OBBTree: A Hierarchical Structure for Rapid Interference Detection

1
OBBTree A Hierarchical Structure for Rapid
Interference Detection

• Gottschalk, M. C. Lin and D. Manocha
• Department of Computer Science, University of N.
  Carolina, Chapel Hill.
• Presenter Tao Ju
• Spring, 2001

2
Introduction

• OBBTree A tight-fitting hierarchical structure
  and an efficient overlap-testing algorithm for
  interference detection amongst complex models
  undergoing rigid motion.
•  
• Bounding Volume OBB (Oriented Bounding Box)
• Overlap test algorithm for OBBs Separating Axis
  Algorithm

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Computing Tight Fitting OBB O(n Log(n))

• Triangulate polygons. ( O(n) )
• Compute the convex hull of the vertices of the
  triangles. ( O(n Log(n)) )
• Orientation of OBB Covariance Matrix. ( O(n) )
• Position and dimension of OBB. ( O(n) )

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Compute Orientation of OBB

• Let the ith triangle of the convex hull have
  vertices pi, qi, and ri. Let the number of
  triangles in the convex hull be n.
• The area of ith triangle is denoted as Ai,
• Ai (pi - qi) (pi - ri) / 2
• The surface area of the convex hull is denoted as
  AH,
• AH ?i Ai

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Compute Orientation of OBB

• The centroid of the ith triangle is denoted by
  vector mi,
• mi ( pi qi ri ) / 3
• The centroid of the entire convex hull is a
  weighted mean of the triangle centroids, denoted
  by mH,
• mH ?i Aimi / AH

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Compute Orientation of OBB

• The elements of the covariance matrix Cnn are
  defined as
• Cnn
• The three eigenvectors of C will be mutually
  orthogonal. After normalization, the three
  eigenvectors become axes of OBB.

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Compute Dimension of OBB

• Find the maximum and minimum extents of the
  original triangle set along each axis, and size
  the OBB.

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Computing Tight Fitting OBB

• Why using covariance matrix?
• Second order statistics summarizing the data
  points.
• Why using convex hull?
• Avoid arbitrary influences from interior
  vertices of the model.
• Why computing areas?
• Infinitely dense sampling.

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Constructing a hierarchical OBBTree Top-down
Approach

• Subdivision method Split the longest axis of an
  OBB with a plane orthogonal to one of its axes,
  partitioning the polygons according to which side
  of the plane their center point lies on.
• The subdivision coordinate along that axis was
  then chosen to be that of the mean point of the
  vertices.

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OBBTree Construction

• Building the OBBTree Recursively partition the
  bounded polygons and bound the resulting groups.

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OBBTree Construction

• Running time Similar to Quicksort.
• Fitting an OBB to n triangles and partitioning
  into two subgroups O(n Log(n))
• Levels of recursion O( Log(n))
• Total computation time O(n Log2(n))

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Fast Overlap Test For OBBs

• Previous algorithms
• Simple test (144 edge-face tests)
• Linear programming
• Closest features computation.
• Performance Two orders of magnitude slower than
  checking two spheres for overlap.

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Fast Overlap Test For OBBs

• Separating axis theorem Two convex polytopes are
  disjoint iff there exists a separating axis
  orthogonal to a face of either polytope or
  orthogonal to an edge from each polytope.
• Separating Axis An axis on which the projections
  of two polytopes dont overlap.

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Fast Overlap Test For OBBs

• Testing 15 axes is sufficient for determining
  overlap status of two OBBs.

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Fast Overlap Test For OBBs

• Radius of interval
• rA ?i aiAiL
• The intervals are disjoint iff
• TL gt rA rB

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Fast Overlap Test For OBBs

• The computation simplifies when L is a box axis
  or cross product of box axes. Worst case run
  time 200 operations.
• ( computed on HP 735/125 )

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Comparison Of Bounding Volumes

• Bounding volumes
• Spheres
• AABBs ( Axis Aligned Bounding Boxes )
• OBBs ( Oriented Bounding Boxes )

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Comparison Of Bounding Volumes

• Cost function of hierarchical structure
• T Nv Cv Np Cp
• where
• Nv of bounding volume pair overlap tests
• Cv cost of testing a pair of bounding volumes
  for overlap,
• Np of primitive pairs tested for
  interference,
• Cp cost of testing a pair of primitives for
  overlap.

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Comparison Of Bounding Volumes

• Cv is one-order of magnitude slower than that for
  sphere trees or AABBs.
• Primary advantage for OBB Low Nv and Np
• In general, OBBs can bound geometry more tightly
  than AABBTrees and sphere trees.

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Comparison Of Bounding Volumes

• Define tightness, ?, of a bounding volume, B,
  with respect to the geometry it covers, G, is Bs
  Hausdorff distance from G, i.e.
• ? maxb ming dist(b, g) b?B, g?G
• Define diameter, d, of a bounding volume with
  respect to the bounded geometry is the maximum
  distance among all pairs of enclosed points on
  the bounded geometry,
• d maxg,h dist(g, h) g, h ?G

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Comparison Of Bounding Volumes

• When bounding low curvature surfaces, AABBTrees
  and spheres have?with linear dependence on d,
  whereas OBBTrees have?with quadratic dependence
  on d.
• To cover a surface patch with volumes to a given
  tightness, if OBBs require O(m) bounding volumes,
  AABBs and spheres would require O(m2) bounding
  volumes.

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Comparison Of Bounding Volumes

• AABBs vs. OBBs Approximation of a Torus This
  shows OBBs converging to the shape of a torus
  more rapidly.

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Experiment Of Bounding Volumes

• Parallel close proximity every point on each
  surface is close to some point on the other
  surface.

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Experiment Of Bounding Volumes

• Point close proximity two nonparallel surfaces
  patches come close to touching at a point.

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Comparison Of Performance

• With AABBs Improvement from 1/7 1/5 of a
  second for computation of all contacts between
  models to 1/75 1/25 of a second. ( on SGI
  Indigo2 Extreme ).
• With spheres Improvement of one order of
  magnitude.

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Interference Detection In Action

• RAPID
• V-COLLIDE
• H-COLLIDE

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Interference Detection In Action

• Interactive Interference Detection on Complex
  Interweaving Pipeline 140, 000 polygons each
  Average time to perform collision query 4.2 msec

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Interference Detection In Action

• Interactive Interference Detection for a Torpedo
  on a Pivot Structure Torpedo has 4780
  triangles Pivot has 44921 triangles Average
  time to perform collision query 100 msec

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Interference Detection In Action

• Interactive Interference Detection for a Complex
  Torus Torus has 20000 polygons Environment has
  98000 polygons Average time to perform collision
  query 6.9 msec

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Conclusion

• New efficient algorithms for hierarchical
  representation using tight-fitting OBBs.
• Use of a separating axis theorem to check two
  OBBs for overlap in about 100 operations on
  average.
• By comparison with AABBs, show that for many
  close proximity situations, OBBs are
  asymptotically much faster.