Collision Detection

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Collision Detection

• Nov. 6. 2001
• SonOu Lee

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Contents

• Introduction
• Basic algorithm and its problems
• Categorizing previous methods
• State of the art
• Reference
• A Survey on Collision Detection, ???, VR Lab
  Tech Memo 98-18, 1998.
• Real-Time Rendering by Tomas Möller and Eric
  Haines
• CS492 Textbook

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Introduction

• What is collision?
• a status that two or more objects have occupied
  or will occupy the same space at the same time
• Why collision detection is important?
• real world vs. virtual world
• Which research domain deals collision detection?
• robotics, computational geometry, computer
  graphics, VR

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DDR?

• Collision Detection
• detects whether two objects collide
• Collision Determination
• finds the actual intersections between a pair of
  objects
• See Real-Time Rendering for more information
• Collision Response
• determines what actions should be taken in
  response to the collision of two objects

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• Collision Detection algorithm should
• achieve interactive rates
• handle polygon soups
• can undergo rigid-body motion
• provide efficient bounding volume
• have static simulation methods

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Basic Algorithm

• Comparison time
• for objects
• NC2NM
• N moving, M static objects
• for features
• K2
• K features per objects
• total comparison time
• (NC2NM) K2

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Problems of Basic Algorithm

• 3 types of weaknessHubbard95
• fixed-time step weakness
• related with ?td
• all-pairs weakness
• related with of objects
• pair-processing weakness
• related with of features

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Categorizing previous methods

• In regard to the CS492 textbook
• broad / narrow / single phase
• In regard to pruning methods VR Lab Tech Memo
  98-18
• global pruning
• local pruning
• time-domain pruning
• Other categorizing features
• accuracy
• speed
• generality

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Global Pruning

• Reducing the number of comparisons for objects
• reducing N of (NC2NM) K2
• Related with all-pairs weakness
• Previous work
• bounding box test
• sweep and prune Barff92Ponamgi95
• space subdivision
• uniform space subdivision Turk88 Kim97a
• octree decomposition Moore88Shaffer92Smith95
  

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Local Pruning

• Reducing the number of comparisons for features
• reducing K of (NC2NM) K2
• Related with pair-processing weakness
• Previous work
• BSP Thibault87
• superquadrics Sclaroff91
• regular grid Garsia-Alonso94
• closest feature Lin93
• OBB tree Gottschalk96b
• hierarchical sphere tree Hubbard95
• hierarchical convex tree Kim97b

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Time-domain Pruning

• Reducing the for given number of comparisons time
• Related with fixed-timestep weakness
• Previous work
• space-time bounds Hubbard96
• with predefined time-variable function
• root finding Canny86
• interval analysis Duff92Snyder93
• without predefined time-variable function
• using bound on velocity and distance for
  increasing ?td Culley86
• 4D structureFoisy90

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State of the Art

• Sweep and Prune Baraff92Ponamgi95
• Incremental Algorithm Lin93
• OBBTree Gottschalk96b
• Space-Time Bounds Hubbard96
• Hierarchical Sphere Tree Hubbard95
• k-DOPTree Klosowski98

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Sweep and Prune

• Known as sort and sweep or sweep and prune
• Using AABBs(Axis-Aligned Bounding Box)
• dynamic AABB
• fixed AABB
• Dimension reduction
• if two bounding boxes collide in 3D, then their
  orthogonal projections on the x,y, and z axes
  must overlap

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• Algorithm overview
• project each AABB onto the x,y, and z axes
• projection results in intervals
• or project each AABB onto the x-y,y-z, and x-z
  planes
• projection results in rectangles
• determine overlaps of these intervals
• a pair of bounding boxes can overlap iff their
  intervals overlap in all three dimensions
• general sorting method
• O(NlogN)
• keep sorted list and using insertion sort or
  bubble sort
• expected O(N)

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Incremental Algorithm

• Lin-Canny Closest Feature Algorithm
• Preprocessing
• Voronoi Region
• a Voronoi region associates with a feature is a
  set of points closer to that feature than any
  others
• Cell
• structure for Voronoi region of single feature
• has a set of constraint plane and pointers to the
  neighboring cells.

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• Closest feature test
• finding closest feature pairs based on Voronoi
  regions
• start with a candidate pair of features
• check whether the closest points line on these
  features
• if true
• calculate distance between two features and
  determine collision
• else
• find neighboring feature of one or both
  candidates and try again

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• Example

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OBBTree

• OBB (Oriented Bounding Box) Tree
• binary tree structure
• known as fastest algorithm practically
• OBB was originally used extensively to speed up
  ray tracing and interference detection
  computationArvo89

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• Building an OBBTree
• constructing tree structure
• top-down approach
• split longest axis of a box with a plane
  orthogonal to one of its axes
• making OBBs
• Simple but erroneous method

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• Using convex hull to get more correct result
• Consider only surface primitives

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• OBBT Example

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• Fast overlap test for OBB Tree
• general method
• 144 times comparison (12edges x 6 faces x 2
  boxed)
• separating axis test
• separating axis theorem Gottschalk96a
• need 15 times separating axis test

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k-DOPTree

• k-DOP?
• Generalization of AABB
• Why k-DOP?
• Has both a faster BV/BV overlap test than the OBB
  and a tighter fit
• How can determine value k?
• Scenario-dependent
• k 18 gave the best execution times
  experimentally

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• Hierarchy Building
• Top-down approach
• Binary tree
• Subdivide criteria
• Min Sum
• Min Max
• Splatter
• Longest Side

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• Updating the k-DOPTree
• Due to a rigid-body motion
• Approximation method
• vertices locations at the corners of the original
  k-DOPs are transformed and used
• Hill climbing method
• Store convex hull information and use local
  updating

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Space-Time Bounds HST

• Overview Hubbard95,96

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• Broad Phase
• upper bound on the objects accelerations to
  build a set of space-time bounds
• gives a conservative estimate of where an object
  could be in the future
• address the fixed-timestep weakness
• Narrow Phase
• progressively refines the accuracy
• sphere-tree
• address the pair-processing weakness

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Space-Time Bounds
0
y
y
0
1
1
t
t
x

• This means that a 3D bound on objects position
  at t is a sphere of radius centered
  at
• parabolic horn shape

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• Simplify parabolic horn shape
• hypertrapezoid
• encloses an objects parabolic horn
• consists of six 4D faces, one for each 3D face of
  its cross-sectional cubes
• each constant-t cross section of a 4D face is an
  isothetic 3D square
• 4D Cutting Plane
• If acceleration is limited to only certain
  direction

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• Space-Time Bounds are conservative
• if two objects collide at simulation time t,
  their space-time bounds must intersect at some ti
  ? t
• Check intersection between two space-time bounds
• let Bi be a space-time bound consisting of
  hypertrapezoid Ti and cutting plane Pi and assume
  that T1 and T2 are disjoint at t0
• intersection between f1 of T1 and f2 of T2
• intersection between f1 of T1 and P2
• intersection between P1 and P2

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Hierarchical Sphere Tree

• Time-critical collision detection
• negotiated graceful degradation
• trade accuracy for speed
• maintain real-time performance
• approximation of objects real surfaces
• Requirement for preprocess
• must be automatic
• each hierarchy is effective search structure
• generate hierarchies in which each level fits the
  object as tightly as possible

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• Medial-axis
• skeleton or stick figure representation of a
  two-dimensional object.
• Medial-axis surface
• extension of medial-axis to 3D
• Building medial-axis surfaces
• not simple
• only exact algorithm by Hoffmann90
• complicate and limited to CSG objects
• approximating medial-axis surface
• using voronoi diagram

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• Building process
• places the set of points P on the polyhedrons
  surface using the algorithm by Turk91
• building voronoi diagram for the points
• identify voronoi vertices
• record vertices adjacency
• Generating spheres
• using each voronoi vertex and its forming
  vertices

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• Merging spheres
• building a hierarchy
• optimization problem
• greedy method (not optimal)
• choose the minimum cost merger which most
  preserves the levels tightness around the
  polyhedron
• cost function
• Hausdorff distance
• maxaminb d(a,b)
• no algorithm for concave polyhedra
• approximating Hausdorff distance

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• Examples

Octree
Medial axis
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Speed up Trick?

• Not all application needs detailed collision
  detection
• Especially in games domain
• Collision detection with rays
• Environment with BSP tree
• Hierarchical collision detection method
• Reject with cheap computation

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General HierarchicalCollision Detection

• Hierarchy building
• Bottom-up
• BOXTREE, HSBT
• Top-down
• OBBT, k-DOPT
• Hierarchy can be created lazily -gt time saving
  for off-line calculation such as path planning ,
  non-real-time animation, and more
• k-ary tree
• Binary tree is welcome
• minimized work to be done when traversing a
  single path from the root to the worst-case leaf
  Klosowski
• easy to compute hierarchy
• Well balanced trees always provide good
  performance?
• No!

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Conclusion

• There is no perfect collision detection algorithm
  for all domain
• Tight bounding volume vs. fast BV overlap test