Tetra-Cubes: An algorithm to generate 3D isosurfaces based upon tetrahedra

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Tetra-Cubes An algorithm to generate 3D
isosurfaces based upon tetrahedra

• BERNARDO PIQUET CARNEIRO
• CLAUDIO T. SILVA
• ARIE E. KAUFMAN
• Department of Computer Science
• Department of Applied Mathematics Statistics
• State University of New York at Stony Brook
• Anais do IX SIBGRAPI(1996)

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Abstract

• Tetra-cubes is similar to the marching-cubes
  technique.
• Its basic building block consists of tetrahedral
  cells instead of cubes.
• Tetra-cubes is simpler than marching-cubes and
  does not have ambiguity configurations.
• Irregular Grid

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Introduction

• There are several methods for the visualization
  of 3D scalar fields.
• The two most common are
• direct volume rendering
• 3D iso-surface extraction
• In this paper we study iso-surface extraction
  techniques.

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Comparison

• Direct volume rendering vs. Iso-surface
  extraction
• Interactivity
• Triangle meshes(speed, simplify)
• Flexibility(especially for handling multiple
  transparent surface)
• Ambiguity problem

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Tetra-Cubes Algorithm

• Following the idea in the marching-cubes.
• Extracting the tetrahedra from the cubes.
• (Each cube can be divided into a collection of
  five tetrahedra.)

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Marching-Tetras vs. Marching-Cubes

• Classification
• 24 16
• By symmetry, we narrow down these cases to only
  three.

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Grid Connection problem

• A problem with the connection among the
  collections of five tetrahedra extracted from
  neighboring hexahedra arises.
• Erroneous Connection among adjacent cubes.

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Grid Connection problem

• Resolution
• We create a new collection that is obtained by a
  ninety degrees rotation of the previous one.
• Zig-zag connection (chess-table configuration)

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No Ambiguities

• The binary approach used to distinguish the cases
  leads to some ambiguities in the marching-cubes
  implementation.
• No ambiguity is found usingtetrahedra as basic
  cells.
• A fast implementation may be accomplished
  combiningthe marching-cubes and thetetra-cubes
  algorithms.
• (Cubes would be used as basic cells.)

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Code

• Data input
• Extraction of tetrahedra
• For each logical cube(hexahedra), the cubes
  vertices are used as the tetrahedrons vertices.
• Accessing the look-up tables
• Triangles creation

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Results

• Clock time to generate the iso-surfaces using
  tetra-cubes algorithm.
• Function density value 128
• Using SGI Indigo2 machine

Size
70 x 70 x 70
19 x 21 x 19
34 x 17 x 34
127 x 67 x 67
40 x 32 x 32
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Results
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Results
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Results
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Conclusion

• Relative to Marching-Cubes
• Our method is simpler to implement due to the
  smaller number of cases.
• It does not have the ambiguity problem.
• One possible shortcoming of our approach is the
  large number of triangles generated.
• Fortunately, this can be solved with
  simplification techniques, such as decimation.

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Decimation of Triangle Meshes

• William J. Schroeder
• Jonathan A. Zarge
• William E. Lorensen
• General Electric Company
• Schenectady, NY
• SIGGRAPH(1991)

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Introduction

• The polygon remains a popular graphics primitive
  for computer graphics application.
• Simple representation
• Rendering of polygons is widely supported by
  commercial graphics hardware and software.
• Rendering speeds and memory requirements are
  proportional to the number of polygons.
• One algorithm that generates many polygons is
  marching cubes.
• Brute force surface construction algorithm.

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The Decimation Algorithm

• The goal of this algorithm
• Reduce the total number of triangles in a
  triangle mesh.
• Preserving the original topology and good
  approximation to the original geometry.

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Overview

• The three steps of the algorithm
• During a pass, perform three steps on every
  vertex on mesh
• 1. Characterize the local vertex geometry and
  topology,
• ( Simple, Complex, Boundary, Interior Edge,
  Corner )
• 2. Evaluate the decimation criteria, and
• If it meets the specified decimation criteria,
  the vertex and all triangles that use the vertex
  are deleted.
• Until some termination condition is met.
• a. Percent reduction of the original mesh.
• b. Maximum decimation value.
• 3. Triangulate the resulting hole.

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Characterizing Local Geometry / Topology

• The outcome of this process determines whether
  the vertex is a potential candidate for deletion.
• Each vertex may be assigned one of five possible
  classifications.
• Complex vertices are not deleted from the mesh.

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Simple vertex and Complex vertex

• simple vertex is
• surrounded by a complete cycle of triangles,
• and each edge that uses the vertex is used by
  exactly two triangles.
• complex vertex
• If the edge is not used by two triangles,
• or if the vertex is used by a triangle not in the
  cycle of triangles.

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Boundary vertex

• A vertex that is on the boundary of a mesh,
• i.e. within a semi-cycle of triangles.

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Feature Edge

• Feature edge exists if the angle between the
  surface normals of two adjacent triangles is
  greater than a user-specified feature angle.

If ? gt ?fa E Feature Edge
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Interior Edge vertex and Corner vertex

• When a vertex is used by two feature edges, the
  vertex is an interior edge vertex.
• If one or more feature edges use the vertex, the
  vertex is classified a corner vertex.

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Evaluating the Decimation Criteria

• Simple vertices use the distance to plane
  criterion.
• If the vertex is within the specified distance to
  the average plane it may be deleted.Otherwise it
  is retained.

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Evaluating the Decimation Criteria

• Boundary and interior edge vertices use the
  distance to edge criterion.
• If the distance to the line is less than
  specified distance, the vertex can be
  deleted.Otherwise it is retained.

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Evaluating the Decimation Criteria

• Corner vertices are usually not deleted to keep
  the sharp features.

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Triangulation

• Deleting a vertex and its associated triangles
  creates one(simple or boundary vertex) or two
  loops(interior edge vertex).
• It is desirable to create triangles with good
  aspect ratio and that approximate the original
  loops as closely as possible.

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Triangulation

• A recursive loop splitting procedure
• (Divide-and-conquer algorithm)
• If two loops do not overlap, the split plane is
  acceptable.
• Each new loop is divided again, until only three
  vertices remain in each loop.

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Triangulation

• Best split plane is determined using an aspect
  ratio.
• The best splitting plane is the one that yields
  the maximum aspect ratio.

Min. distance of the loop vertices to the split
plane
Aspect Ratio
Length of the split line
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Triangulation

• Special Cases
• Closed surface
• Topological holes in the mesh

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Results

• Volume Modeling (Bone surface)
• Size 25625693
• Triangle over 560,000 triangles
• Full Resolution Gouraud Shaded
• 75 decimated Gouraud Shaded
• 75 decimated Flat Shaded
• 90 decimated Flat Shaded

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Results
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Results
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Results
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Results

• Volume Modeling (industrial CT dataset)
• Size 512512300
• Triangle 1.7 million triangles
• 75 decimated Flat Shaded
• 90 decimated Flat Shaded

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Results
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Results