Articulated Meshes Eric Landreneau Scott Schaefer Texas A&M - PowerPoint PPT Presentation

Simplification of Articulated Meshes Eric Landreneau Scott Schaefer Texas A&M University

Introduction Simplification 1,087,716 10,000 faces faces

Introduction Articulated meshes

Introduction Articulated meshes    ˆ v ( M v ) k k k

Introduction Articulated meshes    ˆ v ( M v ) k k k M : Bone Transformation Matrix k

Introduction Articulated meshes    ˆ v ( M v ) k k k M : Bone Transformation Matrix k  : Skin Weights k      1 , 0 k k k

Introduction Unsimplified

Introduction Unsimplified

Introduction

Introduction Static simplification

Introduction Static simplification insufficient for deformable models

Quadric Error Functions Basic QEF equation: : i th vertex p in mesh : normal of m th adjacent face

QEF Edge Collapses Q m = Quadric Error Function (distance to plane on face m) Q m

QEF Edge Collapses Q 2 Q 1 Q 3 Q 0 Q 4 Q 5

QEF Edge Collapses Q v = Q 0 + Q 1 + Q 2 + Q 3 + Q 4 + Q 5 Q 2 Q 1 Q 3 Q v Q 0 Q 4 Q 5

QEF Edge Collapses Q v = Q 0 + Q 1 + Q 2 + Q 3 + Q 4 + Q 5 Q v

QEF Edge Collapses

QEF Edge Collapses Q v0

QEF Edge Collapses Q v1

QEF Edge Collapses Q e =Q v0 + Q v1 Q e Q v0 Q v1

QEF Edge Collapses Q e

Our Method Example Poses

Our Method Modify QEF Equation:

Our Method Modify QEF Equation:    ˆ j j v M v k k k

Our Method Modify QEF Equation: T         )        j j j E ( v , M v Q M v i k k k i k k     j k k    ˆ j j v M v k k k

Our Method Modify QEF Equation: T         )        j j j E ( v , M v Q M v i k k k i k k     j k k Problem : equation is quartic Solution : split into alternating quadratic equations

Our Method Quadratic #1 – Solve for position   T                 T j j j min E ( v ) v M Q M v   i k k i k k     v   j k k Hold weights constant and solve for position v

Our Method Quadratic #2 – Solve for weights min  Hold V constant and solve for weights    j j j  V M v M v M v j 0 1 k

Our Method Quadratic #2 – Solve for weights min     subject to 1 k k Hold V constant and solve for weights    j j j  V M v M v M v j 0 1 k

Our Method Quadratic #2 – Solve for weights min       subject to 1 , 0 k k k Hold V constant and solve for weights    j j j  V M v M v M v j 0 1 k

Our Method Alternating minimization

Results Input Poses 240,448 poly

Results 10,000 poly 5,000 poly 2,000 poly

Results Input Poses 206,672 poly

Results

Results Comparison with previous techniques Original deCoro et al. Mohr et al. Ours

Results

Results

Results DeCoro et al.

Results Ours

Results DeCoro et al. Ours

Results Weight Influences

Results Weight reduction Restriction to n weight influences : • Minimize • Prune down to n largest weights • Minimize again

Results Weight Reduction Unconstrained Constrained up to 11 weights/vertex 5 weights/vertex

Results RMS Error Comparison (METRO)

Results

Conclusions • Minimizes both skin weights and vertex positions • Easy to implement (quadratic minimization) • Requires few example poses • Reduces to a specified number of weights everywhere in the hierarchy

Questions?