Parameterization of Deforming Surfaces

1
Parameterization of Deforming Surfaces

• Jingyu Yan
• Department of Computer Science, UNC-Chapel Hill
• Fall 2002, Comp258 Geometric Modeling,
• Final Project Presentation

2
Introduction

• Parameterization of deforming surfaces
• Parameterization
• A mapping from a 2D domain to a 3D surface
• Lots of methods proposed 1234567
• A naïve way parameterizing the whole mesh after
  each interval when it deforms
• Time-consuming

3
The Objective

• Interactive rate
• To improve the speed of parameterization
• To provide trade-offs between accuracy and speed
  of parameterization

4
Our approaches

• To improve the speed
• Using the coherence of a deforming mesh and its
  parameterization
• To provide trade-offs between accuracy and speed
• Localizing the parameterization operation
• Relaxing the maximum residual tolerance

5
Coherence

• Iterative methods
• All of the parameterization 1234567
  use iterative methods to solve the
  parameterization problem
• A quick overview of iterative methods
• Ax b gt I x (I - A)x b
• x0 , x1 , x2 , xn converges to x
• iteration stops when tolerance gt A xN b
• The closer the initial guess x0 is to x, the
  less iterations are needed

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Coherence

• The observation of a deforming mesh
• a deforming mesh and its parameterization are
  coherent after each small time step.
• Our proposal
• For solving the parameterization at time t, we
  use the parameterization at time t - ?t as the
  initial guess

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Coherence

• The result The iteration number is reduced
• Video

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Our approaches

• To improve the speed
• Using the coherence of a deforming mesh and its
  parameterization
• To provide trade-offs between accuracy and speed
• Localizing the parameterization operation
• Relaxing the maximum residual tolerance

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Localization of parameterization operation

• When the deformation is local
• does parameterizing the deformed region give us
  the same result as parameterizing the whole mesh?
  -------- No.

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Localization of parameterization operation

• The Observation
• The further a vertex is from the deformed region,
  the less its parameters will be affected by the
  deformation.
• We use layers to define
• the distance between a vertex
• and the deformed region

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Localization of parameterization operation

• Our proposal
• Parameterizing an enlargement of the deformed
  region which is defined by the number of layers

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Localization of parameterization operation

• The result
• The number of layers provides a trade-off between
  the accuracy and the efficiency of
  parameterization.
• Video1 layers Video2 localization

0 layer 1 layer 2
layers 3 layers ...
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Our approaches

• To improve the speed
• Using the coherence of a deforming mesh and its
  parameterization
• To provide trade-offs between accuracy and speed
• Localizing the parameterization operation
• Relaxing the maximum residual tolerance

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Relaxing the maximum residual tolerance

• Iterative methods
• iteration stops when tolerance gt AxN b
• Relaxing the maximum residual tolerance can
  further reduce the iteration numbers.
• Another trade-off between accuracy and speed

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Relaxing the maximum residual tolerance

• The result
• Video1
• Video2 the extreme case

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Conclusion

• Three approaches for parameterization of
  deforming surfaces
• Using the coherence of a deforming mesh and its
  parameterization
• Localizing the parameterization operation
• Relaxing the maximum residual tolerance

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References

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