1. 1/23 Vector field reconstruction from sparse samples with applications Marcos Lage, Fabiano Petronetto, Afonso Paiva, Hélio Lopes, Thomas Lewiner and Geovan Tavares Laboratório Matmidia - Departamento de Matemática - PUC-Rio - Rio de Janeiro - Brasil

2. 2/23 Vector fields

3. 3/23 Problem description

4. 4/23 on the whole region Problem description ?

5. 5/23 Motivation Particle Image Velocimetry (Luiz Fernando A. Azevedo, PUC-Rio)

6. 6/23 Smoothed Particle Hydrodynamics (SPH) (F. Petronetto, A. Paiva, T. Lewiner, G. Tavares – PUC-Rio – Sibgrapi 2006) Motivation

7. 7/23 Outline Problem description Motivation Local approximation From local to global approximation Results Future Works

8. 8/23 on the whole region ? Local approximations

9. 9/23 Local approximations Classical least squares method: Ax = B

10. 10/23 Acceleration fitting We can improve the vector field approximation …

11. 11/23... changing the minimization problem: µ user parameter Acceleration fitting

12. 12/23 Robustness Ridge regression: get around zero eigenvalues Least squares numerical instability (A + β I d )x = B β user parameter

13. 13/23 Effects of Ridge Regression With Ridge Regression Without Ridge Regression

14. 14/23 From local to global approximation

15. 15/23 Domain subdivision Compact supports From local to global approximation

16. 16/23 From local to global approximation Partition of Unity: and Kernel functions:

17. 17/23 Results Synthetic field:

18. 18/23 Stable Fluids: (Stam, J. 1999. Stable fluids) Results

19. 19/23 Particle Image Velocimetry: (Luiz Fernando A. Azevedo, PUC-Rio) Results

20. 20/23 Smoothed Particle Hydrodynamics (SPH): (F. Petronetto, A. Paiva, T. Lewiner, G. Tavares – PUC-Rio) Results

21. 21/23 Conclusion & Future Work Fast and accurate reconstruction Fast and accurate reconstruction → Extend to time dependent 2D vector fields Applicable to visualization Applicable to visualization → Extend to time dependent 3D vector fields Minimize quadratic error Minimize quadratic error → More constraints: conservative, divergent-free vector fields reconstructions.

22. 22/23 More results … The Method has already been extended to 3D vector fields

23. 23/23 Thanks !!!