A Linear Order Algorithm for Generating Isolines

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• A Linear Order Algorithm for Generating Isolines
•  
•  by
• Sanjeeb Nanda
• SDS International Inc., Advanced Technologies
  Division
• 3403 Technological Avenue, Suite 7
• Orlando, FL 32817

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Traditional Techniques for Generating
IsolinesThe Basis

• A Convex Hull on a set of DTED vertices

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A Modification to the Basis

• A Convex Hull on a set of DTED vertices

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The Results of Traditional Techniques
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The Drawback of Traditional Techniques

• Pseudo-Code for a Modified Convex Hull Algorithm
  to Generate Isolines
• Let P be a given set of points and S an empty
  stack
• Let p1 be the point in P having minimum
  y-coordinate
• Let p2, p3, , pN P p1, sorted by their
  angle with p1 in counterclockwise order
• TopS ? 0 Push (S, p1) Push (S, p2) Push (S,
  p3)
•  
• for i ? 3 to N do
• while angle between points BelowTopS,
  TopS, pi makes non-left turn, Push (S, pi)
•  
• return S
• Disadvantage This is a O(Nlog2N) algorithm where
  N number of DTED vertices.

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The Drawback of Traditional Techniquesin
Practical Terms

• The previously shown technique requires
  approximately 2 minutes to generate the isolines
  for an area spanning 15 km x 15 km whose
  corresponding elevation data is represented by a
  grid of 513 x 513 DTED vertices on a platform
  with 1 GB system memory and a 3.0 GHz system
  processor.
• That is slow and unacceptable for any field
  application!
• Prognosis
• A faster method with O(N) is required to solve
  this problem!

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A New Technique Using Graphics Hardware

• Basis
• GL provides terrain lighting with considerable
  realism when give the normal map for the grid of
  DTED vertices corresponding to the texture of the
  terrain overlying it. The normal map is comprised
  of a vector corresponding to each vertex in the
  grid that intuitively specifies the average slope
  of the terrain at that point with respect to its
  neighbors.
• Motivation
• Can we create isolines employing such lighting
  furnished by GL/Graphics card?

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A Normal Vector

• Nothing more than a simple cross product.

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Computing a Normal on a DTED Vertex
Notice that the vertex is adjacent to six
vertices wi, 1 i 6. So we take the average of
the cross products (u x wi).
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Displaying Isolines
Consider a stratification of DTED vertices, where
each vertex is placed in a strata with a minimum
and maximum heights of hmin and hmax
respectively. Furthermore each vertex qualifying
into such a strata has its elevation reset to the
value (hmin and hmax)/2. Then we can visualize
the result of this process yielding stacks of
plateaus. The edges of these plateaus would have
a 90o drop to the plateau beneath it.
Furthermore, these edges would have a normal that
is not normal to the x-y plane, where we assume
the z axis points vertically upwards (akin to the
UTM coordinate system). Now if we situate a
point light at infinite distance above the
terrain, the normal for each vertex at the edge
of a plateau would show up as a dark facet. If
the edge vertices are relatively close together,
then we would get a well-delineated contour line.
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Illustration of Normal for Edge Vertex
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Illustrations of the New Technique
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Advantage of the New Technique
The new technique requires the computation of a
normal vector for each DTED vertex by determining
the average of at most 6 normals (to at most 6
neighbors respectively). As a result the process
is O(N), where N is the number of DTED
vertices. In practical terms this advantage
translates to a huge difference. It requires less
than 2 seconds to generate the isolines for an
area spanning 15 km x 15 km whose elevation data
is represented by a grid of 513 x 513 DTED
vertices on a platform with 1 GB system memory
and a 3.0 GHz system processor. In other words,
the new technique is approximately 120 times
faster than the older technique it supplants.
Furthermore such results can be dynamically
computed.
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Uses for this Technique

• 2D Contour 3D Terrain Visualization
• Considerable amount of mission planning takes
  place using 2D contour maps that provide accurate
  scale of representation and avoid the problems of
  foreshortening associated with 3D perspective
  views. So having a tool that associates 2D
  contour maps with their 3D topographical
  counterparts is very useful.