Procedural Texturing
and
Modeling
Advanced Animation and
Motion Capture (AA&MC)
Topics covered:
Fractals & Noise
a. Random
b. Pseudo-random
c. Perlin Noise
d. Iterated functions
e. Spiral Phyllotaxis
Grammars
a. Recursion
b. L-Systems
2D and 3D Textures
a. Surface textures
b. Cellular textures
c. 3D Textures
d. Hypertextures
Procedural Modeling
a. Instancing
b. Replication
c. Placement
Noise
What is randomness?
Randomness:
Some definitions…
“Having no definite aim or purpose; not sent or guided in a particular
direction..” Oxford English Directionary
“Free will is limited to low-level decision making.”
Martin Luther
“God does not play dice with the universe.” Albert Einstein
“Don’t tell God what to do.” Neils Bohr
“[God] gave humans free will. He gave them free will because a
world of mere automata could never love..” C.S. Lewis
Randomness in the universe implies free will.
Michael Noll, 1962. Computer Composition with Lines
Based on Piet Mondrian’s Composition with Lines
Bela Julesz, Hungarian psychologist
Random-dot stereograms. 3D object hidden in random images.
Is it random? What makes it random?
How do we get true randomnesss?
Which of these are random?
a. Throwing dice
b. Roulette wheel
c. Temparature of a specific point in nature
d. The next number I say
e. Digits of PI
Roulette wheel
The Eudaemonic Pie, 1985. Thomas A. Bass.
Describe a group of physics graduates from University of Santa Cruz who
knew that roulette wheels obey Newtonian physics, but is just
very sensitive to initial conditions.
Using miniaturized computers, hidden in special shoes, they could
capture the state of the ball and the wheel, and could increase
their odds by 44%.
Hardware Random Number Genarators
Best sources of physical randomness so far:
- Nuclear decay of atoms.
- Photos traveling through a partial mirror
SSL Accelerator card
Applications to cryptography.
Many security systems are based on a random key.
If the key is not really random, system is not safe.
How do computers generate random numbers?
?
Tn
Tn+1
Numbers are generated as a sequence, starting from some seed point.
Problem is finding the next number so that it has nothing to do with the last.
Why is this hard? Because computers are deterministic. The computer
program to get Tn+1 is an exact sequence of steps. So if you
figure out what is Tn, my program will give you Tn+1.
Computer has no free will to choose a different number.
Plot of the rand() function in C.
int r = rand();
When your computer really needs something random,
how does it do it?
srand ( time() );
// check the second
hand on the clock
int r = rand();
// get random number
This says the seed of the random number generator
is based on the when you tell the program to run.
All numbers after that will look random.
If you start the program over, a different clock time
is used for the sequence.
These are called pseudo-random sequences.
Randomness in Computer Graphics
We actually like pseudo-random sequences in graphics.
Why?
1. Results can be reproduced, while still appearing random.
Useful for testing.
2. Randomness that depends on time will not
jump around from one frame to the next.
e.g. Random placement of trees in a game.
3. Faster to compute
Computer Graphics
Patterns in nature are semi-random.
Wood chips.
Natural and Man-Made Patterns
often have a grain.
Wood – due to process of growth
Metal – due to process of cleaning
Sheet metal
Random noise
Scaled on one axis
Fractals
Nature often exhibits pattens of a fractal structure.
- Exhibit scale invariance.
Same structure
at different scales.
- Exhibit self-similarity.
- Exhibit
Fractal dimensionality.
“Clouds are not spheres, mountains are not cones,
coastlines are not circles, and bark is not smooth..”
Benoit Mandelbrot, French-American (b. 1924)
The Fractal Geometry of Nature, 1982
Perlin Noise
Fractal noise.
Simple idea: Generate random noise at different scales (frequencies)
and add them together.
Ken Perlin, New York University.
Invented in 1982. Academy award for Tron.
Chaos
The property of a non-linear system to be stable at times, but
occasionally enter into a chaotic, seemingly-random state.
Exhibited by many physical systems:
Simple Example – Logistic Map
Based on the logistic map, developed to study population dynamics.
x n+1 = r xn ( 1 – x n)
Creates a number sequence. Where:
0 < Xn < 1
Ratio of existing populaton to maximum possible
population. % population
r
Combined rate for reproduction and starvation.
So we have two competing factors:
r Xn
Says that population will increase based on
the previous population.
r (1-Xn)
Says that population will decrease based due to
starvation based on how close we are to maximum.
Biologist Robert May, 1976. Mathematics first from Pierre Francois Verhulst (1804)
Bifurcation Diagram
Iterated Functions
All these systems share a common feature:
How to Generate a sequence Tn+1 from Tn.
- Noise
Generator is pseudo-random sequence
- Roulette
Generator is the law of Newtonian physics
- Chaos
Generator is a non-linear system /w feedback
- Logistic Map
Generator is competition between
population factors
Iterative functions describe many basic processes in nature.
The previous state of the system determines the next state.
See the chaos game, Michael Barnsley. Method of making geometric fractals.
Spiral Phyllotaxis
Many patterns in nature can be understood with
very simple iterated functions.
Two leaves appear on opposite sides. A third appears between them.
The first two push the 3rd away. The result is the golden angle.
Iteration, adding new leaves, settles into the minimization of energy,
which results in the golden ratio (angle).
Wilhelm Hofmeister, 1868.
Fermat’s Spiral
Phyllotaxis pattern found in sunflowers.
o
A Collision-based Model of Spiral Phyllotaxis, 1992
Deborah Fowler, Przemyslaw Prusinkiewicz, Johannes Battjes
“The Algorithmic Beauty of Plants”
Przemyslaw Prusinkiewicz
Entire book available online:
http://algorithmicbotany.org/papers/#abop
Procedural Textures – 2D and 3D
f ( x, y )
=
image
f ( x, y, z)
= volume
Deals specifically with generated images and volumes
and their application to graphics.
Procedural Textures – 2D and 3D
f ( x, y ) defines an image, or surface texture.
Image f ( x, y )
Object Polygon
World space
Texture space
Texture mapping.
Two-dimensional function is mapped onto the
three dimensional geometry. Process is usually done with
inverse sampling, where you look up the desired value in f.
Texture mapping
Texture appears on surface, and follows surface distortions of the object.
Volumetric Textures
f ( x, y, z ) defines a volume texture
Volume f ( x, y, z )
Object Polygon
World space
f ( x, y, z)
Volume Texture space
Texture is defined at every point in space.
Texture color is selected directly from the 3D volume location.
Wood and marble are naturally
described as volumetric textures.
Modeling and Rendering of Weathered Stone
Julie Dorsey, Alan Edelman, Henrik Jensen, Justin Legakis, Hans Pedersen
Combine Perlin noise with other simluated physical processes,
such as rain or rust, and layer them together.
Texture mapping:
- Faster to compute
- Less memory to store (only if you need to save)
- Texture appears on the surface of object
- Texture distorts with changes in the object.
Volume textures:
- Slower to compute
- More memory to store (only if you need to save)
- Texture remains rigid relative to object.
- Object changes in shape will move through the texture.
- Good for rigid objects with cracks or breaks.
Cellular Textures
There are other ways to generate the texture function f ( x, y )
aside from using random noise.
Generate a set of polygons
in the plane (2D).
Compute the distance to
the nearest edge.
Crocodile skin
Procedural Texturing – Types of mappings
Texture mapping assigns a function f to the color of a
point on an object.
Different features of an object can be modified by procedural textures.
Procedural Texturing – Types of mappings
Bump mapping modifies the surface normal
based on the function f.
The surface normal determines the visible
orientation of the surface for lighting
calculations.
Notice the silouette does
not look bumpy.
The geometry is not changed,
only the surface direction relative
to the light source.
Procedural Texturing – Types of mappings
Environment mapping modifies the reflected
ray color based on function f.
Useful for reflective surfaces.
Notice that the geometry of the surface is still not changed.
The sphere is still smooth.
Procedural Texturing – Types of mappings
Displacement mapping modifies the actual surface position
based on function f.
Most realistic for any rough surface.
Notice silouette is also bumpy.
Can be expensive to compute. May need to increase the geometric
detail of the object in order to give a good appearance.
Grammars and L-Systems
How can we change the structure of something?
With procedural textures there is essentially one
object, the function or system, which can change
over time. Differences are between points in
space or sequences in time.
What if we want to modify the object
itself, change its change or add new shapes?
Example:
L-Systems
Process of replacement can be formalized using a grammar,
or parallel rewriting system, called an L-system.
An L-system is a tuple (set of three things):
G = ( V, w, P )
V = Alphabet. Set of symbols that can be replaced.
w = Start or axiom.
P = Set of production rules.
Aristid Lindenmayer, 1968. Hungarian biologist and botanist.
Variables:
A B
Start:
A
Rules:
A -> B
B -> AB
Process:
1. Take the Start (w) as the current string
2. Apply the production rule to every symbol
in the current string.
3. When finished, call this the new string.
4. Repeat
start:
iteration 1:
iteration 2:
iteration 3:
iteration 4:
A
B
AB
BAB
ABBAB
(rule A->B)
(rule B->AB)
(rules A->B, B->AB)
(rules B->AB, A->B, B->AB)
L-Systems can have a physical interpretation, if we assign meaning to letters.
L-System Plants
L-system Weeds
Procedural Cities
L-systems can be used to describe many objects where the
structure changes iteratively, such as city blocks.
http://www.youtube.com/watch?v=yI5YOFR1Wus
Procedural Buildings
Generative Modeling Language (GML)
“A Generative Theory of Shape”, Michael Leyton. (availabe online)
Procedural Furniture
Parametric modeling allow you to change the size and shape
of various parts of an object continuously.
Instancing and Replication
or.. How to make a complete forest. (First try)
Noise
L-system
Structure
of branches
Noise can tell us where to plant trees.
L-system can tell us how to build a three-dimensional tree.
But do we make one tree and copy it, or do we make lots of different trees?
Some times instancing can look pretty good. Pines trees all look similar.
In other cases, instancing does not produce enough visual variation.
This also depends on the placement algorithm.
1 instance
2 instances
1 instance
8 instances
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Procedural Modeling - Efficiency
Procedural modeling can produce a lot of geometry.
Steps must be taken to reduce the number of polygons.
1. Level of detail – Only create new
geometry where it is needed
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2. Instancing – Copy geometry already
in memory instead of rebuilding
3. Imposters – Use a picture for
distant objects instead of
rendering the whole thing.
http://www.youtube.com/watch?v=rl5p63JabrA
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