Isosurface Extractions II

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Isosurface Extractions (II)
3D Isosurface
2D Isocontour
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Contour Propagation
Basic Idea Given an initial cell that contains
isosurface, the remainder of the isosurface can
be found by propagation
Initial cell A Enqueue B, C Dequeue
B Enqueue D Until the FIFO queue becomes
empty
Breadth-First Search
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Challenges
Need to know the initial cells!
The Key Problem For any given isovalue C, we
need to find the initial cells (seed calls) to
start the propagation
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Solutions

• Extrema Graph (Itoh vis95)
• Seed Sets (Bajaj volvis96)

Problem Statement Given a scalar field with a
cell set G, find a subset S G, such that for
any given isovalue C, the set S contains initial
cells to start the propagation. We need search
through S for an arbitrary, but S is usuall much
smaller than G.
We will talk about extrema graph
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Extrema Graph

• The main idea
• There is at least one extreme point (local min or
  max) inside a closed isosurface, unless there is
  no interior point inside the isosurface
• There is at least one boundary cell intersected
  by an open isosurface

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Extrema Graph

• Idea 1 the arcs connect the extrema points will
  intersect all isocontours (except the open ones)
• Idea 2 the open isocontours will intersect with
  boundary cells

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Seed Set Finding

• Processing the data with the following 3 steps
• Extraction of extreme points
• Construction of an extrema graph
• Arcs that connect extreme points
• Collection of boundary cells

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Extreme Point

• Definition An extreme point is a grid point
  whose scalar value is maximum or minimum in all
  cells that share the grid point

0
4
3
O(N) to find Extreme points Need to visit all
cells
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8
7
11
9
5
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Not a exteme point
extreme point
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Constructing Extreme Graph
E2
E1
Extreme Graph E, A E extrema points
A Arcs conneccts E
a1
a2
a3
E3
E4
a5
An arc consists of cells that connect extrema
points (we also store min/max of the arc)
a4
E7
a7
E5
a6
E6
E8
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Construct Extreme Graph

• You want to minimize the number of cells included
  in the arcs
• Generating straight lines between close extreme
  points
• Polygonal cell traversal

(1)
(2)
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Boundary Cells

• Only the extrema graph is not enough

Some open isocontours will not intersect with
extreme graph arcs
Include all Boundary Cells!!
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Extract Isosurfaces

• Algorithm
• Given an isovalue
• Search the arcs of the extrema graph (to find the
  arcs that have min/max contains the isovalue
• Walk through the cells along each of the arcs to
  find the seed cells
• Start to propagate from the seed cells
• Search the cells along the boundary and find seed
  cells from there
• Propagate open isocontours

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Better Algorithm Needed

• The above algorithm has the following problems
• Processing cost is in proportion to the number of
  extreme points, which can be large
• There can be a large number of boundary cells

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Extreme Skeleton

• Extreme Skeleton A set of cells that connects
  all extreme points while intersecting with all
  disjoint parts of isosurfaces (including open
  isosurfaces)
• Based on a Volume Thinning algorithm

Volume thinning
Image thining
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Volume Thinning

• Start from the entire volume cells (default seed
  set)
• Eliminating cells that will not affect the
  connectivity of its adjacent cells
• Never eliminate cells that touch extreme points
• Topology preserving (considering voids or through
  holes)

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Volume Thinning
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Algorithm

• Classify each cell as c0, c1, c2, c3, c4 (assume
  tetrahedral cells) C0 means eliminated cells,
  and Ci means there are I adjacent non-eliminated
  cells
• Start to eliminate c1 cells, then c2, c3 based on
  certain criteria
• Every time a cell is eliminated, update the
  adjacent cells from Ci to Ci-1
• Loop until no more cells can be eliminated

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Criteria for Elimination

• We can certainly eliminate C1 cells
• C2 cells we try to see whether we can propagate
  from Ci to Cj (the adjacent cells) without going
  through Ck (the cell in question)

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Criteria for Elimination

• C3 cells Dont want eliminate cells on the
  boundary of void or through-hole that will break
  the connectivity of adjacent cells

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Criteria for Elimination
Also, remember to remove cells that form bubbles
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Prick the bubbles
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Example of Thinning