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Deformable Sequence Reconstruction 1 EG 2012 Tutorial: Dynamic - - PowerPoint PPT Presentation

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Deformable Sequence Reconstruction 1 EG 2012 Tutorial: Dynamic Geometry Processing Eurographics 2012, Cagliari, Italy Deformable Shape Matching Basic Principle Eurographics 2012, Cagliari, Italy Example Correspondences? Eurographics 2012,

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Deformable Sequence Reconstruction

EG 2012 Tutorial: Dynamic Geometry Processing

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Eurographics 2012, Cagliari, Italy

Deformable Shape Matching

Basic Principle

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Example

Correspondences?

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What are We Looking for?

Problem Statement: Given:

  • Two surfaces S1, S2 ⊆ ℝ3

We are looking for:

  • A reasonable deformation function f : S1 → ℝ3

that brings S1 close to S2

?

S1 S2

f

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Example

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correspondences? no shape match too much deformation

  • ptimum
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This is a Trade-Off

Deformable Shape Matching is a Trade-Off:

  • We can match any two shapes

using a weird deformation field

  • We need to trade-off:

– Shape matching (close to data) – Regularity of the deformation field (reasonable match)

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Components: Matching Distance: Deformation / rigidity:

Variational Model

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Variational Model

Variational Problem:

  • Formulate as an energy minimization problem:
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Part 1: Shape Matching

Data Term:

  • Objective Function:
  • Distance measures:

– Least-squares (L2) – Reweighted (robustness) – Hausdorf distance – Lp-distances, etc.

  • L2 measure is frequently used (models Gaussian noise)

– Reweighting/truncation for robustness

S2 f (S1)

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Surface Approximation

Basic: Closest point matching

  • “Point-to-point” energy
  • Usually iterated: “Iterated Closest Points (ICP)”

– Establish nearest-neighbor correspondences – Minimize energy (with regularizer)

f (S1)

S2

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Surface Approximation

Improvement: Linear approximation

  • “Point-to-plane” energy
  • Fit plane to k-nearest neighbors

S2

f (S1)

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Robust Least-Squares

Robustness: Reweighting

  • Ignore Outliers

– Large distance – Connection to normal at large angle – Many matches to one point

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Variational Model

Variational Problem:

  • Formulate as an energy minimization problem:
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What is a “nice” deformation field?

  • Elastic deformation

– Volumetric elasticity – Thin shell model (more complex)

  • Intrinsic

– Isometric matching

  • Smooth deformations

– “Thin-plate-splines” (TPS) – Allowing strong deformations, but keep shape

Deformation Model

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What is a “nice” deformation field?

  • Elastic deformation

– Volumetric elasticity – Thin shell model (more complex)

  • Intrinsic

– Isometric matching

  • Smooth deformations

– “Thin-plate-splines” (TPS) – Allowing strong deformations, but keep shape

Deformation Model

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How to Detect Deformations?

S1 V1 f S2 ∇f f (V1)

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How to Detect Deformations?

f

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Elastic Volume Model

Extrinsic Volumetric “As-Rigid-As Possible”

  • Measure orthogonality
  • Integrate over deviation from orthogonality

S1 V1 f S2 ∇f

f (V1)

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Deformable ICP

How to build a deformable ICP algorithm

  • Pick a surface distance measure
  • Pick an deformation model / regularizer

) ( ) ( ) (

) ( ) (

f E f E f E

r regularize match

+ =

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Deformable ICP

Deformable ICP Algorithm

  • Select model: E(match), E(regularizer)
  • Initialize f (S1) with S1 (i.e., f = id)
  • (Non-linear) optimization:

– Newton, Gauss Newton – LBGFS (quick & effective)

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Animation Reconstruction

Reconstructing Sequences of Deformable Shapes

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“Factorization” Approach

t = 0 t = 1 t = 2

data urshape

S f f f

deformation

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Hierarchical Merging S f(S) data f

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Hierarchical Merging S f(S) data f

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Initial Urshapes S f(S) data f

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Initial Urshapes S f(S) data f

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Alignment S f(S) data f

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Alignment S f(S) data f

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Hierarchical Alignment S f(S) data f

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Hierarchical Alignment S f(S) data f

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Global Optimization Energy Function

E(f, S) = Edata + Edeform + Esmooth

Components

  • Edata(f, S)

– data fitting

  • Edeform(f)

– elastic deformation, smooth trajectory

  • Esmooth(S)

– smooth surface

Final Optimization

  • Minimize over all frames
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Deformation Field S f

  • Elastic energy
  • Smooth trajectories
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Additional Terms More Regularization

  • Acceleration:

Eacc = ∫T ∫V|∂t

2 f|2

 Smooth trajectories

  • Velocity (weak):

Evel = ∫T ∫V|∂t f|2

 Damping

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Surface Reconstruction

Data fitting

  • Smooth surface
  • Fitting to noisy data

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S

fi

  • 1(Di)

S fi

  • 1(Di)

S

f

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Results

(Joint work with: Bart Adams, Maksim Ovsjanikov, Alexander Berner, Martin Bokeloh, Philipp Jenke, Leonidas Guibas, Hans-Peter Seidel, Andreas Schilling)

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Edeform(f)

Elastic Deformation Energy S Di f

Regularization

  • Elastic energy
  • Smooth trajectories
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geometry

Discretization Example Approach:

  • Full resolution geometry
  • Subsample deformation

deformation

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Discretization “Subspace” Approach:

  • Sample volume
  • Place basis functions
  • Decouple from resolution of geometry

geometry deformation

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Surface Reconstruction Data fitting

  • Smooth surface
  • Fitting to noisy data

S Di

Esmooth(S)

fi

  • 1(Di)

S fi

  • 1(Di)

S

f

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Example

?

correspondences? no shape match too much deformation

  • ptimum