Automatic Isosurface Propagation Using an
Extrema Graph and Sorted Boundary Cell Lists
Introduction



Visiting non-intersected cells is very
time-consuming in huge volumes.
Elimination of non-intersected cells
outside the process is more effective.
The number of visited cells is less than
O(n)
Propagation Algorithm (1)
Propagation Algorithm (2)
Basic Idea:
Given an starting cell that contains isosurface, the
remainder of the isosurface can be found by propagation
Initial cell: A
C
E
A
B
D
Enqueue: B, C
FIFO Queue
A
BC
Dequeue: B
C
Enqueue: D
CD
…
….
Breadth-First
Search
Propagation Algorithm (3)
Challenges
Need to know the starting cells!
For any given isovalue C, finding
the starting cells to start the
propagation.
You could do a global search,
but …
Solution：Extrema Graph &
Boundary Cell Lists
Extremum Point

Extremum points ard defined as grid-points
whose scalar values are higher or lower than
the values of all adjacent grid-points.
Generating an Extrema Graph
Searching for starting cells:
Find all the local minimum and maximum
points, and connect them together by
straight lines (Arcs).
The closed isosurface is intersected by
at least one of the arcs.
Extrema Graph
Extreme Graph:
{
E, A: E: extrema points
A: Arcs conneccts E
}
Hole
Problem：Holes!!
Generating Boundary Cell Lists
The open isosurface is intersected by visiting boundary cells in order.
Hole
Generating Isosurfaces
Searching for starting cells
Outline of the Algorithm
Void main(){
/* Preprocess */
ExtremumPointExtraction();
GenerateGraph();
GenerateBoundList();
}
/* Isosurfacing process*/
while(1){
Specify an isovalue C;
GenerateSurface(C);
}
O n 
1
3
2
3
O(n )  O(n )
About Arc
Image of Isosurface
Volume Thinning for
Automatic Isosurface Propagation
Topology of an extrema graph and an isosurface
Boundary cells are not
necessary if there is a cycle
around a through-hole
Image thinning method
3
8
2
p
7
6
5
1
(a)
4
p
(b)
(f)
p
(c)
p
p
(d)
(e)
(g)
Volume thinning method
Extremum point
Bubble-like layer
We initially assumes that a seed set of a volume contains all cells in the volume.
The extremum point will never be eliminated from the seed set during the process.
Finally, the seed set form a one-cell-wide skeleton.
Elimination of Bubble-Like Layers of Cells
Visited cells in the isosurfacing process
Preprocess
Outline of the Algorithm
Void main(){
/* Preprocess */
ExtremumPointExtraction();
VolumeThinning();
}
O n 
/* Isosurfacing process */
while(1){
Specify an isovalue;
Extract isosurface cells from the extrema skeleton;
IsosurfacePropagation();
1
}
O(n 3 )
Image of Volume Thinning Process
Image of Volume Thinning Process
Image of Isosurface(1)
Image of Isosurface(2)