jackson Pollock

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Self-Similarity and J S Bach

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This movement depicts the island as a physical object. There are three structural principles at work: 1) reading the silhouette of the island seen from the shore (east to west)as a graph with range/density along the y-axis and time (19')along the x-axis 2) alternating "inbreaths" and "outbreaths" of varying durations calculated by careful measurement of the inlets and outlets of the island's shoreline, as shown on a hand- drawn map 3) a series of soundscapes representing the various environments encountered during a brisk walk around the island (which takes 19').

Self-Similarity within MusicISLAND SYMPHONY: Andrew Hugill

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Fractal: Origin of the word? 30+ years old!

The word Fractal was introduced by B Mandelbrot in the 1970s.

The term fractal is derived from the Latin adjective fractus. The corresponding Latin verb frangere means ‘to break’, to create irregular fragments. In addition to ‘fragmaneted’ fractus should also mean ‘irregular’, both meanings being preseved in fragment

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Intr

oduc

tion

of C

G in

the

70

’s

Fractal: Mandelbrot Sets

“Objets Fractals” – published 1975

“Fractals Form Chanceand Dimension” –published 1977

Zooming into the Mandelbrot Set`

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Fractal: Mandelbrot Sets

Intr

oduc

tion

of C

G in

the

70

’s

“The Fractal Geometryof Nature” – published 1982

Zooming into the Mandelbrot Set`

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Initiator:

Production Rule:

Properties of Fractals:The Von Koch Curve

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Properties of Fractals:The Von Koch Curve

Repeating theProduction ruleagainandagain and again.

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Properties of Fractals:The Von Koch Island

Joining threeCurves gives usan Island.

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Properties of Fractals:The Von Koch Island

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Fundamental properties of a fractal signal

1. It is nowhere differentiable.2. It is infinitely long.3. It is finitely bounded.4. It has a notion of self-similarity at

each level.

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Dimension of a Fractal

The dimension is the rate at which the signal approaches infinite length.

A smooth signal has a lower dimension.

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Dimension of a Fractal

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Box Counting – Dimension

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Fractional Brownian Motion

fBm has zero mean Gaussian distribution increments with variance;

with scaling property

Most famous case when, H=1/2

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Fractional Brownian Motion

fBm is related to Box Counting

So the number of boxes,

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Power Spectrum Method

We consider this a ‘favoured’ system;

We wish to match a power spectrum decay rate, that has similar properties to fBm

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Euclidean Objects in Fourier Space

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Fractal Objects in Fourier Space

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Frequency Spectrum and Log-Log

Frequency spectrum F is proportional to (Frequency)-q or

F = c/k q

where c is a constant and k denotes frequency.

Take logs of both side and we get

log(F) = C - qlog(k) where C = log(c)

Equation describes a straight line with a negative gradient determined by value of q.

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Fractal Signal of Koch Curve and its Log-Log Frequency Spectrum

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The Fractal Dimension D and q (the Fourier Dimension)

The larger the value of the fractal dimension D the smaller the value of q.

D and q must be related in some way.

For fractal functions: q=(5-2D) /2; 1<D<2.

The Fourier Dimension determines the ‘roughness’, ‘texture’ or frequency characteristics or spectrum of the fractal.

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Random Fractals and Nature: Stochastic Noise

Very few natural shapes are regular fractals.

The majority of natural shapes are statistically self-affine fractals.

As we zoom into a random fractal, the shape changes, but the distribution of lengths (texture) remains the same.

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Fundamental Definition of a Random Fractal Signal: Frequency Spectrum

Let f be a random function with spectrum F.

Let n be ‘white’ noise with spectrum N.

If F = N/kq where k is frequency, then a property of f is that

Pr[f(x)] = q Pr[f(x)]

N.B. The equation for statistical self-affinity implies that the spectrum of a fractal obeys an inverse q-power law.

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Randomisation of the Koch Curve

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Aside: all the fractals!

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Summary

Objects

Self-similar Classical

Deterministic Statistical

Random RealPure Chaotic

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Noise: White to Black

Colour

0 White

1 Pink

1.5 Brown

>2 black

qk/1

kFourier Spectrumhas the property,

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Basic property is that the Power Spectrum is proportional to

Basic algorithm is thus:

1. Compute white noise field n.

2. Compute DFT of n to give N (complex).

3. Filter N with filter

4. Compute inverse DFT of result to give u[i] (real part) - fractal signal.

Simulation of Random Fractal Signalsqk/1

2//1 qk

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Fractal Signal for D=1.2 (top) with Log-Log Power Spectrum (bottom)

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Noise: White to Black – Elegant Algorithm

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Noise: Fourier Fractals

Mat

lab

Cod

e

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Noise: Non-Fourier

Subdivide – a grid adding noise at each level

Matlab example

ddnH

n

2/

2

1

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Perlin Noise – a popular alternative

Similar structure but the sum of weighted frequencies.

Very similar to previous technique but values of ‘d’ from a LUT

Perlin Noise from Jia Liu

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Control of Perlin Noise


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Parameters control


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Tensor Gravity Field Analysis


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Tensor Gravity Field Analysis


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Aside: Fractional Brownian Motion

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P=0.50, D=1.621

P=0.75, D=1.609

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P=0.10, D=1.690

P=0.25, D=1.641

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Adding Deterministic Information

How can we incorporate a priori information on the large scale structure of a fractal field and thus Tailor it accordingly?

Replace white noise field n by (1-t)n+tp where p is a user defined function and t (0<t<1) is a ‘transmission coefficient’.

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Example: Sine Wave with Fractal Noise (t=0.1)

0 200 400 600 800 1000 12000.4

0.41

0.42

0.43

0.44

0.45

0.46

0.47

10 0 10 1 10 2 10 3 10 410 -6

10 -4

10 -2

10 0

10 2

10 4

10 6

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Adding Deterministic Info.

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Adding Deterministic Info.

Matlab system from J. Blackledge

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Fractal Clouds: q=1.9

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Novel Methods

Use of Transparency as a texture modifier for an ellipse

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Novel Methods

Use of 3D fractal texture bumpmap

Fractal transparency for rings on a circle

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Divergent fractal fields

Consider the 2D stochastic fractional divergence equation


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Rotational fractal fields

Consider the 2D stochastic fractional rotation equation


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Fractal flow fields

Consider the 2D stochastic fractional flow equation

q2>q1: Flow is in x-direction

q2<q1: Flow is in y-direction


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Fractals in Haptics

Controlling a multi-frequency haptic vibration device.

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Generalisation

Random fractal process: PSDF=ck-q

Ornstein-Uhlenbeck process: ck (k02 +k2)-1

Bermann process: c |k|g (k02+k2)-1

Combining the results we can consider the following generalisation:

c |k|g (k02+k2)-q

where k0 is a ‘carrier frequency’, c is a constant and (q, g) are fractal parameters.

Book on fractal geometry Book on fractal geometry

Theory, applications

and algorithms of

fractal geometry

for image synthesis,

image processing

and computer vision

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Pause…

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Language for Fractals

Lindermayer Systems

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Language for Fractals: Ex 1.

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Language for Fractals: FractInt

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Language for Fractals: FractInt Ex. 1

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Language for Fractals: FractInt Ex. 2 – (Revisited)

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Language for Fractals: Ex. 4+

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Language for Fractals: Ex. 4++

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Language for Fractals: Ex. 4.5++

Simulating and Modeling Lichen Growth – Brett Desbenoit, Eric Galin and Samir Akkouche, LIRIS, CNRS Universite Claude Bernard, Lyon

(Poss. wrong one)

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Language for Fractals: VR

VR World of DMU

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Language for Fractals: VR

VR World of DMU

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Language for Fractals: VR+

VR World of Manchester – Japanese Tuition: Anja Le Blanc

Fractal Plants within a VLE – Virtual Learning Environment: 3D world

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Language for Fractals: VR+

Blueberry3D – lots of LOD tricks and billboarding; circa 2003/4

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Thanks …

Over 10 years of Fractal curiosity:

Allan Evans, Jonathan Blackledge and all at the old ISS - Institute of Simulation Sciences, De Montfort University

The guys at the Virtual Environment Centre at De Montfort University

All the staff and students at the Manchester Visualization Centre at the University of Manchester

Plus many others (whose credits should be there).

Research Computing

The University of Manchester

Web: http://www.rcs.manchester.ac.uk/

Resources at:

http://wiki.rcs.manchester.ac.uk/community/Fractal_Resources_Tutorial

http://www.rcs.manchester.ac.uk/

Fractal Geometry Course

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