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

Preparation:

Warm-up: Understanding Subdivision and Its Analysis

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.
• For rectangular quad meshes this results in quadratic B-spline 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.
• Discrete Fourier transform of the oscillations of the rings of neighboring vertices.
  

Comparing Subdivision Schemes for Surfaces

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!

• Tensor Product Splines
• E. Catmull and J. Clark (1978)
• Cubic quad stencil
• Weight masks for new vertices
• D. Doo and M. Sabin (1978)
• Quadratic quad stencil
• Weight masks for new vertices
• Triangle Splines
• C. Loop (1987);  Construction : Chapter 3;  Analysis: Chapter 4:
• Quartic triangle stencil
• Weight masks for new vertices, regular case
• Weight masks for new vertices, generalization
• Different weights introduced by Warren
• Triangle spline as the shadow of a cube
• Subdivision of shadow cube
• D. Zorin, P. Schroeder, W. Sweldens (1996): Interpolating !
• Triangle stencil
• Weigth mask
  

Evaluation / Analysis of Subdivision Schemes

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): w_ij = (3+2cos(2p(i-j)/n))/4n
• Loop thesis, triangular spline N²²², (chapter 4):
• New extraordinary vertex V^k+1 = a_nV^k + (1-a_n)Q^k, where Q^k is the centroid of the surrounding vertices P^k_i
• Pick a_n for best performance; convergence occurs for -5/8 <  a_n < 11/8.
• Convergence proof in two steps: Show:  V^k  --> Q^k,  and also for each i:  P^k_i --> Q^k
• The explicit point of convergence is:  Q^k = b_nV⁰ + (1-b_n)Q⁰, where b_n = 3 / (11 - 8a_n ).
• Tangent Plane Continuity -- gives narrower bounds on a_n : -0.25 cos 2p/N < a_n  <  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 a_n 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:  a_n = (3/8 + 0.25 cos(2p/N))² + 3/8

Discussion of Your Genus-4 Designs...

Reading Assignments:

Current Homework Assignment:

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

• 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)