123 maart 2006

Surface construction techniques for volumetric objects

How to maintain convex and concave features?

Eddy Loke and Erik Jansen

23 maart 2006 2

Introduction

Discretized Marching Cubes builds a surface• Halfway inbetween object and background• Rounding of edges and corners • Can we maintain sharp features?

23 maart 2006 3

Surface construction from voxel data

• Data defined on grid• Surface construction with

MC (linear approximation)• Binary data• Surface construction on

binary data• Discrete Marching Cubes

(DMC)• Discrete version of MC• Less triangles• Manifold• Rounded volumes• Ambiguous cases

4-boundary 8-boundary 8+

Discrete Marching CubesDiscrete Marching Cubes – alt.Marching cubes

8-connectivity4-connectivity

23 maart 2006 4

Ambiguous cases

X-cell:• MC: asymptotic decider• 4-connected

“background priority”• 8-connected

“object priority”• Subgrid

23 maart 2006 5

Boundary construction cell-by-cell

• Apply configurations• We can also apply it on

subgrid• But then we have some

ambiguous cases

23 maart 2006 6

Ambiguous cases

X-cell:• 4-connected

“background priority”

• 8-connected “object priority”

Boundary connectivity and node connectivity are two different things!

23 maart 2006 7

3D

Connectivity:• 6-neighbors• 18 neighbors• 26 neighbors

23 maart 2006 8

3D “Object priority”

P4a P4b

P4d P4e P4g

P4cP3a

P7a

P5a

P5c P6bP6a

P5b

P6c

Kenmochi et al. 99

23 maart 2006 9

Object vs background priority

Object priority Background priority

P6b P6b

Node connectivity 18/26 Node connectivity 6

23 maart 2006 10

Connectivity

P5a - OP P5a - BPP4g - OP P4g - BP

P6c - OP P6c - BP

Some other cases

23 maart 2006 11

Background priority

P3a

P7a

P4a P4b P4c

P4d P4e P4g P5a P5b

P5c P6b P6cP6a

23 maart 2006 12

Object versus background priority

23 maart 2006 13

Convex vs. concave

Object priority: convex edges Background priority: concave edges

23 maart 2006 14

Object versus background priority

23 maart 2006 15

Discrete Marching Cubes

DMC- n DMC- o

DMC- e DMC- k DMC- l

DMC-p

23 maart 2006 16

• Build a DMC configuration by using subgrid triangulation

Discrete Marching Cubes

DMC- p DMC- p DMC- p

23 maart 2006 17

BP versionDMC-p OP-version

DMC triangulation

DMC has background priority triangulation

23 maart 2006 18

• DMC configurations are background priority

• and triangulation is BP as well!

Discrete Marching Cubes

23 maart 2006 19

Results

DMC standard DMC refined

23 maart 2006 20

Concave

P7a – OP/BP P7a – SBPP6a P6a - SBP

23 maart 2006 21

Concave variants

23 maart 2006 22

Convex extrapolation

• DMC rounded corners

• Feature detection• Subgrid

construction• OP-triangulation

Similar for concave edges

23 maart 2006 23

Results

DMC standard DMC refined

23 maart 2006 24

Results

DMC standard DMC refined

23 maart 2006 25

Conclusions

• Flexible triangulation scheme for convex and concave edges

• DMC extended/refined• Discrete models may be of interest for

high resolution models

23 maart 2006 26

Triangulation

Configurations(cell 2x2x2 nodes)

Other possibilities?Black versus white

diagonals

23 maart 2006 27

Two P6b cells with object priorityTwo P6b cells with background priority

Non-manifold

23 maart 2006 28

DMC- e DMC- l

DMC- n DMC- pDMC- o

DMC- k

DMC configurations with BP-triangulation

23 maart 2006 29

Concave variants

23 maart 2006 30

Filters feature detection

23 maart 2006 31

23 maart 2006 32

23 maart 2006 33

23 maart 2006 34

23 maart 2006 35

23 maart 2006 36

23 maart 2006 37

23 maart 2006 38

23 maart 2006 39

23 maart 2006 40

23 maart 2006 41

23 maart 2006 42

?

?

23 maart 2006 43

23 maart 2006 44

23 maart 2006 45

positive triangulations negative triangulations

23 maart 2006 46

Surface construction from voxel data

• Data defined on grid• Surface construction

with MC (continue)• Surface construction

on binary data

Marching cubes Discrete Marching Cubes4-boundary 8-boundary