Isosurface Extraction and Visualization of Irregular Datasets

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Isosurface Extraction and Visualization of
Irregular Datasets

• Alisa Neeman

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Agenda

• Review
• Isosurface Extraction
• Delaunay Triangulation
• Only Finished Literature Search
• Prototype Interface

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Review
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What is an Irregular Dataset?

• Composed of one or more types of data. Generally,
  the cell vertices are Not located on a regular
  grid.
• In addition to x,y,z values, vertices have an
  additional value representing some physical
  feature such as temperature or density

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What Is Isosurface Extraction?

• If the data is composed of tetrahedrons or voxels
  (cubes), we can extract an isosurface.
• We first select an isovalue which is a number
  that falls in the range the range of our physical
  values. For example, temperature values might
  range from 0-255, and we select 150.

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Isosurface Extraction, contd

• Then we check the values of our cell vertices. If
  two endpoint values straddle the isovalue, then
  an isosurface passes through the cell.
• Using interpolation, we can determine how the
  plane passes through the cell and it will make a
  polygon we can display.

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Delaunay Triangulation

• Why triangulate?
• Creates grid proportional in size to original
  dataset (the density of the grid follows the
  proximity of the data points a regular grid
  introduces storage overhead for a sparse areas)
• Fig.1 Los Angeles Basin1

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Further Motivation

• Can be used for finite element analysis
• The finite element method is a numerical
  technique which enables general numerical
  solutions to be obtained for many problems in
  science and engineering. A given structure is
  discretised into a number of elements and
  equations of motion are formulated using
  knowledge of the behaviour of each individual
  element. 2

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Where Else Triangulation Is Used

• Computational Fluid Dynamics
• Modeling and simulation of fluid behavior
• Traditionally used in aerospace and nuclear
  industries
• Even for medical applications simulating
  bioeletric field created during defibrillation of
  someones heart

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Multi-Component Wing
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Background Review What is a Delaunay
Triangulation?

• Recall from Professor Yins lecture A Delaunay
  Triangulation of vertices V
• Let u and v be any two vertices of V. The
  circumcircle of edge uv is any circle that passes
  through u and v. The edge uv is Delaunay iff
  there exists an empty circumcircle of uv. 1

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In 3 Dimensions..

• A triangulation of V is a set of 3-simplices,
  whose interiors do not intersect each other, and
  whose union is the convex hull of V (if every
  3-simplex intersects V only at its vertices).
• For the triangulation to be Delaunay we must
  ensure for any four points that
• its circumsphere is empty.

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Really Nice Properties

• In 2D this means we get the nicest triangles
• Minimizes cases of very small or very large
  angles
• Long, thin slivers have bad effects for
  numerical methods
• Roundoff error
• Slow convergence when using iterative methods
• IN 3D WE GET NO SUCH GUARANTEES BUT WE DO GET
  NON-INTERSECTING INTERIORS - WHICH IS NECESSARY
  FOR VISUALIZATION

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So How do you do it in 3D?

• Found only secondary sources original papers so
  old they are no longer available.
• But the main idea is this when you add a vertex,
  you check whether it falls within the
  circumsphere of the surrounding triangle faces.
  You delete all tetrahedrons whose circumspheres
  encircle the new vertex, and rebuild them to
  include the new vertex.

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Details From Schmidt et al.

• struct tetrahedron
• int point4 //four corners
• int neighbor_tetra4
• float origin3 //x,y,z
• float radius

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Algorithm Pseudocode

• For i 1 to number_of_points
• Determine tetrahedra to be deleted DTi i1,dt
• For i 1 to dt
• determine neighbors of deleted tetrahedra
• NT_DTjj 1 to nt_dt

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Algorithm, contd

• For j 1 to nt_dt
• Connect Point i to the three points of NT_DTj
  to create new tetrahedron
• Find the 4 new neighbors of the
• new tetrahedron (a shared face)
• Reorganize list of tetrahedra.

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Associated Trivia

• Biggest cost Finding which tetrahedra should be
  deleted. (Number of tetrahedra keeps growing)
• Incorrect results due to floating point round off
  error

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Interface prototype showing wireframe
triangulation and convex hull
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References

• 1 Shewchuck, Jonathan Richard, Lecture Notes on
  Delaunay Mesh Generation, 1999.
• 2http//members.tripod.com/PhilShorter/FEbasics
  .html
• 3 J.A. Schmidt, C.R. Johnson, J.C. Eason, and
  R.S. MacLeod, Applications of Automatic Mesh
  Generation and Adaptive Methods in Computational
  Medicine, Institute for Mathematics and Its
  Applications, Volume 75 Modeling, Mesh
  Generation, and Adaptive Numerical Methods for
  Partial Differential Equations