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Adaptive Space Deformations Based On Rigid Cells

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Adaptive Space Deformations Based On Rigid Cells

Adaptive Space Deformations Based On Rigid Cells
Weekly seminar series
Visual Media Lab, KAIST, Korea

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Adaptive Space Deformations Based On Rigid Cells Adaptive Space Deformations Based On Rigid Cells Presentation Transcript

  • Adaptive Space Deformations Based on Rigid Cells
    Mario Botsch, Mark Pauly, Martin Wicke and Markus Gross
    *Mainly based on the authors’ presentation in EG `07
  • Linear Surface Deformation
    • Minimize quadratic surface energy
    • Variational calculus
    • Solve sparse linear system
    [Sorkine et al, SGP 04]
    [Lipmanet al, SIG 05]
    [Kobbelt et al, SIG 98]
  • Problematic Cases
    Topological
    inconsistencies
    Geometric
    degeneracies
    Highly complex
    models
  • Linear Space Deformation
    • Deform embedding space
    • Deformation complexity << surface complexity
    [Sederberg & Parry, SIG 86]
    [Hsu et al, SIG 92]
    [Ju et al, SIG 05]
  • Linear vs. Nonlinear
    Problems with large deformations
  • Nonlinear Surface Deformation
    • Minimize nonlinear energies
    – Intuitive large-scale deformation
    – Robustness issues
    – Performance issues
    [Au et al, SIG 07]
    [Botsch et al, SGP 06]
  • PriMo [Botsch et al, SGP 2006]
    1. Extrude triangles to prisms / cells
    2. Prescribes position/orientation for cells
    3. Find optimal rigid motions per cell
    4. Update vertices by averaged cell transformations
  • Adaptive Space Deformations Based on Rigid Cells
    Design decisions:
  • Adaptive Space Deformations Based on Rigid Cells
    Design decisions:
    – Realistic behavior ⇒ Physically plausible
  • Adaptive Space Deformations Based on Rigid Cells
    Design decisions:
    – Realistic behavior ⇒ Physically plausible
    – Large deformations ⇒ Nonlinear energy
    nonlinear
    linear
  • Adaptive Space Deformations Based on Rigid Cells
    Design decisions:
    – Realistic behavior ⇒ Physically plausible
    – Large deformations ⇒ Nonlinear energy
    – Robustness ⇒ Rigid cells
  • Adaptive Space Deformations Based on Rigid Cells
    Design decisions:
    – Realistic behavior ⇒ Physically plausible
    – Large deformations ⇒ Nonlinear energy
    – Robustness ⇒ Rigid cells
    – Applicability ⇒ Space deformation
  • AdaptiveSpace Deformations Based on Rigid Cells
    Design decisions:
    – Realistic behavior ⇒ Physically plausible
    – Large deformations ⇒ Nonlinear energy
    – Robustness ⇒ Rigid cells
    – Applicability ⇒ Space deformation
    – Performance ⇒ Adaptive discretization
  • Deformation Pipeline
  • Deformation Energy
    • Continuum mechanics
    – Strain energy defined by displacement’s gradient
    – Local variation of displacement causes stretching
    • Discrete rigid cells
    – Each cell stores rigid motion
    – Local differences of rigid transformations
  • Difference of Transformations
    • Frobenius norm of matrices
    – Geometric meaning of matrix elements?
    • [Pottmann et al, 04]
    – Difference of images of sample points (which?)
  • Nonlinear Energy
    • Integrate over neighboring cells’ interiors
  • Nonlinear Energy
    • Integrate over neighboring cells’ interiors
    • Accumulated global energy
  • Nonlinear Minimization
    • Find rigid motion per cell
    • Generalized shape matching [Botsch et al, SGP 06]
    – Robust geometric optimization
    – Nonlinear Newton-type minimization
  • Nonlinear Minimization
    • Generalized shape matching [Botsch et al, SGP 06]
    – Affine transform approximation (Linearization)
    – Nonlinear Newton-type minimization
    - Linear solve followed by rigid transform projection
  • Robust Optimization
  • Deformation Pipeline
  • Volumetric Discretization
    • Octree-like adaptive voxelization
    – Easy to implement, efficient to compute
    – Analytic integration
    • T-junctions are no problem !
  • Boundary Refinement
  • Energy-Driven Refinement
  • 3D Adaptive Refinement
  • 3D Adaptive Refinement
  • 3D Adaptive Refinement
  • Deformation Pipeline
  • Deformation Interpolation
    Final:
    RBF Interpolation
    Preview:
    Averaging / Skinning
  • Complex Models
  • Triangle Soup
  • Aliasing
  • Conclusion
    • Physically plausible deformation energy
    – Shells & solids, 2D & 3D
    – Arbitrary convex cell arrangements
    – Robust geometric shape matching
    • Adaptive discretization
    – Static and dynamic refinement
    • Smooth space deformation
    – Meshes, triangle soups, point sets, ...
  • 질문은 손들고