Frame Field - Stony Brook University

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Dec 2, 2013 (3 years and 4 months ago)

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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



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|

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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

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|

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

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|

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



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7

|

Kaohsiung, Taiwan

Regular domain parameterization




Hierarchical NURBS representation

Motivation

Regular/ Tensor
-
product

Preserve boundary feature

Continuous/Small DOF



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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



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|

Kaohsiung, Taiwan

Cube Corners


1.
Pick up 8 corners for each cube.

2.

Shortest path between corners

Feature


Extraction

Isosurfaces

Normal
Directions

Cube

Corners



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|

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






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|

Kaohsiung, Taiwan

Arbitrary alignment Singularity




Cube domain decides alignment of normal directions


Normal Direction

Arbitrary alignment

Iso
-
value
-
only alignment



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|

Kaohsiung, Taiwan

Arbitrary alignment Singularity




Cube domain decides alignment of normal directions


Normal Direction

Arbitrary alignment

Iso
-
value
-
only alignment



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|

Kaohsiung, Taiwan

1. Collect Neighbor nodes

2. ICP
-
based registration


1) Local transformation


2) Transformation covariant matrix


3) SVD decomposition


Frame Field Smoothing



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|

Kaohsiung, Taiwan

1. Collect Neighbor nodes

2. ICP
-
based registration


1) Local transformation


2) Transformation covariant matrix


3) SVD decomposition


Frame Field Smoothing



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|

Kaohsiung, Taiwan

1. Collect Neighbor nodes

2. ICP
-
based registration


1) Local transformation


2) Transformation covariant matrix


3) SVD decomposition


Frame Field Smoothing



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|

Kaohsiung, Taiwan

1. Collect Neighbor nodes

2. ICP
-
based registration


1) Local transformation


2) Transformation covariant matrix


3) SVD decomposition


Frame Field Smoothing



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|

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


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|

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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


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|

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Input: frame field ( )

Output:
u,v,w

Energy minimization

Volumetric
Parametrization



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|

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Energy Minimization



H
I
u
H
u
I
=

=



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|

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Energy Minimization

Known

Unknown



Pacific Graphics 2011

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|

Kaohsiung, Taiwan

Energy Minimization

Known

Unknown



Pacific Graphics 2011

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|

Kaohsiung, Taiwan

Energy Minimization

Known

Unknown



Pacific Graphics 2011

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|

Kaohsiung, Taiwan

Energy Minimization

Known

Unknown



Pacific Graphics 2011

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|

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Constraint on
Iso
-
parametric faces




Strong
-
orientation metric

Parametrizations

Less
-
edge
-
length metric



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|

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Trivariate

T
-
Spline

For each
c
ontrol

point :


Knot vectors



Basis functions


Blending function



Formulation





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|

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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


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|

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


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|

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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



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Results



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Results



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Results



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Results



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Results



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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!