Modeling Fuzzy Regions by Delaunay Triangulation

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Modeling Fuzzy Regions by Delaunay Triangulation

• Hechen Liu
• smartlhc_at_ufl.edu

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Outline

• Fuzzy phenomena in nature
• What are fuzzy set theory?
• What are fuzzy spatial data types?
• How to represent fuzzy regions?
• Comparison of approaches
• Delaunay triangulation approach

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The Life looks Simple
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However, it is not as simple as we have imagined
Mountain or Valley?
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U.S winter temperature zones
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Why Fuzzy?

• Many spatial objects cannot be described with
  precision.
• They do not have a sharp boundary, or the
  boundary and the interior cannot be
  differentiated.
• Natural, social, or cultural phenomena can be
  fuzzy
• Examples temperature zones, vegetated area,
  English speaking regions

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Classical Set Theory
The Set A is characterized by a membership
function
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Fuzzy Set Theory

• A has no sharp bolder line. It is a fuzzy subset
  of X, characterized by a membership function
• is the element xs degree of membership
  in A

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Fuzzy Spatial Data Types
fuzzy points
points
fuzzy lines
lines
regions
fuzzy regions
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Representing Fuzzy Regions

• Fuzzy region is the most commonly seen spatial
  data type in nature
• How to represent the indeterminate boundary?
• How to represent the transition of membership
  values?
• Computer systems only provide finite resolution,
  how to handle infinite number of values?

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Models based on 3-value logic

• VASA (Vague Spatial Algebra) (Schneider 1997)
• Egg-Yolk Approach (Cohn and Gotts 1996a, b)
• Broad Boundary Approach (Clementini and di Felice
  1996)
• Rough-Set Model (Roy and Stell 2001)

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Compare with Other Models

• Alpha-cuts Model
• Problem
• the part with higher membership value is always
  surrounded by the part with lower membership
  value
• stepwise jump

• Represent a fuzzy region F as the regular crisp
  set of points whose membership values in F are
  greater than or equal to alpha. The alpha-level
  regions are nested.
• If we selected membership values as
• then,

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Triangulation approach

• The idea comes from height interpolation
• Set of data points A ? R2
• Height (p) defined at each point p in A
• How can we most naturally approximate height of
  points not in A?

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Why Delaunay Triangulation?

• Some triangulations are better than others
• Avoid skinny triangles, i.e. maximize minimum
  angle of triangulation

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Delaunay Triangulation in representing Fuzzy
regions

• First, we have a set of points representing the
  boundary and internal points as input. Each point
  is of type
• Then, we perform a Delaunay triangulation on the
  point set.
• The membership value of any point inside the
  simple fuzzy region is calculated from a linear
  interpolation of the membership values at
  vertices to which the point belongs.

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Steps of Delaunay Triangulation

• Creating a Delaunay triangulation from the input
  vertices
• Inserting missing line segments from the boundary
  and deleting the Delaunay edges that overlap them
• Removing triangles at concavities and holes
• Adding more points in order to improve the
  quality of the triangulation

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Within a single triangle
Given
Whats the membership value of P1 and P2?
Through linear interpolation, we can get
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Summary of Delaunay Triangulation

• Advantages
• Can represent the continuous transition of
  membership values
• Only boundary, flat areas, and a few additional
  points are stored, a save for space
• Unique representation
• Disadvantage triangulation takes a lot of time!

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Questions
?
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Thank you!