CS 284: CAGD
Lecture #13  Tu 10/10, 2006.
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Preparation:
Zorin
et al: "Interpolating Subdivision Meshes with Arbitrary Topology"
Chapters 4 and 5 from:
C.
Loop, "Smooth Subdivision Surfaces Based on Triangles"
Topic: Subdivision (cont.)
Interpolating Subdivision Schemes for Surfaces
Discussion of the Zorin paper:

What are the key ideas: (Section 1, six bullets in right column)
 Triangle stencil (Figure 3).
 The construction of new edge midpoints: (Section 3.2)

what are the actual regions of influence ?

why does it make sense to ignore the neighbors to the right of S1 and S6
in Figure 3b ?

The modified subdivision scheme: (Section 3.3: four cases)

Improvements over Butterfly scheme: (Fig. 4a)

How can such interpolated regions be stitched together ?
 Treatment of boundary edges.
Discussion of new Homework Assignnment: Devising a Subdivision Scheme
How does your scheme deal with the above issues ?
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 ...)

C^{n} continuity can readily be inferred form behavior of the polynomial
or trigonometric functions.

G^{n }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 Bspline 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 ngon;
this ngon is used to define a reference plane (= tangent plane).

Second eigenvalue for w=0 is 0.25; describes
hill/bowllike 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(ij)/n))/4n

Loop thesis, triangular spline N^{222}, (chapter 4):

New extraordinary vertex V^{k+1} = a_{n}V^{k}
+ (1a_{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_{n}V^{0}
+ (1b_{n})Q^{0}, 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 V^{0}.

Tangent plane is defined by ring of neighbors only !

Curvature Continuity 

Rather than explicitly develop the periodic normalcurvature 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 welldefined curvature function around an extraordinary point
! :(

I.e., welldefine Gaussian curvature does not exist at extraordinary points
!

A reasonable choice that gives goodlooking surfaces: a_{n}
= (3/8 + 0.25 cos(2p/N))^{2} + 3/8
Reading Assignments:
Labsik and Greiner: "Interpolatory Root3 Subdivision".
Current Homework Assignment: Devise an Interpolating Subdivision Scheme for QuadMeshes.
Continue with the Warmup Exercise in the same five groups.
Exchange phone numbers, emails, available time slots ... NOW.
Meet as a group at least twice between now and Tuesday to discuss this task.
As a team, email me an interim report (less than 1 page) describing your scheme and its preliminary evaluation.
DUE: TUESDAY 10/10/2006, Noon!:
New Homework Assignment:  Two options to choose from:

Implement your proposed quad subdivision scheme as a team (in the language of your choice), or ...

DUE: TUESDAY 10/17/2006, 2:10pm.
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