L-System and

L-System
Tree Hugger Project
Bauhaus University Weimar
Hasibullah Sahibzada
25-Nov-2013

Agenda
L-System algorithm and its variations
Axial Tree
Brief on Related works ( plastic trees)
Brief on Plastic trees

L-System
Lindenmayer systems (L-systems)
created

Aristid Lyndenmeyer

http://www.avatar.com.au/courses/Lsystems/References.html#ref004

Developmental systems were introduced (Lindenmayer, 1968, 1971) in
order to model morphogenetic (pattern-generating) processes in
growing, multicellular, filamentous organisms.
The emphasis was on plant topology , on the spatial relations
between cells or larger plant modules.
To model multicellular plant growth.
They did not include enough detail to allow for comprehensive modeling of
higher plants.

-- original work

L System Theory
L-Systems are mathematical models to describe the development of plants
in a visually convincing way. This method is preferable to 3D geometry
modelling as it requires much less human interaction and often produces
more convincing results in modeling the small, next to unnoticed details
that a human often would not bother to model.
Central Concept of L-System is that of Re-writing.
Rewriting is a technique for defining complex objects by successively replacing
Snowflaks
parts of a simple initial object using a set of rewriting rules or productions.
Classical Example from Von Koch snowflake

1: One begins with two shapes, an initiator and a generator.
2: The latter is an oriented broken line made up of N equal sides of length r.
3: Thus each stage of the construction begins with a broken line and consists in
replacing each straight interval with a copy of the generator, reduced and displaced
so as to have the same end points as those of the interval being replaced.
B. B. Mandelbrot. The fractal geometry of nature. W. H. Freeman, San Francisco, 1982.

Re-writing
Many Examples are done based on rewriting technique.

1: Rewriting systems on graphs
2: Rewriting systems on rectangular arrays and matrices ( Cellular
automata)
3: Rewriting systems on character strings
History
Begin 19th century: Thue provided first formal definition of a string rewriting system (srw)
1960: Backus and Naur used rewriting notation in the definition of programming language
ALGOL
1968 the biologist A. Lindenmayer introduced L-Systems to model multicellular plant growth
S. Ginsburg and H. G. Rice. Two families of languages related to ALGOL. J. ACM, 9(3):350–371, 1962.

L-System (DOL)
Simplest class of L-systems, those which are deterministic and context-free, called DOL- systems.

Development of a filament of the bacteria Anabaena catenula
1: The symbols a and b represent
cytological states of the cells
2: The subscripts l and r indicate cell
polarity, specifying the positions in which
daughter cells of type a and b will be
produced.
L-system describes development
– startsAxiom(distinguished string) ar
– p1:ar→albr
– p2:al→blar
– p3:br→ar
– p4:bl→al
Under a microscope, the filaments appear
as a sequence of cylinders of various
lengths, with a-type cells longer than btype cells.

Graphical modeling using L-systems

Turtle interpretation
1: A state of the turtle is defined as a triplet
(x, y, α)
2: the Cartesian coordinates (x, y)
represent the turtle’s position,
and the angle α, called the
heading, is interpreted as the between lines)
Gets more complicated ( space
direction in which the turtle is
facing.
3: Given the step size d and the angle
increment δ, the turtle
can respond commands represented by the
following symbols
Approximations of the quadratic Koch island taken
from Mandelbrot’s book [95, page 51].

Video
http://www.youtube.com/watch?v=fq9kx0hjx-U

Turtle Example

http://www.youtube.com/watch?v=xTepbIRGn6o

DOL Example

Axial Trees
A rooted tree has edges that are labeled and directed.

An axial tree is a special type of rooted
tree
1: At each of its nodes, it has most one
outgoing straight segment.
2: All remaining edges are called
lateral or side segments.
3: A sequence of segments is called an
axis, if:

Sub-tree

a: the first segment in the sequence
originates at the root of the tree or
as a lateral segment at some node,
b: each subsequent segment is a
straight segment.
c: the last segment is not followed

by any straight segment in the
tree.

Sample tree generated using a method based on Horton– Strahler
analysis of branching patterns

L System Model
(Ijiri_wiss2005)Sketching L-System: Interface for designing Flactal Stractures by drawing
Axis

L-System
Summary
L-Systems are based on string rewriting using production rules ( graph grammars)
Turtle graphics give them a graphical interpretation

L-Systems serve as models of development in biology, but also in other areas.
Small changes of rules have often surprisingly large impact.
Axial trees are important branching structures in nature (rivers, botanical trees,

Related Work
Early Plant Models growth

Procedural approach

Early models of plants were based on procedural approaches that replicated growth by

repetitive application of a small set of rules to an initial structure to yield very complex results.

Captured Internal properties of trees

branching angle

branch length

Procedural Approach
The forms of nature based on spirals and ramification are generated
not through the use of object data calculated by measurement, but
through the use of an algorithmic structure based on the laws of
nature.
Explained Processes of recreating some forms of nature, including
shells, horns, tusks, claws, and spiral plants.

KAWAGUCHI, Y. 1982. A morphological study of the form of nature. In SIGGRAPH ’82: Proceedings of the 9th annual conference on Computer graphics and interactive techniques, ACM Press, New York, NY, USA, 223–232.

3-D Shapes
Spiral Shells
Spiral Growth into Tendrils
Horn

Mouth of shell
Young plant

Procedural Approach
The whole form of actual trees,therefore, was speculated to be affected
also by their branching angle and relative ratio of their branch lengths
HONDA, H. 1971. Journal of Theoretical Biology 31, 331–338.

Problem

views of trees or their crown

1: Pattern-recognition. How Define a specie from the various forms (same crown )
2: Morphogenesis. ( Gens specify the form of tree)

Described erect trees as repeated branching structures.
The whole form of actual trees seems to be determined by a great
many factors, (branching angles and relative ratios of branch lengths)

Textures
The present work seeks to model trees with sufficient realism that they may
be the subject of animation, rather than simple elements of the landscape.
The model should have a well-defined structure; beneath the bark the limb
should be smooth: leaves should be properly attached to twigs.
Several spline segments
interpolate the data points
(asterisks) with C2 continuity. A
strobe captures a disk as it passes
along the curve.

Surfaces

Order of creation of limbs (red,
then orange, yellow, green, blue,
and white)

BLOOMENTHAL, J. 1985. Modeling the mighty maple. SIG- GRAPH Computer Graphics 19, 3, 305–311.

Textures
Resolution Issues
A tree tends to maximize its surface area to volume ratio.
If the viewpoint is close to a limb, a large number of limbs will be off-screen.
Thus, a method is desired for polygonizing limbs that varies the axial and
circumferential(enclosing boundry) resolutions according to the projection of the
limb onto the screen and that culls off-screen limb sections.

Verisimilar Bark

X-raying

Surface plaster

Mapping

Leaves by Camera
Movable Joints


Botanical Structure & Development
DE REFFYE, P., EDELIN, C., FRANÇON, J., JAEGER, M., AND PUECH, C. 1988. Plant models faithful to
botanical structure and development. In Proceedings of SIGGRAPH ’88, 151–158.

Concern here is the faithfulness of the models to the botanical nature of
trees and plants.
In the model They Integrated Botanical Knowledge of tree architecture
Growth
Mechanism to Occupy space
Location of Leaves, flowers and fruits
Important Thing

Time Integration
Viewing the aging of a tree

Different Pictures
Simulation of death of leaves and branches
Wind and insects

Botanical Structure & Dev

Meristems

Found in zones of a plant where growth
Take place.
Functionality
1: Growth
Time
2: Ramification Diversions from small to big

3: Mortality.
Each Bud is given two probabilities

Break/abort

Ramification


Growth Simulation
Bellow Parameters should be given
-the age,
- the clocks or growth speeds of the axes,
- the number of possible buds at each node, as a function of order,

- the probabilities of death, pause, ramification and reiteration
By experiment
Mathematical Law

Botanical Tree Model
Dynamic Modeling Technique
Dynamic modeling and representation technique
for trees that aims at incorporating aspects of the trees
genotype into our models to allow them to react to the
environment.
Dependency of model

Endogenous

Exogenous

Botanical Tree Model
1: Exogenous ( Environment )

Related Works
B: user-assisted plant modeling
1: complex parametric model.
Weber and Penn [1995]

2: introduce decomposition graphs
[Boudon et al. 2003]

3: Xfrog modeling technique
Lintermann and Deussen [1999]

which combines rule-based and procedural modeling and also allows for creating animated
models.

Not possible for models to dynamically react to their environment.

Related Works
C: Image-based techniques
1:Register Input Images to reconstruct the 3D shape
While Reche-Martinez et al. [2004]

2: loosely arranged images
Neubert et al. [2007]

The main branches are determined by the user and the static model is constructed using a
particle flow system and some botanic heuristics.

D: sketch-based technique

1: Combination of rule-based and image-based techniques on procedural made trees
Ijiri et al. [2006] and Zakaria and Shukri [2007]

2: 2D-3D

Chen et al. [2008]

Bringing Set of biologically motivated branching rules to infer the 3D structure of the
tree from a given

2D sketch.

Related Works
E: Environment
Modification of the environment itself can be used as a way of controlling the

procedural model.

2: Climbing of plants that grow on support structures and are influenced by the
light density.

Arvo and Kirk [1988], Greene [1989], and later Benes and Millan
[2002]

F: Computing light Method
1: Simplified, technique proposed by Rudnick et al. [2007].
2: Radiant energy transfer

[Soler et al. 2003].

Plastic Trees
Trees reaction in Environment changes:

Studied
1: Geometrical
2: Topological

1: Study the environmental conditions

Self Shadowing

2: Natural morphological properties.
3: Construct a procedural model .

Isolated space

Behavior of branch
Controllable by environmental parameters
Morphological parameters

Self Shadowing

Branch Age
Factor

Growth rate of entire tree.

Growth rate in a branch
The growth rate of an individual branch is determined by
how many internodes (segments without buds) a given
branch produces in one season.

But

Nodes Depends on resources
Branch Age =>> Internode length and growth rate
Distance from a given segment to its furthest leaf node,

Threshold dt = 0.2dr

Distance from the root of the tree to its furthest leaf node.

Branch age
Growth rate

Distance from the segment to the root of the tree
Minimum growth rate(all)

Absolute growth rate

Length of internode

Length of branch segment
Parent age

Temporal Light
Conditions
Simulate effect for each leaf
Leaf Cluster cast shadow

Shadow calculated by
Shadow volume that
is attached to shadow caster

point

Leaf translucency

Amount of light

Normalization constant
[0,1] == in coming light

Visibility of hemisphere
From P

Inverse Tropism
A tropism is the tendency of the branches to grow towards or away from some entity.

Need to know Environmental changes + different stages of tree development
Influence of tropisms on tree growth and shape.
unit direction of the tropism

tropism

Tropism strength

Tropism
1: Phototropism
2: Gravitropism

Phototropism
The tendency of a given branch to grow towards the light direction.
Calculated on All branches using Temporal
Light Model

Gravitropism
Gravitropism controls bending of the branches either away from or towards gravity

Set the angle of branches

Tropism

New Direction

Linear combination of all tropisms

Normalized direction

weights

weight of the original direction of branch segment

Branch segment lenght

Inverse tropism
To Compute the effect of tropism on a input tree.

Inverse tropism

d0

Pruning estimation
Natural pruning influences the tree structure
Apical meristematic cells in a bud produce
wood or plant organs

If no light then die

Sum of Leaf node distances
Resource by their child leave cluster

<

Pruning Factor

Normalized amount
of light

Leaf Cluster

radius

Pruning estimation

User controllable parameter = 0.8
Local pruning Factor
5th percentile of local pruning factor

topiary

Dynamic Interaction
After growth behavior and pruning

Learn

1: Amount of changes in environment.
2: Tree response according to the changes

1: Tree Graph transformation
2: Modeling of leaf-Clusters
3: Types of Interaction

Tree Graph
Transformation
Changes in tree growth
according

Age, Light conditions

1: binding
2: Rotation of a branch === > transferred to parent
3: Leaf Cluster changes ( shadow )
4: Pruning

Modeling a leaf cluster
1: Response to light
Amount of light

a. Branch creation, and branchlets ( Cluster Density )
b. Branch orientation
c. Number of leaves per branch

Initial density

Normalize density
For leaf cluster

Incoming light

Modeling a leaf cluster
1: Response to Obstacles

Intersection of leaf
New tree

overlap

Collide and then pruned.
Hull cluster

1

Black = input tree color
Red=binding
Blue = pruned
Exception for small branches

2

Types of Interaction
As seen before
interaction

A: Tree-Obstacle
B: Tree-Tree
A: Global Light

Computation of transformation

Implementation and
Results
Tree Modeling’s main focus is on the branches rather then leaf cluster

Thickness Threshold set

Big

Rendered
LOD ( Level of Detail) technique is created

The amount of produced geometery
Cluster Size
Light situation
LOD stage
Depends on Camera Location too.

Small

Leaf Cluster

Implementation and
Results
Main branching structure
Stored

Tree self similarities
Branchlets ( patches)

CPU Mem
Mapped
GPU / VBO

Texture buffer
GPU

Results
Models from
1: Xfrog
2: Open L-System
3: LiDar

Plastic Tree:
1: Representation Model
2: Interaction model
3: Influence in tropism
Limitation:
1: Prodxuction of new branches
2: Soil change effect.
3: Influence in tropism

References
HONDA, H. 1971. Journal of Theoretical Biology 31, 331–338.

Ijiri_wiss2005 Sketching L-System: Interface for designing Flactal Stractures by drawing Axis
R. E. Horton. Hypsometric (area-altitude) analysis of Erosional topology. Bull. Geol. Soc. America, 63:1117–1142,
1952.
OPPENHEIMER, P. E. 1986. Real time design and animation of fractal plants and trees. SIGGRAPH Comput. Graph. 20, 4, 55– 64.
BOUDON, F., PRUSINKIEWICZ, P., FEDERL, P., GODIN, C., AND KARWOWSKI, R. 2003. Interactive design of bonsai tree
models. Computer Graphics Forum. Proceedings of Eurographics 22, 3, 591–599.
DE REFFYE, P., EDELIN, C., FRANÇON, J., JAEGER, M., AND PUECH, C. 1988. Plant models faithful to botanical structure
and development. In Proceedings of SIGGRAPH ’88, 151–158.
KAWAGUCHI, Y. 1982. A morphological study of the form of nature. In SIGGRAPH ’82: Proceedings of the 9th annual conference on Computer graphics and interactive techniques, ACM Press, New York, NY, USA, 223–232.

http://www.avatar.com.au/courses/Lsystems/References.html#ref004

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