pde-based geometric modeling and applications
TRANSCRIPT
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PDE-based Geometric Modelingand Interactive Sculpting
for Graphics
Hong QinCenter for Visual Computing
Department of Computer ScienceSUNY at Stony Brook
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Geometric Modeling
Geometric modeling techniques
Shape representations
Geometric applications
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Background Review• Introduction to PDEs• PDE techniques and applications
Geometric modeling, visualization, simulation, animation, image processing, ……
• Other modeling techniquesFree-form splines, implicit functions, physics-
based techniques, medial axis extraction
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PDE Techniques and Applications• Elliptic PDEs for geometric modeling• Level set method• Diffusion equations• Other applications
– Simulation and animation– Image processing……
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PDEs for Geometric Modeling• Geometric objects are defined by a set of
PDEs• PDE objects are controlled by a few
parameters• Powerful numerical techniques to solve
PDEs are available• PDE is related to energy optimization• PDE models can potentially unify geometric
and physical aspects
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Geometric Modeling• Shape representations
–Explicit model• Defines objects by positions• free-form splines, parametric PDE model, Subdivision model, ……
–Implicit model• Defines objects by level set of scalar functions• CSG model, level-set model, splines,……
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Geometric Applications• Shape design
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Geometric Applications• Shape design• Object deformation
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Level Set Illustration
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Geometric Applications• Shape design• Object deformation• Model reconstruction
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Geometric Applications• Shape design• Object deformation• Model reconstruction• Shape blending ……
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A PDE Example• PDE (Partial Differential Equation)
– Order r– , g(u,v) : control functions– : unknown function of u,v
( ) ( ) ),(,,0 0,
, vugvufvu
vur
n
nml
mlml
n
ml =∑ ∑∂∂∂
=
=+
≥α
( )vuml ,,α( )vuf ,
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Related Work of Physics-based Modeling[Terzopoulos et al. 87][Terzopoulos and Fleisher 88][Celniker and Gossard 91][Qin and Terzopoulos 94, 96][Koch et al. 96][Mandal et al. 98, 99][Dachille et al. 99]……
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Background Summary• Geometric PDE techniques• Level set method• Diffusion equations with applications• PDE-based simulation and image
processing• Implicit models• Physics-based techniques• Medial axis extraction
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PDE Techniques for Graphics• Using differential properties• Various applications
– Image processing
[Bertalmio et al. 00]
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Image Inpainting• Inpainting:
– Modify images in an undetectable way– Damage recovery, selected area removal
• Use gradient information, especially around the boundary of selected regions
• Propagate information from the surrounding areas using certain PDEs of gradient vectors
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[Bertalmio et al.00]
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Modeling Fracture•
[O’Brien and Hodgins99]
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Fluid Dynamics• Navier-Stokes equations
– u: velocity field– p: pressure field– ρ: density– v: kinematic viscosity of the fluid– f: external force–
( )⎪⎩
⎪⎨⎧
+∇+∇−∇⋅−=∂∂
=⋅∇
(b)1(a)0
2 fupuuuu
vt ρ
( ) ( )zyxyx ∂∂∂∂∂∂=∇∂∂∂∂=∇ ,,,,
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Applications of Fluid Dynamics• Gas simulation• Water simulation• Explosions• Nature texturing……
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Simulating Gaseous Phenomena
[Foster and Metaxas97b]
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Animating Explosions
[Yngve et al.00]
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PDE Techniques for Graphics• Using differential properties• Various applications
– Image processing– Simulation– Visualization– Geometric modeling
[Schneider and Kobbelt 00]
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αj pi
Surface Fairing• Curvature flow
∑ −+Α
=j
jijj ))(cot(cot41 ppn βακ
pj βj
nx κ−=&
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PDE Approach for Surface Fairing• PDE approach can solve the fairing problem
directly• Fairing based on geometric invariants• Construct surfaces based on discrete data
with subdivision connectivity of regular patches
0=Δ HB
( ) ( )2
22
2
2
2
2 ,1
,1
Δ=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=ΔvuvuvuB λλ
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Surface Fairing• Taubin
• Membrane or thin-plate energy
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Outline• Motivation and contributions• Related work• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
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Outline• Motivation and contributions• Related work• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
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Motivation: Why PDE Techniques?• Formulate natural physical process• Satisfy continuity requirements• Minimize energy functionals• Define objects using boundary information• Provide intuitive and natural control• Unify geometric and physical attributes• Employ powerful numerical techniques
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Motivation: Limitations of Prior Work• Indirect manipulation for PDE objects• Limit constraints for geometric PDE objects• Lack of local control for regional shape sculpting• No intuitive manipulation with physical properties• Limitations of acceptable shape representations
for PDE models• Lack of integration framework of different types of
PDEs• Limit applications of geometric PDE modeling
system
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Motivation: A General PDE Framework• General modeling framework for geometric
objects of different data formats• Direct manipulation and interactive sculpting
with global/local control• Integration of physical properties for realistic
modeling• Comprehensive toolkits for various modeling
functionalitiesDesign, reconstruction, abstraction, manipulation ……
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Applications:
Contributions: Overview
……MorphingModel Abstraction
Shape Sculpting
Object Reconstruction
Shape Design
PDE-based Geometric Modeling System
Modeling Representations and Techniques:Free-form
SolidsDynamic
ModelImplicit
FunctionsArbitrary Meshes
Parametric Surfaces
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Contributions: System Functions• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit PDE shape design and manipulation– PDE-based free-form modeling and deformation
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Contributions: System Components
PDE-based Geometric Modeling System
Physics-based PDE
Surfaces
Free-Form PDE Modeling with Intensity
Implicit PDE Model
Arbitrary PDE
Meshes
Iso-Surface Extraction
Scattered Datasets
Intensity DistributionBoundary
Surface Sculpting
Embedded Datasets
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Outline
• Motivation and contributions• Related work• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
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Another Classification of PDEs• Initial value problem
– Given information at , the solution will propagate forward in time
• Boundary value problem– Given boundary
information of the region of interest of variables
– Solution will be a static function within the region
0t
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PDE types Hyperbolic PDE
Parabolic PDE
Elliptic PDE
Initial value problem
Wave equation
Diffusion equation
Boundary value problem
Poisson equation
Summary of PDE Classifications
• Poisson equation: Geometric modeling, image processing……
• Diffusion equation: Texture synthesis, image processing, ……
• Wave equation: Fluid simulation, nature texturing, ……
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PDE-based Geometric Modeling• PDE surfaces and solids:
– Blending problem [Bloor and Wilson 89]– Free-form surfaces [Bloor and Wilson 90b]– B-spline approximation [Bloor and Wilson 90a]– Functionality design [Lowe et al. 90]– PDE solids [Bloor and Wilson 93]– Interactive design [Ugail et al. 99]
• Variational models:Surface fairing [Schneider and Kobbelt 00]
……
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PDE Surfaces and Solids
( ) ,2
2
22
2
2
0X =⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂ vu
va
u
( ) ,,2
2
2
2
2
2
0X =⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂ wvu
wvu
• PDE surface formulation
• PDE solid formulationBiharmonic equation if a=1
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Level Set Method• Originally defined for front propagation
• Prior work of level set method– Front propagation
[Osher and Sethian88], [Adalsteinsson and Sethian95]– Shape reconstruction
[Zhao et al.00], [Zhao et al.01]– Shape transformation
[Breen and Whitaker01]– Shape modeling and editing
[Barentzen and Christensen02], [Museth et al.02][SIGGRAPH’02 Course Notes 10]……
}.0),(|{ :set level-zero theisfront Moving; to from distance theis )( ),()0,(
function; speed theis ,0
0
==ΓΓ±==
=∇−
txxxxdxdtx
FF
t
t
φφ
φφ
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Classification of PDEs
• 3 types of PDEs based on characteristics– B²-AC>0: hyperbolic
• Wave equation
– B²-AC=0: parabolic• Diffusion equation
– B²-AC<0: elliptic• Poisson equation
2
22
2
2
xuv
tu
∂∂
=∂∂
⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=∂∂
xuD
xtu
( )yxyu
xu ,2
2
2
2ρ=
∂∂
+∂∂
GFuyuE
xuD
yuC
yxuB
xuA =+
∂∂
+∂∂
+∂∂
+∂∂
∂+
∂∂
2
22
2
22
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Shape Reconstruction Using Level Set Method
[Zhao et al.00]
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Shape Morphing Using Level Set Method
[Breen and Whitaker 01]
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Diffusion Equations• Reaction-diffusion textures
[Witkin and Kass91], [Turk91]• Tensor field visualization
[Kindlmann et al.00]• Vector field visualization
[Diewald et al.00]• Image processing
Image enhancement, filtering,……
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Reaction-Diffusion Texture• Synthesizing natural textures
• Reaction-diffusion system– Diffusion of morphogens– Nonlinear PDEs– Biological patterns
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Reaction-Diffusion Equation• Diffusion, dissipation, reaction• Reaction-diffusion equation:
function. reaction theis n,dissipatiofor constant rate theis
diffusion,for constant rate theis
,
, of derivative time theis ,
2
2
2
22
22
Rba
yC
xCC
CCRbCCaC
∂∂
+∂∂
=∇
+−∇=&
&
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By type: Isotropic, multi-orientation, and diffusion mapped
1.reptile, giraffe, coral, scalloped.
2.spiral, triweave, twisty maze, repli-cation, purple thing3.sand, maze, zebra haunch, radial4.space giraffe, zebra, stucco, beats us, weave
Reaction-diffusion texture buttons
[Witkin and Kass 91]
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Vector Field Visualization
Different time steps of the anisotropic diffusion for both principal curvature directions
[Diewald et al. 00]
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Other Applications of PDEs• Modeling fracture
[O’Brien and Hodgins99], [O’Brien00]• Simulating gas
[Foster and Metaxas97b]• Water simulation
[Kass and Miller90], [Foster and Metaxas96,97a], [Stam99]• Modeling explosion
[Yngve et al.00]• Image processing
[Bertalmio et al.00], [Pérez et al.03]…….
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Implicit Model• Implicit surfaces and solids:
{(x,y,z)| f(x,y,z)=c}, {(x,y,z)| f(x,y,z)≤ c}
• Techniques to model implicit objects– Particle based implicit surface sculpting
[Witkin and Heckbert 94]
– Trivariate B-splines for implicit models[Raviv and Elber 99], [Hua and Qin01,02]
– Level set method [Zhao et al.00,01]
– Variational implicit functions [Turk and O’Brien 99,02], [Morse et al.01]
…….
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Example of Implicit Models
[Turk and O’Brien02]
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Physics-based Modeling• Combines physical properties with geometric
models• Leads to deformable models• Allows direct manipulation of objects via
forces• Creates natural-looking motions through
simulation • Can be integrated with general PDE
framework
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Deformable Models• Controlled by Lagrangian equations of
motion
– r (a,t): position of particle a at time t– μ (a): mass density of the body at a– γ (a): damping density– f (r, t): external force– ε (r): measures the potential energy of the
elastic deformation of the body
( )ttt
,)(t
rfrrrr
=∂
+∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂ δεγμ
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Discretized Mass-Spring Model
• M: mass matrix• D: damping matrix• K: stiffness matrix• f: external force• p: discrete sample points
fKppDpM =++ &&&
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Applications of Physics-based Modeling• Dynamics NURBS (DNURBS)
[Qin and Terzopoulos 94, 95, 96]• Physics-based subdivision
[Mandal et al. 98, 99], [McDonnell et al. 00, 01]• Cloth simulation and animation
[Carignan et al. 92], [Baraff and Witkin 98]• Facial simulation
[Lee et al. 95], [Koch et al. 96]• Physics-based implicit functions
[Jing and Qin 01,02]……
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Medial Axis Extraction• Locus of all centers
of circles/spheres inside the object
• Collection of points with more than one closest points on the boundary
• Set of singularities of signed distance function from boundary
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Medial Axis Extraction Techniques• Thinning
[Arcelli and Baja85][Lee and kashyap94][Manzanera etal.99]• Distance functions
[Arcelli and Baja92][Leymarie and Levine92][Bitter et al.01]• Voronoi skeletons
[Goldak et al.91][Ognievicz93][Amenta et al.01]• Level set method
[Kimmel et al.95][Ma et al.03]• Direction testing
[Bloomenthal and Lim99]• Hybrid techniques
[Bouix and Siddiqi00]
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Outline
• Motivation and contributions• Related work• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
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Outline
• Motivation and contributions• Background review• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
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System Outline
PDE-based Geometric Modeling System
Physics-based PDE
Surfaces
Free-Form PDE Modeling with Intensity
Implicit PDE Model
Arbitrary PDE
Meshes
Boundary Surface
Sculpting
Iso-Surface Extraction
Embedded Datasets
Scattered Datasets
Intensity Distribution
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PDE Surfaces• PDE surface formulation:
– a(u,v): blending coefficient function controlling the contributions of the parametric directions
( ) ( )
( ) [ ]T
2
2
22
2
2
),(),(),(,
0,,
vuzvuyvuxvu
vuv
vuau
=
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
X
X
![Page 62: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/62.jpg)
PDE Surface Displacements• Displacements
• PDE surface displacement formulation:( )
( ) ( ) 0O
OXX
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+=
vuv
vuau
vuvuvu
,,
,,),(),(2
2
22
2
2
0
![Page 63: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/63.jpg)
Finite Difference Method• Divides the working space into discrete grids
![Page 64: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/64.jpg)
Finite Difference Method• Divides the working space into discrete grids • Samples the PDE at grid points with
discretized approximations
22,1,1,,1,11,11,11,11,1
22,
4
4,1,1,2,2,
4,
4
4,,1,1,2,2
4,
4
4)(2
,644
,644
vuvu
vv
uu
jijijijijijijijijiji
jijijijijiji
jijijijijiji
ΔΔ
++++−+++=
∂
∂Δ
+−−+=
∂
∂Δ
+−−+=
∂
∂
+−+−+++−−+−−
+−+−
+−+−
XXXXXXXXXX
XXXXXX
XXXXXX
0,
2
2
22,2
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
jiji va
uX
![Page 65: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/65.jpg)
Finite Difference Method• Divides the working space into discrete grids • Samples the PDE at grid points with
discretized approximations• Forms a set of algebraic equations
• Enforcing additional constraints
• Physics-based discrete PDE model:
zHX =
cc zXH =
cc zf)XH(KXDXM +=+++ &&&
![Page 66: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/66.jpg)
Discretized Approximations• Displacement model:
• Iterative techniques and multi-grid algorithm to improve the performance
• Easy for local control
OXX
zOH
+=
=0
cc
![Page 67: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/67.jpg)
Flexible Boundary Conditions• Generalized boundary conditions
where and , and and are isoparametric curves
– Hermite-like boundary constraints– Coons-like boundary constraints– Gordon-like boundary constraints
)(g),()(f),(uvuvvu
jj
ii
==
XX
10 ≤≤ iu 10 ≤≤ jv ( )vif ( )ujg
![Page 68: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/68.jpg)
Hermite-like Boundary Constraints
Boundary curves
Derivative curves
![Page 69: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/69.jpg)
Coons-like Boundary Constraints
u=0
u=1
v=1v=0
![Page 70: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/70.jpg)
Gordon-like Boundary Constraints
u=0
v=0.5
v=0.75
v=0.25
v=0
![Page 71: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/71.jpg)
Joining Multiple PDE Surfaces• Boundary curves: • Three connected
surfaces:
![Page 72: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/72.jpg)
PDE Surface Manipulations• Boundary sculpting• Blending coefficient control• Direct manipulation
– Point based sculpting: position, normal, curvature
– Curve deformation– Region manipulation
• Displacement deformation
![Page 73: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/73.jpg)
Boundary Curve Sculpting
![Page 74: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/74.jpg)
Effect of a(u,v)• The value of at each can be changed interactively.
( )ji vua , ( )ji vu ,
• a(u,v)=3.1 • a(u,v)=5.2 at yellow part • a(u,v)=5.2
![Page 75: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/75.jpg)
Point-based Manipulation
ccji zXHzHXpX =⇒=⇒=,
![Page 76: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/76.jpg)
Curve and Region Sculpting
![Page 77: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/77.jpg)
Local Sculpting
![Page 78: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/78.jpg)
Curve Editing1. Select an arbitrary source curve on the
PDE surface by picking points on the u-vdomain;
2. Define a cubic B-spline curve with desired shape as the destination curve;
3. Map the source curve to the shape of the destination curve, i.e. put the constraints into the system;
4. Solve the constrained equations to get the new surface.
![Page 79: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/79.jpg)
Curve Editing
![Page 80: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/80.jpg)
Region Sculpting1. Select an area on the PDE surface;2. Define a cubic B-spline patch with the
same number of sample points of the source region;
3. Map the source region to the shape of the destination patch, i.e. put those constraints into the linear equation system;
4. Solve the constrained equations to get the new surface.
![Page 81: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/81.jpg)
Region Sculpting
![Page 82: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/82.jpg)
Interface of PDE Surfaces
![Page 83: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/83.jpg)
Point Editing on Displacements
![Page 84: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/84.jpg)
Displacement Curve and Region Sculpting
![Page 85: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/85.jpg)
B-spline Approximation• B-spline surfaces
• Obtain B-spline control mesh from PDE surfaces
• B-spline approximation for dynamic models
∑∑= =
=k
i
l
jjidjci vBuBvu
1 1,,, )()(),( pX
XBBPBXBP
ΤΤ =
=
![Page 86: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/86.jpg)
B-Spline Formulation• B-Spline curve and surface:
• B-Spline basis function:
Knots sequence:
( )∑=i
nii uNu ds )(
( )
[ )
[ )
( ) ( ) ( ) ( )⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
>−
−+−
−
∉=
∈=
=
+++
−+
+++
−
+
+
0,
;, and 0 if ,0
;, and 0 if ,1
11
11
1
1
1
1
ruuuNuu
uuuNuu
uuur
uuur
uN
iri
ri
riiri
ri
i
ii
ii
ri
( ) ( )∑ ∑=i j
mj
niji vNuNvu ,),( ds
[ ]L,, 10 uu
![Page 87: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/87.jpg)
NURBS Formulation
( )( )∑
∑==
=ni
pii
ni
piii
uNwuNwu
0
0)( PC
( ) ( )( ) ( )∑ ∑
∑ ∑=
= =
= =ni
mj
qj
piji
ni
mj
qj
pijiji
vNuNwvNuNw
vu0 0 ,
0 0 ,,),(P
S
Weights: jii ww ,,
![Page 88: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/88.jpg)
Free-Form Splines• Piecewise polynomials with certain
differentiability constraints• Local control and extra DOF for
manipulation• NURBS
– Non-Uniform Rational B-Splines– Industrial standard
![Page 89: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/89.jpg)
Pros and Cons of NURBS• Model both analytic
and free-form shapes• Local control• Clear geometric
interpretations• Smooth objects• Powerful modeling
toolkits• Invariant under various
manipulations
• Extra storage for traditional objects
• Too many degrees of freedom
• Difficult to model intersection, overlapping
• Less natural and counter-intuitive
• Strong mathematics• Difficult for arbitrary
topology
![Page 90: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/90.jpg)
Examples of B-spline Approximation
![Page 91: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/91.jpg)
Physics-based PDE Surfaces• Physics-based PDE surface and
displacement model• Flexible boundary conditions• Global manipulations
Joining multiple surfaces, boundary sculpting• Direct local sculpting
Coefficient control, point, curve, region sculpting, displacement manipulation, material property modification
• B-spline approximation
![Page 92: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/92.jpg)
System Outline
PDE-based Geometric Modeling System
Physics-based PDE
Surfaces
Free-Form PDE Modeling with Intensity
Implicit PDE Model
Arbitrary PDE
Meshes
Boundary Surface
Sculpting
Iso-Surface Extraction
Embedded Datasets
Scattered Datasets
Intensity Distribution
![Page 93: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/93.jpg)
Outline• Motivation and contributions• Background review• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
![Page 94: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/94.jpg)
Interface
![Page 95: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/95.jpg)
PDE-based Arbitrary Mesh Model• Traditional PDE surfaces
– Defined on regular domain– Difficult to model arbitrary topological surfaces
• Polygonal meshes– Define surfaces as collection of points and their
relations– Shape of arbitrary topology
• Goal: use PDE techniques to model arbitrary polygonal meshes
![Page 96: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/96.jpg)
PDE Approximation of Arbitrary Meshes• Umbrella operator for discrete Laplacian:
• Approximating formulations:,1
)(2
1∑ ∈
−≈∇iNj iji n
ppp ( )∑ ∈−≈∇
)( ,2
1
2iNj jiiji e
Eppp
∑∈
∇−∇≈∇∇=∇
=∇=∇
)(
22224
42
1
1)(
,
iNjijii n
pppp
FpFp
![Page 97: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/97.jpg)
Direct Manipulation of Arbitrary Meshes• Take input meshes as general constraints• Use umbrella operators to approximate the
PDEs• Point-based manipulation for shape
deformation• Local control by selecting regions of
interests• Possible to integrate with subdivision
models
![Page 98: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/98.jpg)
Direct Manipulation of Arbitrary Meshes
![Page 99: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/99.jpg)
PDE-based Medial Axis Extraction• Compact representation of arbitrary
polygonal meshes• Diffusion-based equations to simulate
grassfire process• Approximates medial axes for manipulation
purposes• Facilitates skeleton-based shape sculpting
for arbitrary meshes
![Page 100: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/100.jpg)
Formulations• Diffusion-based PDE
• Normal approximation
• Gaussian curvature approximation
SNDpS 2),(),(∇=
∂∂ κ
tt
∑∑−
=
−
=
×=×=1
0
1
021
2sin2cosn
jj
n
jji n
jnj ppttN ππ
∑∑
∑∑
−
=
−
=
−
=
−
=−
=−
=1
0
1
0
1
0
1
0
31,
31
2n
j j
n
j jin
j j
n
j ji
AA
φπκ
φπκ
![Page 101: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/101.jpg)
Medial Axis Extraction Algorithm• Initialization
– Approximate surface normal and other differential properties
• Skeletonization– Compute evolving surface– Collision detection to find skeletal points– Surface optimization
• User interaction– User-defined skeleton, local skeletonization
![Page 102: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/102.jpg)
Progressive Medial Axis Extraction
![Page 103: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/103.jpg)
Local Region Skeletonization
![Page 104: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/104.jpg)
User-defined Skeleton
![Page 105: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/105.jpg)
Skeleton-based Sculpting
![Page 106: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/106.jpg)
Skeleton-based Sculpting
![Page 107: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/107.jpg)
Curvature Manipulation
![Page 108: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/108.jpg)
PDE-based Arbitrary Mesh Modeling• Direct manipulation on polygonal meshes• Diffusion-based medial axis extraction
– Progressive visualization– User interaction– Local region skeletonization
• Skeleton-based shape manipulation
![Page 109: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/109.jpg)
Outline
• Motivation and contributions• Background review• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
![Page 110: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/110.jpg)
Implicit PDE Modeling• Implicit elliptic PDE formulation• General boundary constraints• Radial Basis Function (RBF) for initial guess• Direct manipulation in the implicit working
space• Interactive sculpting of implicit PDE objects
![Page 111: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/111.jpg)
RBF Method (1)• RBF: Radial Basis Function• Solving interpolation problems by
minimizing thin-plate energy in 3D
• Basis function:
• Interpolation function:
∫∫ ++ dxdyfff yyxyxx222 2
( ) 3xx =φ
)()()(1
xcxx Pwfk
jjj +−= ∑
=
φ
![Page 112: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/112.jpg)
RBF Method (2)• Interpolation constraints:
• Linear equation system:)()()(
1i
k
jjijii Pwfh cccc +−== ∑
=
φ
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
=
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
0000
0000...0000...0000...00001...11
1...
1...1...
2
1
3
2
1
0
2
1
21
21
21
21
22222221
11111211
kk
zk
zz
yk
yy
xk
xx
zk
yk
xkkkkk
zyxk
zyxk
h
hh
ppppw
ww
ccccccccc
ccc
cccccc
MMMMMMMOMM
φφφ
φφφφφφ
![Page 113: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/113.jpg)
Interface of Implicit PDE Model
![Page 114: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/114.jpg)
Blending Coefficient Manipulation
![Page 115: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/115.jpg)
System Outline
PDE-based Geometric Modeling System
Physics-based PDE
Surfaces
Free-Form PDE Modeling with Intensity
Implicit PDE Model
Arbitrary PDE
Meshes
Boundary Surface
Sculpting
Iso-Surface Extraction
Embedded Datasets
Scattered Datasets
Intensity Distribution
![Page 116: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/116.jpg)
Implicit PDE Model• Implicit PDE formulations:
• Generalized boundary constraints– Initial guess
• RBF (Radial Basis Function) interpolation• Distance field approximation
– Smoothing
( ) ( ) ( ) ( ) 0,,,,,,,,2
2
22
2
22
2
22 =⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂ zyxd
zzyxc
yzyxb
xzyxa
( ) ( ) ( ) ( ) 0,,,,,,,, 2
22
2
22
2
22 =⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂ zyxd
zzyxc
yzyxb
xzyxa
![Page 117: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/117.jpg)
Boundary Constraints for Implicit PDE• Traditional boundary conditions (cross-
sectional constraints)• Boundary constraints for shape blending• Arbitrary sketch curves
– Initial guess: variational interpolation (RBF)• Unorganized scattered data points
– Initial guess: distance field approximation
![Page 118: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/118.jpg)
Traditional Boundary Conditions
![Page 119: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/119.jpg)
Traditional Boundary Conditions
![Page 120: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/120.jpg)
Boundary Constraints for Shape Blending
![Page 121: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/121.jpg)
Boundary Conditions of Sketch Curves
![Page 122: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/122.jpg)
Local RBF Method for Complex Model
![Page 123: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/123.jpg)
Boundary Conditions of Scattered Points
![Page 124: PDE-based geometric modeling and applications](https://reader034.vdocuments.net/reader034/viewer/2022051507/58a2df0a1a28aba5548b7f3d/html5/thumbnails/124.jpg)
Manipulation of Implicit PDE Objects• Sketch curve sculpting
– Shape, intensity, and gradient directions• Blending control coefficient manipulation• Direct manipulations
– Iso-contour– Region intensity– CSG tools– Gradient sculpting– Curvature manipulation
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Sculpting of Sketch Curves
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Changing Gradient Directions
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Direct Intensity Manipulations
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Direct CSG Manipulations
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Gradient and Curvature Approximation
• Gradient approximation
• Curvature approximationMean curvature:
)2
,2
,2
(),,( 1,,1,,,1,,1,,,1,,1
zdd
ydd
xdd
zyxd kjikjikjikjikjikji
Δ
−
Δ
−
Δ
−≈∇ −+−+−+
),,( zyxd∇⋅∇
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Gradient Manipulations
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Curvature Manipulations
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Summary of Implicit PDE Modeling• General boundary constraints for shape
design, reconstruction, blending, and recovery
• RBF method or distance field approximation for initial guess with generalized constraints
• Manipulation of implicit PDE objects– Sketch curve sculpting– Blending coefficient manipulation– Direct manipulation of implicit objects
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Outline• Motivation and contributions• Background review• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
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Free-Form PDE Modeling and Deformation• Formulations
– Geometry– Intensity Integration
• Boundary constraints and manipulations• Direct sculpting of PDE solid geometry• Intensity-based free-form modeling and
deformation
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Interface of PDE Solids
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Numerical Techniques• Spectral approximation• Finite-element method (FEM)• Finite-difference method (FDM)• Solving linear equation system
– Iterative method– Multi-grid improvement
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Finite Element Method• Approximate the infinite problem by
interpolation functions over sub-domains– Discretize the domain into sub-domains– Select the interpolation functions– Formulate the system of equations– Solve the equations for coefficients of the
interpolation to approximate the solution
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Typical Finite Elements
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Finite Difference Method• Divides the working space into discrete grids • Samples the PDE at grid points with
discretized approximations• Forms a set of algebraic equations• Uses iterative techniques and multi-grid
algorithm to improve the performance
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Working Space Discretization
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Difference Equation Approximation
22,1,1,,1,11,11,11,11,1
22,
4
4,1,1,2,2,
4,
4
4,,1,1,2,2
4,
4
42222
,644
,644
vufffffffff
vuf
vfffff
vf
ufffff
uf
jijijijijijijijijiji
jijijijijiji
jijijijijiji
ΔΔ
+−−−−+++=
∂
∂Δ
+−−+=
∂
∂Δ
+−−+=
∂
∂
+−+−+++−−+−−
+−+−
+−+−
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Finite Difference Method• Simple and easy for implementation• Allows flexible and generalized boundary
conditions and additional constraints• Enables local control and direct
manipulation• Guarantees an approximate solution• Time performance depends on resolution of
discretization of working space
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Solving Linear Equations• Iterative methods
– Gauss-Seidel iteration
– SOR iteration• Difference between approximation and the real
solution
• Multi-grid method improvement– Starting from coarsest grids, linear
interpolating the coarse solution to get initial guess of finer resolution
( ) ( ) bXAXA
bXAXAAAAbAX
+=
+=−==−1
,,n
rn
d
rdrd
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System Outline
PDE-based Geometric Modeling System
Physics-based PDE
Surfaces
Free-Form PDE Modeling with Intensity
Implicit PDE Model
Arbitrary PDE
Meshes
Boundary Surface
Sculpting
Iso-Surface Extraction
Embedded Datasets
Scattered Datasets
Intensity Distribution
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Free-Form PDE Solid Geometry• PDE formulation:
• Free-form deformation for explicit model:
( ) ( ) ( ) ( ) ,,,,,,,,2
2
22
2
22
2
22 0X =⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂ wvu
wwvuc
vwvub
uwvua
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Free-Form PDE Solid Geometry• Boundary conditions:
– Surfaces:
– Curve network:
( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ,1,, ,,0,,
,,,1, ,,,0,,,,,1 ,,,,0
10
10
10
vuvuvuvuwuwuwuwuwvwvwvwv
WXWXVXVXUXUX
======
( ) ( ) { } { }( ) ( ) { } { }( ) ( ) { } { } 1,0or 1,0 ,,,
;1,0or 1,0 ,,,
;1,0or 1,0 ,,,
∈∈=∈∈=
∈∈=
srrssr
lkkllk
jiijji
vuwwvuwuvwvu
wvuwvu
WXVX
UX
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PDE Solid from Boundary Surfaces
Boundary surfacesv
w
u
Corresponding PDE solid
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Corresponding PDE solid
Boundary surfaces
v
w
u
PDE Solid from Boundary Surfaces
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PDE Solids from Boundary Curves
w
u
v
w
u
v
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Boundary Surface Sculpting
Curve editing on the boundary surface of u=1
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Direct Manipulation of PDE Solids
Modifying selected regions on an embedded dataset
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Direct Manipulation of PDE Solids
Directly moving a point on an embedded dataset
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Geometric Free-Form Deformation
From a PDE solid cube From a PDE solid sphere
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Integrating Intensity Attributes
• Formulation:
( ) ( ) ( ) ( )
( )
( ) ),,(0
0),,(,,
, ),,(),,(
,,
,,,,,,,,,
x
2
2
22
2
22
2
22
⎥⎦
⎤⎢⎣
⎡=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
wvuawvua
wvu
wvudwvu
wvu
wvuw
wvuv
wvuu
wvu
d
a
XP
0Pcba
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Intensity Initialization
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Intensity Initialization
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Arbitrary Shape Blending
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Arbitrary Shape Blending
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Iso-surface Deformation
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Intensity Field Modification
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Intensity Field Modification
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Free-Form PDE Modeling Summary• Boundary surfaces or curve network as boundary
constraints• Boundary surfaces manipulation for solid
deformation• Free-form deformation for embedded datasets• Sculpting toolkits for direction manipulation• Integrating with implicit PDE for more general
modeling– Arbitrary shape blending based on intensity– Intensity-based shape manipulation and deformation
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Outline• Motivation and contributions• Related work• PDE-based geometric modeling system
– Physics-based PDE surfaces/displacements– PDE-based arbitrary mesh modeling– Implicit elliptic PDE model– PDE-based free-form modeling and deformation
• Conclusion
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Conclusion• Integrated PDE modeling system for parametric
objects, arbitrary meshes, and implicit models• Incorporation of popular geometric modeling
techniques and representations• Information recovery from partial input• Physical properties for dynamic behavior • Various modeling toolkits for direct manipulation
and interactive sculpting• Shape design, recovery, abstraction, and
modification in a single framework
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Future Work• Geometric modeling
– Shape design, morphing, reconstruction• Image processing and medical imaging
– Enhancement, denoising, medical data reconstruction
• Simulation and animation– Natural phenomena simulation, medical
simulation
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Related Publications• Haixia Du and Hong Qin. Dynamic PDE-Based Surface Design Using Geometric
and Physical Constraints. Accepted by Graphical Models, 2003.• Haixia Du and Hong Qin. A Shape Design System Using Volumetric Implicit PDEs.
Accepted by CAD Special Issue of the ACM Symposium on Solid Modeling and Applications, 2003.
• Haixia Du and Hong Qin. PDE-based Free-Form Deformation of Solid Objects. In preparation for journal submission, 2004.
• Haixia Du and Hong Qin. PDE-based Skeletonization and Propagation for AbitraryTopological Shapes. In preparation for journal submission, 2004.
• Haixia Du and Hong Qin. Medial Axis Extraction and Shape Manipulation of Solid Objects Using Parabolic PDEs. Accepted by The Nineth ACM Symposium on Solid Modeling and Applications 2004.
• Haixia Du and Hong Qin. Interactive Shape Design Using Volumetric Implicit PDEs. In Proceedings of The Eighth ACM Symposium on Solid Modeling and Applications 2003, p235-246.
• Haixia Du and Hong Qin. Integrating Physics-based Modeling with PDE Solids for Geometric Design. In Proceedings of Pacific Graphics 2001, p198-207.
• Haixia Du and Hong Qin. Dynamic PDE Surfaces with Flexible and General Constraints. In Proceedings of Pacific Graphics 2000, p213-222,
• Haixia Du and Hong Qin. Direct Manipulation and Interactive Sculpting of PDE Surfaces. In Proceedings of EuroGraphics 2000, pC261-C270.
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Acknowledgements• Committee members• Members of VisLab
http://www.cs.sunysb.edu/~dhaixia