Pose-independent Simplification of Articulated Meshes

1
Pose-independent Simplification of Articulated
Meshes
Christopher DeCoro Szymon Rusinkiewicz
2
Motivation

• Decimation enables complex scenes
• Interactive rendering
• Automatic Level of Detail
• Non-static geometry crucial for movies and games
• Goal single decimated mesh for multiple poses
• Lower cost than view / pose-dependent LOD
• Framework for geometric processing on non-static
  geometry

3
Pose-Independent Simplification

• Current methods consider a single pose
• Do not preserve enough detail for some poses
• Coloration based on triangle area, note uniform
  color

Reference from Surface Simplification using
Quadric Error Metrics, M. Garland, P. Heckbert,
1997
4
Pose-Independent Simplification

• Heuristics for adapting static simplification
• Bending joints then applying
• Preserve high detail near joints
• Drawbacks manual intervention,excessive
  preservation of detail

5
Pose-Independent Simplification

• Adapt decimation to considererror across all
  poses
• Preserve detail where necessary to allow
  deformation
• Preserve detail only where necessary

6
Simplifying Articulated Models

• Observation need to specify meaningful, but
  restrictedclass of deformation
• Kinematic skeletons with linear blend skinning
• Probability distribution constraining
  deformations
• Major result define pose-independent quadric
  containing information about all poses
• Computed as a preprocess
• Used by standard iterative decimation method
• Generalizes quadric from union of multiple planes
  tounion over multiple planes and poses

7
Framework

• Preprocess
• Monte Carlo sampling of pose probability
  distribution
• Build pose-independent quadrics
• Run-Time
• Standard iterative contraction
• Update skinning weight vectors

8
Outline

• Framework
• Background
• Kinematic skeletons
• Linear blend skinning
• Pose probability distributions
• QSlim
• Algorithm
• Results

9
Kinematic Skeletons

• Heirarchy of affine linear transformations
  (bones)
• Each non-root bone defined in frame of unique
  parent
• Changes to parent frame affect all descendent
  bones

10
Linear Blend Skinning

• Each vertex of skin potentially influenced by all
  bones
• Normalized weight vertex wv gives influence of
  each bone transform
• When bones move, influenced vertices also move
• We can compute a transformation Mv for a skinned
  vertex
• For each bone
• Compute transformation Nb from bind pose to bone
  coordinates
• Compute global bone transformation Mb from parent
  transformation
• For each vertex
• Take a linear combination of bone transforms
• Apply transformation to vertex in original pose

11
Assigning Probability Distributions

• Simplest Box functions
• Range of values for which probability is non-zero
• More control Gaussian distributions
• Assign preferred angle (mean), stiffness
  (standard deviation)
• Corresponds to intuitive quantities
• Detailed control sampled functions
• Pre-defined animations
• Motion capture
• All of these are high-dimensional functions
  withcomplex correlations
• Can reduce dimensionality by letting Mb
  TranslateRotateScale

12
QSlim Simplification Algorithm

• Combine framework, primitive, and metric QSlim
• Framework Greedy selection
• Primitive Edge collapse
• Metric Quadric error
• Compute initial quadrics Qv for every vertex v
• Compute optimal error for each edge collapse
• Place into min-priority queue keyed on collapse
  cost
• While the queue is not empty
• Collapse the edge on the top of the queue to a
  single vertex
• Update the costs of all edges in the affected
  neighborhood

Reference from Surface Simplification using
Quadric Error Metrics, M. Garland, P. Heckbert
13
Quadric Error Metric

• Compute sum-of-squared distances over all planes
• Goal is to approximate distance of simplified to
  original mesh
• Each vertex has associated set of constraint
  planes
• We can factor out vertices from the summation
• Factor planes into error quadrics (symmetric 4x4
  matrices)
• Avoids requirement to keep list of planes plane
  union as quadric addition
• Common math technique to represent quadratic form
  as a matrix multiplication

14
Outline

• Framework
• Background
• Algorithm
• Expectations
• Overview
• Pose-independent Quadric Derivation
• Blend-weight Update Rule
• Results

15
Intuitive Expectations

• Suppose we have a leg model
• Joint bends at the knee, 90 degree range of
  motion
• What do we expect to see?
• Areas away from the joints dramatically
  simplified
• Crease at the bottom preserved in high detail
• Rounded top maintains mid-level of detail

16
Incorporating Multiple Poses

• Mohr Gleicher 03 Arbitrary Deformation
• Consider error in each pose for every collapse
• Our Method Linear Blend Skinning
• Consider all poses in a preprocess
• Collapses are independent of pose

Foreach collapse Foreach pose Compute
error Perform collapse
Foreach vertex Foreach pose Compute
quadric Combine quadrics from all poses Foreach
collapse Use single quadric to compute
error Perform collapse
17
Pose-independent Metric

• Each vertex v corresponds to vertex vp in pose P
• The quadric for v in pose P, Qv(P), can be
  computed directly
• Expected point-to-plane distance over all poses,
  weighted by ?
• Our goal will be to factor the vertices out of
  the integral, and define a pose-independent
  quadric that incorporates all poses

18
Algorithm Overview

• Integrated over every pose P
• Compute initial quadrics Qv(P) for every vertex v
• Map quadrics into reference coordinate system
• Compute optimal error for each edge collapse
• Place into min-priority queue keyed on collapse
  cost
• While the queue is not empty
• Collapse the edge on the top of the queue to a
  single vertex
• Compute new skin influence vector
• Update the costs of all edges in the affected
  neighborhood
• Similar structure to QSlim
• Makes large changes to initial quadric
  computation
• We define a metric, which is independent of
  framework

19
Remapping Quadrics

• A vertex v in pose P is transformed by Mv(P)
• Suppose we compute error d(v) integrated over all
  poses
• We can let vP Mv(P)v, then factor out Qv
  independent of pose
• Equivalent to applying quadric update rule with
  Mv-1

20
Monte Carlo Integration

• Integral over poses can not be evaluated
  analytically
• Standard quadrature exponential in dimensionality
• We have up to 12 degrees of freedom per bone
• Evaluate pose-independent quadrics withMonte
  Carlo sampling
• Recursively stratified to reduce variance
• In practice good results with 10-20 poses

21
Weight Update Rule

• New vertices must be given a bone influence
  vector
• Each parent exerts an influence on the child per
  bone
• Directly proportional to bone influence on parent
  vertex
• Influence falls-off based on distance
• Use the following linear interpolation system
• Strong empirical justification

22
Outline

• Framework
• Background
• Algorithm
• Results
• Intuitive results
• Weight-update rule validation
• Simplification results
• Quantitative analysis
• Timing

23
Intuitive Results

• Previously, we discussed our intuitive
  expectations
• We show various levels of simplification (26,
  13, 5, 2.5)
• We can see these were achieved in the results
• Crease kept in highest detail
• Simplified in time 32 longer than Qslim

colored randomly
coloredby area
Less simplified ? More simplified
24
Simplification Results

• Standard QSlim is on the left, our method is on
  the right
• 64k polygons, 16 poses, 25 longer than standard
  QSlim

25
Simplification Results

• Our method better approximates than
  QSlim/Straight Pose
• Leg was simplified, then bent. 0.5 resolution,
  35 time penalty

26
Changing Probability Distribution

• Left knee has larger range of motion, thus
  greater detail
• Note that region at crease is flat under
  deformation

27
Quantitative Comparison

• Used the Metro tool to compare generated meshes
• Computes the approximate Hausdorff distance
• We reduce variance of approximation across poses

28
Future Work

• Apply to more general classes of deformations
• Linear Free Form Deformation is a natural choice
• When geometry is known, but not skeleton, we
  require fitting transforms
• Potentially fit linear transforms to non-linear
  deformations
• Determine the effect of importance sampling
• Does it effectively reduce variance?
• Analyze under which conditions this is
  qualitatively effective
• Subtle effect for the meshes seen
• Certain conditions may result in more drastic
  benefits

29
(No Transcript)
30
Algorithm Timings

• Compares favorably to standard QSlim
• Considering 16 poses only 25 more than single
  pose

31
Algorithm Timings

• Compares favorably to standard Qslim
• About 25 overhead w/ 16 samples, less with fewer
  samples
• For larger models, QSlim iterative contraction
  dominates
• Our preprocess is O(kn), the contraction is O(kn
  n log n)

32
Weight-update Rule Tests

• We show both procedural and automatic weights
• Procedural weights are ground truth
• Our method is virtually indistinguishable from
  ground truth