Energy-minimizing Curve Design Gang Xu Zhejiang University
Ouyang Building, 20-December-2006 2 ContentContent Application
Background(5min) Traditional energy-minimizing curves(30min)
Internal energy External energy Energy-minimizing curves in
manifolds(15min) Energy-minimizing curve networks (15min) Our
recent work(15min) Summary and outlook(5min) 3 4 Application
Background(1) Energy minimizing is the favourite of nature! 5
Application Background(2) Fairing curves 6 Application
Background(3) Computer vision and image processing geodesics and
active contours 7 Application Background(4) Medicine (Path planning
of surgical sutures) 8 Application Background (5) Robotic Snake
robots Motion design Traditional Energy-minimizing Curve Internal
energy External energy 10 Internal energy of curves(1) Stretch
energy length Strain (bend) energy spline and fairness 11 Internal
energy of curves(2) Energy in tension Variation of curvature
circle-like 12 Internal energy of curves(3) Jerk and load Energy of
3D curves 13 Interpolating curves with gradual changes in
curvature(CAD,1987) H. Meier and H. Nowacki, Germany Interpolating
Solve linear system 14 Method to approximate the space curve of
least energy and prescribed length (CAD, 1987) M Kallay This paper
presents a numerical method for computing the curves of least
strain energy, given the positions and directions of the endpoints
and the total length. Discrete method 15 Variational design of
rational Bezier curves (CAGD, 1991) H Hagen and GP Bonneau Describe
a calculus of variation approach to design the weights of a
rational curve in a way as to achieve a smooth curve in the sense
of an energy integral Method? 16 Minimum curvature variation curves
(PhD, 1992) (1) HP. Moreton, CH. Sequin Method: numerical
integration gradient descent 17 Minimum curvature variation curves
(PhD, 1992) (2) Space curves 18 Variational subdivision curves
(TOG, 1998) Leif Kobbelt Interpolating variational subdivision
curves Approximation variational subdivision curves (Hofer and
Pottmann, TVC, 2002) 19 Interpolating Method(1) Objective function
(open) 20 Interpolating Method(2) Solve a linear system 21
Interpolating Method(3) 22 Approximating Method(1) Objective
function 23 Approximating Method(2) Solve a linear system 24
Approximating Method(3) 25 Approximating Method(4) 26
SummarySummary Fair and smoothness Numerical method Not geometric
method! Traditional Energy-minimizing Curve Internal energy
External energy 28 Interactive design of constrained variational
curves (CAGD, 1995) W. Wesselink, RC. Veltkamp Motivation
constrained condition energy function (global) Edit using control
points Not flexible! Not variational design! 29 Solution Solution
30 ConstraintsConstraints Point interpolation Normal (tangent)
interpolation 31 External energy operators(1) Director 32 External
energy operators(2) Point attractor 33 External energy operators(3)
Curve attractor 34 Combing the energy terms 35
Computation(1)Computation(1) 36 Computation(2)Computation(2) 37
Computation(3)Computation(3) 38 ExamplesExamples 39
LimitationLimitation 40 Modeling 3D curves of minimal energy( EG,
1995) Generalize 2D to 3D Differences(1) 41
Difference(2)Difference(2) 42 Difference(3) constraints
Point-in-planePoint-in-plane Point-on-linePoint-on-line 43
Difference (4) Plane attractor 44 Difference (5) Director 45
Difference (6) Profiler 46 Difference (7) Point repellor 47
SummarySummary Unify of smoothness and interaction Generalization
to surface is easy! Numerical method Not geometric method! 48
Energy minimizing splines in manifolds (Siggraph, 2004) Hofer,
Pottmann, Wallner CharacterizationComputationApplication 49
Variational interpolation in curved geometries Find curve as
solution of a variational problem Use energy functions from spline
theory, but restrict curve to surface Any surface representation,
dimension & co-dimens. 50 InputInput Points p 1,..., p N on
d-dimensional surface S in R n, parameter values u 1,..., u N p1p1
pipi pNpN p2p2 S 51 Geodesics on surfaces Minimize L 2 norm of 1 st
derivative on surface S: Shortest connecting curve c on surface
traced with const. speed Pieces of c have 2 nd derivative vectors
orthogonal to S S 52 Counterparts to cubic splines on surfaces
Minimize L 2 norm of 2 nd derivative: Interpol. C 2 curve c on S 4
th derivative vectors of c are orthogonal to surface Existence
[Bohl 1999, Wallner 2004] S 53 Counterparts to splines in tension
on surfaces(1) Minimize: C 2 curve such that is orthogonal to
surface S 54 Counterparts to splines in tension on surfaces(2) 55
Energy minimizing splines in manifolds
CharacterizationComputationApplication 56 Computation (1)
Non-linear problem Even for simple surfaces no explicit solution
Numerical algorithm for various surface representations and
dimensions 57 Computation (2) Discretize curve on S in R low to
polygon P View P as a point X in high-dim space R high Constraint
manifold in R high is set of Xs for which vertices of P are
contained in S P R low S R high X 58 Computation (3) Quadratic
functional Discretization Quadratic function P... Minimizer of F in
R high P*... Minimizer of F on Matrix Q of F determines a Euclidean
metric R high P P*P* 59 Computation (4) Solution P* of our problem
is normal footpoint of P on in the metric given by Q Iterative
algorithm with geometrically motivated stepsize control P R high
P*P* X0X0 60 Computation (5) Iterative algorithm with geometrically
motivated stepsize control Algorithm needs: initial value x 0
tangent space Projection onto surface 61 Spline curves on various
surface representations parametric implicit triangle mesh point
cloud 62 Energy minimizing splines in manifolds
CharacterizationComputationApplication 63 Cyclic motion minimizing
cubic spline energy E 2 64 Cyclic motion minimizing tension spline
energy E t 65 Cyclic motion minimizing kinetic energy E 1 66
Splines avoiding obstacles in 3D / on bounded surfaces 67
Variational motion design in the presence of obstacles 68 Energy
minimizing curve networks Mininum variation networks Mininum
variation networks Fair webs on surface Fair webs on surface 69
Minimal variational networks (Siggraph, 1992) Moreton, Sequin
Problem Method: constraints and energy minimizing 70
MotivationMotivation obtain high quality surface 71
ExamplesExamples 72 Energy minimizing curve networks Mininum
variation networks Mininum variation networks Fair webs on surface
Fair webs on surface 73 Fair Webs (TVC, 2007) Wallner, Pottmann,
Hofer Contribution: a variational approach to the design of energy
minimizing curve networks that are constrained to lie in a given
surface or to avoid a given obstacle 74 InputInput Connectivity 75
EnergyEnergy 76 PropertiesProperties 77 Fair polygon networks 78
Application (1) Aesthetic remeshing 79 Application (2) Fair
parameterization 80 Application (3) Surface restoration and
approximation 81 Application (4) Fair surface design in the
presence of obstacles 82 Thanks! Congratulations on 7th Anniversary
of Macao's Return!
