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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

2020/23/23

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 !!!