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Interactive and Anisotropic Geometry Processing Using the Screened Poisson Equation Ming Chuang and Misha Kazhdan Johns Hopkins University.

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Presentation on theme: "Interactive and Anisotropic Geometry Processing Using the Screened Poisson Equation Ming Chuang and Misha Kazhdan Johns Hopkins University."— Presentation transcript:

1 Interactive and Anisotropic Geometry Processing Using the Screened Poisson Equation Ming Chuang and Misha Kazhdan Johns Hopkins University

2 Goal Extend gradient-domain image processing techniques to processing of high-res geometry. SharpeningSmoothing

3 Goal Homogenous Filtering: Space: [Taubin ’95] [Desbrun et al. ’99] Frequency: [Guskov et al. ’99] [Vallet and Levy ’08] Anisotropic Filtering: [Clarenz et al. ’00] [Meyer et al. ’02] [Bajaj et al. ’02] [Tasdizen et al. ’02] In real time! [Taubin 1995] [Clarenz et al. 2002]

4 Outline Introduction Approach – Variational Formulation – Discretization Implementation Results Conclusion

5 Approach Extend the image-processing formulation of [ Bhat et al. ’08 ]: [Bhat et al. 2008]  >1 value fidelitygradient modulation FoFo F new

6 Approach Extend the image-processing formulation of [ Bhat et al. ’08 ]: This also works for signals on meshes. FoFo F new  >1 value fidelitygradient modulation

7 Approach Extend the image-processing formulation of [ Bhat et al. ’08 ]: Three different parameters: 1.The original signal 2.The gradient modulation 3.The metric FoFo F new  >1 value fidelitygradient modulation

8 1) Original Signal Support geometry processing by using the embedding:  >1 FoFo F new value fidelitygradient modulation

9 2) Gradient Modulation Support inhomogenous edits by allowing the modulation to be spatially varying.  =0  =2 FoFo F new value fidelitygradient modulation

10 3) Metric Support direction-aware editing by letting the metric scale anisotropically. value fidelitygradient modulation  <1 FoFo F new

11 Defining a Metric General Formulation: Choose a basis for each tangent plane Prescribe a symmetric positive definite matrix: |u||u| w q v1(q)v1(q) v2(q)v2(q) r v1(r)v1(r) v2(r)v2(r) p v2(p)v2(p) v1(p)v1(p)

12 Defining a Metric Feature-Aware Formulation [Clarenz et al. ’00]: Use principal directions as a basis Prescribe a diagonal positive definite matrix p v1(p)v1(p) v2(p)v2(p) q v1(q)v1(q) v2(q)v2(q) r v1(r)v1(r) v2(r)v2(r)

13 Defining a Metric Feature-Aware Formulation [Clarenz et al. ’00]: – Area grows with scale – Derivatives shrink with scale value fidelitygradient modulation p v1(p)v1(p) v2(p)v2(p) q v1(q)v1(q) v2(q)v2(q) r v1(r)v1(r) v2(r)v2(r)

14 Outline Introduction Approach – Variational Formulation – Discretization Implementation Results Conclusion

15 Discretization Finite Elements: – Choose a basis {B 1 (p),…,B n (p)} – Discretize the system

16 Discretization In our system: – Modulate gradient – Scale metric system matrixsystem constraints To make the system interactive: 1.Fast integration 2.Fast solver w/o pre-factorization

17 Outline Introduction Approach Implementation – Finite Elements – Integration Results Conclusion

18 Grid-Based Finite Elements Imposing regularity [Chuang et al. 2009]

19 Grid-Based Finite Elements Advantages: 1.Supports multigrid  System can change at run-time Relax Solve Down- Sample Up- Sample Relax

20 Grid-Based Finite Elements Advantages: 1.Supports multigrid 2.Controllable dimension

21 Grid-Based Finite Elements Advantages: 1.Supports multigrid 2.Controllable dimension 3.Supports parallelization

22 Parallel Gauss-Seidel Relaxation Use “safe-zones” [Weber et al. 2008]: ` ` ``

23 Parallel Gauss-Seidel Relaxation I.Function assignment Thread 1 Thread 2

24 Parallel Gauss-Seidel Relaxation I.Function assignment II.Safe-zone expansion Thread 1 Thread 2

25 Parallel Gauss-Seidel Relaxation I.Function assignment II.Safe-zone expansion Thread 1 Thread 2

26 Parallel Gauss-Seidel Relaxation I.Function assignment II.Safe-zone expansion Thread 1 Thread 2

27 Parallel Gauss-Seidel Relaxation I.Function assignment II.Safe-zone expansion Thread 1 Thread 2

28 Parallel Gauss-Seidel Relaxation I.Function assignment II.Safe-zone expansion III.Eroding relaxation Thread 1 Thread 2

29 Iteration #1 Iteration #2Iteration #3 Iteration #2Iteration #3 Parallel Gauss-Seidel Relaxation Done! I.Function assignment II.Safe-zone expansion III.Eroding relaxation Thread 1 Thread 2

30 Outline Introduction Approach Implementation – Finite Elements – Integration Results Conclusion

31 Fast System Computation Requirements: – Differentiable basis functions – Integrable scale and modulation   i and  can be piecewise constant B(p)B(p)B(p)B(p)  i (p)|  (p)

32 Integration is done per-element  Integrals be computed off-line. Fast System Computation B(p)B(p)B(p)B(p) (p)(p)(p)(p) Pre-processing Run-time

33 Outline Introduction Approach Implementation Results Conclusion

34 Our Interactive System Three editing interfaces:

35 Our Interactive System Three editing interfaces: – Slider – Spray can – Curvature profile system matrixsystem constraints  =0  =1  >1

36 Our Interactive System Three editing interfaces: – Slider – Spray can – Curvature profile system matrix  >1 system constraints

37 Our Interactive System Three editing interfaces: – Slider – Spray can – Curvature profile system matrixsystem constraints  <0  =0  >0  <1  >1

38 Demo

39 VerticesFunctions Frames/Second Solver Speed-Up RelaxConstraintMatrix Neptune 7.0 x x x Armadillo 1.7x x x Dragon 4.4x x x Forma Urbis 1.0x x x David Head 2.0x x x Performance

40 Conclusion Anisotropic geometry processing – Generalizes image processing – Extends to anisotropic editing Real-time system – Fast integral – Parallel multigrid solver

41 Conclusion Future Work – General metrics Spatially varying Non-diagonal – Surface deformation

42 ~ Thank You~ Code & Executable:

43

44 Limitation Coarser resolution reveal the p.-linear nature of the basis. (This disappears with second order splines.)

45


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