OBBTree

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OBBTree

• Han, Gabjong
• hkj84_at_postech.ac.kr
• POSTECH VR Lab
• 2006. 4. 12.

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Outline

• Introduction
• Previous Methods
• Hierarchical Methods
• Other Approaches
• Previous OBB Methods
• OBBTree Building
• Overlap Test for OBBs
• OBB vs. Other Volumes
• Implementation and Performance
• Robustness and Accuracy
• Performance
• Comparison with Others
• RAPID
• Cost Evaluation
• Conclusion

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Introduction - Concepts

• What is collision detection?
• Checks whether two objects overlap in space
• What is proximity?
• Information about the relative configuration of
  two objects

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Introduction - Concepts

• What is OBB?
• Synonym for oriented bounding box
• Rectangular bounding box at an arbitrary
  orientation in 3D space

OBB
AABB
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Introduction - Problems

• Problems in collision detection
• Model Complexity
• Unstructured Representation
• Close Proximity
• Accurate Contact Determination

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Introduction - Contributions

• OBBTree Contributions
• New hierarchical representation for OBB models
• Fast overlap test for two OBBs
• Comparison with other representations
• Robust and interactive implementation and
  demonstration

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Hierarchical Methods

• Simplest algorithms in collision detection
• Problems in hierarchical methods
• Poor performance in extreme cases
• Close proximity
• Multiple contacts

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Hierarchical Methods - Examples

• Examples
• Bounding volumes
• AABBs and Spheres
• Hierarchical structures
• Cone trees, k-d trees and octrees
• Sphere trees, R-trees
• Spatial representations
• BSP and multi-space partitions
• Space-time bounds or four-dimensional testing

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Other Approaches

• Theoretically efficient algorithms
• Problems of this kind
• Unclear practical utility
• Limited models
• Limited motion
• Slow performance

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Previous OBB Methods

• Used to speed up ray-tracing and other
  interference computation
• Issues in OBBs
• Computing tight-fitting OBBs
• Testing two boxes for overlap

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Previous OBB Methods- Tight-fitting Issues

• Minimal enclosing box algorithm
• O(n3)
• Simple Incremental algorithm
• O(n), but constant factor is very high
• Using Hierarchical model
• Cannot be applied for large unstructured models

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Previous OBB Methods- Overlap Test Issues

• Edge by edge intersection test
• Linear programming algorithm
• General purpose test

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OBBTree Building OBB Data Structure

• Midpoint
• C
• 3 basis vectors
• bu, bv, bw
• 3 half-dimensions
• hu, hv, hw

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OBBTree Building Covariance Matrix

• What is covariance matrix?
• A matrix of covariances between elements of a
  vector

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OBBTree Building

• How to get the midpoint of OBB?
• Assume the centroid of all polygons as midpoint
• How to get the orientation of OBB?
• Using the covariance matrix
• Eigenvectors of the matrix is basis

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OBBTree Building- Simple Approach

• Triangulate all polygons
• Obtain the centroid with all triangles
• Compute the covariance matrix
• Get the basis using the eigenvectors of the
  matrix
• Compute maximum and minimum extents
• Get the center and the half-dimensions of the OBB

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OBBTree Building

• Problem with Simple Approach
• Sampling problem
• Interior vertices
• Dense collection of vertices
• Solution
• Using a convex hull
• Integrating over the surface of each triangle

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OBBTree Building - Building Steps of OBB

• Triangulate all polygons
• Compute the convex hull (added)
• Calculate the centroid (revised)
• Compute the covariance matrix (revised)
• Get the eigenvectors
• Compute the extreme veritces
• Take the points

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OBBTree Building

• Triangulate all polygons

• Compute the convex hull

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OBBTree Building

• Calculate the centroid
• The area of a triangle
• The centroid of the triangle
• c (pqr) /3
• The centroid of the entire convex hull

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OBBTree Building

• Compute the covariance matrix
• Get the eigenvectors
• Covariance matrix is diagonalizable

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OBBTree Building

• Compute the extreme vertices along each axis
• Take the points
• A midpoint is the mean of extreme values
• A half-dimension value is distance between the
  midpoint and one extreme value

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OBBTree Building - Method

• Top down methods of OBBTree Building

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OBBTree Building

• Building steps of OBBTree
• Begin with a group of all polygons
• Subdivide recursively until all leaf nodes are
  indivisible
• Subdivision rule
• Split longest axis to shortest one until it can
  split

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Overlap Test for OBBs - Concepts

• Separating axis theorem
• Two convex polytopes are disjoint iff there exist
  a separating axis orthogonal to a face of either
  polytope or orthogonal to an edge from each
  polytope

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Overlap Test for OBBs

• Separating axis test for two boxes
• T distance between center of A and B
• L separating axis
• Ai half-dimension
• ai basis vector

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Overlap Test for OBBs

• Possible separating axis
• Face to face, face to edge, face to vertex
• Each face can be a separating plane
• Edge to edge
• Two edges can create one plane
• Vertex to vertex, edge to vertex
• They can be expressed by other plane

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Overlap Test for OBBs

• Each box has three unique face normals and three
  unique edges
• 3 faces 3 faces
• 3 edges 3 edges
• 15 candidates for a separating axis

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Overlap Test for OBBs

• Projection intervals are disjoint iff
• Simplifed computation example

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Overlap Test for OBBs

• Degenerate OBBs
• An OBB become a rectangle or a line segment
• Much simpler test

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Overlap Test for OBBs

• OBBs with infinite extents
• One term should be removed by one infinity
  condition
• if a2 is infinite,

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Overlap Test for OBBs

• Performance of overlap testing algorithms
• Two previous algorithms are optimized for general
  convex polytopes
• All these algorithms are much faster than 144
  edge-face tests
• New algorithm is about 10 times faster than
  previous algorithms

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OBB vs. Other Volumes

• What is Hausdorff distance?
• distance from set A to set B is a maximin
  function, defined as

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OBB vs. Other Volumes

• Tightness
• Diameter
• Aspect ratio

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OBB vs. Other Volumes Analytical Performance
Comparison

• Aspect ratio and volume changes in hierarch

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OBB vs. Other Volumes - First experiment

• Test for parallel close proximity
• Use two concentric spheres
• Change radius of sphere
• One is 1 and another is 1e

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OBB vs. Other Volumes- Second experiment

• Test for point close proximity
• Use two same-size spheres
• Change distance of e

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OBB vs. Other Volumes

• Analysis
• High curvature
• No advantages for OBBs
• Coarse tessellation
• Have to traverse entire hierarchies
• No advantages for OBBs
• Quadratic convergence property is useful in
  parallel close proximity

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Robustness and Accuracy

• Robustness
• Unstructured polygonal method
• Degenerate polygons
• No adjacency information
• No special cases for overlap test
• Easily implementable in hardware
• Accuracy
• Numerically stable
• Error margin guards missing intersection

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Performance

• SGI Reality Engine
• 90 MHZ R8000 CPU, 512MB

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Performance - Pipe scenario

• Smaller object is entirely embedded
• Long thin triangles cannot efficiently
  approximated in general

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Performance - Torus scenario

• The spikes prevent large bounding boxes
• The dimples provide concavities
• The wrinkles and the twisting make it impractical
  to decompose into convex polytopes

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Comparison with Others

• No standard benchmark available
• Environments
• Based on line-stabbing, AABB and sphere tree
  algorithms
• On SGI Indigo2 Extreme
• About 10000 polygon models
• About 1/71/5 a second
• Compute all the contacts
• Result
• One order of magnitude improvement

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RAPID

• Rapid and Accurate Polygon Interference Detection
  (RAPID)
• Implementation of OBBTree

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Cost Evaluation - Equation

• T Nv Cv Np Cp
• T total cost function
• Nv number of bounding volume pair overlap test
• Cv cost of testing a pair of bounding volumes
  for overlap
• Np the number primitive pairs tested for
  interference
• Cp cost of testing a pair of primitives for
  interference

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Cost Evaluation - Result

• Cv is one-order magnitude slow
• Nv is asympototically low
• Np is asympototically low
• It is asympototically fast for close proximity

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Conclusion

• OBBTree
• A hierarchical data structure for rapid and exact
  collision detection between polygonal models
• Efficient and fast
• Accurate and robust

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Reference

• S. Gottschalk, M. C. Lin, and D. Manocha,
  "OBBTree A Hierarchical Structure for Rapid
  Interference Detection, inProceedings of the
  ACM SIGGRAPH Conference,1996, pp. 171-180.
  (Errata)
• RAPID Homepagehttp//www.cs.unc.edu/geom/OBB/OBB
  T.html

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QnA