CS 284: CAGD
Lecture #10  Tue 9/25, 2012.
PREVIOUS
<     > CS
284 HOME <     > CURRENT
<     > NEXT
Preparation: Read:
1.) The seminal paper by E. Catmull and J. Clark:"Recursively generated Bspline surfaces on arbitrary topological meshes"
2.) Paper by D. Doo and M. Sabin: "Behavior of recursive division surfaces near extraordinary points"
3.) Chapters 2 and 3 from:
C.
Loop, "Smooth Subdivision Surfaces Based on Triangles"  Some
errata found in this thesis.
Recursive Subdivision (cont.)
Key points about useful subdivision schemes:
 There are two components to any subdivision scheme, a topological one and a geometrical one:
 topology: In a fixed way split the parametric domain of an edge or a face;
 geometry: Move some of the old and newly created vertices to new locations (that promise to yield a smoother shape).
 The number of points (line segments) must grow at a geometrical
rate with each generation.
 The newly introduced points should have a smoothing effect and
converge towards a limit function.
 This can typically be achieved with affine mapping schemes described
with a subdivision matrix.
 The infinite application of this matrix then leads directly to
a point on the curve or surface.
 The scheme should be independent of any coordinate system.
The Classical Subdivision Surfaces by Catmull & Clark
Presentation by: Laura and Eric
 This is a foundation for most modern subdivision algorithms

A generalization of the Bspline scheme

Can handle vertices with valences different from 4

Can handle meshes with facets other than quadrilaterals

One subdivision iteration calculates:
 new vertices at the centers of all faces,
 new vertices "below" the centers of all edges,
 a new vertex position for each old vertex.
(This could be described as a matrix transformation on all the old vertices.)

After the first iteration step, there will only be quadrilateral mesh facets;
 and from then on, there will be a constant number of extraordinary points:
 vertices of valence <>4 for each such original vertex.
 vertices of valence <>4 for each non quadrilateral mesh in the original
control polyhedron.

After the second iteration, any facet contains at most one
extraordinary vertex.

The extraordinary points will not disappear, but they will become more
and more "isolated",
 being surrounded by ever shrinking irregular regions,
 and being separated by more and more quadrilateral meshes joining in
valence4 vertices,

In these ever more dominating "regular" regions, the surface will approach
a Bspline.
Doo, Sabin Paper: Focus on a Quadratic Subdivision Surface
Presentation by: Michael

Extension of Chaikin's Corner Cutting algorithm (1974) to surfaces.

For rectangular quad meshes this results in quadratic Bspline surface.
 Gain an understanding of the role of the subdivision matrix and its eigenvalues.

Careful analysis and treatment of irregular points and convergence behavior around it.

A more detailed analysis of what happens at irregular points and of the convergence behavior around it;
 this paper also introduces matrices into the subdivision process,
 and the analysis of the eigenstructure to understand the behavior of
the limit surface.
 Discrete Fourier transform of the oscillations of the rings of neighboring vertices.
Charles Loop, Master Thesis: A Trianglebased Subdivision Scheme
Presentation by: Soham and Brandon
 Also allows to cover arbitrary topologies,
 but now with the more flexible triangle meshes.
 An approximating scheme, based on triangular spline N^{222};
 the resulting polynomials are of degree 4 (quartic).
 Also gives curvature continuity, except at the extraordinary points;
 but still yields a welldefined tangent plane (G1continuity) at these points.
How to efficiently and effectively read a paper to prepare for these discussions:
 Read: "Introduction" and "Conclusions;" look at figures and read captions.
This should typically tell you WHAT the authors have done, and WHY.  Now try to find out HOW they did it:
Look for the appropriate sections that describe the key techniques.  Also try to get an understanding what the limitations and caveats are of the described approach.
Might there be things that the authors do NOT tell you ?  You can probably skip the section on "Previous Work" or just skim it.
 Write down a few of the key points that you have learned,
as well as some questions that are good for discussion in class.
Reading Assignments for Thu. Sep. 27, 2012:
Zorin
et al: "Interpolating Subdivision Meshes with Arbitrary Topology"
Optional: if you are interested in the details of analyzing a particulart subdivision scheme:
Chapters 4 and 5 from:
C.
Loop, "Smooth Subdivision Surfaces Based on Triangles"
PREVIOUS
<     > CS
284 HOME <     > CURRENT
<     > NEXT
Page Editor: Carlo H. Séquin