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

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, Cagliari, Italy What are We Looking for? ? Problem Statement: f Given: S 2 • Two surfaces S 1 , S 2 ⊆ ℝ 3 S 1 We are looking for: • A reasonable deformation function f : S 1 → ℝ 3 that brings S 1 close to S 2

Eurographics 2012, Cagliari, Italy Example ? correspondences? no shape match too much deformation optimum

Eurographics 2012, Cagliari, Italy 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)

Eurographics 2012, Cagliari, Italy Variational Model Components: Matching Distance: Deformation / rigidity:

Eurographics 2012, Cagliari, Italy Variational Model Variational Problem: • Formulate as an energy minimization problem:

Eurographics 2012, Cagliari, Italy Part 1: Shape Matching Data Term: • Objective Function: S 2 f ( S 1 ) • Distance measures: – Least-squares (L 2 ) – Reweighted (robustness) – Hausdorf distance – L p -distances, etc. • L 2 measure is frequently used (models Gaussian noise) – Reweighting/truncation for robustness

Eurographics 2012, Cagliari, Italy Surface Approximation S 2 f ( S 1 ) Basic: Closest point matching • “Point-to-point” energy • Usually iterated: “Iterated Closest Points (ICP)” – Establish nearest-neighbor correspondences – Minimize energy (with regularizer)

Eurographics 2012, Cagliari, Italy Surface Approximation S 2 f ( S 1 ) Improvement: Linear approximation • “Point-to-plane” energy • Fit plane to k -nearest neighbors

Eurographics 2012, Cagliari, Italy Robust Least-Squares Robustness: Reweighting • Ignore Outliers – Large distance – Connection to normal at large angle – Many matches to one point

Eurographics 2012, Cagliari, Italy Variational Model Variational Problem: • Formulate as an energy minimization problem:

Eurographics 2012, Cagliari, Italy Deformation Model 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

Eurographics 2012, Cagliari, Italy Deformation Model 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

Eurographics 2012, Cagliari, Italy How to Detect Deformations? ∇ f f V 1 f (V 1 ) S 1 S 2

Eurographics 2012, Cagliari, Italy How to Detect Deformations? f

Eurographics 2012, Cagliari, Italy Elastic Volume Model Extrinsic Volumetric “As-Rigid-As Possible” • Measure orthogonality • Integrate over deviation from orthogonality ∇ f f V 1 f (V 1 ) S 1 S 2

Eurographics 2012, Cagliari, Italy Deformable ICP How to build a deformable ICP algorithm • Pick a surface distance measure • Pick an deformation model / regularizer = + ( match ) ( regularize r ) E ( f ) E ( f ) E ( f )

Eurographics 2012, Cagliari, Italy Deformable ICP Deformable ICP Algorithm • Select model: E ( match ) , E ( regularizer ) • Initialize f (S 1 ) with S 1 (i.e., f = id) • (Non-linear) optimization: – Newton, Gauss Newton – LBGFS (quick & effective)

Eurographics 2012, Cagliari, Italy Animation Reconstruction Reconstructing Sequences of Deformable Shapes

Eurographics 2012, Cagliari, Italy “Factorization” Approach t = 0 t = 1 t = 2 data f f deformation f urshape S

Eurographics 2012, Cagliari, Italy Hierarchical Merging data f ( S ) f S

Eurographics 2012, Cagliari, Italy Hierarchical Merging data f ( S ) f S

Eurographics 2012, Cagliari, Italy Initial Urshapes data f ( S ) f S

Eurographics 2012, Cagliari, Italy Initial Urshapes data f ( S ) f S

Eurographics 2012, Cagliari, Italy Alignment data f ( S ) f S

Eurographics 2012, Cagliari, Italy Alignment data f ( S ) f S

Eurographics 2012, Cagliari, Italy Hierarchical Alignment data f ( S ) f S

Eurographics 2012, Cagliari, Italy Hierarchical Alignment data f ( S ) f S

Eurographics 2012, Cagliari, Italy Global Optimization Energy Function E( f , S ) = E data + E deform + E smooth Components • E data ( f , S ) – data fitting • E deform ( f ) – elastic deformation, smooth trajectory • E smooth ( S ) – smooth surface Final Optimization • Minimize over all frames

Eurographics 2012, Cagliari, Italy Deformation Field f S • Elastic energy • Smooth trajectories

Eurographics 2012, Cagliari, Italy Additional Terms More Regularization E acc = ∫ T ∫ V | ∂ t 2 f | 2 • Acceleration:  Smooth trajectories E vel = ∫ T ∫ V | ∂ t f | 2 • Velocity (weak):  Damping

Eurographics 2012, Cagliari, Italy Surface Reconstruction f S Data fitting -1 ( D i ) -1 ( D i ) f i f i • Smooth surface • Fitting to noisy data S S 34

Eurographics 2012, Cagliari, Italy Results (Joint work with: Bart Adams, Maksim Ovsjanikov, Alexander Berner, Martin Bokeloh, Philipp Jenke, Leonidas Guibas, Hans-Peter Seidel, Andreas Schilling)

Eurographics 2012, Cagliari, Italy Eurographics 2012, Cagliari, Italy

Eurographics 2012, Cagliari, Italy Eurographics 2012, Cagliari, Italy

Eurographics 2012, Cagliari, Italy Eurographics 2012, Cagliari, Italy

Eurographics 2012, Cagliari, Italy Eurographics 2012, Cagliari, Italy

Eurographics 2012, Cagliari, Italy Elastic Deformation Energy D i E deform ( f ) f S Regularization • Elastic energy • Smooth trajectories

Eurographics 2012, Cagliari, Italy Discretization geometry deformation Example Approach: • Full resolution geometry • Subsample deformation

Eurographics 2012, Cagliari, Italy Discretization geometry deformation “Subspace” Approach: • Sample volume • Place basis functions • Decouple from resolution of geometry

Eurographics 2012, Cagliari, Italy Surface Reconstruction D i E smooth ( S ) f S Data fitting -1 ( D i ) -1 ( D i ) f i f i • Smooth surface • Fitting to noisy data S S

Eurographics 2012, Cagliari, Italy Example ? correspondences? no shape match too much deformation optimum