A Tetrahedron Based Volume Model Simplification Algorithm

1
A Tetrahedron Based Volume Model Simplification
Algorithm
JinJin Hong, Lixia Yan, Jiaoying Shi (State
Key Lab. of CADCG, Zhejiang University)
2
Motivation

• Tetrahedron mesh is one of the most popular
  representations of volume model.
• Huge amount of tetrahedrons lead to problems on
  data storage, rendering and computation.

3
Previous Work

• Few researches on volume data simplification have
  been developed until now.
• While many researches on surface data reduction
  have been developed.

• all these algorithms are only available for
  surface simplification either by merging elements
  or by resampling vertices of the original object.

4
Our Work

• Provides a new method to simplify the tetrahedron
  mesh of a volume model.
• The key advantages of our algorithm
• available for volume data
• simple to implement
• high reduction rates and excellent results
• a multi-resolution representation.

5
The Volume Data (1)

• Three types of tetrahedrons are defined
• T0-tetrahedron
• T1-tetrahedron
• T2-tetrahedron.

6
The Volume Data (2)

• The element in a volume data set is a polyhedron.
• A polyhedron can be divided into several T0, T1
  and/or T2 tetrahedrons.
• An example.

7
Input Data accepted (1)

• The input objects that our algorithm can accept
  and process are layered tetrahedrons.
• They are gained from 3D reconstruction of layered
  scanning images (MRI or CT).
• A layered tetrahedron is defined as a tetrahedron
  with vertices only on two adjoining planes
  parallel with each other.

8
Input Data Accepted (2)

• A layered tetrahedron model.
• All of the tetrahedrons are classified into two
  categories
• border tetrahedrons
• non-border (internal) tetrahedrons.

9
Algorithm Description

• The main loop
• Surface simplification
• Hexahedron mesh construction
• Filling the resulting hole

10
The Main Loop (1)

• We adopt a layered simplification approach.
• fetch M layers of tetrahedrons from the input
• manipulate these M layers of tetrahedrons
• output one layer of newly-generated
  tetrahedrons
• when all of the original tetrahedral data are
  processed, we finish.

11
The Main Loop (2)

• How to calculate M?

12
The Main Loop (3)

• How to manipulate these M layers of tetrahedrons?
• a vertex removal approach is introduced to
  simplify border tetrahedrons.
• a number of hexahedrons will be substituted for
  non-border tetrahedrons.
• the resulting hole between the simplified surface
  and substituent hexahedrons is filled with
  tetrahedrons.
• More Detailed Discussion...

13
Surface Simplification (1)

• The border tetrahedron simplification is a
  typical surface simplification algorithm.
• it starts with the original surface and
  successively simplifies it.
• vertices from Layer1 to Layer(M-1) are removed
  and the resulting holes are re-triangulated until
  no further vertices can be removed.
• the triangle mesh left, with all vertices in
  Layer0 or Layer(M) is the simplified surface that
  we need.

14
Surface Simplification (2)

• Vertex removing

removing vertex Vr and re-triangulate the
remaining hole.
the simplified surface
15
Hexahedrons Construction (1)

• We substitute regular hexahedrons for the
  internal tetrahedrons.
• construct a closing box for M layers of original
  tetrahedrons.
• divide the closing box into N sub-hexahedrons.
• adopt all C-hexahedrons, discard A- and
  B-hexahedrons.
• subdivide each of C-hexahedrons into 6
  tetrahedrons as the simplified non-border
  tetrahedrons.

16
Hexahedrons Construction (2)

• How to calculate N?

17
Hexahedrons Construction (3)

• Whats the three types of sub-hexahedrons?
• A-hexahedron
• does not includes any tetrahedrons of original
  model.
• B-hexahedron
• at least includes one border tetrahedron.
• C-hexahedron
• only includes non-border tetrahedrons.

18
Hexahedrons Construction (4)

• An Example

19
Filling The Resulting Hole (1)

• Holes between the simplified surface and
  hexahedrons we built.

20
Filling The Resulting Hole (2)

• How to fill the hole with tetrahedrons?
• a complicated task
• solved by keeping track of the correspondence
  between the simplified surface and hexahedrons.

21
Filling The Resulting Hole (3)

• More detailed discussion...

• start with an arbitrary T0-triangle
• get a triangle-set unit B0
• find an arris nearest to B0
• now that we have got a polyhedron.
• divide it into several T0, T1 and T2-tetrahedrons
  to fill the hole.
• indicate B0 to be used.

22
Filling The Resulting Hole (3)

• More detailed discussion (continued)
• get the next triangle-set B1
• find an arris nearest to B1
• the next polyhedron is got and divided into
  tetrahedrons
• indicate B1 to be used
• to avoid tetrahedron intersecting, each search
  must counterclockwise and resume the previous
  search from the previous ending position.

23
Filling The Resulting Hole (3)

• More detailed discussion (continued)
• a pentahedron composed by that two arrises is
  also divided into three tetrahedrons to fill the
  hole.
• repeat all the previous steps until B0 is reached
  again.

24
Filling The Resulting Hole (4)

• How to compute the distance between an arris and
  a triangle-set?

25
Filling The Resulting Hole (5)

• Decomposition of a polyhedron

26
Results
27
Results
28
Conclusion

• The strengths of our method
• works for volume data
• can preserve sharp edges
• establish a multi-resolution volume data
• is easy to implement.

29
Further Research

• Apply the algorithm to our virtual surgery
  simulation system.
• Use of multi-resolution object hierarchies in
• collision detection
• cutting
• suturing.

30
Acknowledgement