Dual Marching Cubes: An Overview

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Dual Marching Cubes An Overview

• Paper by Gregory M. Nielson

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Purpose

• Improve existing Marching Cube algorithm
• Eliminate or reduce poorly shaped triangles
• Eliminate or reduce wonky specular highlights

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Motivation

• Duality Principal of Projective Geometry
• All propositions occur in dual pairs.
• One can infer the corresponding proposition of a
  pair by interchanging the words point and line.
• Dual Polyhedra
• For every polyhedron there is another polyhedron
  where faces and vertices occupy complementary
  locations.

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Examples
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What Is Needed

• Develop a correspondence between the surface
  elements of a mesh and the vertices of some other
  mesh.
• Both meshes must have specific properties for
  this to occur.

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Creating a Dual Surface

• Create a patch surface realized by eliminating
  the edges of the interior surface to the voxels.
• Make each patch of the new surface polygon
  bounded.
• Include each vertex of the original MC surface in
  exactly four patches.
• Call this surface S.

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Creating a Dual Surface, pt. 2

• The dual surface will consist of quad patches.
• The connectivity of the patches is the same as
  the connectivity of the vertices of the surface S
• Now exploit duality principal.

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Marching Cubes Defined

• Input Fi,j,k F(i?x, j?y, k?z)
• Denote the voxel (i?x, j?y, k?z) to ((I1)?x,
  (j1)?y, (k1)?z)
• By Bijk
• Separate lattice points with Fi,j,k gt a

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Defining the Patch

• The patch surface has a fine structure given that
  it consist of three mutually orthogonal planar
  curves.
• Let Fj where j 1, , M denote the patches of S
  and
• Fj (Vj1, V,j2 , Vj3) where
• j 1, , M
• denote the vertices of each patch.

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Defining the Patch, cont.

• Each vertex of S lies on the edge of the lattice.
• Four voxels share each edge of the lattice.
• Thus each vertex, Vj, has exactly four patches of
  S that contain it.

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Defining the Patch, cont.

• Let X(X) be the space of surfaces produced by
  allowing the vertices of S to slide anywhere
  along their respective lattice edges.
• The edges (of the lattice) joining a point in X
  to points not in X are the same edges containing
  the vertices of the MC surface.

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Defining the Dual

• For S in X(X) S? is a surface comprised of quad
  patches with the following properties
• For each patch Fj of S there is a vertex of S?
  lying in the interior of the voxel containing Fj
• For every vertex of S there is one quad patch Pj
  of S?
• For every edge of S there is an associated edge
  of S?

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Four Adjacent Patches
Dual and Patch Surface
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Summary of Definitions

• The topology for both the patch space, X(X), and
  the dual space ?(X) is completely determined by
  the subset X of the lattice L
• The connectivity of the vertices of X(X) is the
  same as the connectivity of the patches of ?(X)
  and vice versa.
• There is a one-to-one coorespondance between the
  edges of S and S?. The edge joining two vertices
  of S cooresponds to the common edge of their
  associated quad patches.

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Producing the Dual Directly

• It is not necessary to produce the patch surface
  and then the dual surface afterwards.
• There is a triangulation (quadulation?) table for
  computing the dual surface directly.

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MC Triangulator
Dual Rectangulator
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Examples of the Dual Surface

• Splatting

Mesh Viewing
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Improvement of Specular Lighting
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Viewing Segmented Volumes

• The original volume with the samples of the
  function is not needed in order to apply the
  algorithm.
• Using the separating midpoint surface we can
  still view the mesh using the algorithm

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Utilizing the Duality Principal

• Since the correspondance is one-to-one we have an
  inverse mapping back into the patch space
• Triangulation of Quadrilaterals will need to
  occur.

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To Infinity and Beyond

• Repeated application of the v operator leads to
  increasing approximation of the original object.
• Eventually the v operator has no effect on the
  surface produced.
• This staged is called the fixed-point shroud.

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Summary Pros

• A definite refinement over the standard MC
  algorithm.
• Reduces visual artifacts from elongated triangles
  and improves specular lighting.
• Allows the user to define the level of precision
  in the final mesh.

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Summary Cons

• Not all surfaces produced are manifolds.
• Normals, Curvature, etc
• Simplification, Multiresolution, etc.

• Quad rendering is somewhat slower.

• Complexity?
• Parallelism?