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

2/23 Vector fields

3/23 Problem description

4/23 on the whole region Problem description ?

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

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

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

8/23 on the whole region ? Local approximations

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

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

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

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

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

14/23 From local to global approximation

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

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

17/23 Results Synthetic field:

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

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

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

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.

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

23/23 Thanks !!!