1. (c) University of Wisconsin
Last Time
B-splines
Recall a property of B-spline surfaces – the control point grid must be rectangular
Clarification: For non-uniform B-splines, there is one control point for every n-d knots, regardless of whether the knot values are equal
0,0,0,1,2,2,2 requires 7 – 3 = 4 control points for a cubic B-spline
12/03/02
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2. (c) University of Wisconsin
Today
Subdivision schemes
Introduction to Global Illumination
12/03/02
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3. B-splines as Approximation
B-splines were developed as approximation functions
Given a set of points – the control points – the B-spline approximates them with a smooth curve or surface
Ideally, we would like to be able to provide a polygonal mesh, and then smooth it out with a B-spline surface
The 3D equivalent of smoothing a 2D poly-line with a B-spline
But we can’t do this in general, because B-spline control meshes must be rectangular
Can’t even do simple cases like a sphere: You can’t wrap a sphere is a rectangular sheet without singularities
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Subdivision Schemes
Basic idea: Start with something coarse, and refine it into smaller pieces, smoothing along the way
We have seen how subdivision may be used to render parametric curves and Bezier surfaces
We will see how it can be used for modeling specific objects, and as a modeling scheme in itself
In this lecture:
Subdivision for tessellating a sphere
Subdivision for fractal surfaces
Subdivision for B-spline patches
General subdivision surfaces
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5. (c) University of Wisconsin
Tessellating a Sphere
Spheres are best parameterized in polar coordinates:
Note the singularity at the poles
Tessellation: The process of approximating a surface with a polygon mesh
One option for tessellating a sphere:
Step around and up the sphere in constant steps of  and 
Problem: Polygons are of wildly different sizes, and some vertices have very high degree
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Subdivision Method
Begin with a course approximation to the sphere, that uses only triangles
Two good candidates are platonic solids with triangular faces: Octahedron, Isosahedron
They have uniformly sized faces and uniform vertex degree
Repeat the following process:
Insert a new vertex in the middle of each edge
Push the vertices out to the surface of the sphere
Break each triangular face into 4 triangles using the new vertices
Octahedron
Isosahedron
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The First Stage
Each new vertex is degree 6, original vertices are degree 4
Each face gets split into 4:
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8. Sphere Subdivision Advantages
All the triangles at any given level are the same size
Relies on the initial mesh having equal sized faces, and properties of the sphere
The new vertices all have the same degree
Mesh is uniform in newly generated areas
This is a property we will see later in subdivision surfaces
Makes it easier to analyze what happens to the surface
The location and degree of existing vertices does not change
The only extraordinary points lie on the initial mesh
Extraordinary points are those with degree different to the uniform areas
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Fractal Surfaces
Fractals are objects that show self similarity
The word is overloaded – it can also mean other things
Landscapes and coastlines are considered fractal in nature
Mountains have hills on them that have rocks on them and so on
Continents have gulfs that have harbors that have bays and so on
Subdivision is the natural way of building fractal surfaces
Start with coarse features, Subdivide to finer features
Different types of fractals come from different subdivision schemes and different parameters to those schemes
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Fractal Terrain (1)
Start with a coarse mesh
Vertices on this mesh won’t move, so they can be used to set mountain peaks and valleys
Also defines the boundary
Mesh must not have dangling edges or vertices
Every edge and every vertex must be part of a face
Also define an “up” direction
Then repeatedly:
Add new vertices at the midpoint of each edge, and randomly push them up or down
Split each face into four, as for the sphere
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11. Fractal Terrain Example
A mountainside
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12. Fractal Terrain Details
There are options for choosing where to move the new vertices
Uniform random offset
Normally distributed offset – small motions more likely
Procedural rule – eg Perlin noise
Scaling the offset of new points according to the subdivision level is essential
For the subdivision to converge to a smooth surface, the offset must be reduced for each level
Colors are frequently chosen based on “altitude”
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13. (c) University of Wisconsin
Fractal Terrains
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Terrain, clouds generated using procedural textures and Perlin noise
tool is called Terragen
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15. (c) University of Wisconsin
Terrain, clouds generated using procedural textures and Perlin noise
tool is called Terragen
12/03/02
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16. Fractal Terrain Algorithm
The hard part is keeping track of all the indices and other data
Same algorithm works for subdividing sphere
Split_One_Level(struct Mesh terrain)
Copy old vertices
for all edges
Create and store new vertex
Create and store new edges
for all faces
Create new edges interior to face
Create new faces
Replace old vertices, edges and faces
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17. Subdivision Operations
Split an edge, create a new vertex and two new edges
Each edge must be split exactly once
Need to know endpoints of edge to create new vertex
Split a face, creating new edges and new faces based on the old edges and the old and new vertices
Require knowledge of which new edges to use
Require knowledge of new vertex locations
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18. (c) University of Wisconsin
Data Structure Issues
We must represent a polygon mesh so that the subdivision operations are easy to perform
Questions influencing the data structures:
What information about faces, edges and vertices must we have, and how do we get at it?
Should we store edges explicitly?
Should faces know about their edges?
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19. General Subdivision Schemes
Subdivision schemes can also be used where there is no “target” surface
They aim to replace a polygonal mesh with a smooth surface that approximates the coarse mesh
There are many schemes:
Butterfly scheme (for triangular meshes)
Catmull-Clark subdivision (for mostly rectangular meshes, converges to B-splines in uniform regions)
Loop’s scheme (for triangular meshes)
Modified butterfly scheme (for triangular meshes)
Many more…
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Butterfly Scheme
Subdivides the same way we have been discussing
Each edge is split
Each face is split into four
Rules are defined for computing the splitting vertex of each edge
Basic rule for a uniform region
Splitting an edge with endpoints that have degree 6
As before, all new interior vertices will have degree 6
Take a weighted sum of the neighboring vertices
Weights define rules
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21. (c) University of Wisconsin
Butterfly Scheme (1) c b c a a d d c b c
Multiply each vertex by its weight and sum them up
w is a control parameter – determines how closely the shape conforms to the original mesh
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22. Modified Butterfly Scheme
The butterfly scheme must be modified to deal with edges with an endpoint of degree  6
In that case, compute new vertex based only the neighbors of the extraordinary vertex
If an edge has two extraordinary endpoints, average the results from each endpoint to get the new endpoint
The modified butterfly scheme is provably continuous about extraordinary vertices
Proof formulates subdivision as a matrix operator and does eigen-analysis of subdivision matrix
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23. Modified Butterfly Schemev
eN-1
eN-3
eN-2
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24. Modified Butterfly Example
Notes:
The mesh is uniform everywhere except the original vertices
It interpolates the original vertices
It has smoothed out the underlying mesh
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Shading Revisited
Some applications are intended to produce pictures that look photorealistic, or close to it
The image should look like a photograph
A better metric is perceptual: the image should generate a target set of perceptions
Applications include: Film special effects, Training simulations, Computer games, Architectural visualizations, Psychology experiments, …
To achieve the goal of photorealism, we must think carefully about light and how it interacts with surfaces
What you should take away: The various aspects of light interaction and how algorithms capture or ignore them
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Light Transport
Light transport problems are concerned with how much light arrives at any surface, and from what direction
The physical quantity of interest is radiance: How much light (power) is traveling along a line in space per unit foreshortened area per unit solid angle
We will not go into the theory - it takes 3 hours just to give the definitions and equations
CS779 will cover this material in detail
Similar problems arise in radiated heat transport (i.e. satellites), where some of the technology was originally developed
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Light Transport
Which surface gets more light? Why?
How much light reaches point “a”?
If the walls are black?
If the walls are mirrors? a a b
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Reflectance Modeling
Reflectance modeling is concerned with the way in which light reflects off surfaces
Clearly important to deciding what surfaces look like
Also important in solving the light transport problem
Physical quantity is BRDF: Bidirectional Reflectance Distribution Function
A function of a point on the surface, an incoming light direction, and an outgoing light direction
Tells you how much of the light that comes in from one direction goes out in another direction
General BRDFs are difficult to work with, so simplifications are made
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29. (c) University of Wisconsin
Simple BRDFs
Diffuse surfaces:
Uniformly reflect all the light they receive
Sum up all the light that is arriving: Irradiance
Send it back out in all directions
A reasonable approximation for matte paints, soot, carpet
Perfectly specular surfaces:
Reflect incoming light only in the mirror direction
Rough specular surfaces:
Reflect incoming light around the mirror direction
Diffuse + Specular:
A diffuse component and a specular component
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30. (c) University of Wisconsin
Light Sources
Sources emit light: exitance
Different light sources are defined by how they emit light:
How much they emit in each direction from each point on their surface
For some algorithms, “point” lights cannot exist
For other algorithms, only “point” light can exist
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31. Global Illumination Equation
The total light leaving a point is given by the sum of two major terms:
Exitance from the point
Incoming light from other sources reflected at the point
Exitance
Sum
BRDF
Incoming
light
Light leaving
Incoming light reflected at the point
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32. Photorealistic Lighting
Photorealistic lighting requires solving the equation!
Not possible in the general case with today’s technology
Light transport is concerned with the “incoming light” part of the equation
Notice the chicken and egg problem
To know how much light leaves a point, you need to know how much light reaches it
To know how much light reaches a point, you need to know light leaves every other point
Reflectance modeling is concerned with the BRDF
Hard because BRDFs are high dimensional functions that tend to change as surfaces change over time
12/03/02
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