JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron...
-
Upload
cecilia-shepherd -
Category
Documents
-
view
215 -
download
0
Transcript of JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron...
![Page 1: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/1.jpg)
JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University)
A Tetrahedron Based Volume Model Simplification Algorithm
![Page 2: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/2.jpg)
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.
![Page 3: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/3.jpg)
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.
![Page 4: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/4.jpg)
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.
![Page 5: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/5.jpg)
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
![Page 6: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/6.jpg)
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
![Page 7: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/7.jpg)
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.
![Page 8: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/8.jpg)
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 .
![Page 9: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/9.jpg)
Algorithm Description
The main loop; Surface simplification; Hexahedron mesh construction; Filling the resulting hole;
![Page 10: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/10.jpg)
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.
![Page 11: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/11.jpg)
The Main Loop (2)
How to calculate M?
S
L
ii
TST
L
iiSS
TTL
LL
rTTL
T
T
/)1
(/
*)(*
0
0
ST LLM /
![Page 12: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/12.jpg)
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...
![Page 13: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/13.jpg)
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.
![Page 14: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/14.jpg)
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
![Page 15: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/15.jpg)
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.
![Page 16: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/16.jpg)
Hexahedrons Construction (2)
How to calculate N?
sTN
![Page 17: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/17.jpg)
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.
![Page 18: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/18.jpg)
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
![Page 19: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/19.jpg)
Filling The Resulting Hole (1)
Holes between the simplified surface and hexahedrons we built.
Hexahedron mesh
Simplifed surface
Resulting hole
![Page 20: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/20.jpg)
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.
![Page 21: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/21.jpg)
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.
![Page 22: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/22.jpg)
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.
![Page 23: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/23.jpg)
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.
![Page 24: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/24.jpg)
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
![Page 25: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/25.jpg)
Filling The Resulting Hole (5)
V00
V01 V02
V10 V11
M0i
M1i
V02
V11V13
M0j
M1jB0 B1
V12
V03
Decomposition of a polyhedron:
![Page 26: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/26.jpg)
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
![Page 27: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/27.jpg)
Results
![Page 28: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/28.jpg)
Conclusion
The strengths of our method:– works for volume data;– can preserve sharp edges;– establish a multi-resolution volume data;– is easy to implement.
![Page 29: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/29.jpg)
Further Research
Apply the algorithm to our virtual surgery simulation system.
Use of multi-resolution object hierarchies in:– collision detection;– cutting;– suturing.
![Page 30: JinJin Hong, Lixia Yan, Jiaoying Shi (State Key Lab. of CAD&CG, Zhejiang University) A Tetrahedron Based Volume Model Simplification Algorithm.](https://reader035.fdocuments.net/reader035/viewer/2022062805/5697bfe11a28abf838cb38bd/html5/thumbnails/30.jpg)
Acknowledgement