Lect. 3&4

Lectures 3&4

Plants


A stream with vegetation (Deussen et al SIGGRAPH 98)	Bali 2005, Vue 5 Infinite demo. image

1. Lindenmeyer Grammars (L-systems) for Plant Modelling

References:

Prusinkiewicz, P, Lindenmeyer, A, Hanan, J.
"Developmental Models of Herbaceous Plants for Computer Imagery Purposes"
Computer Graphics, Vol 22 No. 4, August 1988 (SIGGRAPH 88), ACM Press, pp141-150

Prusinkiewicz, P, Lindenmeyer, A.
"The Algorithmic Beauty Of Plants",
Springer Verlag, 1990 [Available in PDF format]

Prusinkiewicz P, Hammel M, Mjolsness E.
"Animation Of Plant Development"
Computer Graphics, SIGGRAPH 93, ACM Press, pp351-360

"Visual Models of Morphogenesis" (website)

Examples on the Algorithmic Botany website in animation format.

What are L-Systems?

A developmental model of plant growth: a grammar is used to specify the changing topology of the growing structure.
The grammar does not account for external factors governing the growth of plants eg. phototropism, physical collisions. Extensions have been proposed which attempt to deal with this deficiency.
A plant is viewed as a branching structure whose components are modules.

How is an L-grammar specified?

An L-System consists of:

V, a set of symbols called the alphabet of the grammar representing the modules of the plant;

W, an axiom or initial string consisting of symbols from the alphabet representing the first module of the plant;

P, a set of production rules to be applied in parallel to the axiom and symbols of the subsequent strings. Production rules replace a symbol, the predecessor module, with zero, one or several successor modules.

Grammars proceed over discrete time intervals at which each module in the string is transformed by the production rules.
The development of the plant model is therefore described in discrete steps also.
The state of the string at each step is interpreted visually by reference to a set of geometric rules.
Symbols in the alphabet of the grammar may be interpretted by the geometric rules as:
- lengths of tree limb segments
- instructions to branch left or right
- instructions to bud or produce a leaf
These may be understood in terms of movements for a graphics 'turtle' like that used by Logo.
A branching structure may be represented using L-systems by incorporating brackets [ and ] or ( and ) into the alphabet to denote branches.

Now, how about some examples?

An example of Monopodial branching

K = {V, W, P}
V = {0, 1, [, ] }
W = 0

P =
p1: 0 -> 11[0]0
p2: 1 -> 1
p3: [ -> [
p4: ] -> ]

Step#	String
0	0
1	11[0]0
2	11[11[0]0]11[0]0
3	11[11[11[0]0]11[0]0]11[11[0]0]11[0]0

FIG. A sample visual representation of the string: 11[0]0

An example of Dichotomous branching

K = {V, W, P}
V = {0, 1, [, ] }
W = 0

P =
p1: 0 ->11[0][0]
p2: 1 -> 1
p3: [ -> [
p4: ] -> ]

Step#	String
0	0
1	11[0][0]
2	11[11[0][0]][11[0][0]]
3	11[11[11[0][0]][11[0][0]]][11[11[0][0]][11[0][0]]]

FIG. A sample visual representation of the string: 11[0][0]

A further example...


A visual representation of a tree production P.	Application of P to edge S in tree 1.

Context-Sensitive Grammars

Context-free grammar: production rules are used to replace an individual predecessor with its successor by reference only to the predecessor itself (as in the examples above).

Context-sensitive grammar: production rules are used to replace an individual predecessor with its successor by reference to the predecessor and its neighbouring modules, that is the context of the strict predecessor.

A predecessor in a context sensitive grammar consists of:

a path: l	(the left context)
an edge: S	(the strict predecessor)
an axial tree: r	(the right context)

The asymmetry between l and r reflects the asymmetry in the neighbourhood of a module. Ie. There is only one path leading from the root of the tree to a given module, but each module may be the parent of a complete subtree.

FIG. The satisfaction of a context-sensitive grammar.

Non-Deterministic L-Systems

Where exactly one production matches each possible instance of a symbol (or in the case of context-sensitive grammars, set of symbols) we have a deterministic L-System. (As in all the examples above.)

Development of Shepherd's Purse. (Prusinkiewicz et. al 1988)

Shepherd's Purse can be represented by a non-deterministic, parametric L-system as follows.

K =	{V, W, P}
V =	{A, B, F, I, L [, ] }
W =	A
P =	p1: A -> I0 [ L0 ] A p2: A -> I0 [ L0 ] B p3: B -> I0 [ L0 F0 ] B p4: Xi -> Xi+1 where i>=0 and X is an element of {I, L, F}

Production p1 describes the initial vegetative growth of (L)eaves and (I)nternodes from apex A.

At some point in time as determined by one of the methods below, p1 is replaced by p2. This changes the Apex to its (F)lowering state.

From this point on, flowers are produced by p3 instead of leaves as previously.

The consecutive steps of I, L and F represent the parametized stages of development of an individual Internode, Leaf and Flower respectively.

An L-system can be used to represent the development of leaves, flowers and other plant organs by encoding the lengths, curves, number of petals, leaf veins etc. in alphabetic symbols of the grammar.

Eg. A vector may be repetitively scaled using a production rule: X -> SX

Controlling Non-determinstic L-Systems

A delay mechanism

Ambiguity is avoided in the selection of production rules by adding a mechanism to the grammar which, after some number of iterations of a rule (or set of rules), triggers a switch to invalidate one of the productions and replace it with another.
A stochastic mechanism

Ambiguity is avoided by specifying a probability with which each ambiguous rule in a particular instance may operate in precedence over all others.
Environmental change

The entire set of production rules may be altered to another set after some external factor triggers the change.

Further Variations on the (non-deterministic) Model

Timed L-systems

Extensions of the model representing continuous development have been implemented at a basic level.
Differential L-systems

Differential equations have also been used to completely separate the continuous development of the organism from the discrete events required to animate its growth.
Map L-systems

Model the development of two-dimensional cellular layers.
Cellwork L-systems

Model three-dimensional cellular structures.
Environmentally Sensitive L-systems

Extend parametric L-systems by incorporating a mechanism by which, once the position of the turtle is known for that step, it's position, orientation etc. may be fed back into the grammar as parameters.

2. Fractal Methods for Plant Modelling

L-Systems may be used to model fractal plants, however, simpler fractal/graftal models of plants are available.

References:

Oppenheimer P.E.,
"Real Time Design and Animation of Fractal Plants and Trees"
Computer Graphics, Vol 20 No. 4, August 1986 (SIGGRAPH 86), ACM Press, pp55-64

A Simple Tree Construction Method

A tree consists of a branch attached to a (sub)tree.
This is just a "normal tree" in the sense of the recursively constructed and traversed data-structure.
Each branch is depicted as a cylinder or other geometric object.
When the structure has recursed deeply enough, the branch is truncated, possibly by drawing a leaf, fruit or flower.
In addition to producing a regular tree, we may alter (randomly or regularly) characteristics of the branches at each level of recursion.

Some characteristics to alter include:
- angle between parent branch and its children
- size ratio between parent branch and its children
- rate at which a branch tapers
- helical twist of child relative to parent
- number of children for each parent

a = 1.0
b = 0.8
c = 0.4

stem/stem ratio
= b/a
= 0.8

branch/stem ratio
= c/a
= 0.4

branching angle = 60 degrees

A variation on this method by Weber
SIGGRAPH 1995

3. Voxel (Space) Automata for Plant Modelling

Reference:

Greene, Ned
"Voxel Space Automata: Modeling With Stochastic Growth Processes In Voxel Space"
Computer Graphics Vol 23, July 1989 (SIGGRAPH 89), ACM Press, pp175 - 184

Accounting for Environmental Effects

Phototropic effects determine rate / direction of growth.
Growth occurs around obstacles (rocks, walls, tree limbs and creepers)
The calculations above may be accelerated by avoiding analytic geometry and relying on simple tests in 'voxel space'.
A 3D space may be partitioned into a uniform array of cubes (voxels / volume elements).
The process of identifying and labelling voxels that are intersected by a primitive object is called tiling.
Voxel representation of an environment simplifies intersection testing & measurement of proximity of objects.

Greene (SIGGRAPH 1989)

A simulation proceeds...

The voxel space is initially empty or tiled with an environmental model.
The plant is grown from seed points and rendered as geometric objects (eg. cylinders) satisfying a set of rules (given shortly).
From an active node, a potential direction and distance of growth is determined by employing the rules.
The direction of the previous segment is perturbed according to some distribution function. Eg. a bias might helically twist the new segment upward.

Sample Rules for Growth

Seed Locations

seed 0.5 -0.89 0.56
seed 0.35 -0.89 0.56

Skeleton Parameters

random tree {

length	2.4	limb length in voxel widths
branch_age_range	1->3	1,2 or 3 iterations before branching (random)
branch_angle	60	degrees
vertical_bias	0.08	slight bias to grow upward
no_of_trials_max	300	300 randomly perturbed trials before giving up
no_of_trials_min	150	150 trials before selecting best proximity fit
seekprox	1.5	pick trial with average proximity closest to 1.5 voxel widths
maxprox	3.0	reject trials with average proximity greater than 3 voxel widths

}

Illumination Parameters

light {

blend sun	0.8	blend sun exposure
blend sky	0.2	blend sky exposure
exposure_min	0.3	nodes with <30% exposure become inactive
exposure_max	1.0	nodes with up to 100% exposure may be active
boost	1.8	100% exposed nodes grow 1.8x faster than those with 30% exposure

}

Leaf Element Parameters

element {

number_of_trials

attempt placement 30 times before giving up

expected frequency

1.5

try to place 1.5 leaves per branch segment

model {

polygon(x,y,z)(x,y,z)
polygon(x,y,z)(x,y,z)

}

co-ordinates of polygonal leaf model
...

}

A new (potential) segment is accepted if it satisfies tests for...

Collisions against existing objects in the voxel array.
Proximity to other elements of the structure or the environment.
The amount of light falling on the cell occupied by the segment (see below).

Light and Growth

The amount of light falling into a voxel consists of two components.

1. The sky exposure for a voxel:

Cast rays from the cell towards a hemisphere representing the sky.
The fraction of a sample of rays which reach the sky without intersecting a voxel already tiled by an object, gives the sky_exposure of that voxel.

2. The sun exposure for a voxel:

Cast rays from the cell towards a point on a 180 degree arc representing the Sun's trajectory across the sky.
The fraction of a sample of rays which reach the arc without intersecting a voxel already tiled by an object, gives the sun_exposure of that voxel during a full day.

The different effects of Sun/sky exposure are weighted according to whether or not the plant responds well to diffuse or direct light.

Other Points of Interest

Greene (SIGGRAPH 1989)

Heliotropism may be simulated by incorporating a Sun-seeking bias into the rule set.
A plant may be made to grow to fit an existing rough geometric model tiled in voxel space.
Regions of the model may alter the growth characteristics of the plant.

For example, paving stones may forbid growth over their surface, gables may encourage growth along their length or arches may encourage vines to crawl around their edge.

4. Fractal Terrain

...a place for plants to grow

References:

Watt, Watt
"Advanced Animation and Rendering Techniques, Theory and Practice"
Addison-Wesley 1992, Section 7.1, pp203-205

Foley, van Dam, Feiner, Hughes,
"Computer Graphics, Principles and Practice"
2nd edn, Addison-Wesley, 1987, Section 20.3, pp1020-1026

Kajiya, J.T,
"New Techniques for Ray Tracing Procedurally Defined Objects"
SIGGRAPH 83, pp91-102

The ever-popular fractal landcape may be simply constructed...

Triangulate the selected polygon representing the terrain by bisecting each of its sides.
Randomly perturb each of these midpoints upwards by some amount proportional to the triangle's side-length.
Go to step 1 (until you have reached a fine enough resolution such that the peturbations result in detail finer than will be rendered)

A height map may be used to add the effect of rock strata or snow-capped peaks. For example...

Snow on top,

rock beneath,

foliage further beneath

and sand at the base.

Foliage may also be simulated by appropriately colouring polygons with their normals within some range of the vertical (and their height below the snow-line).

5. Particle Systems

Useful for virtual rain to make virtual trees grow, and even for modelling leaf clusters or grass. (Not forgetting to mention its use for modelling fireworks!)

Watt, Watt
"Advanced Animation and Rendering Techniques, Theory and Practice"
Addison-Wesley 1992, Section 18.1, pp415-417

Foley, van Dam, Feiner, Hughes,
"Computer Graphics, Principles and Practice"
2nd edn, Addison-Wesley, 1987, Section 20.5, pp1031-1034

Image from The Adventures of Andre and Wally B. (Lucasfilm) 1984. Leaves and grass generated by particle systems.

For class discussion...
- What might a particle system be?
- How might one be constructed?
- How might it be used?


Particle emission from circular generation shape.	Particle emission from point (or spherical) generation shape.