Topological Hierarchy Sorting Algorithm

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Topological Hierarchy Sorting Algorithm

• Patrice Arruda

Supervised by Dr. Michiel Smid
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Content

• Background problem
• Overall view of the topological hierarchy sorting
  algorithm
• Techniques
• Data Structures
• Implementation and quick demo

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Background

• Layered Manufacturing builds a freeform object
  layer by layer
• The author of the paper A topological hierarchy
  - sorting algorithm for layered manufacturing
  (written by S.H. Choi and K.T. Kwok) developed a
  hierarchy-sorting algorithm to simplify a
  particular problem
• Dr. Smid found a way to improve the algorithm
• The algorithm can be applied for the following
• Virtual simulation of LM
• Medical imaging
• Building objects made out of multiple materials

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Problem

• Building a hierarchy tree such that the tree
  follows the following property

The node u is the parent of the node v if the
polygon at v is nested immediately within the
polygon stored at u.
v
u
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Example
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Example
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Example
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Example
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The Algorithm

• Concept
• Each polygon is represented by a simple structure
• Sweep line technique
• A modified version of the Skip List to store
  edges that are crossing the sweep line
• When the new polygon P is discovered by the sweep
  line, we find the parent polygon for P

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The Algorithm (Continued)

• Click here to run the demo.

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Example of a Fake Parent

• When the sweep line discovers a new polygon, it
  will search for the closest top and bottom edges.
  Since these two edges are from the same polygon,
  the new polygon will have the parent P. However,
  this is not true. To fix this problem we
  introduce a technique called Internal and
  External Edges.

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Internal and External Edges

• Take an edge and look at its top. If the top of
  the edge is touching the outside area of the
  polygon, then this is an exterior edge. Otherwise
  the edge is an interior one.
• To determine whether the edge is interior or
  exterior Starting from the left most point, move
  the polygon counter clockwise. If point .x lt
  .x ,then this is an interior edge.
  Otherwise this is an exterior edge.

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Back to the Example of the Fake Parent

• We again search for a top and bottom edge. The
  top edge must be external whereas the bottom edge
  must be internal. The example shows the opposite
  case the parent of the new polygon is the parent
  of polygon P.

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All Possible Cases
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Keeping Track of Edges that are Intersecting the
Sweep Line Skip List

• Another way to look at it How to jump in a
  linked list.
• Insert, remove, and search method run in O(logn)
  expected time

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Modified Version of the Skip List

• Key Element for all the nodes in the skip list.
• The key is the end points of an edge
• The left most column nodes key are replaced with
  the topmost edge of the root polygon in a
  specific layer
• The right most column nodes key are replaced with
  the bottommost edge of the root polygon in a
  specific layer
• The search method is performed by test turns
• Two new search methods are added
  searchForTopEdge and searchForBottomEdge
• The delete method receives the point p and
  deletes any edge that contains point p

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Example of Modified Version of the Skip List
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And if We Rotate the Skip List...
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How to Move the Sweep Line

• We extract all the vertices of a polygon and
  assign each vertex an action Insertion, Removal
  and Update Point. Each vertex has a pointer to
  the edge that is incident. Each edge has a
  pointer to the polygon.
• We do this for all polygons, and place all the
  points in the array. Then we sort the array. The
  leftmost point of the array is the first point
  that the sweep line will touch.

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Dealing with Vertical Edges

• Vertical edges are a pain. We just ignore them
  (do not add them to the skip list). However, we
  must be careful with each vertex action.

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Compilation

• For each layer, the algorithm goes through two
  stages
• Preprocess
• Convert each contour to a polygon
• Extract all the vertices, add to an array A, and
  sort the array. This array will make the sweep
  line move.
• Create a root polygon to contain all the polygons
• Building the Hierarchy Tree
• Initialize the skip list
• Process each vertex of the array
• If the vertex v is the leftmost point of a new
  polygon, assign the parent to the new polygon by
  going through all the cases in slide x.
• If the vertex v is an update point, find the edge
  in the skip list that is incident to v, and
  replace it with the new edge.
• If the vertex v is a removal point, find all the
  edges in the skip list that are incident to v,
  and remove all of them.
• Running time O(nlogn) time.

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Implementation and Demo

• Implemented the algorithm in java using Eclipse
  IDE
• Tested the algorithm with a sample dataset of a
  human pelvis
• Click here for the demo

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Questions
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