Surface Reconstruction: Part II - Computer graphicsgraphics.stanford.edu/.../LectureSlides/04_Surface_Reconstruction.pdf · Surface Reconstruction: Part II. Surface ReconstructionSurface

SurfaceSurfaceSurface Surface Reconstruction:Reconstruction:

Part IIPart II

Surface ReconstructionSurface ReconstructionSurface ReconstructionSurface Reconstruction

Today:

Scanning

Today:

Scanning devices

RegistrationImplicit

reconstruction

physicalmodel

acquiredpoint cloud

reconstructedmodelp

2

Input DataInput Datapp

• Set of range scans– Each scan is a regular quad- or tri-

meshNormal vectorsNormal vectors are well defined–– Normal vectors Normal vectors are well defined

– Scans registered in common coordinate system

• Set of irregular sample pointsg p p– Typically without normal vectorswithout normal vectors– More general

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Shape RepresentationsShape RepresentationsShape RepresentationsShape Representations

• Explicit representationp p– Image of parametrization

• Implicit representation– Zero set of distance function

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Implicit RepresentationsImplicit RepresentationsImplicit RepresentationsImplicit Representations

• Level set of 2D function defines 1D curveLevel set of 2D function defines 1D curve

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Explicit / ImplicitExplicit / ImplicitExplicit / ImplicitExplicit / Implicit• Explicit representation

– Image of parameterization– Easy to find points on shape

C d f bl– Can defer problems to paramdomain

• Implicit representation– Zero set of distance function

Easy in/out/distance test– Easy in/out/distance test–– Easy to handle different topologiesEasy to handle different topologies

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• Easy to handle different topologiesEasy to handle different topologies

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• General implicit pfunction:– Interior: F(x,y,z) < 0( ,y, )– Exterior:F(x,y,z) > 0– Surface: F(x,y,z) = 0Surface: F(x,y,z) 0

• Special case– Signed distance function

(SDF)

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Explicit Reconstruction MethodsExplicit Reconstruction Methodspp

• Connect sample points by triangles

• Exact interpolation of sample pointsp p p

• Bad for noisy or misaligned data

• Can lead to holes or non manifold situations• Can lead to holes or non-manifold situations

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Implicit Reconstruction MethodsImplicit Reconstruction Methodspp

• Estimate signed distance function (SDF)

• Extract zero iso-surface

• Approximation of input points

• Watertight manifold results by construction• Watertight manifold results by construction– Can have spurious components

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Input Implicit Explicit

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Implicit Function Approach

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• Define a function 3:f R R→

with value < 0 outside the shape and > 0

:f R R→

the shape and > 0 inside

< 0 > 0014


• Define a function 3:f R R→

with value < 0 outside the shape and > 0 inside

:f R R→

the shape and > 0 inside

• Extract the zero set• Extract the zero-set

{ : ( ) 0}x f x =< 0 > 00

{ : ( ) 0}x f x =

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Signed Distance FunctionSigned Distance FunctionSigned Distance FunctionSigned Distance Function

• Construct SDF from point samplesp p– Distance to points is not enough

– Need inside/outside information/

– Reconstruct normal vectors first

+ -+

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Normal EstimationNormal EstimationNormal EstimationNormal Estimation

• To find normal ni for each sample point pip1. Examine local neighborhood for each point

• Set of k nearest neighborsSet of k nearest neighbors

2. Compute best approximating tangent plane• Covariance analysis• Covariance analysis

3. Determine normal orientation• MST propagation pi• MST propagation pi

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• To find normal ni for each sample point pip1. Examine local neighborhood for each point



3. Determine normal orientation• MST propagation• MST propagation

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Local NeighborhoodLocal NeighborhoodLocal NeighborhoodLocal Neighborhood

• Find k nearest neighbors (kNN) of a pointFind k nearest neighbors (kNN) of a point– Brute force: O(n) complexity

• Use BSP tree– Binary space partitioning treey p p g

– Recursively partition 3D space by planes

– Tree should be balanced, put plane at medianTree should be balanced, put plane at median

– log(n) tree levels, complexity O(log n)

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BSP Closest PointsBSP Closest PointsBSP Closest PointsBSP Closest Points

PA A

DF B C

PA

PBED

PC

F G

E

PC

G

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APA

DF B C

PA

ED F GPB

P

E

PC

G

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APA

B CD

F

PA

ED F GPB

P

E

PC

G

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APA

B CD

F

PA

ED F GPB

P

E

PC

G

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APA

B CD

F

PA

ED F GPB

P

E

PC

G

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Node::dist(Point x, Scalar& dmin)( ){if (leaf_node())

for each sample point p[i]dmin = min(dmin, dist(x, p[i]));

else{

d = dist_to_plane(x);if (d < 0) {{

left_child->dist(x, dmin);if (|d| < dmin) right_child->dist(x, dmin);

} else {{

right_child->dist(x, dmin);if (|d| < dmin) left_child->dist(x, dmin);

}}

}}

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More TreesMore TreesMore TreesMore Trees

Quad-tree (oct-tree)Cells are squares (cubes)

Kd-tree Cells are axis-aligned boxes

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• Find normal ni for each sample point pip1. Examine local neighborhood for each point




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Principal Component AnalysisPrincipal Component AnalysisPrincipal Component AnalysisPrincipal Component Analysis

• Fit a plane with center c and normal n to a set of points {p1, ..., pm}

Mi i i l t• Minimize least squares error

• Subject to non-linear constraintn

Subject to non linear constraintc pi

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Principal Component AnalysisPrincipal Component AnalysisPrincipal Component AnalysisPrincipal Component Analysis

• Plane center is barycenter of pointsy p

• Normal is eigenvector w.r.t. smallest geigenvalue of point covariance matrix

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• Find normal ni for each sample point pip1. Examine local neighborhood for each point




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Naïve Orientation PropagationNaïve Orientation PropagationNaïve Orientation PropagationNaïve Orientation Propagation

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Normal OrientationNormal OrientationNormal OrientationNormal Orientation

• Build graph connecting neighboring pointsu d g ap co ec g e g bo g po s– Edge (ij) exists if pi ∈ kNN(pj) or pj∈ kNN(pi)

• Propagate normal orientation through graph• Propagate normal orientation through graph– For neighbors pi,pj: Flip nj if niTnj < 0– Fails at sharp edges/corners

• Propagate along “safe” paths (parallel normals)Minimum spanning tree with angle based edge weights– Minimum spanning tree with angle-based edge weights wij = 1- | niTnj |

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SDF from Point & NormalsSDF from Point & NormalsSDF from Point & NormalsSDF from Point & Normals

• Signed distance from tangent planes– Points + normals determine local tangent planes

– Use distance from closest center’s tangent planeg p

– Linear approximation in Voronoi cell

– Simple and efficient, but SDF is only C-1Simple and efficient, but SDF is only Cx( )F x

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Smooth SDF ApproximationSmooth SDF ApproximationSmooth SDF ApproximationSmooth SDF Approximation

• Scattered data interpolation problemScattered data interpolation problem– On-surface constraints dist(pi) = 0

Avoid trivial solution dist ≡ 0– Avoid trivial solution dist ≡ 0– Off-surface constraints dist(pi + εni) = ε

11

1

0

0

0 01

10

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Smooth SDF ApproximationSmooth SDF ApproximationSmooth SDF ApproximationSmooth SDF Approximation

• Scattered data interpolation problemScattered data interpolation problem– On-surface constraints dist(pi) = 0

Avoid trivial solution dist ≡ 0– Avoid trivial solution dist ≡ 0– Off-surface constraints dist(pi + εni) = ε

• Radial basis functions– Well suited for smooth interpolationWell suited for smooth interpolation

– Sum of shifted, weighted kernel functions

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Radial Basis FunctionsRadial Basis Functions InterpolationInterpolationRadial Basis FunctionsRadial Basis Functions InterpolationInterpolation

How do we find the weights?

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Radial Basis FunctionsRadial Basis FunctionsRadial Basis FunctionsRadial Basis Functions

2n equations2n equations

2n variables

The on- and off-surface points are the centers ci

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2n equations2n equations

2n variables

The on- and off-surface points are the centers ci

0

00εK w dε

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• Solve the system Kw = d for w• Implicit function is defined as:

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RBF Basis FunctionsRBF Basis FunctionsRBF Basis FunctionsRBF Basis Functions

• Tri-harmonic basis functions

– Globally supported

– Leads to dense symmetric linear systemy y

– C2 smoothness

– Provably smoothProvably smooth

– Works well for highly irregular sampling

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Other Radial Basis FunctionsOther Radial Basis FunctionsOther Radial Basis Functions Other Radial Basis Functions • Polyharmonic spline

• Multiquadratic

• Gaussian

• B-Spline (compact support)

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How Big isHow Big is εε??How Big is How Big is εε??

Without normal length lid ti

With normal length lid tivalidation validation

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ExamplesExamplesExamplesExamples

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ComparisonComparisonComparisonComparison

Distance Compact RBF Global RBFDistanceto plane

Compact RBF Global RBFTriharmonic

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Complexity IssuesComplexity IssuesComplexity IssuesComplexity Issues

• Solve the linear system for RBF weightsSolve the linear system for RBF weights– Hard to solve for large number of samples

• Compactly supported RBFs– Sparse linear system– Efficient solvers

• Greedy RBF fittingGreedy RBF fitting– Start with a few RBFs only– Add more RBFs in region of large errorAdd more RBFs in region of large error

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Extracting the SurfaceExtracting the SurfaceExtracting the SurfaceExtracting the Surface

F(x) = 0 surface

olog

y.co

m

F(x) < 0 inside

e fr

om: w

ww

.farf

ield

tech

no

F(x) > 0 outside

Imag

e46

Sample the SDFSample the SDFSample the SDFSample the SDF

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Sample the SDFSample the SDFSample the SDFSample the SDF

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2D: Marching Squares2D: Marching Squares

?

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3D: Marching Cubes3D: Marching Cubes3D: Marching Cubes3D: Marching Cubes

• Classify grid nodes as inside/outsideClassify grid nodes as inside/outside

• Classify cell: 28 configurations

i i l i l d• Linear interpolation along edges

• Look-up table for patch configuration– Disambiguation more complicated

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Marching CubesMarching CubesMarching CubesMarching Cubes

• Cell classification:Cell classification:– Inside

Outside– Outside

– Intersecting

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Marching CubesMarching Cubes256 cases 15 cases

InversionRotationMarching CubesMarching Cubes

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Marching CubesMarching CubesMarching CubesMarching Cubes

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Marching Cubes ProblemsMarching Cubes ProblemsMarching Cubes ProblemsMarching Cubes Problems

• AmbiguityH l– Holes

• Generates HUGE meshes– Millions of polygons

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AmbiguityAmbiguityAmbiguityAmbiguity

Separate pink

Separate blue

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The Inversion ProblemThe Inversion ProblemThe Inversion ProblemThe Inversion Problem

• Reduction from 256 to 15 cases includesReduction from 256 to 15 cases includes inversion

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The Inversion ProblemThe Inversion ProblemThe Inversion ProblemThe Inversion Problem

Inversion

InversionMismatch

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Ambiguity SolutionAmbiguity SolutionAmbiguity SolutionAmbiguity Solution• 256 cases 23 cases

– Rotation only– Rotation only– Always separate same “color”– Ambiguous faces triangulated consistently

• 8 new cases

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Without InversionWithout InversionWithout InversionWithout Inversion

Match

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Ambiguity

NoAmbiguity

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Ambiguity No Ambiguity

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Marching Cubes IssuesMarching Cubes IssuesMarching Cubes IssuesMarching Cubes Issues

• Grid not adaptiveGrid not adaptive

• Many polygons required to represent small featuresfeatures

Images from: “Dual Marching Cubes: Primal Contouring of Dual Grids” by Schaeffer et al.

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Beyond RBFBeyond RBFBeyond RBFBeyond RBF

• Poisson reconstructionPoisson reconstruction[Kazhdan 06]

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Beyond RBFBeyond RBFBeyond RBFBeyond RBF

• Interactive approachInteractive approach[Sharf, 2007]

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Documents

Surface Reconstruction: Part II - Computer graphicsgraphics.stanford.edu/.../LectureSlides/04_Surface_Reconstruction.pdf · Surface Reconstruction: Part II. Surface ReconstructionSurface