Marching Cubes: A High Resolution 3D Surface Construction Algorithm

1
Marching CubesA High Resolution 3D Surface
Construction Algorithm

• William E. Lorenson
• Harvey E. Cline
• General Electric Company
• Corporate Research and Development, SIGGRAPH 1987
• Presented by Mark Blackburn, Fall 2005

2
Organization

1. Motivation
2. Related Work
3. Information Flow
4. Algorithm
5. Short Demo
6. Results
7. Conclusion Future Work

3
Motivation

• 3D visualization is critical to medical field
• Existing algorithms inadequate
• lack detail
• may introduce artifacts
• Create polygonal representation of constant
  density surfaces from 3D array of data

4
Related Work

• Existing methods of 3D surface generation
• Trace contours within each slice then connect
  with triangles ( cf. topography map)
• Create surfaces from cuberilles (voxels)
• Ray casting to find the 3D surface
• Use hue-lightness to shade surface
• Use gradient to shade
• Display density volumes

5
Related Work

• Shortcomings of Existing Techniques
• Throw away useful information in the original
  data
• Cuberilles uses thresholding to represent
  surface
• Ray casting uses depth shading alone or
  approximates shading using unnormalized gradient
• Some lack hidden surface removal
• Volume models display all values and rely on
  motion to produce a 3D sensation

6
Related Work

• Benefits of Marching Cubes approach
• Uses all information from source data
• Derives inter-slice connectivity, surface
  location, and surface gradient
• Result can be displayed on conventional graphics
  display systems using standard rendering
  algorithms

7
Information Flow

• Information Flow for 3D Medical Algorithms
• Data Acquisition collect a series of 2D slices
  of information
• Image Processing find structures with 3D data or
  filter / bin data
• Surface Construction Create surface model of
  voxels or polygons
• Viewing Display Display the surface using ray
  casting, depth or color shading.

8
Algorithm Overview

• The Marching Cubes Algorithm
• Consists of 3 basic steps
• Locate the surface corresponding to a
  user-specified value.
• Create triangles.
• Calculate normals to the surface at each vertex.

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Step 1Surface Intersection

• To locate the surface, it uses a logical cube
  created from eight pixels (Four each from 2
  adjacent layers)

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Step 1 Surface Intersection

• Binary vertex assignment (p (i, j, k) gt TU ? 1
  0)
• Set cube vertex to value of 1 if the data value
  at that vertex exceeds (or equals) the value of
  the surface we are constructing
• Otherwise, set cube vertex to 0
• If a vertex 1 then it is inside the surface
• If a vertex 0 then it is outside
• Any cube with vertices of both types is
  intersected by the surface.

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Step 2 Triangulation

• For each cube, we have 8 vertices with 2 possible
  states each (inside or outside).
• This gives us 28 possible patterns 256 cases.
• Enumerate cases to create a LUT
• Use symmetries to reduce problem from 256 to 15
  cases.

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Step 2 Triangulation

• Use vertex bit mask to create an index for each
  case based on the state of the vertexes.
• Using the index to tell which edge the surface
  intersects, we can then can linearly interpolate
  the surface intersection along the edge.
• In our previous example of green pixels, v0 and
  v3 were outside the surface and thus our index
  would 11110110 246

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Step 3 Surface normals

• To calculate surface normal, we need to determine
  gradient vector, g (derivative of the density
  function).
• To estimate the gradient vector at the surface of
  interest, we first estimate the gradient vectors
  at the vertices and interpolate the gradient at
  the intersection.
• The gradient at cube vertex (i , j, k), is
  estimated using central differences along the
  three coordinate axes by

D (i, j, k) is the density at pixel (i, j) in
slice k. ?x, ?y, ?z are lengths of the cube edges
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Step 3 Surface normals

• Dividing the gradient by its length produces the
  unit normal at the vertex required for rendering.
• Then the algorithm linearly interpolates this
  normal to the point of intersection.

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Algorithm Summary

1. Scan 2 slices and create cube
2. Calculate index for cube based on vertices
3. Use index to lookup list of edges intersected
4. Use densities to interpolate edge intersections
5. Calculate unit normal at each edge vertex using
   central differences. Interpolate normal to each
   triangle vertex
6. Output the triangle vertices and vertex normals
7. March to next position and repeat.

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Demo

• Short Demo / tutorial
• http//www.essi.fr/lingrand/MarchingCubes/applet.
  html

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Enhancements

• Efficiency
• Take advantage of pixel-to-pixel, line-to-line,
  and slice-to-slice coherence by keeping previous
  calculations.
• Functional
• Added solid modeling capability
• Boolean operations permit cutting and capping of
  solid models as well as multiple surface
  extraction.

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Results
Bone Surface
Soft Tissue Surface
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Results The Visible Human
Inside skeleton view
Torso / bowels
Images courtesy Marching Through the Visible
Man http//www.essi.fr/lingrand/MarchingCubes/a
ccueil.html)
20
Results The Visible Man
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Results
Human brain surface reconstructed by using
marching cubes (128,984 vertices and 258,004
triangles
Magnified display of brain surface
Images courtesy IF of Polytech' Nice-Sophia
http//www.essi.fr/lingrand/MarchingCubes/accueil
.html)
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Conclusion

• Marching Cubes is an algorithm for 3D surface
  reconstruction well suited to series of 2D
  medical images (slices).
• Algorithm summary ( 3 steps)
• Locate the surface corresponding to a
  user-specified value using cube vertex tests
• Create triangles based on vertex states of the
  cube
• Calculate normals to the surface at each vertex.

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Future Work

• Dividing Cubes
• An algorithm that generates points rather than
  triangles. As the resolution of the data
  increases, the number of triangles approaches the
  number of pixels.
• Dual Marching Cubes
• creates isosurfaces very similar to Marching
  cubes surfaces, but they are comprised of quad
  patches that eliminate some of the problems of
  poorly shaped triangles that can occur in this
  algorithm.

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Evaluation

• Advantages
• Uses all information from source data
• Derives inter-slice connectivity, surface
  location, and surface gradient
• Result can be displayed on conventional graphics
  display systems using standard rendering
  algorithms
• Allows Solid modeling capability ( cutting and
  capping)
• Disadvantages
• Requires user input
• Mainly limited to medical images with clear
  contiguous intensity boundaries (constant
  density)
• Is performing a modified form of thresholding.

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References

• W. Lorensen and H. Cline. Marching cubes A High
  Resolution 3D Surface Construction Algorithm,
  Proceedings of SIGGRAPH 1987, pages 163-169,
  1987.
• The Visible Human http//www.nlm.nih.gov/research
  /visible/visible_human.html
• Marching Cubes Demo/Tutorial http//www.essi.fr/
  lingrand/MarchingCubes/applet.html
• Nielson, Gregory M. Dual Marching Cubes, IEEE
  Visualization archive
• Proceedings of the conference on Visualization
  '04, Pages 489 - 496