Marching Cubes

1
Marching Cubes

• A High Resolution 3D Surface Construction
  Algorithm

2
Slice Data to Volumetric Data(1)
3
Slice Data to Volumetric Data(2)
4
Marching Cube

• Create cells (cubes)
• Classify each vertex
• Build an index
• Get edge list
• Based on table look-up
• Interpolate triangle vertices
• Obtain polygon list and do shading in image space

5
Cube
Consider a cube defined by 8 data values, 4
from slice k, and another 4 from slice k1
6
Classify each vertex

• Label 1 or 0 as to whether it lies inside or
  outside the surface

0
1
0
0
Match!!!
0
0
1
1
7
Build an index
Create an index of 8 bits from the binary
labeling of each vertex.
8
Get edge list
Give an index, store a list of edges. Because
symmetry 256/2128rotation 128/816256
cases are reduced to 14 cases.
9
Interpolate triangle vertices
(iso_value - D(i))

• X i

(D(i1) - D(i))
20
10
i
i1
X
i
i1
X
iso_value18
iso_value14
10
Problems about MC

• Empty cells
• 30-70 of isosurface generation time was spent in
  examining empty cells.
• Speed
• Ambiguity

11
The Asymptotic Decider

• Resolving the Ambiguity in Marching Cubes

12
Ambiguity Problem (1)

• Ambiguous Face a face that has two diagonally
  oppsed points with the same sign

13
Ambiguity Problem (2)

• Certain Marching Cubes cases have more than one
  possible triangulation.

Mismatch!!!
Hole!


Case 6
Case 3


14
Ambiguity Problem (3)

• To fix it

Match!!!


Case 6
Case 3 B


The goal is to come up with a consistent
triangulation
15
Asymptotic Decider (1)

• Based on bilinear interpolation over faces

B00 B01 B10 B11
1-t t
B11
B(s,t) (1-s, s)
B01
B00(1- s)(1- t) B10(s)(1- t) B01(1-
s)(t) B11(s)(t)
(s,t)
The contour curves of B (s,t) B(s,t) a
are hyperbolas
B00
B10
16
Asymptotic Decider (2)
(Sa, Ta)
(1,1)
(0,0)
(Sa, Ta)
Asymptote
Not Separated
17
Asymptotic Decider (3)
(Sa, Ta)
(1,1)
(0,0)
Asymptote
(Sa, Ta)
Separated
18
Asymptotic Decider (4)
B( Sa , 0) B( Sa , 1)
(S1 , 1)
B( 0, Ta) B( 1 , Ta)
(1 , T1)
Sa B00 - B01 B00 B11
B01 B10 Ta B00 B10
B00 B11 B01 B10 B(Sa,Ta) B00 B11
B10 B01 B00 B11 B01 B10
(0 , T0)
(Sa, Ta)
(S0 , 0)
19
Asymptotic Decider (5)

• case 3, 6, 12, 10, 7, 13
• (These are the cases with at least one ambiguious
  faces)