Marching Cubes

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Marching Cubes
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Isosurface Definition

• Given a scalar field, f Rd ? R, and a scalar
  value s, a level set is the set of points p
  f(p) s .

Given a sampling of a scalar field, f Rd ? R,
and a scalar value s, construct a (piecewise
linear) approximation (isosurface) to the level
set p f(p) s .
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Isosurface Reconstruction Algorithms

• Cuberille Draw cubes (voxels) and smooth
• Herman and Liu. Comp Graph Image Proc, 1979
• Chen, Herman, et. al. IEEE Comp. Graph Appl.,
  1985
• Artzy, Frieder, et. al. Comp. Graph Image
  Proc., 1981
• Taubin, Siggraph 95, 1995
• Marching cubes algorithm
• Lorensen and Cline. Comput. Graphics, 1987
• Montani, Scateni, Scopigno. Visual. Comp. 1994
• Nielson and Hamann, Visualization 91, 1991
• Weigle and Banks, Visualization 96, 1996
• Bhaniramka, Wenger, Crawfis, Isosurfacing in
  Higher Dimensions, Vis 00, 2000
• Dual contouring algorithms
• Gibson, Surface Nets, Visualization 98, 1998.
• Ju, Losasso, et. al. Dual contouring of Hermite
  data, Trans on Graphics, 2002
• Nielson, Dual marching cubes, Visualization
  04, 2004

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Regular Grid Samples of a Scalar Field
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Regular Grid
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(/-) Grid
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(/-) Midpoints
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Marching Squares
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Marching Squares
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Marching Squares
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Marching Squares
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Marching Squares
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Isosurface
Level Set x f(x,y,z) a
Polygonal Approximation Isosurface
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Isosurface Construction

• Identify midpoints of (,-) edges
• Draw isosurface through midpoints.

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Isocontour Construction

• Identify midpoints of (,-) edges
• Draw isocontour through midpoints.

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2D Cases
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3D Cube
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3D Cube
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3D Cases
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3D Cases
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Marching Cubes (Lorensen Cline)

• Build cube lookup table
• For each cube, retrieve surface fragment from
  lookup table.

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Marching Cubes (Lorensen Cline)

• Read four slices into memory
• Scan slices and create cubes
• Calculate cube index based on 8 vertex scalars
• Look up edge (triangle) list from table
• Interpolate surface edge intersections
• Calculate unit normals using central differences
• Output triangles and vertex normals

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Ambiguity
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Ambiguity
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Ambiguity
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3D Ambiguity
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3D Ambiguity



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3D Ambiguity
Durst. Comput. Graph. 1988.
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3D Correct Isosurface
Montani, Scateni, Scopigni, Visual. Comput.
1994. Nielson and Hamann. Visualization 91, 1991.