VOTS VOlume doTS as Point-based Representation of Volumetric Data S. Grimm, S. Bruckner, A. Kanitsar and E. Gröller Institute of Computer Graphics and

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.


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


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


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


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


OutlineOutline


Generation of a VOTGeneration of a VOTTaylor series expansion:

N

PPfPPfPPf

)(

~

!

1)(

~)(

For practical reasons: N = 3

)( and (P) ),(~

),(~

~~ PTHPfPfff


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


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?



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

~


Taking the partial derivatives with respect to the 20 unknowns


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~


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

QQ

QQ

Q

QQ

QQ

Q

M

6/3

1

6/3

1

Inversion of M produces the coefficients


OutlineOutline


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


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


OutlineOutline


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


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


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


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


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

Thank you for your attentionThank you for your attention

Institute of ComputerGraphics and Algorithms

Sponsored by:

Tiani MedGraph AG

Documents

VOTS VOlume doTS as Point-based Representation of Volumetric Data S. Grimm, S. Bruckner, A. Kanitsar and E. Gröller Institute of Computer Graphics and