Marching Cubes:
A 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
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Organization
1.
2.
3.
4.
5.
6.
7.
Motivation
Related Work
Information Flow
Algorithm
Short Demo
Results
Conclusion & Future Work
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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
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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
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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
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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
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Information Flow
Information Flow for 3D Medical Algorithms
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–
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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.
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Algorithm Overview
•
The Marching Cubes Algorithm
Consists of 3 basic steps:
1. Locate the surface corresponding to a userspecified value.
2. Create triangles.
3. Calculate normals to the surface at each
vertex.
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Step 1:Surface 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) >= 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
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•
•
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.
Δx, Δy, Δz 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.
2.
3.
4.
5.
Scan 2 slices and create cube
Calculate index for cube based on vertices
Use index to lookup list of edges intersected
Use densities to interpolate edge intersections
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/accueil.html)
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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)
1. Locate the surface corresponding to a user-specified
value using cube vertex tests
2. Create triangles based on vertex states of the cube
3. 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