1. Isosurface Extractions (II)
2D Isocontour
3D Isosurface

2. Isosurface cell search
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.

3. 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 < C < bk; C=isovalue}

5. Efficient search methods
Spatial subdivision (domain search)
Value subdivision (range search)
Contour propagation

6. 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))

7. Range Search (1) Subdivide the cells based on their min/max ranges
Global minimum
Global maximum
Hierarchically subdivide the cells based on their min/max ranges
Isovalue

8. Range Search (2)
Within each subinterval, there are more than one cells
To further improve the search speed, we sort them.
Sort by what ?
Min and Max values G1
Max
Min
M5 M2 M6 M4 M1 M3 M7 M8 M11 M10 M9
m5 m1 m6 m3 m8 m7 m2 m9 m11 m4 m10 G2
Isosurface cells = G1 G2

9. Range Search (3) A clean range subdivision is difficult …
Difficult to get an optimal speed ?

10. 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?

11. Span Space What are the isosurface cells? max How to search them? min

12. 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
each of the dimensions at each level of the tree

13. NOISE Algorithm (K-d tree)
Median point
Min
Construction
Max
left
right ?
max up …
down … …
* One node per cell
min

14. NOISE Algorithm (Query)
Median point
Min
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.
Max
left
right ? up …
down … …
Complexity = O( N + k)

15. Span Space Search (2) Search Method (2): ISSUE O(log(N/L)) O(1)
Complexity = ? ?
O(log(N/L))

16. … … Back to Range Search Sort all the data points Interval Tree:
(x1,x2,x3,x4,…. , xn)
Let d = x (mid point)
n/2 Id
I left
I right
We use d to divide the cells into three
sets Id, I left, and I right
Id : cells that have min < d < max
I left: cells that have max < d
I right: cells that have min > d … …

17. … … Interval Tree Id I right I left
Now, given an isovalue C
If C < d
If C > d
3) If C = d Id
I left
I right … …
Complexity = O(log(n)+k)
Optimal!!
Id : cells that have min < d < max
I left: cells that have max < d
I right: cells that have min > d

18. 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!!!

19. Contour Propagation Basic Idea:
Given an initial cell that contains isosurface, the remainder of the isosurface can be found by propagation
FIFO Queue A
B C C
C D ….
Initial cell: A
Enqueue: B, C
Dequeue: B
Enqueue: D … A B D C E
Breadth-First
Search

20. Challenges Need to know the initial cells!
For any given isovalue C, finding
the initial cells to start the propagation is almost as hard as
finding the isosurface cells.
You could do a global search, but …

21. Solutions Extrema Graph (Itoh vis’95) 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 smaller
than G.
We will only talk about extrema graph due to time constraint

22. Extrema Graph (1)

23. 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.

24. Extrema Graph (3) E1 E2 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 only store
min/max of the arc though) a4 E7 a7 E5 a6 E6 E8

25. Extrema Graph (4) 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 ….
There is something more needs to be done…

26. 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)

27. Extrema Graph (5) Boundary Cells!! These open isocontours will
Contours missed
These open isocontours will
intersect with ?? cells

28. Extrema Graph (6) Algorithm (continued) 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