An Intuitive Framework for Real-Time Freeform Modeling

Transcription

1 An Intuitive Framework for Real-Time Freeform Modeling Leif Kobbelt

2 Shape Deformation Complex shapes Complex deformations

3 User Interaction Very limited user interface 2D screen & mouse Intuitive metaphor needed Control points / handles

4 Shape Deformation Move control handle to edit surface S = S + B (δc) Complex modification B (δc)? Either complex user interaction δc Or complex basis functions B

5 Shape Deformation Keep user interaction simple Non-expert users Limited user interface Need custom-tailored basis functions B Arbitrary support Smoothness Surface stiffness & bending

6 Overview Introduction Related Work Boundary constraint modeling Results Conclusion

7 Related Work NURBS & Subdivision surfaces Freeform Deformation Distance-Based Deformation Boundary Constraint Modeling

8 NURBS & Subdivision Basis functions are smooth bumps Simple B, complex δc Fixed support

9 NURBS & Subdivision Basis functions are smooth bumps Simple B, complex δc Fixed support

10 NURBS & Subdivision Basis functions are smooth bumps Simple B, complex δc Fixed support

11 NURBS & Subdivision Fixed basis functions Support Number & position Bound to control points Initial patch layout crucial Requires experts! De-couple deformation basis from surface representations!

12 Freeform Deformation Deform space around object Tri-variate tensor-product spline Simple B, complex δc

13 Freeform Deformation Deform space around object Tri-variate tensor-product spline Simple B, complex δc

14 Freeform Deformation Deform space around object Tri-variate tensor-product spline Simple B, complex δc

15 Distance-Based Propagation Propagate handle transformation Euclidean / geodesic distance Surface properties? Interpolation constraints?

16 Distance-Based Propagation? Distance Partition of Unity Smooth interpolation

17 Boundary Constraint Modeling Constrained energy minimization Moreton & Sequin 1992 Welch & Witkin 1992 Kobbelt et al Boundary element method James & Pai 1999 Alternative detail representation Sorkine et al Yu et al. 2004

18 Overview Introduction Related Work Boundary constraint modeling Results Conclusion

19 Boundary Constraint Modeling Prescribe boundary constraints Vertex positions Vertex continuities Constraint energy minimization E k (S) = F k (S u k, S u k 1 v,..., S v k) Euler-Lagrange PDE: k (S) = 0

20 Energy Functionals Membrane S = 0 Thin-Plate 2 S = 0 3 MVS S = 0

21 Modeling Metaphor [Kobbelt et al. 98] Support region (blue) Handle regions (green) Fixed vertices (gray)

22 Linear System h = {h 1,..., h H } [Kobbelt et al. 98] (k rings only) p = {p 1,..., p P } f = {f 1,..., f F } (k rings only) k 0 I F 0 p f = 0 f h h 0 0 I H

23 Industrial Evaluation [Kobbelt et al. 98] targeted primarily at conceptual design More control for engineering applications: Specify boundary smoothness Anisotropic bending behavior

24 Boundary Smoothness How smooth does the deformed region blend with the fixed part? C 0 /C 2 C 2 /C 0 C 2 /C 2

25 Boundary Smoothness Δ k surfaces can do up to C k-1 Real-valued smoothness c(p) [0, k 1] Adjust recursive Laplace definition 3 (p) = (λ 2 (p) (λ 1 (p) (p))) 0 [0, 1] C 0+λ 1(p) [0, 1] 1 C 1+λ 2(p)

26 Boundary Smoothness C 2 C 0.2 C 0 C 1 Segment-wise boundary smoothness

27 Anisotropic Bending Isotropic bending for standard Laplace Not intuitive for anisotropic support region Bending should adapt to support s shape isotropic anisotropic

28 Anisotropic Bending S = 0 isotropic

29 Anisotropic Bending S = 0 isotropic LSCM Φ : Ω IR 3 Ω

30 Anisotropic Bending S = 0 isotropic LSCM Ω scale Φ : Ω IR 3 Ω

31 Anisotropic Bending S = 0 isotropic LSCM Ω scale Ω Φ = 0 anisotropic

32 Real-Time Modeling k 0 I F I H p f h = 0 f h Solve linear system each frame: Too slow even for multi-grid solvers (#p>20k) System changes if certain constraints change Precompute per-handle basis function

33 Precomputed Basis Functions System to be solved k 0 I F 0 } 0 0 {{ I H } =:L p f h = 0 f h Columns of inverse are bases p f h = L 1 0 f 0 + L h

34 Precomputed Basis Functions Simple user interaction Handle is transformed affinely only Represent handle points w.r.t. affine frame h = Q [a, b, c, d] T

35 Precomputed Basis Functions p f = L 1 0 f + L h 0 h

36 Precomputed Basis Functions p f = L 1 0 f + L [a, b, c, d] T. h 0 Q

37 Precomputed Basis Functions p f = L 1 c 0 f + L 1 B 0 0 [a, b, c, d] T. h 0 Q Solve linear system 3 times (const. part) 4 times per handle c IR N 3 B IR N 4 Moderate precomputation times 10k: 1 sec, 50k: 9s, O(#p)

38 Precomputed Basis Functions Real-time per-frame solution p = c + B [a, b, c, d ] T Custom-tailored basis function S = S + B [δa, δb, δc, δd] }{{} δc T

39 Custom-Tailored Basis Function Energy functional (stiffness)

40 Custom-Tailored Basis Function Energy functional (stiffness) Support & handle (fullness)

41 Custom-Tailored Basis Function Energy functional (stiffness) Support & handle (fullness) Boundary smoothness

42 Custom-Tailored Basis Function Energy functional (stiffness) Support & handle (fullness) Boundary smoothness Isotropic / anisotropic

43 Custom-Tailored Basis Function Energy functional (stiffness) Support & handle (fullness) Boundary smoothness Isotropic / anisotropic Arbitrary affine handle transformation

44 Overview Introduction Related Work Boundary constraint modeling Results Conclusion

45 Multiresolution Modeling Freeform modeling builds smooth surfaces Real-world models have fine details Modify existing models Integrate into multiresolution framework

46 Multiresolution Modeling Decomposition Freeform Modeling Detail Information Reconstruction

47 Sillboard C 1 /C 1 isotropic C 1 /C 0 isotropic C 1 /C 0 anisotropic

48 Stretching the Hood Multiple handles Wheel houses stay circular

49 Conclusion Custom-tailored basis functions Simple user interaction Boundary constraint modeling Boundary smoothness Anisotropic bending Precomputed basis functions Real-time deformations

50 Future Work Topological flexibility Anisotropy requires parameterization Disk shaped support Numerical improvements Robustness & efficiency Remeshing approach, SGP 2004 Dynamic remeshing