CS 284: CAGD  Lecture #11 -- Mon 10/5, 2009.

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Preparation:

Paper by Doo and Sabin on subdivision surfaces

Chapters 2 and 3 from: C. Loop, "Smooth Subdivision Surfaces Based on Triangles"
This is also a nice review of B-spline subdivision!
Some errata found in this thesis.

Topic: 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.

Doo, Sabin Paper: Focus on a quadratic subdivision surface

• Extension of Chaikin's Corner Cutting algorithm (1974) to surfaces.
• 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.
• Discrete Fourier transform of the oscillations of the rings of neighboring vertices.

Comparing Subdivision Schemes for Surfaces

Tensor-product patches, e.g., cubic tensor-product Bezier patches;
readily do in the "u" and in "v" directions what we have learned about curves.

We can also use with triangular patches, but need a different interpolation scheme:
Barycentric coordinates: three numbers, but with constraint that they must sum to 1.0.
DeCasteljau evaluation technique can also be applied to triangular patches!

Evaluation / Analysis of Subdivision Schemes

How do we know whether a particular interpolation or subdivision scheme is any good ?

Testing / Evaluation by Visual Inspection

• Subject your scheme to many tough test cases: -- ideally move control points interactively and continuously,
because "transition cases" (e.g., extra inflection points) often show the weaknesses of a scheme.

Formal Analysis of Blending / Subdivision Methods

• If curve is formed with analytical functions (e.g., for Bezier, Lagrange, Circle Splines ...)
• Cn continuity can readily be inferred form behavior of the polynomial or trigonometric functions.
• Gn continuity needs a separate analysis; perhpas a bound on curvature can be established;
or it may be sufficient to show that the velocity cannot get to zero (Circle Spline paper, section 3.4)

• Subdivision curves/surfaces are harder to analyze:
• How do you prove that final curve points do not have small fractal oscillations ?
• or that the tangents converge to a well defined value at every point ?

• Doo & Sabin, extraorinary points in quadratic B-spline surfaces:
• Do not analyze the behavior of individual points, but of the whole ring of vertices around an extraordinary point.
• Do a discrete Fourier analysis of this ring of vertices; needs frequencies from w=0 to n/2 (n=valence) to capture all DoF.
• Repeated application of the subdivision matrix converges to a vector corresponding to largest eigenvector of the matrix.
• For the regular (valence 4) vertex we observe this behavior:
• Largest eigenvalue for w=0 is 1.0;
• this guarantees translation invariance of the process (i.e., does not "run away").
• Largest eigenvalue for w=1 is 0.5;
• this implies that the region around this vertex shrinks towards an affinely distorted regular n-gon;
this n-gon is used to define a reference plane (= tangent plane).
• Second eigenvalue for w=0 is 0.25; describes hill/bowl-like behavior at this point.
• Largest eigenvalue for w=2 is 0.25; describes the amount of warping (into a saddle) at this point.
• Doo&Sabin found subdivision coefficients for the extraordinary cases that also give these eigenvalues,
• and thus also guarantee tangent plane continuity at these points (page 360): wij = (3+2cos(2p(i-j)/n))/4n
• Loop thesis, triangular spline N222, (chapter 4):
• New extraordinary vertex Vk+1 = anVk + (1-an)Qk, where Qk is the centroid of the surrounding vertices Pki
• Pick an for best performance; convergence occurs for -5/8 <  an < 11/8.
• Convergence proof in two steps: Show:  Vk  --> Qk,  and also for each i:  Pki --> Qk
• The explicit point of convergence is:  Qk = bnV0 + (1-bn)Q0, where bn = 3 / (11 - 8an ).
• Tangent Plane Continuity -- gives narrower bounds on an : -0.25 cos 2p/N < an  <  0.75 + 0.25 cos 2p/N.
• Again, use discrete Fourier transform to capture the behavior of all edges converging in V0.
• Tangent plane is defined by ring of neighbors only !
• Curvature Continuity --
• Rather than explicitly develop the periodic normal-curvature function around an extraordinary vertex,
study the rate of change of the tangent function with respect to the subdivision process !
• Analysis shows: No choice of an can assure a well-defined curvature function around an extraordinary point !   :-(
• I.e., well-defined Gaussian curvature does not exist at extraordinary points !
• A reasonable choice that gives good-looking surfaces:  an = (3/8 + 0.25 cos(2p/N))2 + 3/8

Zorin et al: "Interpolating Subdivision Meshes with Arbitrary Topology"

Optional, if you are interested in the details of analyzing a particulart subdivision scheme:
C
hapters 4 and 5 from:  C. Loop, "Smooth Subdivision Surfaces Based on Triangles"

Current Homework Assignment:

Design the Control Mesh for a Smooth Genus-4 Surface.

The goal is to design a highly-symmetrical control mesh for a closed genus-4 subdivision surface based on triangles
(which could be later used for experiments in surface-energy minimization studies).
Following an iterative design process, we will do this in stages:
• MON 10/5:  Hand in a sketch of the rough geometry of the object that you plan to construct,
• and a paragraph that outlines your plan for constructing the actual control mesh.

• MON 10/12:  Complete assignment due. Hand in a printout of a smooth Loop surface;
• and submit your SLIDE file electronically.

Paper Presentation Assignments (for the future)

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