Subdivision Surfaces Geris Game (1989) : Pixar Animation Studios - - PowerPoint PPT Presentation

Subdivision Surfaces

Nathan Carr, N. Nikolaidis et al

Geri’s Game (1989) : Pixar Animation Studios

Smooth versus General

• Polygon meshes are very general, but hard to

model with

– In a production context (film, game), creating a dense, accurate mesh requires lots of work – Biggest problem is smoothness

• We desire a way to “smooth out” a polygonal

mesh

– We can model at a coarse level, and automatically fill in the smooth parts

• Subdivision surfaces are part of the answer

Subdivision Schemes

• Basic idea: Start with something coarse, and

refine it into smaller pieces, smoothing along the way

– Stop whenever the required smoothness is achieved (usually 3-5 iterations)

• Starting mesh is called control mesh

Subdivision Surfaces

• Approach Limit Curve Surface through an Iterative Refinement

Process.

Refinement 1 Refinement 2 Refinement ∞ Note: Limit Curve/Surface not known!

Subdivision in 3D

• Same approach works in 3D

Refinement

Types of Subdivision

• Interpolating Schemes

– Limit Surfaces/Curves will pass through original set of data points. – Each iteration generates only new vertices, does not move old

• nes

– Curve/Surface interpolates vertices from the previous step

• Approximating Schemes

– Limit Surface will not necessarily pass through the original set of data points.

Interpolating vs Approximating schemes

• Approximating:

– Create fair surfaces (smooth bends) – Converge faster – Shape is lowpass filtered, often shrinks!

• Potential remedy: use denser and carefully selected control mesh
• Interpolating:

– Less fairness, might create unnatural undulations – Creates mesh that usually is more “faithful” to the control mesh

• Differences almost disappear in meshes with many

triangles or smooth

Types of Subdivision

• Triangle based
• Polygon based
• Uniform: same update/subdivision rules for all

vertices/edges

• Nonuniform: different update/subdivision rules for

different vertices/edges (e.g. on boundary edges or irregular vertices)

• Stationary: same subdivision rule in all subdivision steps
• Non-stationary: different subdivision rule in different

subdivision steps

Two phase process (conceptually)

• In each subdivision step:

– Refinement phase: creates new vertices and connects them to create new, smaller triangles/polygons – Smoothing phase: moves new and perhaps old vertices

A Primer: Chaiken’s Algorithm

2 1 3 2 1 2

4 1 4 3 4 3 4 1 P P Q P P Q + = + =

3 2 5 3 2 4

4 1 4 3 4 3 4 1 P P Q P P Q + = + =

1 1 1

4 1 4 3 4 3 4 1 P P Q P P Q + = + =

1 1 2 1 2

4 1 4 3 4 3 4 1

+ + +

+ = + =

i i i i i i

P P Q P P Q

Apply Iterated Function System Limit Curve

In each step, points are doubled Corners are cut: lowpass filtering

P0 P1 P2 P3 Q0 Q1 Q2 Q3 Q4 Q5

Subdivision Operations Example

• 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

Definitions

• Valence of a vertex: the number of neighboring vertices
• f this vertex (the ones connected by edges)

– Number of edges emanating from the vertex

• In triangular meshes:
• Vertex of valence 6: regular
• Valence<>6: irregular, extraordinary

1-ring Rectangular meshes: regular vertices are usually those of valence 4

3D Surfaces Loop Subdivision

• Works on triangular meshes
• Is an Approximating Scheme
• Guaranteed to be smooth everywhere except at

extraordinary vertices.

Loop’s scheme (‘87)

Edge Vertex

8 3

8 1 8 3

8 1

2

1 5 3 2 2 cos 8 8 8 n n π β ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ = − + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

1 nβ − β β β β β β

Loop’s scheme (‘87)

• Edge mask: position of new vertex
• Vertex mask: new position of the existing vertex
• Both are applied on the values of the existing

vertices

• Weights in each mask sum to one: new/updated

point within the neighborhood of the weighted points

• C2 continuity (at the limit) except for irregular

vertices in the control mesh (its C1 there)

Loop’s scheme - Boundary

1 2 1 2 1 8 3 4 1 8

Vertex point Edge point

• The boundary is a cubic B-Spline curve
• The curve only depends on the control points on the boundary
• Good for connecting 2 meshes
• C0 for irregularities near boundary

Loop’s scheme - Summary

• Triangles – better use valence 6 (no special rules for irregular

vertices: uniform in that respect)

• Approximation
• Face Split
• C2 on regular meshes
• Only C1 on extraordinary points
• C0 for irregularities near boundary
• The scheme generates three directional box Splines at the limit
• Closed formulas for vertices and tangents at the limit exist (also for
• ther schemes)

Piecewise smooth subdivision

• Subdivision produces smooth continuous surfaces, not

always desirable.

• How can “sharpness” and creases be controlled in a

modeling environment? ANSWER: Define new subdivision rules for “creased” edges and vertices.

• 1. Tag sharp edges.
• 2. If an edge is sharp, apply new

sharp subdivision rules.

• 3. Otherwise subdivide with

normal rules.

Sharp Edges…

• 1. Tag Edges as “sharp” or “not-sharp”
• 2. Count number s of sharp edges coming in

at a vertex

• s = 0 – “not sharp-smooth”
• s=1 dart
• s=2 crease
• s>2 corner

During Subdivision,

• 3. Near “sharp” edges, use “sharp”

subdivision rules, according to the type of vertices.

• 4. Use normal rules otherwise

Subdivision as Matrices

• Subdivision can be expressed as a matrix Smask of

weights w.

– Smask is very sparse – Never Implement this way! – Allows for analysis

• Curvature
• Limit Surface

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ =

2 1 1 11 10 01 00

ˆ ˆ ˆ ˆ ˆ p p p p p p p w w w w w P P S

n nj mask

M M L O M M L L

Smask Weights Old Control Points New Points

What About Continuity and Curvature..

• Subdivision mask weights w are derived from splines,

such as B-Splines.

– In many cases subdivision surfaces converge to spline surfaces with C2 continuity everywhere.** – Too lengthy to cover here, but there is lots of literature.

Subdivision Methods for Geometric Design Joe Warren, Henrik Weimer. (2002) **Math works out except at “Extraordinary Vertices”. Most Subdivision Schemes have and “ideal” valence for which it can be shown that the limit surface will converge to a spline surface.

Ordinary and Extraordinary

Loop Subdivision Valence 6 Catmull-Clark Subdivision Valence 4

• Subdividing a mesh does not add extraordinary vertices.
• Subdividing a mesh does not remove extraordinary vertices.

How should extraordinary vertices be handled?

• Make up rules for extraordinary vertices that keep the surface

“smooth”.

Modified Butterfly scheme – Zorin(’96)

• Interpolating scheme: vertices of the previous

step are maintained.

• Connectivity as in Loop’s algorithm: insert a new

vertex in each edge, connect new vertices and split each triangle into four new ones

Modified Butterfly scheme – Zorin(’96)

a a b b c c c c d d

w d w c w b w a : 16 1 : 2 8 1 : 2 1 : : Weights − − + −

• 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 (Moller book: w=0, d vertices do not participate)

Modified Butterfly Scheme –Cont’

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ≥ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − = N j N j N e v N e e e e v N e e e v N

j

π π 4 cos 2 1 2 cos 4 1 1 : , 4 3 : : 5 : , 8 1 : , : , 8 3 : , 4 3 : : 4 12 1 : , 12 1 : , 12 5 : , 4 3 : : 3 : Weights

3 2 1 2 1

v e0 e1 e2 e3 eN-1 eN-2 eN-3

Extraordinary Vertex New Edge vertex 1 ring neighborhood

• 1. (Valence 6) X (valence 6): prev. case
• 2. (Valence 6) X (valence != 6): see

formulas on the right (N=valence)

• 3. (Valence != 6) X (valence != 6): (can

happen only in the first iteration) compute temporary vertices using formulas above, average to get new vertex.

• 4. Boundary vertices (see Moller)

Modified Butterfly Scheme - Summary

• Triangles – use any valence
• Interpolation
• Face Split
• C1 everywhere
• Not C2 even on regular meshes
• Non-uniform

Catmull-Clark scheme ’78

1

1

m i i

f p m

=

=

∑

Face Point Vertex Point

2 2 1 1

1 1 2

n n i i i i

n v f e p n n n

= =

− = + +

∑ ∑

1 2 1 2

4 p p f f e + + + =

Edge Point

( 3) 2 p n Q R v n n n − = + +

• Q – Average of face points
• R – Average of edge points
• P – old vertex

Keep subdividing…

• After 1 iteration, every new face is a rectangular.
• Extraordinary vertices are forever. Valence is retained.
• Ultimately, at the limit, the surface will be a standard bicubic B-spline

surface at every point except at these “extraordinary points”, and therefore at all but extraordinary points.

• Catmull & Clark did not prove or guarantee continuity at the

extraordinary points but note that trials indicate this much.

• Later on it was proven to be C1

) 2 (

C

Catmull-Clark - Summary

• Any topology – better use quad faces and vertices with valence 4
• Approximation
• Generalization of bicubic B-Spline
• Face Split
• C2 everywhere. C1 on extraordinary vertices

Visual Comparison

Visual Comparison – Cont’

Visual Comparison – Cont’

Subdivision: pros and cons

Pros:

• Many of the pros of splines
• Flexible wrt. topology/mesh
• No gaps or seams!
• Features are ”easy”
• Simple to implement
• Efficient to visualize
• Simple/intuitive to manipulate – modeling
• Hierarchical modeling with offsets
• Good for deformation – animation
• Built-in LOD for rendering

Cons:

• Ck , k>1 is hard (not good for CAD)
• Evaluation not straight forward (but…)
• Artefacts (ripples etc.)

Geri’s Game (Pixar studios) Splines in Toy Story 1

Adaptive Subdivision

• Not all regions of a model

need to be subdivided.

• Idea: Use some criteria and

adaptively subdivide mesh where needed.

– Curvature – Screen size ( make triangles < size of pixel ) – View dependence

• Distance from viewer
• Silhouettes
• In view frustum

– Careful! Must ensure that “cracks” aren’t made crack subdivide View-dependent refinement of progressive meshes Hugues Hoppe. (SIGGRAPH ’87)

Displaced Subdivision Surfaces

• Subdivision surfaces lack detail-smooth
• Displaced subdivision surfaces!
• Subdivision of a control mesh along with

displacement along the normals using a displacement map (having scalar displacement values)

Displaced Subdivision Surfaces

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?

Conclusions and Future Work

• Currently there is no standard data structure for

handling Subdivision Surface schemes

• Hardware (GPU) implementations do not exist
• Subdivision of higher dimensional surfaces

References

• "Recursively generated B-Spline surfaces on arbitrary topological meshes", by E Catmull and J
• Clark. Computer Aided Design, Vol. 10, No. 6, pp 350-355, November 1978.
• "Behavior of recursive division surfaces near extraordinary points", by D Doo and M Sabin.

Computer Aided Design, Vol. 10, No. 6, pp 356-360, November 1978.

• "Geometric Modeling with Splines, An Introduction" by Cohen, Riesenfeld, Elber: Section 7.2.1,

Section 13.4, Chapter 20.

• ZORIN, D., SCHR ¨ODER, P., AND SWELDENS, W. Interpolating Subdivision for Meshes with

Arbitrary Topology. Computer Graphics Proceedings (SIGGRAPH 96) (1996), 189–192.

• DYN, N., LEVIN, D., AND GREGORY, J. A. A Butterfly Subdivision Scheme for Surface

Interpolation with Tension Control. ACM Trans. Gr. 9, 2 (April 1990), 160–169.

• LOOP, C. Smooth Subdivision Surfaces Based on Triangles. Master’s thesis, University of Utah,

Department of Mathematics, 1987.

• Subdivision for Modeling and Animation – Siggraph 2000 course notes