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
1/23
Vector fields
2/23
Problem description
3/23
Problem description
?
on the whole region
4/23
Motivation
Particle Image Velocimetry
(Luiz Fernando A. Azevedo, PUC-Rio)
5/23
Motivation
Smoothed Particle Hydrodynamics (SPH)
(F. Petronetto, A. Paiva, T. Lewiner, G. Tavares – PUC-Rio – Sibgrapi 2006)
6/23
Outline
Problem description
Motivation
Local approximation
From local to global
approximation
Results
Future Works
7/23
Local approximations
?
on the whole region
8/23
Local approximations
Classical least squares method:
Ax = B
9/23
Acceleration fitting
We can improve the vector field approximation …
10/23
Acceleration fitting
... changing the minimization problem:
µ user parameter
11/23
Robustness
Least squares numerical instability
Ridge regression: get around zero eigenvalues
(A + βId)x = B
β user parameter
12/23
Effects of Ridge Regression
Without
Ridge
Regression
With Ridge
Regression
13/23
From local to global approximation
14/23
From local to global approximation
Domain subdivision
15/23
Compact supports
From local to global approximation
Partition of Unity:
and
Kernel functions:
16/23
Results
Synthetic field:
17/23
Results
Stable Fluids: (Stam, J. 1999. Stable fluids)
18/23
Results
Particle Image Velocimetry: (Luiz Fernando A. Azevedo, PUC-Rio)
19/23
Results
Smoothed Particle Hydrodynamics (SPH):
(F. Petronetto, A. Paiva, T. Lewiner, G. Tavares – PUC-Rio)
20/23
Conclusion & Future Work
•
Fast and accurate reconstruction
→ Extend to time dependent 2D vector fields
•
Applicable to visualization
→ Extend to time dependent 3D vector fields
•
Minimize quadratic error
→ More constraints:
conservative, divergent-free vector fields reconstructions.
21/23
More results …
The Method has already
been extended to
3D vector fields
22/23
Thanks !!!
23/23