PG 2011
Pacific Graphics 2011
The
19
th
Pacific
Conference
on
Computer
Graphics
and
Applications
(Pacific
Graphics
2011
)
will
be
held
on
September
21
to
23
,
2011
in
Kaohsiung,
Taiwan
.
Feature

Aware Reconstruction of Volume
Data via
Trivariate
Splines
Bo Li and Hong Qin
Stony Brook University
(
State University of New York at Stony
Brook
)
Pacific Graphics 2011
2

Kaohsiung, Taiwan
Volume data Regular domain parameterization
Continuous representation

Accuracy

Feature shape preserving

Structured regularity (Less singularity)

More compactness (Less degree

of

freedoms)
Goal
Pacific Graphics 2011
3

Kaohsiung, Taiwan
Applications
Compression
/Data
reducing
Physical analysis
Vector field/Texturing
Registration
GPU computing
Iso

surface rendering
More…
[Zhang et al. CMAME 2007]
Isogeometric
analysis
[Martin et al. TVCG 2011 ]
Iso

surface rendering
GPU computing
[Kalbe et al. CGF 2008]
Pacific Graphics 2011
4

Kaohsiung, Taiwan
Superspline
/
Boxspline
/…
Hierarchical bounding box
Hexahedral Mesh
Previous Work for Reconstruction
[
Rossl
et al. Visualization 2003]
[
Finkbeiner
et al. C&G 2009]
[Lamar et al. Visualization 1999]
[Wang et al. TVCG 2011]
[Zhang et al. SPM 2004]
[Shepherd 2007]
Irregular/no tensor

product
Pacific Graphics 2011
5

Kaohsiung, Taiwan
Superspline
/
Boxspline
/…
Hierarchical bounding box
Hexahedral Mesh
Previous Work for Reconstruction
[
Rossl
et al. Visualization 2003]
[
Finkbeiner
et al. C&G 2009]
[Lamar et al. Visualization 1999]
[Wang et al. TVCG 2011]
[Zhang et al. SPM 2004]
[Shepherd 2007]
Irregular/no tensor

product
No boundary feature/ continuity
Pacific Graphics 2011
6

Kaohsiung, Taiwan
Superspline
/
Boxspline
/…
Hierarchical bounding box
Hexahedral Mesh
Previous Work for Reconstruction
[
Rossl
et al. Visualization 2003]
[
Finkbeiner
et al. C&G 2009]
[Lamar et al. Visualization 1999]
[Wang et al. TVCG 2011]
[Zhang et al. SPM 2004]
[Shepherd 2007]
Irregular/no tensor

product
No boundary feature/ continuity
No parameter/ Large DOF
Pacific Graphics 2011
7

Kaohsiung, Taiwan
Regular domain parameterization
Hierarchical NURBS representation
Motivation
Regular/ Tensor

product
Preserve boundary feature
Continuous/Small DOF
Pacific Graphics 2011
8

Kaohsiung, Taiwan
Outline
Frame
Field
Volumetric
Mapping
Spline
Fitting
Input
Output
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
9

Kaohsiung, Taiwan
Outline
Frame
Field
Volumetric
Mapping
Spline
Fitting
Input
Output
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
10

Kaohsiung, Taiwan
Outline
Frame
Field
Volumetric
Mapping
Spline
Fitting
Input
Output
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
11

Kaohsiung, Taiwan
Outline
Frame
Field
Volumetric
Mapping
Spline
Fitting
Input
Output
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
12

Kaohsiung, Taiwan
Outline
Frame
Field
Volumetric
Mapping
Spline
Fitting
Input
Output
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
13

Kaohsiung, Taiwan
Outline
Frame
Field
Volumetric
Mapping
Spline
Fitting
Input
Output
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
14

Kaohsiung, Taiwan
Outline
Frame
Field
Volumetric
Mapping
Spline
Fitting
Input
Output
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
15

Kaohsiung, Taiwan
Desirable features: boundary of segmentation
Goal: Integrate feature into frame filed
1. Extract
iso

surfaces
2. Take normal directions on
iso

surfaces
Feature Extraction
[
Haralick
et al. IEEE Systems 2007]
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
16

Kaohsiung, Taiwan
Cube Corners
1.
Pick up 8 corners for each cube.
2.
Shortest path between corners
Feature
Extraction
Isosurfaces
Normal
Directions
Cube
Corners
Pacific Graphics 2011
17

Kaohsiung, Taiwan
Generate a frame field ( )
Requirement: In order to preserve the shape feature,
on the
iso

surface, one of 3 direction
must align with the normal direction.
Frame Field
Pacific Graphics 2011
18

Kaohsiung, Taiwan
Generate a frame field ( )
Requirement: In order to preserve the shape feature,
on the
iso

surface, one of 3 direction
must align with the normal direction.
Frame Field
Pacific Graphics 2011
19

Kaohsiung, Taiwan
Generate a frame field ( )
Requirement: In order to preserve the shape feature,
on the
iso

surface, one of 3 direction
must align with the normal direction.
Frame Field
Pacific Graphics 2011
20

Kaohsiung, Taiwan
Generate a frame field ( )
Requirement: In order to preserve the shape feature,
on the
iso

surface, one of 3 direction
must align with the normal direction.
Frame Field
Pacific Graphics 2011
21

Kaohsiung, Taiwan
Arbitrary alignment Singularity
Cube domain decides alignment of normal directions
Normal Direction
Arbitrary alignment
Iso

value

only alignment
Pacific Graphics 2011
22

Kaohsiung, Taiwan
Arbitrary alignment Singularity
Cube domain decides alignment of normal directions
Normal Direction
Arbitrary alignment
Iso

value

only alignment
Pacific Graphics 2011
23

Kaohsiung, Taiwan
1. Collect Neighbor nodes
2. ICP

based registration
1) Local transformation
2) Transformation covariant matrix
3) SVD decomposition
Frame Field Smoothing
Pacific Graphics 2011
24

Kaohsiung, Taiwan
1. Collect Neighbor nodes
2. ICP

based registration
1) Local transformation
2) Transformation covariant matrix
3) SVD decomposition
Frame Field Smoothing
Pacific Graphics 2011
25

Kaohsiung, Taiwan
1. Collect Neighbor nodes
2. ICP

based registration
1) Local transformation
2) Transformation covariant matrix
3) SVD decomposition
Frame Field Smoothing
Pacific Graphics 2011
26

Kaohsiung, Taiwan
1. Collect Neighbor nodes
2. ICP

based registration
1) Local transformation
2) Transformation covariant matrix
3) SVD decomposition
Frame Field Smoothing
Pacific Graphics 2011
27

Kaohsiung, Taiwan
1. Collect Neighbor nodes
2. ICP

based registration
1) Local transformation
2) Transformation covariant matrix
3) SVD decomposition
Frame Field Smoothing
T
R U V
T
A U SV
Pacific Graphics 2011
28

Kaohsiung, Taiwan
1. Collect Neighbor nodes
2. ICP

based registration
1) Local transformation
2) Transformation covariant matrix
3) SVD decomposition
Frame Field Smoothing
T
R U V
T
A U SV
Pacific Graphics 2011
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Kaohsiung, Taiwan
Input: frame field ( )
Output:
u,v,w
Energy minimization
Volumetric
Parametrization
Pacific Graphics 2011
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Kaohsiung, Taiwan
Energy Minimization
H
I
u
H
u
I
=
=
Pacific Graphics 2011
31

Kaohsiung, Taiwan
Energy Minimization
Known
Unknown
Pacific Graphics 2011
32

Kaohsiung, Taiwan
Energy Minimization
Known
Unknown
Pacific Graphics 2011
33

Kaohsiung, Taiwan
Energy Minimization
Known
Unknown
Pacific Graphics 2011
34

Kaohsiung, Taiwan
Energy Minimization
Known
Unknown
Pacific Graphics 2011
35

Kaohsiung, Taiwan
Constraint on
Iso

parametric faces
Strong

orientation metric
Parametrizations
Less

edge

length metric
Pacific Graphics 2011
36

Kaohsiung, Taiwan
Trivariate
T

Spline
For each
c
ontrol
point :
Knot vectors
Basis functions
Blending function
Formulation
Pacific Graphics 2011
37

Kaohsiung, Taiwan
Recursive Fitting
•
Goal:
–
High accuracy
–
Eliminate superfluous points by T

junction
–
Adaptive control
•
Fitting Minimization Equation:
2
0
  ( ) ( )  
n
H
i i
i
F u I u
Pacific Graphics 2011
38

Kaohsiung, Taiwan
Recursive Fitting
•
Goal:
–
High accuracy
–
Eliminate superfluous points by T

junction
–
Adaptive control
•
Fitting Minimization Equation:
Evaluation
Origin
2
0
  ( ) ( )  
n
H
i i
i
F u I u
Pacific Graphics 2011
39

Kaohsiung, Taiwan
Recursive Fitting
2
0
  ( ) ( )  
n
H
i i
i
F u I u
All control points: Traverse knot vectors
Spline
Fitting: Solve linear equation
All control grid: Compute fitting error
Subdivide control grid with large error
Pacific Graphics 2011
40

Kaohsiung, Taiwan
Results
Pacific Graphics 2011
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Kaohsiung, Taiwan
Results
Pacific Graphics 2011
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Kaohsiung, Taiwan
Results
Pacific Graphics 2011
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Kaohsiung, Taiwan
Results
Pacific Graphics 2011
44

Kaohsiung, Taiwan
Results
Pacific Graphics 2011
45

Kaohsiung, Taiwan
Results
Pacific Graphics 2011
46

Kaohsiung, Taiwan
Conclusion
Volume data Regular domain parameterization
Continuous representation

Accuracy

Feature shape preserving

Structured regularity (Less singularity)

More compactness (Less degree

of

freedoms)
Future Work

Physical
analysis (
Iso

geometric analysis)

Hard

ware acceleration.

More model: complex interior material.
Kaohsiung, Taiwan
Pacific Graphics 2011
.
47

Questions?
Thank You!
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