informatik a multi-scale approach to 3d scattered data interpolation with compactly supported basis...
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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
•Multi-scale InterpolationMulti-scale Interpolation
•Results and ProblemsResults and Problems
INFORMATIKINFORMATIK
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.
INFORMATIKINFORMATIK
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)
INFORMATIKINFORMATIK
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.
INFORMATIKINFORMATIKContentsContents
•Single-scale InterpolationSingle-scale Interpolation
•Multi-scale InterpolationMulti-scale Interpolation
•Results and ProblemsResults and Problems
•Single-scale InterpolationSingle-scale Interpolation
•Multi-scale InterpolationMulti-scale Interpolation
•Results and ProblemsResults and Problems
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
INFORMATIKINFORMATIKContentsContents
•Single-scale InterpolationSingle-scale Interpolation
•Multi-scale InterpolationMulti-scale Interpolation
•Results and ProblemsResults and Problems
•Single-scale InterpolationSingle-scale Interpolation
•Multi-scale InterpolationMulti-scale Interpolation
•Results and ProblemsResults and Problems
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
INFORMATIKINFORMATIK
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
INFORMATIKINFORMATIK
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
INFORMATIKINFORMATIK
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
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