Advanced Iso- Surfacing Algorithms Mengxia Zhu, Fall 2007
Jan 02, 2016
Advanced Iso-Surfacing Algorithms
Mengxia Zhu,
Fall 2007
2D Contour Display Common method for displaying a result across a
surface Contour lines: represent a constant value across
the surface
10000 ft 9000 ft
8000 ft
Topographic Map Weather Map
2D Contour Lines
2D contour (0)
Remember bi-linear interpolation
p2 p3
p0 p1
P =?
p4 p5
To know the value of P, we can first compute p4 andP5 and then linearly interpolateP
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
p0 p1
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
Iso-contour (2)
p2 p3
p0 p1
We can solve the problem based on linear interpolation
(1) Identify edges that contain points P that have value f(P) = C
(2) Calculate the positions of P
(3) Connect the points with lines
Iso-contouring – Step 1
(1) Identify edges that contain points P that have value f(P) = C
v1 v2
If v1 < C < v2 then the edge contains such a point
Iso-contouring – Step 2
(2) Calculate the position of P
Use linear interpolation:
P = P1 + (C-v1)/(v2-v1) * (P2 – P1)v1 v2
Pp1 p2
C
Iso-contouring – Step 3
p2 p3
p0 p1
Connect the points with line(s)
Based on the principle of linear variation, all the points on the line have values equal C
Cases of 2D Cells (Squares)
case 0 case 1 case 2 case 3 case 4 case 5 case 6 case 7
case 8 case 9 case 10 case 11 case 12 case 13 case 14 case 15
By complementary and rotational symmetries, the number of the basic cases is reduced to 4
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
3D Iso-surface Example
How many cases for 3D?
Now we have 8 vertices
So it is: 2 = 2568
How many unique topological cases?
Case Reduction (1)
Value Symmetry
+
+
_ _
_
__
_
+
+
_
_
+
+
+
+
Case Reduction (2)
Rotation Symmetry
+
+
_ _
_
__
__
_
+
+
_ _
__
By inspection, we can reduce 256 14
Iso-surface Cases
Total number of cases: 14 + 1
Marching Cubes Algorithm(1)
A Divide-and-Conquer Algorithm
v1 v2
v3v4
v5v6
v7v8 Vi is ‘1’ or ‘0’ (one bit) 1: > C; 0: <C (C= iso-value)
Each cell has an index mapped to a value ranged [0,255]
Index = v8 v7 v6 v5 v4 v3 v2 v1
Marching Cubes (2)Given the index for each cell, a table lookup is performedto identify the edges that has intersections with the iso-surface
0
1
2
3
14
e1, e3, e5…
Index intersection edges
e1
e2
e3
e4
e5
e6
e7
e8
e9 e10
e11 e12
Pre-defined look-up table enumerates a) how many triangles will make up the isosurface segment passing through the cubeb) which edges of the cubes contain vertices of triangles, and in what order
Marching Cubes (3)
+
+
+
+ _
_
_
_• Perform linear interpolations at the edges to calculate the intersection points
• Connect the points
Interpolation of Triangle Vertices For each triangle, find an vertex location along the edge using
linear interpolation of the values at the edge’s two end points
x x(i) fac xy y(i) fac yzz(i) fac z
where( )
( 1) ( )isoS S i
facS i S i
10
40
0
30
30
10 20
20
(x(i) a /2,y(i),z(i))
(x(i),y(i) a /4,z(i))
(x(i),y(i),z(i) a /4)
t3 =
t8 =
t4 =t3
t4
t8
Vertices of triangle
a
Why is it called marching cubes?
Linear search through cells •Row by row, layer by layer•Reuse the interpolated points for adjacent cells
Steps in Marching Cubes
Select a cell
Classify the inside/outside state of each vertex
Create an index
Get edge list from table
Interpolate the edge location
Go to the next cell
Marching Cubes at various Isovalues
www.erc.msstate.edu
Dividing Cubes Algorithm Generate isosurface using dense cloud points Use point primitive instead of triangles in MC Conditions:
Large number of points Density of points>= screen resolution Lighting and shading calculation
H. Cline, W. Lorensen, S. Ludke, C. Crawford, and B. Teeter, “Two algorithms for the three-dimesnional reconstruction of tomographs” Medical Physics, vol. 15, no. 3, May 1988
Find Intersecting Voxel Select a voxel (cell) and determine
whether the isosurface passes through it Whether there are scalar values at vertices
both above and below the iso-value
Inside isosurface
Subdivide Voxel The voxel is subdivided into a regular grid of n1 n2
n3 subvoxels ni = wi /R, where R is screen resolution and wi is width of the
voxel
zy
x n2
n3
n1
Generate Points Scalar values at the subpoints
are generated using the interpolation function
Find whether the isosurface passes through each sub-voxel
If it does, generate a point at the center of the subvoxel and compute its normal
Collection of all such points compose the Dividing Cubes’ isosurface
Recursive Implementation Recursively divide the voxel as in
octree decomposition
Scalar values at the new points are interpolated
Process repeats for each sub-voxel if the isosurface passes through it
This process continues until the size of the subvoxel =< R
A point is generated at the center of the sub-voxel
Hierarchy of spatial subdivisions to form an octree
Dividing Squares’ Contour
Dividing Cubes’ Image
Image of human head Image with voxel subdivision into 4x4x4 cubeswww.cs.umbc.edu
Iso-surface cells: cells that contain iso-surface.
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 cell search
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
average number of the
iso-surface cells is O(n x n)
(ratio of surface v.s. volume)n
nn
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 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 (3)
?
A clean range subdivision is difficult …
Difficult to get an optimal speed
Range Search: Interval Tree
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 super fast –
two orders 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 iso-surface, the remainder of the iso-surface 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
Challenges
Need to know the initial cells!
For any given iso-value C, findingthe 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 smallerthan G.
Ambiguity !
2D Ambiguous Cases Ambiguous cases:
5, 10
Contour ambiguity arises when adjacent vertices in different states but diagonal vertices in the same state
Break contour Join contour
Both are valid
or
Break contour(two loops)
Join contour(single loop)
3D Ambiguity Problem
Certain Marching Cube cases have more than onepossible triangulation
Case 6 Case 3
Mismatch!!!
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+
+
+
Hole!
3D Ambiguous Cases
Ambiguous cases: 3, 6, 7, 10, 12,
13 Adjacent vertices in different
states, but diagonal vertices in the same state
Ambiguity cases may cause holes
hole
case 3 case 6c
Isosurface polygons are disjoint across the common element surface
The Problem
Ambiguous Face: a face that has two diagonally opposingpoints with the same sign
+
+
Connecting either way is possible
To fix it …
Case 6 Case 3 B
Match!!!
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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
B01
B00 B10
B11
(s,t)B(s,t) = (1-s, s)
B00 B01B10 B11
1-t t
The contour curves of B:
{(s,t) | B(s,t) = } are hyperbolas
Asymptotic Decider (2)
(0,0)
(1,1)
Where the hyperbolasgo through the cell depends on the valuesat the corners, I.e., B00, B01, B10, B11
Asymptotic Decider (3)
(0,0)
(1,1)
Asymptote
(ST
If B(ST
Asymptotic Decider (4)
(1,1)
Asymptote
(ST
(0,0)
If B(ST
Asymptotic Decider (5)
(1,1) (ST
(0,0)
S B00 - B01 B00 + B11 – B01 – B10
T B00 – B10 B00 + B11 – B01 – B10
B(ST 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).
References
This set of slides are developed from the lecture slides used by Prof. Han-Wei Shen at Ohio State University,
Also from Prof. Karki at Louisiana State University