isosurface extractions 2d isocontour 3d isosurface
Post on 25-Dec-2015
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Isosurfaec cells: cells that contain isosurface.
min < isovalue < max Marching cubes algorithm performs a
linear search to locate the isosurface cells – not very efficient for large-scale data sets.
Isosurface cell search
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
nn
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 < C < bk; C=isovalue}
Efficient search methods
1. Spatial subdivision (domain search)
2. Value subdivision (range search) 3. Contour propagation
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))
Range Search (1)
Subdivide the cells based on their min/max ranges
Global minimum Global maximum
Isovalue
Hierarchically subdivide the cells based on their min/max ranges
Range Search (2)
Within each subinterval, there are more than one cellsTo further improve the search speed, we sort them.
Sort by what ? Min and Max values
Max
Min
M5 M2 M6 M4 M1 M3 M7 M8 M11 M10 M9
m5 m1 m6 m3 m8 m7 m2 m9 m11 m4 m10
G1
G2
Isosurface cells = G1 G2
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?
Span Space Search (1)
With the point representation, subdividing the space ismuch 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 each of the dimensions at each level of the tree
NOISE Algorithm (K-d tree)
left right
up down
… … …
Construction
min
max
* One node per cell
Min
Max
?
Median point
NOISE Algorithm (Query)
Complexity = O( N + k)
left right
up down
… … …
Min
Max
?
Median point
If ( isovalue < root.min )• check the ?? Subtree
If (isovalue > root.min) • Check the ?? Subtree • Don’t forget to check the root ‘s interval as well.
Back to Range Search
Interval Tree:
I
I left I right
Sort all the data points(x1,x2,x3,x4,…. , xn)Let = x mid point)n/2
We use to divide the cells into threesets II left, and I right
Icells that have min < max
I left: cells that have max < I right: cells that have min >
… …
Interval Tree
I
I left I right
… … Icells that have min < max
I left: cells that have max < I right: cells that have min >
Now, given an isovalue C
1) If C < 2) If C > 3) If C =
Complexity = O(log(n)+k)
Optimal!!
Range Search Methods
In general, range search methods all are superfast –
two order of magnitude faster than the marching cubesalgorithm in terms of cell search
But they all suffer a common problem …
Excessive extra memory requirement!!!
Basic Idea:
Given an initial cell that contains isosurface, the remainder of the isosurface can be found by propagation
Contour Propagation
A
BD
CE
Initial cell: A
Enqueue: B, C
Dequeue: B
Enqueue: D
…
FIFO Queue
A
B C
C
C D
….
Breadth-First Search
ChallengesNeed to know the initial cells!
For any given isovalue C, findingthe initial cells to start the propagation is almost as hard as finding the isosurface cells.
You could do a global search, but …
Solutions
(1)Extrema Graph (Itoh vis’95)(2)Seed Sets (Bajaj volvis’96)
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, but S is usually (hopefully) much smallerthan G.
We will only talk about extrema graph due to time constraint
Extrema Graph (2)
Basic Idea:
If we find all the local minimum and maximum points (Extrema), and connect them together by straight lines (Arcs), then any closed isocontour is intersect by at leat one of the arcs.
Extrema Graph (3)
E1 E2
E3E4
E7
E5E6 E8
a2
a3
a4a5
a6a7
a1
Extreme Graph:
{ E, A: E: extrema points A: Arcs conneccts E }
An ‘arc’ consists of cells that connectextrema points (we only store min/max of the arc though)
Extrema Graph (4)
Algorithm:
Given an isovalue1) Search the arcs of the extrema graph (to find the
arcs that have min/max contains the isovalue2) Walk through the cells along each of the arcs to
find the seed cells3) Start to propagate from the seed cells 4) ….
There is something more needs to be done…
We are not done yet …
What ?!
We just mentioned that all the closed isocontours will intersect with the arcs connecting the extrema points
How about non-closed isocontours? (or called open isocontours)
Extrema Graph (5)
Contours missed
These open isocontours will intersect with ?? cells
Boundary Cells!!
Extrema Graph (6)
Algorithm (continued)Given an isovalue1) Search the arcs of the extrema graph (to find the
arcs that have min/max contains the isovalue2) Walk through the cells along each of the arcs to
find the seed cells3) Start to propagate from the seed cells
4) Search the cells along the boundary and find seed cells from there
5) Propagate open isocontours
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