VOTS VOlume doTS as Point-based Representation of Volumetric Data S. Grimm, S. Bruckner, A. Kanitsar...
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Transcript of VOTS VOlume doTS as Point-based Representation of Volumetric Data S. Grimm, S. Bruckner, A. Kanitsar...
VOTSVOTS VOVOlume dolume doTSTS as Point-based as Point-based Representation of Volumetric Representation of Volumetric
DataData
S. Grimm, S. Bruckner, A. Kanitsar and E. GröllerS. Grimm, S. Bruckner, A. Kanitsar and E. Gröller
Institute of Computer Graphics and AlgorithmsInstitute of Computer Graphics and AlgorithmsVienna University of TechnologyVienna University of Technology
Vienna, AustriaVienna, Austria
Sören Grimm Vienna University of Technology
Motivation (1/3)Motivation (1/3)Volumetric data: Volumetric data: Processing is sampled basedProcessing is sampled based Given on grid structure, e.g. regular gridGiven on grid structure, e.g. regular grid
Advantages:Advantages: Efficient spatial addressing Efficient spatial addressing Efficient processing, such as rendering, Efficient processing, such as rendering,
segmentation, etc.segmentation, etc.
Sören Grimm Vienna University of Technology
Motivation (2/3)Motivation (2/3)Rigid shapeRigid shape of grids is a of grids is a limitation factor:limitation factor: Sizes are enormously increasingSizes are enormously increasing Often just parts are of interestOften just parts are of interest Difficult to perform deformationsDifficult to perform deformations Difficult to analytically analyze the dataDifficult to analytically analyze the data
Sören Grimm Vienna University of Technology
Motivation (3/3)Motivation (3/3) Information dependent storage requirementInformation dependent storage requirement Allow to leverage resourcesAllow to leverage resources
Inhomogeneous→ grid is efficient
Partially inhomogeneous→ grid is inefficient
Sören Grimm Vienna University of Technology
What is a VOT (1/2)What is a VOT (1/2)A VOT is basically a thick Volumetric PointA VOT is basically a thick Volumetric Point Represents a region by polynomialRepresents a region by polynomial It consists of:It consists of:
Coefficients of Taylor series, position, Coefficients of Taylor series, position, and a validity areaand a validity area
100 samples 7 VOTS
Sören Grimm Vienna University of Technology
What is a VOT (2/2)What is a VOT (2/2)
P
QP
N
PPf
PPfQ
)(
~
!
1
)(~
)(VOT
VOT Evaluation: Evaluation of Taylor series expansion
Sören Grimm Vienna University of Technology
Generation of a VOTGeneration of a VOTTaylor series expansion:
N
PPfPPfPPf
)(
~
!
1)(
~)(
For practical reasons: N = 3
)( and (P) ),(~
),(~
~~ PTHPfPfff
Sören Grimm Vienna University of Technology
Cell to VOT conversionCell to VOT conversion
xyz
yzxz
yzxy
xzxy
jiP
jiP
kiP
kiP
kjP
kjP
f
ff
ff
ff
ff
ff
ff
ijij
kiki
jkjk
~0
~~
~0
~
~~0
2
1
4
1
,,
,,
,,
01
01
01
)(~Pf
)(~ PHf
)(~ PTf
ijkP
ijk
f
P
8
18
1P
)(~Pf
001Pf
011Pf
010Pf
000Pf
111Pf
101Pf
110PfCell
100Pf
Sören Grimm Vienna University of Technology
Point cloud to VOT conversion (1/4)Point cloud to VOT conversion (1/4)
),(jQj fQm scattered data points
VOT:
3
)(~
!
1)(
~
PPfPPf
Function Fitting?
Sören Grimm Vienna University of Technology
Point cloud to VOT conversion (2/4)Point cloud to VOT conversion (2/4)
Original data valuesReconstructed values by Taylor series
Linear regression: Minimizing sum-of-squares
m
j
E1
2) ()( )(~
jQf jQf
20 unknowns, due to symmetry
→fff ijkiji ~,
~,
~
3
)(~
!
1
PPff
~
Sören Grimm Vienna University of Technology
Taking the partial derivatives with respect to the 20 unknowns
Point cloud to VOT conversion (3/4)Point cloud to VOT conversion (3/4)
m
j
yj
xjQj
xyz
m
j
yj
xjQj
xy
m
j
xjQj
x
m
jQj
QQfQPff
E
QQfQPff
E
QfQPff
E
fQPff
E
j
j
j
j
1 3
2
1 3
1 3
1 3
))(~1
(2~
))(~1
(2~
))(~1
(2~
))(~1
(2~
Sören Grimm Vienna University of Technology
Point cloud to VOT conversion (4/4)Point cloud to VOT conversion (4/4)Setting derivatives to zero, leads to a system of linear equations:
m
j
yj
xjQ
m
j
yj
xjQ
m
j
xjQ
m
j Q
xxy
xy
x
QQf
QQf
Qf
f
f
f
f
f
M
j
j
j
j
1
2
1
1
1
~
~
~
~
~
~
~
~
T
yj
xj
yj
xj
xj
yj
xj
yj
xj
xj
Q
Q
M
6/3
1
6/3
1
Inversion of M produces the coefficients
Sören Grimm Vienna University of Technology
Grid to VOTS conversion (1/2)Grid to VOTS conversion (1/2)Growing approach:Growing approach: For all sample positions For all sample positions growgrow
a VOTa VOT Size of VOT bounded by Size of VOT bounded by
specified max errorspecified max error Outcome: VOT Outcome: VOT candidatescandidates A small A small subsetsubset of these of these
VOTS is chosen, so that they VOTS is chosen, so that they completely cover the completely cover the underlying volumetric dataunderlying volumetric data
Sören Grimm Vienna University of Technology
Grid to VOTS conversion (2/2)Grid to VOTS conversion (2/2) Goal: small number of large VOTS covering Goal: small number of large VOTS covering
entire volume (small overlap)entire volume (small overlap)
Sort VOT candidates according to sizeSort VOT candidates according to size
While space not coveredWhile space not covered
Select largest VOT candidateSelect largest VOT candidate
Update size of remaining candidatesUpdate size of remaining candidates
Re-sort VOT candidates Re-sort VOT candidates
Sören Grimm Vienna University of Technology
Maximum Intensity Projection (1/2)Maximum Intensity Projection (1/2)Maximum value along a ray:Maximum value along a ray: Regular gridRegular grid →→ sample basedsample based determined determined VOTS VOTS →→ analyticallyanalytically determined determined
Sören Grimm Vienna University of Technology
Maximum Intensity Projection (2/2)Maximum Intensity Projection (2/2) For all VOTS For all VOTS
MIP textures are createdMIP textures are created Send to graphics hardwareSend to graphics hardware
Graphics hardware is used to perform Graphics hardware is used to perform maximum-blendingmaximum-blending
Viewingdirection
Sören Grimm Vienna University of Technology
ResultsResults
-error-error 0.01%0.01% 0.1%0.1% 1%1% 10%10%
Head: #VOTsHead: #VOTs 570 K570 K 538 K538 K 333 K333 K 35 K35 K
Lobster: #VOTsLobster: #VOTs 114 K114 K 114 K114 K 112 K112 K 60 K60 K
Head 1.7 M samples Lobster 500K samples
VOTS Distribution & Maximum Intensity Projection
Sören Grimm Vienna University of Technology
ConclusionConclusionWe proposed a new representation of We proposed a new representation of
volumetric data: volumetric data: VOTSVOTS Intuitive and constructive representationIntuitive and constructive representation Allow user-centric importance samplingAllow user-centric importance sampling Allow to leverage resources Allow to leverage resources Information dependent storage requirementInformation dependent storage requirement Allow to analytically process the dataAllow to analytically process the data No connectivity for reconstruction is needed No connectivity for reconstruction is needed
Sören Grimm Vienna University of Technology
Future WorkFuture Work Conversion of other grid/data structuresConversion of other grid/data structures Sparse volumesSparse volumes More sophisticated conversion techniqueMore sophisticated conversion technique Blending between VOTSBlending between VOTS Efficient rendering methodEfficient rendering method Exploit derivative informationExploit derivative information New visualization techniquesNew visualization techniques Investigate filter possibilitiesInvestigate filter possibilities