Optimal Sampling of Parametric Surfaces
Yufei Li1, Wenping Wang1 and Changhe Tu2
1The
University of Hong Kong
2Shandong University
Surface Sampling

How to compute the optimal sampling of a parametric
surface?
1
?
0
1
Parameter
domain
(a) Non-uniform
sampling of the
surface
(b) Uniform
sampling of the
surface
Motivation

Uniformly sampling parameter domain may not
produce quad faces of a fixed aspect ratio.
Uniform Sampling: of the
parameter domain:
(a)
Non-uniform Sampling of
the parameter domain:
(b)
Optimal Curve Parameterization [Farouki, 1997]


For a parametric curve r(t) of unit arc length and
over the interval [0,1], the unit-speed / arc-length
parameterization is defined as:
Measure of “closeness” to
arc-length parameterization:
Polynomial
parameterization
Optimal
parameterization
Optimal Surface Parameterization [Yang et al, 2006]


The arc-length condition for curve case is imposed
on four boundaries of surface patch.
Rational linear reparameterization is employed.
Counter example
J=0
Formulation

Find optimal sampling via reparameterization
P(u,v)
u = u (s)
v = v (t)
Reparameterization
Q(s,t)
Optimal Surface Parameterization

Definition: P(u,v) is ideal if ||Pu(u,v)|| = k||Pv(u,v)|| for
some constant k>0.
For an ideal parameterization P(u,v), u and v intervals are
uniformly sampled to produce mesh faces of aspect ratio
equal to 1.
Input
Uniformity Metric


Metric D(P) measures how far P(u,v) deviates from
ideal parameterization:
D(P) =0 for an ideal parameterization P(u,v)
Parameterization Optimization

Compute u(s) and v(t) to minimize D(Q)
?
?
Results


Cubic polynomial transformation is used
Objective function D(Q) is minimized using SQP
Input
Output
Results
Left: input Right: output
Summary


Compute optimal sampling of parametric surfaces
by means of reparameterization.
Future work.
?
u = u (s)
u = u (s,t)
v = v (t)
v = v (s,t)
Current work
Future work
Thank You