INFORMATIK A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis...

INFORMATIKINFORMATIK

AA　　 Multi-scale Approach to Multi-scale Approach to 3D Scattered Data Interpolation with 3D Scattered Data Interpolation with

Compactly Supported Basis FunctionsCompactly Supported Basis FunctionsYutaka OhtYutaka Oht

ake ake 　　Yutaka OhtYutaka Oht

ake ake 　　Alexander Alexander

Belyaev Belyaev 　　

Alexander Alexander

Belyaev Belyaev 　　

Hans-PeterHans-PeterSeidelSeidel 　　

Hans-PeterHans-PeterSeidelSeidel 　　

INFORMATIKINFORMATIKObjectiveObjective

Convert scattered points Convert scattered points into implicit representations into implicit representations f(x,y,z)=0f(x,y,z)=0..Convert scattered points Convert scattered points into implicit representations into implicit representations f(x,y,z)=0f(x,y,z)=0..

f(x,y,z)=0 f(x,y,z)=0 thatthatinterpolates pointsinterpolates pointsScattered pointsScattered points

ConvertConvert

INFORMATIKINFORMATIKImplicit Representation Implicit Representation

Surface: Surface: f(x,y,z)=0f(x,y,z)=0(implicit surface) (implicit surface)

Inside: Inside: f(x,y,z)>0f(x,y,z)>0

Outside: Outside: f(x,y,z)<0f(x,y,z)<0

A cross-section A cross-section of of f(x,y,z)f(x,y,z)

A polygonization A polygonization of of f(x,y,z)=0f(x,y,z)=0

INFORMATIKINFORMATIKAdvantages of ImplicitsAdvantages of ImplicitsConstructive Solid GeometryConstructive Solid Geometry• Union, intersection, difference, blending, embossing, …Union, intersection, difference, blending, embossing, …

Constructive Solid GeometryConstructive Solid Geometry• Union, intersection, difference, blending, embossing, …Union, intersection, difference, blending, embossing, …

==//

blendingblending

==

INFORMATIKINFORMATIKAdvantages of ImplicitsAdvantages of Implicits

Filling missing part of the objectsFilling missing part of the objects• Zero sets of Zero sets of f(x,y,z)f(x,y,z) represents a closed surface. represents a closed surface.

Filling missing part of the objectsFilling missing part of the objects• Zero sets of Zero sets of f(x,y,z)f(x,y,z) represents a closed surface. represents a closed surface.

DEMO

INFORMATIKINFORMATIKPrevious WorksPrevious Works

Using Radial Basis Functions (RBF) Using Radial Basis Functions (RBF) • Muraki et al. 1991Muraki et al. 1991

– Blobby model Blobby model • Savchenko et al. 1995, Turk et al. 1999Savchenko et al. 1995, Turk et al. 1999

– Thin-plate splinesThin-plate splines• Morse et al. 2001Morse et al. 2001

– Compactly supportedCompactly supported piecewise polynomial RBF piecewise polynomial RBF

• Carr et al. 2001Carr et al. 2001– Biharmonic splines and Biharmonic splines and

truncated series expansions truncated series expansions

Using Radial Basis Functions (RBF) Using Radial Basis Functions (RBF) • Muraki et al. 1991Muraki et al. 1991

– Blobby model Blobby model • Savchenko et al. 1995, Turk et al. 1999Savchenko et al. 1995, Turk et al. 1999

– Thin-plate splinesThin-plate splines• Morse et al. 2001Morse et al. 2001

– Compactly supportedCompactly supported piecewise polynomial RBF piecewise polynomial RBF

• Carr et al. 2001Carr et al. 2001– Biharmonic splines and Biharmonic splines and

truncated series expansions truncated series expansions

Can processCan processlarge point setslarge point sets

INFORMATIKINFORMATIKCompactly Supported RBFsCompactly Supported RBFs

Fast, but have several drawbacks.Fast, but have several drawbacks.• Require uniform samplingRequire uniform sampling• Fail to fill holesFail to fill holes• It can be defined in narrow band of original data.It can be defined in narrow band of original data.

(not solid)(not solid)

Fast, but have several drawbacks.Fast, but have several drawbacks.• Require uniform samplingRequire uniform sampling• Fail to fill holesFail to fill holes• It can be defined in narrow band of original data.It can be defined in narrow band of original data.

(not solid)(not solid)

Irregular samplingIrregular sampling Narrow bandNarrow band holesholes

INFORMATIKINFORMATIKProblem of CSRBFsProblem of CSRBFs

We can recognize inside/outside information We can recognize inside/outside information only near the surface. only near the surface.We can recognize inside/outside information We can recognize inside/outside information only near the surface. only near the surface.

??????(Out of support)(Out of support)

InsideInside

OutsideOutside

INFORMATIKINFORMATIKOur ApproachOur Approach

Multi-scale approachMulti-scale approachMulti-scale approachMulti-scale approachPointsPoints manymany

Support sizeSupport size smallsmall

fewfew

largelarge

INFORMATIKINFORMATIKContentsContents

•Single-scale InterpolationSingle-scale Interpolation• Polynomial Basis RBFPolynomial Basis RBF

•Multi-scale InterpolationMulti-scale Interpolation

•Results and ProblemsResults and Problems

•Single-scale InterpolationSingle-scale Interpolation• Polynomial Basis RBFPolynomial Basis RBF




On-surface pointOn-surface point

0

0 0

0 0)( xf

Standard RBF InterpolationsStandard RBF Interpolations

)(||)(||)(

points surface-/off-on

xxx lpfip

ii

),...,2,1()( Nivpf ii Solve linear equations about unknown coefficients Solve linear equations about unknown coefficients

Off-surface pointOff-surface point1

1

1

1

INFORMATIKINFORMATIKBasic Idea of InterpolationBasic Idea of Interpolation1.1. Define local shape implicit functions Define local shape implicit functions 2.2. Blend the functions (weighted sum)Blend the functions (weighted sum)• Solving a sparse linear system.Solving a sparse linear system.

1.1. Define local shape implicit functions Define local shape implicit functions 2.2. Blend the functions (weighted sum)Blend the functions (weighted sum)• Solving a sparse linear system.Solving a sparse linear system.


0),()( vuhwg x

),( vuhw

p

vu,

w

Local Shape FunctionLocal Shape Function

Height functionHeight function in implicit form in implicit form

22),( vcuvbuavuh

n

Least square fittingLeast square fittingto near pointsto near points

),( vuhw :Shift

0)( xg

INFORMATIKINFORMATIKFormulationFormulation

Pp

iii

i

pgf ||)(||)()( xxx

Local shape function in implicit form

Compactly supportedradial basis (blending) function

else 0

1 if 1)(4r)1()

4 rrr

Introduced by Introduced by Wendland 1995Wendland 19952D Graph of 2D Graph of )()( xx ig

UnknownUnknown(Shift amount)(Shift amount)


Results of Results of single-level interpolation single-level interpolation

35K points35K points5 sec.5 sec.

134K points134K points47 sec.47 sec.

Holes remainHoles remainNarrow bandNarrow band

domain domain

INFORMATIKINFORMATIKResults for Irregular SamplingResults for Irregular Sampling

Irregularly sampled Irregularly sampled pointspoints

Many holes remain because of Many holes remain because of small support of basis functions,small support of basis functions,

but large support leads to but large support leads to inefficient computation procedure.inefficient computation procedure.


•Single-scale InterpolationSingle-scale Interpolation






INFORMATIKINFORMATIKAlgorithmAlgorithm1. Construction of a point hierarchy.1. Construction of a point hierarchy.

2. Coarse-to-fine interpolations.2. Coarse-to-fine interpolations.

1. Construction of a point hierarchy.1. Construction of a point hierarchy.

2. Coarse-to-fine interpolations.2. Coarse-to-fine interpolations.

INFORMATIKINFORMATIKConstruction of Point HierarchyConstruction of Point Hierarchy

•Uniform octree based down sampling.Uniform octree based down sampling.• Coordinates and normals are the average of leaf nodes.Coordinates and normals are the average of leaf nodes.

•Final level is decided Final level is decided according to density of points. according to density of points.

•Uniform octree based down sampling.Uniform octree based down sampling.• Coordinates and normals are the average of leaf nodes.Coordinates and normals are the average of leaf nodes.

•Final level is decided Final level is decided according to density of points. according to density of points.

Level 1Level 1(2(233 cells) cells)

Level 2Level 2 Level 3Level 3 Level 4Level 4 Level 5Level 5 Level 6Level 6 Given Given pointspoints

AppendedAppendedto hierarchyto hierarchy

INFORMATIKINFORMATIKCoarse-to-fine interpolationCoarse-to-fine interpolation

0)(1 xkf

)(xko

Level Level k-1k-1 Level Level kk 0)( xkf

1)(),()()x( 01 xxx foff kkk

k

i

k

Ppi

ki

ki

k pgo ||)(||)()( xxx

Same form Same form f f ((xx) ) as in the single scaleas in the single scale

Lkk 75.0,2/1

Diameter ofDiameter ofobjectobject

Level 9(final level)Level 9(final level) Level 8Level 8　　Approximation (error < 2Approximation (error < 2-8-8))544K points544K points

901K functions901K functions

19 min.19 min.332Mbyte332MbytePentium 4 Pentium 4

1.6 GHz1.6 GHz

7.5 min.7.5 min.198Mbyte198Mbyte

363 K functions363 K functions


Comparison with methodComparison with method by Carr[SIG01] (FastRBF) by Carr[SIG01] (FastRBF)

Our method 7 sec.Our method 7 sec.FastRBF 30 sec.FastRBF 30 sec.

OriginalOriginal13K points13K points

Points with normals form Points with normals form a merged mesh by VRIPa merged mesh by VRIP

(Stand scan only)(Stand scan only)

Noise come from Noise come from noisy boundarynoisy boundary

INFORMATIKINFORMATIKIrregular Sampling DataIrregular Sampling Data

90% decimated90% decimatedJoint parts are smoothJoint parts are smooth


Feature Based Feature Based Shape Reconstruction Shape Reconstruction

FeaturesFeatures(ridges and (ridges and

ravines)ravines) Only feature pointsOnly feature pointsare keptare kept

ReconstructionReconstructionresultresult

Inter-Inter-polationpolation

Points with normalsPoints with normalsfrom meshfrom mesh

Points with Points with noisy normalsnoisy normals

PolygonizationPolygonizationf=0f=0


ComplicatedComplicatedTopological ObjectTopological Object

Point Point set set

surfacesurface

Level1Level1

Level6Level6Level5Level5Level4Level4

Level3Level3Level2Level2

INFORMATIKINFORMATIKExtra Zero-setExtra Zero-set

If the object has very thin parts, If the object has very thin parts, extra zero-sets may appear. extra zero-sets may appear.• Octree based down-sampling is not sensitive topological changes.Octree based down-sampling is not sensitive topological changes.• A smart down-sampling procedure is required.A smart down-sampling procedure is required.

If the object has very thin parts, If the object has very thin parts, extra zero-sets may appear. extra zero-sets may appear.• Octree based down-sampling is not sensitive topological changes.Octree based down-sampling is not sensitive topological changes.• A smart down-sampling procedure is required.A smart down-sampling procedure is required.

No extra zero-setNo extra zero-setinside the bounding boxinside the bounding box

Extra zero-sets appear Extra zero-sets appear near thin parts.near thin parts.

INFORMATIKINFORMATIKSharp FeaturesSharp Features

Original meshOriginal meshwith sharp featureswith sharp features

The proposedThe proposedmethodmethod

FastRBFFastRBF(bi-harmonic)(bi-harmonic)

INFORMATIKINFORMATIKShape TexturesShape Textures

From two bunny’s From two bunny’s range datarange data Too smoothToo smoothHoles are filled, Holes are filled,

butbut

INFORMATIKINFORMATIKSummarySummary

•Multi-scale approach to CS-RBFsMulti-scale approach to CS-RBFs• Simple and fast.Simple and fast.• Robust to Robust to

– Irregular samplingIrregular sampling– Quality of normalsQuality of normals

•Future WorkFuture Work• Avoiding extra zero-setsAvoiding extra zero-sets• Sharp featuresSharp features• Shape texture reconstructionShape texture reconstruction

•Multi-scale approach to CS-RBFsMulti-scale approach to CS-RBFs• Simple and fast.Simple and fast.• Robust to Robust to

– Irregular samplingIrregular sampling– Quality of normalsQuality of normals

•Future WorkFuture Work• Avoiding extra zero-setsAvoiding extra zero-sets• Sharp featuresSharp features• Shape texture reconstructionShape texture reconstruction

INFORMATIK A Multi-scale Approach to 3D Scattered Data Interpolation with Compactly Supported Basis...

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