1/23 vector field reconstruction from sparse samples with applications marcos lage, fabiano...
TRANSCRIPT
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Vector field reconstruction from Vector field reconstruction from sparse samples with sparse samples with
applicationsapplicationsMarcos LageMarcos Lage, Fabiano Petronetto, Afonso Paiva, Hélio Lopes, , Fabiano Petronetto, Afonso Paiva, Hélio Lopes,
Thomas Lewiner and Geovan TavaresThomas Lewiner and Geovan Tavares
Laboratório Matmidia - Departamento de Matemática - PUC-Rio - Rio de Janeiro - Laboratório Matmidia - Departamento de Matemática - PUC-Rio - Rio de Janeiro - BrasilBrasil
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Vector Vector fieldsfields
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Problem Problem descriptiondescription
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on the whole region on the whole region
Problem Problem descriptiondescription
??
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MotivationMotivationParticle Image Particle Image
Velocimetry Velocimetry (Luiz Fernando A. Azevedo, PUC-Rio)(Luiz Fernando A. Azevedo, PUC-Rio)
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Smoothed Particle Hydrodynamics Smoothed Particle Hydrodynamics (SPH)(SPH)
(F. Petronetto, A. Paiva, T. Lewiner, G. Tavares – PUC-Rio – Sibgrapi (F. Petronetto, A. Paiva, T. Lewiner, G. Tavares – PUC-Rio – Sibgrapi 2006)2006)
MotivationMotivation
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OutlineOutlineProblem descriptionProblem description
MotivationMotivation
Local approximationLocal approximation
From local to global From local to global approximationapproximation
ResultsResults
Future WorksFuture Works
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on the whole region on the whole region
??
Local approximationsLocal approximations
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Local approximationsLocal approximations
Classical least squares Classical least squares method:method:
Ax = Ax = BB
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Acceleration fittingAcceleration fitting
We can improve the vector field We can improve the vector field approximation …approximation …
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... changing the minimization ... changing the minimization problem:problem:
µ user µ user parameterparameter
Acceleration fittingAcceleration fitting
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RobustnessRobustness
Ridge regression: get around zero Ridge regression: get around zero eigenvalueseigenvalues
Least squares numerical Least squares numerical instabilityinstability
(A + (A + ββIIdd)x = )x = BBββ user user parameterparameter
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Effects of Ridge Effects of Ridge RegressionRegression
With With Ridge Ridge
RegressioRegressionn
Without Without Ridge Ridge
RegressioRegressionn
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From local to global approximationFrom local to global approximation
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Domain Domain subdivisionsubdivision
Compact Compact supportssupports
From local to global approximationFrom local to global approximation
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From local to global approximationFrom local to global approximation
Partition of Partition of Unity:Unity:
andand
Kernel functions:Kernel functions:
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ResultsResults
Synthetic Synthetic field:field:
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Stable Fluids: Stable Fluids: (Stam, J. 1999. Stable fluids)(Stam, J. 1999. Stable fluids)
ResultsResults
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Particle Image Velocimetry: Particle Image Velocimetry: (Luiz Fernando A. Azevedo, (Luiz Fernando A. Azevedo, PUC-Rio)PUC-Rio)
ResultsResults
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Smoothed Particle Hydrodynamics (SPH):Smoothed Particle Hydrodynamics (SPH):(F. Petronetto, A. Paiva, T. Lewiner, G. Tavares – PUC-Rio)(F. Petronetto, A. Paiva, T. Lewiner, G. Tavares – PUC-Rio)
ResultsResults
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Conclusion & Future WorkConclusion & Future Work• Fast and accurate reconstructionFast and accurate reconstruction
→ → Extend to time dependent 2D vector fieldsExtend to time dependent 2D vector fields
• Applicable to visualizationApplicable to visualization→ → Extend to time dependent 3D vector fieldsExtend to time dependent 3D vector fields
• Minimize quadratic errorMinimize quadratic error→ → More constraints: More constraints:
conservative, divergent-free vector fields conservative, divergent-free vector fields reconstructions.reconstructions.
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More results …More results …
The Method has already The Method has already been extended tobeen extended to3D vector fields3D vector fields
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Thanks !!!Thanks !!!