JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University)

A Tetrahedron Based Volume Model Simplification Algorithm

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.

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.

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.

The Volume Data (1)

Three types of tetrahedrons are defined:– T0-tetrahedron;

– T1-tetrahedron;

– T2-tetrahedron.T0-tetrahedron T1-tetrahedronT2-tetrahedron

Plane E0

Plane E1

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.

hexahedron pentahedron

tetrahedron

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.

Input Data Accepted (2)

A layered tetrahedron model.

All of the tetrahedrons are classified into two categories:– border tetrahedrons;

– non-border (internal) tetrahedrons.

L a y e r N

T e t r a h e d r o n s

L a y e r N + 1

L a y e r N - 1

T e t r a h e d r o n s

V 3

V 1

V 2

V 4

V 5V 6

V 7

V 8

4321 VVVV : n o n - b o r d e r ( i n t e r n a l ) t e t r a h e d r o n ;

8765 VVVV : b o r d e r t e t r a h e d r o n .

Algorithm Description

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

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.

The Main Loop (2)

How to calculate M?

S

L

ii

TST

L

iiSS

TTL

LL

rTTL

T

T

/)1

(/

*)(*

0

0

ST LLM /

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...

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.

Surface Simplification (2)

Vertex removing:

V1 V2

V3

V4V5

V6

Layer(n-1)

Layer(n+1)

Layer(n)

Vr

V1 V2

V3

V4V5

V6

Layer(n-1)

Layer(n+1)

Layer(n)

removing vertex Vr and re-triangulate the remaining hole.

the simplified surface

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.

Hexahedrons Construction (2)

How to calculate N?

sTN

Hexahedrons Construction (3)

What’s 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.

Hexahedrons Construction (4)

An Example:

N*N sub-

hexahedrons

The closing

box of the

orignial data

C CC C

B B B BB

BB B B B

BB

Filling The Resulting Hole (1)

Holes between the simplified surface and hexahedrons we built.

Hexahedron mesh

Simplifed surface

Resulting hole

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.

Filling The Resulting Hole (3)

More detailed discussion...

V00V01

V02

V10V11

M0i

M1i

M0j

M1j

– 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.

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.

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.

Filling The Resulting Hole (4)

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

),(),,(),( 0100010 BMdBMdBMMd iiii Max

2101

2

000

0

0

),(

),(

iT

Bpi

iT

Bpi

vpBMd

vpBMd

Max

Max

Filling The Resulting Hole (5)

V00

V01 V02

V10 V11

M0i

M1i

V02

V11V13

M0j

M1jB0 B1

V12

V03

Decomposition of a polyhedron:

Results

Table : Simplification of a pelvis mode

Index Simplificationrate

Layers Vertices Tetrahedrons

(a) Originalmodel

23 2764 16104

(b) 50% 23 1674 7302

(c) 75% 12 661 2671

(d) 90% 8 344 1227

(e) 95% 6 232 714

(f) 99% 3 84 179

Results

Conclusion

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

Further Research

Apply the algorithm to our virtual surgery simulation system.

Use of multi-resolution object hierarchies in:– collision detection;– cutting;– suturing.

Acknowledgement