A Hierarchical Segmentation of Articulated Bodies

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A Hierarchical Segmentation of Articulated Bodies

• Based on an article by Goes et al.

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Segmentation for Articulation

• This paper proposes a segmentation method for
  effective rigging of 3D articulated bodies.
• The segmentation is hierarchical
• Coarsest level might indicate arms and legs
• Finest level might indicate bone structure.
• The paper introduces two new concepts
• Diffusion distance
• Medial structures

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Related Work

• Exploring concavity and geodesic measures
• Katz and Tal 03
• Liu and Zhang 04
• Detecting feature points
• Hilaga et al. 01
• Liu and Zhang 07
• Convex decomposition
• Lien and Amato 04
• Kreavoy et al. 07

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Contribution

• A novel segmentation method
• Robust to noise
• Isometric invariant
• Discussion on the benefits of diffusion distance
  in this context.
• Introducing a type of medial structures.
• Describing a simple and iterative algorithm.

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Abstract of Approach

• Using a metric on manifolds called the Diffusion
  distance, which is a robust and effective metric.
• A hierarchical set of regions subdividing a
  manifold M is created each region contains
  another set of subregions, etc.
• The subregions of each region are identified by
  creating a medial structure, which identifies the
  segments.

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Diffusion Distance

• A metric on a surface, in the scale-space t
• Reflexive
• Symmetric
• Holds triangle inequality

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Diffusion distance Contd

• Defined by a probability (Markov)
• field , which denotes the
  probability of a particle to leave x and reach y
  in time t.

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Diffusion Distance Contd

• The probability is equivalent to the heat kernel
  on manifolds
• The heat kernel is the amount of
  heat at point y by time t when a unit heat source
  is initially placed at point x.

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Heat Kernel Spectral Decomposition

• The heat kernel can be expressed using the
  eigenvalues and eigenfunctions of the
  Laplace-Beltrami operator on M

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Diffusion Distance Contd

• The diffusion distance can be expressed in terms
  of the eigenvalues then
• If every vertex is mapped to a space of infinite
  dimension thus (using a diffusion map)
• Then the diffusion distance can be expressed as
  the Euclidean distance in that space

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Infinite Dimensions Are Too Much to Handle

• Fortunately, the eigenvalues increase rapidly.
• The authors chose to truncate the sums at
• The diffusion distance is
• Robust to noise and topological short-circuits.
• Invariant to isometric deformations (as the LB
  operator is).
• Multi-scale (depends on the parameter t).
• Enhances concavities

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Average Diffusion Distance

• ADD (Average Diffusion Distance) in a region is
  defined as an average from a point to all other
  points in the regions

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Segments-gtMedial Structures

• A medial point in a region (segment)
• is defined as the set of points with minimal
• ADD to all other points of R
• The medial points correspond to the most distant
  points from the concavities.
• The parameter is chosen according to the
  scale of the mesh (to be shown)

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Medial structures -gtSegments

• Given a set of subsegments , and their
  medial structures , the subsegments can be
  redefined with competing fronts
• The medial structures are the front at distance
  0.
• The boundaries of the new subsegments are the
  vertices of the mesh where the propagating fronts
  collide.

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The Algorithm

• Find eigenvalues and eigenvectors of mesh
• Most computationally-heavy task. Authors Use a
  referenced fast solver.

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The Algorithm Contd

• Input A region , initially the
  entire manifold
• Compute time scale
• Scale invariance using fiedler (first nonzero)
  eigenvalue

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Initialization

• Initialize medial structure
• Medial structure of R -
• Recursively add points which are farthest from
• If distance is farther than ,
  where D is the average distance of vertices to
  .

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Refinement Steps

• Refining Segments
• Via competing fronts
• Creating new medial
• structures
• Computing new ADDs
• Stopping when the ratio
• of vertex changes is
• low (0-5)

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Competing Fronts

• Using a heap, containing pairs (f,i), where f is
  a candidate face for front .
• Pairs are ordered and extracted by the maximum
  distance between vertices of f and medial
  structure

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Computing ADDs

• Updating ADDs, their average, and the farthest
  distance D for each updated (initialized)
  segments can be done in O(V), where V is the size
  of the region.
• Medial structure may be unstable due to low mesh
  resolution
• A tolerance is added
  around the minimum value.

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Results

• Pose Invariance

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Results contd

• Three different cuts of the segmentation
  hierarchy tree

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Results Robustness

• Robustness to noise
• Short circuit

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Large Handles

• The algorithm fails with large handles
• The diffusion distance is greatly altered

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More Examples
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Extracting 1-Skeletons

• Connecting the barycenters of segments

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Complexity of Algorithm

• Most difficult getting the eigenpairs
• Using a solver, linear in the number of
  eigenpairs acquired
• The worst case main loop is O(nlogn)

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Solidity of Paper

• The paper makes reasonable assumptions
• Concavities are natural segment boundaries
• Geodesic distances might not be sensitive enough
• Noise in the input greatly affects most
  algorithms
• Geometric noise
• Short circuits
• Poor mesh resolution

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Novelty of The Solution

• Diffusion distance in computer graphics is a very
  new and very useful topic.
• The paper gives a novel and simple application
• Usage of medial structures is relatively novel
• A generalization of Voronoi structures on
  surfaces
• Is very general and avoid pitfalls of previous
  algorithms.

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Soundness of Algorithm

• The algorithm uses diffusion distances, which
  makes it
• Robust to noise relying on global quantities
• Isometry-invariant
• Easy to reproduce, using spectral decomposition
• The algorithm fails on higher genus objects
• Spectral decomposition might be very expensive
  for very big meshes (gt500k)

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Evaluation of solution

• Many empirical proof of concepts
• Examples of all traits of the algorithm
• A time table
• No theoretical guarantees are provided for
• Convergence
• robustness

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Level of Writing

• Paper is clearly and concisely written
• Reference list seems complete
• Novel concepts are explained thoroughly
• The algorithm can be easily reproduced
• All parameters and formulas are supplied
• More visual examples of the steps of the
  algorithm are in order
• Mainly of ADDs and medial structures