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Vector Field Based Shape Deformations
Vector Field Based Shape Deformations Wolfram von Funck / Holger Theisel / Hans-Peter SeidelMPI Informatik
ACMSIGGRAPH 2006Computer Graphics Lab.SoHyeon Jeong2007/04/16
1ContentsIntroduction Constructing the vector field Modeling metaphors Implementational DetailsEvalutation and Comparison
221. Introduction33Shape Deformation
Original shapeNew Deformed shapePerformanceDetail & Feature preservationVolume preservationAvoidance of self-intersectionsDeformation metaporTransformation with Constraints44Deformation MetaphorsFree movement of certain handlesSingh and Fiume 1998Bendels and Klein 2003Pauly et al. 20039-dof object Botsch and Kobbelt 2004
Two Handed metaphorLlamas et al. 2003
New Metaphor Implicit tools
?
55Modeling Metaphors: Implicit ToolsIdeaUse simple implicit objects as deformation tools
6 , 6Deformation ApproachesMapping problemFinding a mapping transformation between the original and the new deformed shape
Finding path problemFinding continous path that certain point should follow
7 A B , ? A B ? , .
, ? . 7Finding Paths ProblemIntegration of vector field at time TSimilar to flow of fluid [Foster and Fedkiw 2001]scalingTranslationRotation8 T . , , .
8The Main Idea to SolveConstructing vector fields that produce useful deformation Computing deformation by integrating using vector fields
FlexibleVariety of different deformationsTranslations & rotationsSimple Fast computation Interactivity & large mesh deformation9, .
, , . 9Properties of Vector FieldsSimple 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 , .
102. Constructing the Vector Fields v1111Piecewise Region FieldInner regionWell-defined regionOuter regionNo deformationIntermediate regionBlending between Inner & Outer regionDivergence-freeC1 continuity
12Intermediate region deformation non-deformation .12Piecewise Region FieldRegion Field : Inner region Intermediate region Outer region
444444444433333334432222234432111234432101234432111234432222234433333334444444444
1313Terms Scalar field
Gradient
Co-gradient
Divergence
2D 3D1414Constructing the Deformation Vector Field VConstructing a divergence-free vector field
2D Co-gradient field of a scalar field [Davis 1967] :
3D Cross product of gradients of two scalar fields
1515Constructing the Deformation Vector Field V (3D)Define scalar field in terms of region field
Construct divergence-free field using defined scalar fields
: Berstein polynomials
16 inner region outer region intermediate region interpolation , .
3 2 divergence-free 2 .16Blending intermediate regionInner & outer region should be connected smoothly It requires C1 continuity Scalar fields : C2 continuity Vector field : C1 continuity
e(x)0
17 , e(x) zero deformation , . , linear interpolation , c0 continuity . c1 continuity bezier curve .
17Blending: 2D Example
inner regionv constantouter regionv = 0intermediateregion
10-1
10-1
10-1
10-1
10-118182D Example Region Field
19
u,v = (-1,0)19Special Deformations - TranslationTranslation vector field : A constant vector fieldThe center point c : to determine DOF
: The center point
20C degree of freedom(u,w,v) 20Special Deformation - RotationRotation vector field : linear vector field v and rA center point : An Axis :
Ristrected as a cylinder
2121Vector Field Translation Rotation
22223. Modeling Metaphor2323Deformation CycleUsually 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 cDefine scalar field e(x), f(x)with orthogonal vector u, w, a center c and an axis aUpdate vIntegrate point of the shape with v
24Step 24Implicit ToolsPoint toolsPoints in the inner regionat the beginning followthe movement of the toolOther points never enter the inner region no self-intersection
Line tools
2525Deformation PaintThe tool is moved along a path on the surface : the location of the point on the shape at a certain time , = choosen interactively
2626Moving Point SetsMultiple isolated point set the shape : Smooth approximated distance function to this point set , : interactively choosen : Barycenter of all points
2727Collision ToolsAn 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
65433225432211432110032100112101122101223310123442828Collision Tools : Example 1
2929Collision Tools : Example 2
3030Twisting & BendingLinear and quadratic vector fieldTwsiting : linear : direction of the twisting axis : on the twising axis
3131Twisting & BendingBendingUsing a rotation
,
320.614 linear interpolation 0.5 , ri ro . 32Twisting & Bending
twistingbending3333Feature PreservationDetails on the surface are preserved during deformation
34344. Implementation3535Integration with adaptive stepsizeBest tradeoff between speed and accuracy [Nielson et al. 1997]4th order Runge-Kutta integration with adaptive stepsize
3636RemeshingLarge deformation causes unpleasing artifactsUndersample Volume changing
IdeaRemeshing both the original and deformed objectNew vertices undergo same deformation as the original vertices
It Requires remeshing 3737RemeshingM : 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 MNew 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 collapsed38385. Evaluation and Comparision3939Visual QualityThe twisting of a box
[Yu et al 2004][Proposed][Lipman et al. 2005][Zhou et al. 2005]40 40Visual QualityBending a sylinder
[Proposed][Laplacian surface][Poisson Mesh][Zhou et al. 2005][Botsch and Kobbelt 2004]41 laplacian surface editing self-intersection 41SpeedFactorsVertex # 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 stamping
Deformation 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 CPU42 Integration paraleelizable GPU 42SpeedImplementation EnvironmentAMD Opteron 152(2.6 GHz) 2GB RAM GeForce 6800 GT GPU
43Outer region , intermediate region .Gpu 10 43AccuracyAccuracy in volume Discrete surface points produces slight changes of the vlumeBut is tolerable
44 .446. Conclusion and Future Work4545ConclusionAlternative approach to shape deformation Time-dependent divergence-free vector field volume-preservingSelf-intersection Sharp features Realtime deformationAccuracy in volume preserving is high4646Future WorkPreformance 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 surfaces4747