Advanced Iso-Surfacing
Algorithms
Jian Huang, CS594, Spring 2002
This set of slides are developed and used by Prof.
Han-Wei Shen at Ohio State University.
Iso-contour/surface Extractions
2D Iso-contour
3D Iso-surface
Iso-contour (0)
Remember bi-linear interpolation
p2
p3
P =?
p4
p0
p5
p1
To know the value of P,
we can first compute p4 and
P5 and then linearly interpolate
P
Iso-contour (1)
Consider a simple case: one cell data set
The problem of extracting an iso-contour is an
inverse of value interpolation. That is:
p2
p3
Given f(p0)=v0, f(p1)=v1, f(p2)=v2, f(p3)=v3
Find the point(s) P within the cell that have values
F(p) = C
p0
p1
Iso-contour (2)
We can solve the problem based on linear interpolation
p2
p3
(1) Identify edges that contain points P that have
value f(P) = C
(2) Calculate the positions of P
p0
p1
(3) Connect the points with lines
Iso-contouring – Step 1
(1) Identify edges that contain points P that have value
f(P) = C
If v1 < C < v2 then the edge contains such a point
v1
v2
Iso-contouring – Step 2
(2) Calculate the position of P
p1
P
p2
v1
C
v2
Use linear interpolation:
P = P1 + (C-v1)/(v2-v1) * (P2 – P1)
Iso-contouring – Step 3
p2
p0
p3
p1
Connect the points with line(s)
Based on the principle of linear variation, all
the points on the line have values equal C
Inside or Outside?
Just a naming convention
1. If a value is smaller than the iso-value, we call it “Inside”
2. If a value is greater than the iso-value, we call it “Outside”
-
+
p2
p3
p0
p1
outside cell
-
p2
p3
p0
p1
inside cell
Iso-surface Extraction
Extend the same divide-and-conquer algorithm to
three dimension
• 3D cells
• Look at one cell at a time
• Let’s only focus on voxel
Divide and Conquer
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(2 triangles)
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+
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How many cases?
Now we have 8 vertices
So it is: 2 8= 256
How many unique topological cases?
Case Reduction (1)
Value Symmetry
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Case Reduction (2)
Rotation Symmetry
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By inspection, we can reduce 256
14
Iso-surface Cases
Total number of cases: 14 + 3
Marching Cubes Algorithm
A Divide-and-Conquer Algorithm
v8
v4
v6
Each cell has an index mapped to a
value ranged [0,255]
v3
v5
v1
v7
Vi is ‘1’ or ‘0’ (one bit)
1: > C; 0: <C (C= iso-value)
v2
Index = v8 v7 v6 v5 v4 v3 v2 v1
Marching Cubes (2)
Given the index for each cell, a table lookup is performed
to identify the edges that has intersections with the
iso-surface
Index intersection edges
0
e7
e11
e8
e12
e6
e3
e5
e4
e9
e2
1
2
3
e10
e1
14
e1, e3, e5
…
Marching Cubes (3)
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+
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• Perform linear interpolations
at the edges to calculate the
intersection points
• Connect the points
Why is it called marching cubes?
Linear search through cells
•Row by row, layer by layer
•Reuse the interpolated points
for adjacent cells
Iso-surface cell search
• Iso-surface cells: cells that contain isosurface.
min < iso-value < max
• Marching cubes algorithm performs a linear
search to locate the iso-surface cells – not
very efficient for large-scale data sets.
Iso-surface Cells
• For a given iso-value, only a smaller portion of
cells are iso-surface cell.
• For a volume with
n x n x n cells, the
n
average number of the
iso-surface cells is O(n x n)
(ratio of surface v.s. volume)
n
n
Efficient iso-surface 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=iso-value}
Efficient search methods
1. Spatial subdivision (domain search)
2. Value subdivision (range search)
3. Contour propagation
Domain search
•
•
Subdivide the space into several sub-domains, check the
min/max values for each sub-domain
If the min/max values (extreme values) do not contain the
iso-value, 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
Isovalue
Global maximum
Hierarchically subdivide the cells based on
their min/max ranges
Range Search (2)
Within each subinterval, there are more than one cells
To further improve the search speed, we sort them.
Min and Max values
Sort by what ?
G1
Max M5
M2
M6
M4
M1 M3 M7
M8 M11 M10 M9
G2
Min m5
m1
m6
m3
m8 m7 m2
m9 m11 m4 m10
Isosurface cells = G1
G2
Range Search (3)
A clean range subdivision is difficult …
?
Difficult to get an optimal speed
Range Search: Span Space
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
What are the iso-surface cells?
max
How to search them?
C
min
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
NOISE Algorithm (K-d tree)
Median point
Min
Construction
left
right
Max
?
max
up
…
down
… …
* One node per cell
min
NOISE Algorithm (Query)
Median point
Min
left
right
Max
?
up
…
down
… …
Complexity = O( N + k)
If ( iso-value < root.min )
• check the ?? Sub-tree
If (iso-value > root.min)
• Check the ?? Sub-tree
• Don’t forget to check the
root‘s interval as well.
Span Space Search (2)
Search Method (2): ISSUE, discretized span space
O(log(N/L))
O(1)
Complexity = ?
?
O(log(N/L))
Range Search: Interval Tree
Interval Tree:
Id
I left
…
I right
…
Sort all the data points
(x1,x2,x3,x4,…. , xn)
Let d = x n/2 (mid point)
We use d to divide the cells into three
sets Id, I left, and I right
Id : cells that have
I left: cells that have
I right: cells that have
min < d
max < d
min >
d
< max
Interval Tree
Now, given an isovalue C
1)
Id
2)
If C < d
If C > d
3) If C = d
I left
I right
…
Id : cells that have
I left: cells that have
I right: cells that have
…
Complexity = O(log(n)+k)
min < d
max < d
min >
d
< max
Optimal!!
Range Search Methods
In general, range search methods all are super fast –
two orders of magnitude faster than the marching cubes
algorithm in terms of cell search
But they all suffer a common problem …
Excessive extra memory requirement!!!
Contour Propagation
Basic Idea:
Given an initial cell that contains iso-surface, the remainder of
the iso-surface can be found by propagation
Initial cell: A
C
E
A
B
D
FIFO Queue
A
Enqueue: B, C
BC
Dequeue: B
C
Enqueue: D
CD
…
….
Breadth-First
Search
Challenges
Need to know the initial cells!
For any given iso-value C, finding
the initial cells to start the propagation
is almost as hard as
finding the iso-surface 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 iso-value 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
Extrema Graph (1)
Basic Idea:
If we find all the local minimum and
maximum points (Extrema), and connect them
together by straight lines (Arcs), then any closed
Iso-contour is intersect by at least one of the arcs.
Extrema Graph (2)
Extrema Graph (3)
Extreme Graph:
E2
E1
a1
a2
a3
E3
E4
a5
a4
E7
a7
E5
E6
a6
E8
{
E, A:
E: extrema points
A: Arcs conneccts E
}
An ‘arc’ consists of cells that connect
extrema points (we only store
min/max of the arc though)
Extrema Graph (4)
Algorithm:
Given an iso-value
1) Search the arcs of the extrema graph (to find the arcs that have
min/max contains the iso-value
2) Walk through the cells along each of the arcs to find the seed
cells
3) 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 iso-contours will
intersect with the arcs connecting the extrema points
How about non-closed iso-contours? (or called open isocontours)
Extrema Graph (5)
Contours missed
These open iso-contours will
intersect with ?? cells
Boundary Cells!!
Extrema Graph (6)
Algorithm (continued)
Given an iso-value
1) Search the arcs of the extrema graph (to find the arcs that have
min/max contains the iso-value
2) Walk through the cells along each of the arcs to find the seed
cells
3) Start to propagate from the seed cells
4) Search the cells along the boundary and find seed cells from
there
5) Propagate open iso-contours
Extrema Graph
• Efficiency - Number of cells visisted:
– extrema graph - N0.33
– boundary - N 0.66
– Iso-surface - N 0.66
• based on tetrahedra - will create more surface triangles ...
• should extract the same number of cells/ triangulation as
Marching Cubes
Ambiguity Problem
Certain Marching Cube cases have more than one
possible triangulation
Mismatch!!!
Hole!
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Case 6
Case 3
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The Problem
Ambiguous Face: a face that has two diagonally opposing
points with the same sign
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Connecting either way is possible
To fix it …
Match!!!
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Case 6
Case 3 B
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The goal is to come up with a consistent triangulation
Solutions
There are many solutions available – we present a method called:
Asymptotic Decider by Nielson and Hamann (IEEE Vis’91)
Asymptotic Decider
Based on bilinear interpolation over faces
B11
B01
(s,t)
B00
B(s,t) = (1-s, s)
B10
B00 B01
B10 B11
1-t
t
The contour curves of B:
{(s,t) | B(s,t) = a } are hyperbolas
Asymptotic Decider (2)
(1,1)
(0,0)
Where the hyperbolas
go through the cell
depends on the values
at the corners, I.e.,
B00, B01, B10, B11
Asymptotic Decider (3)
(1,1)
(Sa, Ta)
If a
(0,0)
Asymptote
< B(Sa, Ta)
Asymptotic Decider (4)
(1,1)
(Sa, Ta)
If a
(0,0)
Asymptote
>
B(Sa, Ta)
Asymptotic Decider (5)
(1,1)
(Sa, Ta)
Sa =
B00 - B01
B00 + B11 – B01 – B10
Ta=
B00 – B10
B00 + B11 – B01 – B10
(0,0)
B(Sa,Ta) = B00 B11 + B10 B01
B00 + B11 – B01 – B10
Asymptotic Decider (6)
Based on the result of asymptotic decider, we
expand the marching cube case 3, 6, 12, 10, 7, 13
(These are the cases with at least one ambiguious
faces).