Efficient Distance Computation Between NonConvex Objects

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Efficient Distance Computation Between Non-Convex
Objects

• By Sean Quinlan
• Reviewed by Mehboob A. Nazarani

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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

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Assumptions

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

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Basic Structure

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

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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.

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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

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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.

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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

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The Bounding Tree for an Object
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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

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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

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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.

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Execution Time

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

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Computing the Distance

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

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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

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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

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Relative Error

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

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Typical Configuration of Chess Pieces
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Search Size v. Relative Error