geometric partial differential equations methods in...

Geometric Partial Differential Equations Methods inGeometric Design and Modeling

Reporter: Qin Zhang1

Collaborator: Guoliang Xu,2 C. L Bajaj1, Dan Liu2

1CVC, University of Texas at Austin, TX, USA

2LSEC, AMSS of CAS, Beijing, China

09/03/2008

Qin Zhang ( CVC) GPDEs Methods in Geometric Design & Modeling UT 1 / 40

Outline

1 Intoduction

2 The Construction of GPDE

3 Numerical Solutions of GPDEsGeneralized Finite Difference MethodsMixed Finite Element MethodsLevel-Set Methods

4 Conclusion


1. Introduction

What is a GPDE?

A PDE which controls the motion of curves orsurfaces and is merely formulated by the geometricentries is known as Geometric PDE.


Examples

1 Mean Curvature Flow (MCF) (1956, Mullins)

∂x∂t

= 2H

2 Willmore Flow (WF) (1923, Thomsen)

∂x∂t

= −[∆sH + 2H(H2 − K )]n

3 Minimal Mean-Curvature-Variation Flow (MMCVF) (2006, Xu &Zhang)

∂x∂t

= (∆2sH + 2(2H2 − K )∆sH + 2〈∇sH,♦H〉 − 2H‖∇sH‖2)n


Several Differential Geometric Operators of First Order

Assume

S = x(u, v) ∈ R3 : (u, v) ∈ Ω ⊂ R2,f : S → R

Tangential Gradient Operator of Surface S

∇sf = [xu, xv ][ gαβ ][fu, fv ]T ∈ R3,

The Second Tangential Operator

♦f = [xu, xv ][ K bαβ ][fu, fv ]T ∈ R3.

Tangential Divergence Operator

div(v) =1√

g

[∂

∂u,

∂

∂v

] [√g [ gαβ ] [xu, xv ]T v

].


Several Differential Geometric Operators of SecondOrder

Laplace-Beltrami Operator

∆sf = div(∇sf ).

Giaquinta-Hildebrandt Operator

f = div(♦f ).


Why GPDE is important?

Theory aspect: Relate intimately togeometric analysismanifold theorytopologycomplex analysisPDEcalculus of variationgeometric measure theorycritical point theory

Application aspect: Relate intimately tophysics,chemistry,biology,computational geometry,computer graphics,image processing


What can GPDE do in geometric design andmodeling?

Surface ProcessingSurface ReconstructionSurface RestorationSurface BlendingFree-Form DesignBiomolecular DesignImage ProcessingInterface Simulation· · · · · · · · ·


What’s about the result of GPDE?

Some optimal property, e.g.,minimize area

∫S dA = min

minimize the sum of square of principal curvatures∫S(κ2

1 + κ22)dA = min

minimize the variation of mean curvature∫S ‖∇sH‖2dA = min

etc.∫S f (H, K )dA = min

The surfaces designed by GPDEsClear geometric sense

Boundary condition with prescribed smoothness

Fairness

Aesthetic demanding


Surface Processing

input Mean curvature flow

(From Bajaj, Xu 2004)


Surface Reconstruction

scattered data reconstructed surface(Level set method)

(From Bajaj, Xu, Zhang 2006)


Surface Blending

initial input MCF (2nd order)

WF (4th order) MMCVF (6th order)


Free-Form Surface Design I

input initial surface surface diffusion flow


Free-Form Surface Design II

input skeleton PDE surface of 4th order


Surface Restoration I

input head without jaws MCF (2nd order)

WF (4th order) MMCVF (6th order)


Surface Restoration II

MCF SDF 6th order flow


Biomolecular Design

van der Waals surface SES

(from Bajaj, Xu, Zhang 2006)


Image Processing

initial input denoising process with MCF


Comparison Results Among 2,4,6-th Order Flows

MCF — G0 WF — G1 MMCVF — G2

2nd order flows usually used4th order flows often used6th order flows occasionally used


GPDE vs. Classical PDE

The domains of classical PDEs are usually fixed, but GPDE’s arealways changeable.

GPDE is geometrically intrinsic, independent of the parameterrepresentation of surface.

GPDE is highly nonlinear, which makes theoretical analysischallenging.

Classical methods can not be used directly and classical theoriesare not effective.


2. The Construction of GPDE

How to construct GPDE?

Manual approach

∂, ∇s, divs, ∆s, , ♦, · · · ,

⇓E(x) = 0, x ∈ S

For example, ∆2sx = 0, ∆2

sH = 0, · · · .

⇓∂x∂t = ±E(x)n, x ∈ S(t),

S(0) = S0.

Energy based variational approach


Energy Based Variational Approach to ConstructGPDE

Energy E (S) - Variational CalculusE ′(S)

- Construct Flow∂x∂t = −E ′

Energy∫S

dA - −2Hn - ∂x∂t = 2Hn


More Details on the Construction (Example)

Define inner product space:

S = x(u, v) ∈ R3 : (u, v) ∈ Ω is smooth.

f, g ∈ C2(S, R3)

define inner product

(f, g) =

∫S〈f, g〉dA


Three Steps to Construct GPDE

1 Define energy functional

E (S) =

∫S

dA

2 Compute first order variation

δ(E (S),Θ) =

∫S〈E ′

c(S),Θ〉dA,

here, E ′c(S) is named as the L2 gradient of E (S).

3 Construct GPDE of weak form and general form.∫S〈∂x∂t

,Θ〉dA = −∫S〈E ′

c(S),Θ〉dA, ∀Θ ∈ C∞0 (S, R3).

This is the starting point of finite element method. Then GPDE is∂x∂t

= −E ′c(S) ∈ R3.


A General Approach

First-order energy (second-order equation):

E1(S) =

∫S

h(x, n)dA

Second-order energy (fourth-order equation):

E2(S) =

∫S

f (H, K)dA

Third-order energy (sixth-order equation):

E3(S) =

∫S‖∇sg(H, K)‖2dA

and their combinations:

E (S) = αE1(S) + βE2(S) + γE3(S)


GPDEs of General Functional — 1st Order Energy

For first-order energy

E1(S) =

∫S

h(x, n)dA,

we obtain second-order equation∂x∂t = −∇xh − divs(∇nh)n−∇sn∇nh + divs(h∇sx),S(0) = S0.


GPDEs of General Functional — 2nd Order Energy

For the second-order energy

E2(S) =

∫S

f (H, K )dA,

we obtain fourth-order equation∂x∂t = −(fK n)− 1

2∆s(fHn) + divs(fH∇sn) + divs((f − 2KfK )∇sx),S(0) = S0,


GPDEs of General Functional — 3rd Order Energy

For the third-order energy

E3(S) =

∫S‖∇sg(H, K )‖2dA,

we obtain sixth-order equation

∂x∂t

=[∆s(gH ∆sgn) + 2(gK ∆sgn)− 2divs[gH ∆sg∇sn

− 2KgK ∆sg∇sx] + 2divs[R∇sg(R∇sg)T ]− divs[‖∇sg‖2∇sx]],

where

R =1√

g[−xv , xu] [xu, xv ]T .


Special Examples

1. Mean Curvature Flow [Mullins, 1956], (Minimal surface, Euler,1744).

f (H, K ) = 1

E ′n(S) = −2H

2. Surface Diffusion Flow [Mullins, 1957] .f (H, K ) = 1

E ′n(S) = 2∆sH

—- in the sense (·, ·)H−1 .3. Sixth-order Flow [Xu, Pan and Bajaj, 2006].

f (H, K ) = 1

E ′n(S) = −2∆2

sH

—- in the sense (·, ·)H−2 .


Special Examples (Cont.)

4. Weighted Mean Curvature Flow [cf. Zhang and Xu, 2005].h(x, n) = γ(n)

E ′n(S) = ∇sn:(∇2

nnγ)

5. Gaussian Curvature Flow [Firey, 1974].f (H, K ) = H

E ′n(S) = K

6. Willmore Flow. f (H, K ) = H2

E ′n(S) = ∆sH + 2H(H2 − K )

Willmore functional is initially introduced by Thomsen in 1923........


3. Numerical Solutions of GPDEs

Explicit solutions of the GPDEs are hard to obtain. Numerical solutionsare necessary and freasible.

Generalized Finite Difference Methods

Finite Element Methods

Level-Set Methods


Outline

1 Intoduction



4 Conclusion


Generalized Finite Difference Methods

Two Scenarios:

Normal motion equation

nT ∂x∂t

= 2H,

All directions motion equation

∂x∂t

= ∆sx,

Two Pivot Problems:

Discretization of differential geometry operatorsBoundary treatment


Outline

1 Intoduction



4 Conclusion


Mixed Finite Element Methods

Two Scenarios:

Total variation form

Normal variation form

Two Pivot Problems:

Construction of the function spaces of finite element

Boundary treatment


Outline

1 Intoduction



4 Conclusion


Level-Set Methods

Key Techniques:

Narrow band technique

Runga-Kutta method with adaptive step size

Fast computation of higher-order level-set function

Fast reinitialization


Finite Element Methods vs. Finite Difference Methods

Finite Difference MethodsEasy to implementLow costDepends on the discretization of differential geometric operators

Finite Element MethodsMore stableSolid mathematical foundationDepends on the construction of the space of finite element


4. ConclusionInterdisciplinary, GPDEs Methods relate to

differential geometry, manifold, equationsvariational calculusfunction approximationcomputational mathematicscomputer graphics

Challenging, GPDEs deep intogeometry analysis, manifold theorytopologygeometric measure theorycritical theory

Practical, GPDEs closely relate toPhysics and chemistry settings, the motion of interfaces problems,e.g., dissolution, combustion, erosionBiology field, biomembrane vesicle problem, the construction ofprotein surfaceImage processing, edge detection, noise removal, imagerestoration......


Thank you!

Email:[email protected], [email protected]


geometric partial differential equations methods in...

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