Vector Field Based PowerPoint Presentation

Vector Field Based

Shape Deformations

Wolfram von Funck / Holger Theisel / Hans-Peter SeidelMPI Informatik

ACMSIGGRAPH 2006

Computer Graphics Lab.SoHyeon Jeong2007/04/16

Contents

Introduction Constructing the vector field Modeling metaphors Implementational DetailsEvalutation and Comparison

2

1. Introduction

3

Shape Deformation

Original shape

New Deformed shape

Performance

Detail & Feature preservation

Volume preservation

Avoidance of self-intersections

Deformation metapor

Transformation with Constraints

4

Deformation Metaphors

Free movement of certain handlesSingh and Fiume 1998Bendels and Klein 2003Pauly et al. 2003 9-dof object Botsch and Kobbelt 2004

Two Handed metaphorLlamas et al. 2003New Metaphor Implicit tools

?

5

Modeling Metaphors: Implicit Tools

IdeaUse simple implicit objects as deformation tools

6

Deformation Approaches

Mapping problemFinding a mapping transformation between the original and the new deformed shapeFinding path problemFinding continous path that certain point should follow

7

Finding Paths Problem

Integration of vector field at time T Similar to “flow of fluid” [Foster and Fedkiw 2001]

scaling

Translation

Rotation

8

The Main Idea to Solve

Constructing vector fields that produce useful deformation Computing deformation by integrating using vector fieldsFlexibleVariety of different deformationsTranslations & rotationsSimple Fast computation Interactivity & large mesh deformation

9

Properties of Vector Fields

Simple local properties of vector

Global and local properties of shape deformation

Divergence-free C1 continuity Time-dependent path integration

Volume preservation[Davis 1967] Smooth deformation No self-interaction[Theisel et al. 2005]

10

2. Constructing the Vector Fields v

11

Piecewise Region Field

Inner regionWell-defined regionOuter regionNo deformationIntermediate regionBlending between Inner & Outer regionDivergence-freeC1 continuity

12

Piecewise Region Field

Region Field :

Inner region Intermediate region Outer region

4

4

4

4

4

4444433333334432222234432111234432101234432111234432222234433333334444444444

13

Terms

Scalar field Gradient Co-gradient Divergence

2D 3D

14

Constructing the Deformation Vector Field V

Constructing a divergence-free vector field 2D Co-gradient field of a scalar field [Davis 1967] :3D Cross product of gradients of two scalar fields

15

Constructing the Deformation Vector Field V (3D)

Define scalar field in terms of region fieldConstruct divergence-free field using defined scalar fields

: Berstein polynomials

16

Blending intermediate region

Inner & outer region should be connected smoothly It requires C1 continuity Scalar fields : C2 continuity  Vector field : C1 continuity

e(

x

)

0

17

Blending: 2D Example

inner region

v

constant

outer region

v

= 0

intermediate

region

1

0

-1

1

0

-1

1

0

-1

1

0

-1

1

0

-1

18

2D Example

Region Field

19

Special Deformations - Translation

Translation vector field : A constant vector fieldThe center point c : to determine DOF

: The center point

20

Special Deformation - Rotation

Rotation vector field : linear vector field v and rA center point : An Axis :Ristrected as a cylinder

21

Vector Field

Translation

Rotation

22

3. Modeling Metaphor

23

Deformation Cycle

Usually r(x) : the distance to a certain point c : the center of the inner region u, w, a : determined by interactive input device(mouse)IntegrationIf tool moves , the integration inside the inner region moves the points by The step size of the path line integration is chosen so that the path line follows the path of the tool

Define region field r(x) with ri, ro and c

Define scalar field e(x), f(x)with orthogonal vector u, w, a center c and an axis a

Update v

Integrate point of the shape with v

24

Implicit Tools

Point toolsPoints in the inner regionat the beginning followthe movement of the toolOther points never enter the inner region no self-intersectionLine tools

25

Deformation Paint

The tool is moved along a path on the surface : the location of the point on the shape at a certain time , = choosen interactively

26

Moving Point Sets

Multiple isolated point set the shape : Smooth approximated distance function to this point set , : interactively choosen : Barycenter of all points

27

Collision Tools

An arbitrary closed tool shape for which a repeated collistion detection with the deformed shapeFind collision region using Bounding box hierarchy Setting Collision detected points :r = smooth approximated distance function along with ri = 0Inner region is constant It follows the path of the input device

6

5

4

3322543221143211003210011210112210122331012344

28

Collision Tools : Example 1

29

Collision Tools : Example 2

30

Twisting & Bending

Linear and quadratic vector fieldTwsiting : linear : direction of the twisting axis : on the twising axis

31

Twisting & Bending

BendingUsing a rotation ,

32

Twisting & Bending

twisting

bending

33

Feature Preservation

Details on the surface are preserved during deformation

34

4. Implementation

35

Integration with adaptive stepsize

Best tradeoff between speed and accuracy [Nielson et al. 1997]4th order Runge-Kutta integration with adaptive stepsize

36

Remeshing

Large deformation causes unpleasing artifactsUndersample Volume changingIdeaRemeshing both the original and deformed objectNew vertices undergo same deformation as the original vertices

 It Requires

remeshing

37

Remeshing

M : original mesh, M’ : deformed mesh, P : deformation path M and P are storedAll edges of the M’ are tested for refinementlength(edge) > thresholdAngle between the normals of the end-vertices is large  Edge split on both M and M’New vertices of M are deformed using PDiffusion of the vertices Guarantee a uniform distribution of the verticesVertices moves to the barycenter of its 1-ringVertex is projected back onto the surface of the undiffused mesh Repeated a fixed number of stepsDecimation steplength(edge) < thresholdSmall anglePerform step 3 again for collapsed points

 collapsed

38

5. Evaluation and Comparision

39

Visual Quality

The twisting of a box

[Yu et al 2004]

[Proposed]

[Lipman et al. 2005]

[Zhou et al. 2005]

40

Visual Quality

Bending a sylinder

[Proposed]

[Laplacian surface]

[Poisson Mesh]

[Zhou et al. 2005]

[Botsch and Kobbelt 2004]

41

Speed

FactorsVertex # in inner, intermediate regionVertex # in intermediate effects more than vertex # of inner region Modeling metaphorRegion field r Simple r gives a higher performanceCollision detection step in shape stampingDeformation is highly parallelizable using GPU4th order Runge-Kutta path line integration of points Read-back of the computed points drops performces But still 10 times faster than CPU

42

Speed

Implementation EnvironmentAMD Opteron 152(2.6 GHz) 2GB RAM GeForce 6800 GT GPU

43

Accuracy

Accuracy in volume Discrete surface points produces slight changes of the vlumeBut is tolerable

44

6. Conclusion and Future Work

45

Conclusion

Alternative approach to shape deformation Time-dependent divergence-free vector field volume-preservingSelf-intersection Sharp features Realtime deformationAccuracy in volume preserving is high

46

Future Work

Preformance can futher be increased Multi-processor parallelization of the integration Integration of vertices is carried out independentlyApplcation to point-based shape representationDoes not rely on any connectivity information of the meshModeling metaphor can be extendedFull and zero deformation can be marked explicitly on the surfaces

47