Cellular Texture

Generation

• Scales

• Feathers

• Thorns

We are interested in making images of surfaces covered with interacting geometry elements such as :

There are a few challenges in making images of these type :

• It requires a representation with more detail then texture-mapping

but are inconvenient to model with hand-crafted geometry.

•It is often difficult to map appropriate texture

coordinates onto the global geometry and topology .

•The placement, orientation, coloration, and shape of

the individual elements may depend on :

- neighboring elements

- surface characteristics such as local curvature .

- global phenomena such as sunlight .

Key Words – particle system , developmental models ,

data amplification , constraints ,

texture mapping , bump mapping ,

displacement mapping .

Cellular textures are modeled by using small 3d cells

constrained to lie on a surface . The cells interact to form

cellular textures : surface textures with 3d geometry ,

orientation and color .

This approach combines properties of particle systems ,

developmental models,and reaction diffusion methods

into one system .

Development Approach

For generating organic patterns it is natural to consider a biological–based simulation .

biological development model is used to simulate and study patterns generated by the motions and interaction of discrete cells.

These artificial cells can move about , grow , and divide in a

simulated petri dish which is called extracellular environment .

The ability to form a variety of interesting patterns with the

system is the reason for exploring its application to geometric

texture generation .

The textures modeled, are formed from many interacting

geometric elements .

Scales feathers or thorns can be created from a single cell , or

multiple cells .

There are three main software modules :•Cellular particle simulator with surface constraints :

Automatically grow cellular textures by simulating

discrete cells on surfaces . Computes location ,

orientations , sizes and other parameters of cells

based on behavioral specification.

•Parameterized particle to geometry :

Converts cell position , orientations and other

parameters into shape and appearance parameters.

•Render

renders the scene .

Cellular Particle SimulatorAllows the user to specify cell behaviors such as ‘go to surface’ and ‘align with neighbors’ by combining modular cell programs, which are first-order differential equation terms that modify the cell’s state .

For a particular simulation the user defines :

• the cell state variables .

• the extracellular environment .

• the cell programs.• initial placements and other initial conditions.

The user can also stop the simulation ,

change the cell programs and , restart it with the new

modifications .

DefinitionsA cell is an entity that has position, orientation, shape, and other parameters.

It is a generalization of a particle in a particle system .

Cell state variable :

S = ( p , q , r , Sdie , Ssplit ,

Sc0 , Sc1 , Sc2 , … , Sa0 , Sa1 , Sa2 ,

… )

p - position .

q - orientation .

r - size .

Sdie - if exceeds a threshold θdie , the cell will die.

Ssplit - if exceeds a threshold θsplit , the cell will split .

Sci - concentrations of chemicals within the cell .

Sa0 - concentrations of chemicals in the cell membrane .

Extracellular EnvironmentParameters which describe everything the cell can sense from

its current position .

Λ = (Λa0 , Λa1 , Λa2 , ··· , Λp0 , Λp0 , Λp1 , Λp1 , ··· ,

Λv0 , Λv1 , ··· ,

Λwx , Λwy , Λwz , Λu0 , Λu1 ,··· )

Λai – represent the amounts of membrane chemicals that

are bound to membrane chemicals on neighboring cells.

Λpi , Λpi – represents value and gradient of potential field.

Λvi , Λui – other scalar and vector fields .

Λwi – this vector describes the rotation that would align this

cell’s ei axis with the average orientation of the adjacent

cells. Example :

Λwx = ∑ cos-1(exb · ex

c) n ||ex

b × exc||

c neighbors

b – the cellc – its neighbors

Cell programs Each cell has several cell programs , which are first order

equations describing how its state changes over time.

A cell program is a function of the cell’s current state S and

its environment as expressed by Λ .

Different types of cells use different cell programs or

different combination of the same cell program to define

their behaviors .

The entire system of differential equations to be solved is

obtained by superposing ordinary differential equations from

the cell program for every cell .

Examples of cell’s programs

Go to a surface :

Implements a constraint to keep a cell on the implicit surface f(x) = 0 .

As the the simulation runs a cell with this program will descend the

gradient and come to rest on the surface. The parameter k determines the

speed with which the particle approaches the surface.

Die if too far from surface :

Will cause Sdie to rise towards the threshold quickly if the cell is

greater than a certain distance d from the surface.

P' += -k f(p)f(p)

S'die += f(p)θdie - sdie d

Align with a vector field :

In this example the cell’s y-axis , ey , is aligned with a given vector field v(x) ,

Which is evaluated at the cell’s current location , p .

Align with neighbors :

The three orientation constraints on the cell’s x y and z axes fully

constrain its orientation .

Constraining two axes would be sufficient in most cases .

wv = k cos-1(ey · )

||ey × v(p)|| ||v(p)||

q' += (1/2)wvq

q' += (1/2)Λwiq

Maintain unit quaternion

Add a constraint to ensure that the quaternion does not stay too far from a unit quaternion during the integration of differential equations .

q' += 4k(1 – q · q)q

Adhere to other cells

Causes the variable sa2 (representing the surface chemical a2) to

approach and stay at the value 1.0 .

A pair of cells expressing a2 will stick together once they come in

contact , and force is required to pull them apart .

S'a2 += 1.0 – sa2

Divide until surface is covered

Divide until the amount of bound surface chemical a2

reaches the level γ . Λa2 reports the total amount bound

from all cells that are in contact , which gives the cell a

means of determining how many neighbors it has .

The value of will near one for (b >> a) and near zero

for (a << b).

Ssplit += (γ , Λa2) θsplit - ssplit

(a , b) (tanh((b-a) +1)/2

Set size relative to surface feature size

Relates the cell size to the sizes of features in the polygon database .

Achieved by providing the cells with a value Λu0 that represents

the area of the nearest triangle .

Could also be used to capture surface curvature .

r' += Λu0 - r

Example of reaction-diffusion in discrete cellsThe full derivation of this set of equations is beyond the scope of this presentation .

2 2s'c0 += -20Λa0 / Λa2 + 10sc0/(1 + sc0 ) – 0.5 sc0 +13

s'a1 += 0.95Λa0 / Λa2

s'a1 += (3, sc0)sc0

s'a1 += -sa1

s'a0 += 5 – sa0

s'a2 += 1 – sa2

Sc0 is the activator and sa1 is the inhibitor , which is propagated by the activity of membrane chemicals .

Determine interactions of membrane chemicals that lead to an effective diffusion of the value of sa1 among the cells .the value sc0 can then be used to determine the final rendered shape of the cell .

Defines a generic switch that tends to drive sc0 towards one of two values , depending on the influence of the term Λa0 / Λa2 .

}

Surface constraintsThere are several types of surface classes :

• polygonal mesh surfaces .

• Implicit function surfaces .

• isosurfaces of volume data surfaces. The surface constraint cell program, evaluates an implicit

function to enable the cell to find and stay on the surface.

Because the surface could be a mesh, a rough approximation

to an implicit function is created for these meshes .

Any implicitization method will work , and in fact it doesn’t

have to be very exact .

We implement this function by constructing an approximate

kd-tree for the triangular mesh .

Particle converter Converts information about the particles and their

environment into geometry and appearance parameters for

rendering .

It enables us to do a variety of useful operations :

1. Choosing an appropriate

representation for each cell

based on its screen size .

as you can see in the picture

the thorny spheres at further

distance are rendered with

fewer polygons.

2. smoothly changing the appearance of a cell based on

continuously varying parameters.

3. using the cell position to generate spatial subdivision .

4. using the cell orientation to compute a flow on the surface .

5. experimenting with various colorations and geometries

using the same simulation dataset .

Results

Scalescell program used:•Divide until the surface is covered •Stay on the surface•Die if pushed to far from the surface.•Soft constraint to align y-axes with the gradient of the surface implicit function•Align there x and z axes with their neighbors .

A knotty problemCapable of creating textures for surfaces with unusual topologies.

Thorny head ( the first pic )

Main point - Changing cell size to match surface features .

If you look closely you can see there is a finer texture and

geometry around the eyes and mouth .

The cell size is related to the detail level in the underlying

polygonal model.

Thorny spheres

Several important capabilities :

• the creation of a simple reaction-diffusion patterns on a

surface .

• the use of the concentrations of cell chemicals to change

parameters of the rendered geometry .

• the ability to restart simulations from an previous state with

new cell programs , causing new behaviors to occur .

A bear of a surface A fur-covered model of a bear defined as an isosurface of

sampled volume data .•The bear on the left , started from a single cell and used a

set of rules similar to the rules used in the scales picture .•The bear on the right started from about 2000 cells ,

each cell choused one neighbor to align with , and cells

didn’t try to align with neighbors that were oriented in the

directly opposing direction .

conclusionThe combination of particle system constraint techniques with developmental models enables the generation of a variety of cellular textures as shown in the figures .

Some difficulties:

Shapes : the particle converter might make objects with

undesirable intersections .

This can be minimized by a careful choice of cell

geometry .

A more robust solution is to use the desired

geometric shape directly in the cellular particle

simulation .

Experience with writing cell programs :

Writing cell programs can be difficult, it Requires a

Different intuition than other types of programming ,

and often takes a while to get right .

(gets easier with practice) .Simulation speed :

Can be very slow (many hours) for some kinds

of cell programs , or very fast (a few seconds)

for others.

Generally , performance degrades as the

differential equations get stiff .

Data explosion :

The data produced both by the simulation and by

the particle converter can get very large .

Future work

• continue extending and refining the cell programs to generate

more complex cellular textures .

• running simulations on objects as they move and change shape .

• modeling the motion of feathers on the wings of a flying bird .

• implementing more sophisticated cell geometries in the particle

simulator in order to generate more realistic placement of detail

and avoid self-intersections in the rendering .

• explore the possibilities of creating shapes directly from the

fundamental interactions of the cells , without the surface

constraint .

Bibliography

•Computer Graphics Proceedings , Annual Conference

Series 1995

The End