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Efficient Distance Computation Between NonConvex Objects

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Title: Efficient Distance Computation Between NonConvex Objects


1
Efficient Distance Computation Between Non-Convex
Objects
  • By Sean Quinlan
  • Reviewed by Mehboob A. Nazarani

2
Overview
  • Model objects as the union of a set of convex
    components
  • Construct a hierarchical bounding representation
    based on sphere from this model
  • Simple search routine that uses the hierarchical
    bounding representation of each object and
    determines pairs of components to compare with a
    convex distance algorithm.
  • Compute the distance between pairs of convex
    components using preexisting techniques
  • Accept a relative error in the returned result

3
Assumptions
  • Underlying model is a surface representation
    consisting of a set of convex polygons
  • Collision not detected when one object completely
    contains another object.

4
Basic Structure
  • Bounding representation is based on Spheres.
  • Representation consists of a balanced binary tree
  • Each node of the tree contains a single sphere

5
Tree Properties
  • The Union of leaf spheres completely contains the
    surface of the object.
  • The sphere of each node completely contains the
    spheres of its descendant leaf nodes.

6
Idea Behind Bounding Representation
  • Leaf spheres closely approximate the surface of
    the object.
  • Interior nodes represent an approximation of
    descendant leaf spheres
  • Nodes determine the lower bound for the distance
    to any of the descendant leaf nodes

7
Building the Tree First Step
  • Cover surface with leaf nodes
  • Regular grid of equal-sized spheres covers the
    polygon with the center of each sphere lying in
    the plane of the polygon
  • Label each sphere with the polygon for which it
    was created.

8
Building the Tree Second Step
  • Build interior nodes through divide and conquer
    technique
  • Divide set of leaves into two subtrees
  • Build tree for each subset and combine them into
    a single tree by creating a new node with each
    subtree as children
  • Build subtrees through recursion until the set
    consists of single leaf node

9
The Bounding Tree for an Object
10
Splitting
  • No optimal methods for splitting known
  • Objective is to split set of leaf nodes into two
    subsets so that the bounding sphere will be small
  • Bounding Rectanguloid

11
Splitting, contd.
  • Determine Min and Max value for 3 coordinates for
    the position vectors
  • Select axes along which bounding box is longest
    and divide leaf nodes using the average value
    along these axes as the discriminating node
  • Build a tree for each subset
  • Determine a bounding sphere

12
Computing the Sphere
  • First Heuristic Method
  • Find bounding sphere that contains spheres of 2
    children nodes.
  • Works well near the leaves.
  • Second Heuristic Method
  • Directly consider the leaf spheres. Select a
    center by using the average position of the
    centers of the leaf spheres.
  • Examine each descendant leaf sphere to determine
    minimum radius
  • Works well closer to the root.

13
Execution Time
  • Depth logn
  • Expected execution time O(nlogn)
  • O(n2), worst case
  • Precomputation

14
Computing the Distance
  • d is the distance between two objects
  • Goal is to show
  • Objects are d distance apart, or
  • Objects intersect

15
General Outline of Algorithm
  • d 8
  • Examine the pair of nodes in DFS manner starting
    with the root nodes of the two trees
  • if the distance between the two nodes is ? d
  • then do nothing
  • else
  • Examine the children of the nodes

16
Algorithm, contd.
  • Case 1 Both nodes are from interior
  • Split one of the nodes into two children
  • Recursively split the pair
  • Case 2 One interior and one leaf node
  • Split the interior node
  • Recursively search the pair
  • Case 3 Two leaf spheres less than d distance
    apart
  • Compute the distance between the underlying
    polygon
  • Return new d

17
Relative Error
  • Relative Error ?
  • d such that d lt d
  • And d d lt ?d

18
Typical Configuration of Chess Pieces
19
Search Size v. Relative Error
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