Isosurface Extractions II

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Isosurface Extractions (II)
3D Isosurface
2D Isocontour
2
Isosurface cell search

• Isosurfaec cells cells that contain isosurface.
• min lt isovalue lt max
• Marching cubes algorithm performs a linear search
  to locate the isosurface cells not very
  efficient for large-scale data sets.

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Isosurface Cells

• For a given isovalue, only a smaller portion of
  cells are isosurface cell.
• For a volume with
• n x n x n cells, the
• average number of the
• isosurface cells is nxn
• (ratio of surface v.s. volume)

n
n
n
4
Efficient isosurface cell search

• Problem statement
• Given a scalar field with N cells, c1, c2, ,
• cn, with min-max ranges (a1,b1), (a2,b2), ,
• (an, bn)
• Find Ck ak lt C lt bk Cisovalue

5
Efficient search methods

• Spatial subdivision (domain search)
• Value subdivision (range search)
• Contour propagation

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Domain search

• Subdivide the space into several subdomains,
  check the min/max values for each subdomain
• If the min/max values (extreme values) do not
  contain the isovalue, we skip the entire region

Min/max
Complexity O(Klog(n/k))
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Range Search

• Organize the cells based on their min/max ranges
• Find cells that have ranges intersect with the
  isovalue

Global minimum
Global maximum
Isovalue
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Range Search by min/max sort

• Sort the cells based on their min/max values

Max gt isovalue
G1
Min lt isovalue
G2
Isosurface cells G1 G2

• Efficient intersection tests are needed

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Sweeping Simplicies

• Keep two lists min and max lists

0 0 0 0 0 0
0 0 0 0 0
Max
M5 M2 M6 M4 M1 M3 M7 M8
M11 M10 M9
Min
m5 m1 m6 m3 m8 m7 m2 m9
m11 m4 m10
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Sweeping Simplicies

• Given an isovalue, find the cells which have
  smaller minimum values, and then set the
  corresponding bits in the max list

1 0 1 0 1 1
0 1 0 0 0
Max
M5 M2 M6 M4 M1 M3 M7 M8
M11 M10 M9
Min
m5 m1 m6 m3 m8 m7 m2 m9
m11 m4 m10
isovalue
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Sweeping Simplicies

• Then starting from the first cell with maximum
  value greater than isovalue, check the bits

1 0 1 0 1 1
0 1 0 0 0
Max
M5 M2 M6 M4 M1 M3 M7 M8
M11 M10 M9
Min
m5 m1 m6 m3 m8 m7 m2 m9
m11 m4 m10
isovalue
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Sweeping Simplicies

• If the isovalue changes, only need to flip the
  bits between the old and the new isovalues

1 0 1 0 1 1
0 1 0 0 0
Max
M5 M2 M6 M4 M1 M3 M7 M8
M11 M10 M9
Min
m5 m1 m6 m3 m8 m7 m2 m9
m11 m4 m10
isovalue
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Sweeping Simplicies

• Hierarchical subdivision of the value ranges

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Range Search (4)
Span Space Instead of treating each cell as a
range, we can treat it as a 2D point at (min,
max) This space consists of min and max axes is
called span space
Any problem here?
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Span Space
What are the isosurface cells?
max
How to search them?
min
C
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Span Space Search (1)
With the point representation, subdividing the
space is much easier now. Search method 1
K-D tree subdivision (NOISE algorithm)

• K-d tree
• A multi-dimensional version of binary tree
• Partition the data by alternating between
• each of the dimensions at each level of the
  tree

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NOISE Algorithm (K-d tree)
Median point
Min
Construction
Max
left
right
min
max
up
down



One node per cell
min
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NOISE Algorithm (Query)
Median point

• If ( isovalue lt root.min )
• Prune the right subtree
• Check the left subtree

Min
Max
left
right
?
up
down



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NOISE Algorithm (Query)
Median point

• If (isovalue gt root.min)
• Check both subtrees, but now the left subtree
  guarantees to satisfy the min condition
• Dont forget to check the root s interval as
  well.

Min
Max
left
right
?
up
down



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Lattice subdivision of span space
Search Method (2) ISSUE
Complexity ?
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Interval Tree (Optimal)
Sort all the data points (x1,x2,x3,x4,. ,
xn) Let d x (mid point)
Interval Tree
n/2
We use d to divide the cells into three sets
Id, I left, and I right (recursively) Id
cells that have min lt d lt max I left
cells that have max lt d I right cells that
have min gt d


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Interval Tree
Interval Tree
Within Id, We keep two sorted lists - One by
cell min values in ascending order (AL)
- The other by cell max values in descending
order (DR)


Id cells that have min lt d lt max I
left cells that have max lt d I right cells
that have min gt d
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Interval Tree
Interval Tree
Within Id, We keep two sorted lists - One by
cell min values in ascending order (AL)
- The other by cell max values in descending
order (DR)
Query isovalue q If q d


Id cells that have min lt d lt max I
left cells that have max lt d I right cells
that have min gt d

• All the cells in Id are reported.

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Interval Tree
Interval Tree
Within Id, We keep two sorted lists - One by
cell min values in ascending order (AL)
- The other by cell max values in descending
order (DR)
Query isovalue q If q lt d


Id cells that have min lt d lt max I
left cells that have max lt d I right cells
that have min gt d

• Scan the AL list to find cells with min lt q
• Prune the right subtree
• Visit left subtree recursively

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Interval Tree
Interval Tree
Within Id, We keep two sorted lists - One by
cell min values in ascending order (AL)
- The other by cell max values in descending
order (DR)
Query isovalue q If q gt d


Id cells that have min lt d lt max I
left cells that have max lt d I right cells
that have min gt d

• Scan the DR list to find cells with max gt q
• Prune the left subtree
• Visit right subtree recursively

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Interval Tree

• Now, given an isovalue C
• If C lt d
• If C gt d
• 3) If C d

Complexity O(log(n)k) Optimal!!
Id cells that have min lt d lt max I
left cells that have max lt d I right cells
that have min gt d
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Range Search Methods
In general, range search methods all are
superfast two order of magnitude faster than
the marching cubes algorithm in terms of cell
search But they all suffer a common problem
Excessive extra memory requirement!!!