Surface Reconstruction: Part II - Computer...

SurfaceSurfaceSurface Surface Reconstruction:Reconstruction:

Part IIPart II

Surface ReconstructionSurface ReconstructionSurface ReconstructionSurface Reconstruction

Today:

Scanning

Today:

Scanning devices

RegistrationImplicit

reconstruction

physicalmodel

acquiredpoint cloud

reconstructedmodelp

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

Shape RepresentationsShape RepresentationsShape RepresentationsShape Representations

• Explicit representationp p– Image of parametrization

• Implicit representation– Zero set of distance function

Implicit RepresentationsImplicit RepresentationsImplicit RepresentationsImplicit Representations

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

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

– Image of parameterization– Easy to find points on shapeC d f bl– Can defer problems to paramdomain

• Implicit representation– Zero set of distance functionEasy in/out/distance test– Easy in/out/distance test

–– Easy to handle different topologiesEasy to handle different topologies


• Easy to handle different topologiesEasy to handle different topologies


• 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)

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

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

Input Implicit Explicit

Implicit Function Approach


• Define a function 3:f R R→

with value < 0 outside the shape and > 0

:f R R→

the shape and > 0 inside

< 0 > 00


• 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 =

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

+ ‐+

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


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



3. Determine normal orientation• MST propagation• MST propagation

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)

BSP Closest PointsBSP Closest PointsBSP Closest PointsBSP Closest Points

PA A

DF B C

PA

PBED

PC

F G

E

PC

G


APAD

F B C

PA

ED F GPBP

E

PC

G


APA

B CD

F

PA

ED F GPBP

E

PC

G


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);

}}

}}


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




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

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


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




Naïve Orientation PropagationNaïve Orientation PropagationNaïve Orientation PropagationNaïve Orientation Propagation

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 wij = 1– Minimum spanning tree with angle‐based edge weights wij = 1-| niTnj |

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

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

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

Radial Basis FunctionsRadial Basis Functions InterpolationInterpolationRadial Basis FunctionsRadial Basis Functions InterpolationInterpolation

How do we find the weights?

Radial Basis FunctionsRadial Basis FunctionsRadial Basis FunctionsRadial Basis Functions

2n equations2n equations

2n variables

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


2n equations( ) ( )( ) 0j i i j i j idist w wϕ ϕ= = − =∑ ∑p p p c 2n equations( ) ( )

( )( )

( ) 0

( )

j i i j i j ii i

j j i j j i

dist w w

dist w

ϕ ϕ

ε ϕ ε ε+ = + − =

∑ ∑

∑

p p p c

p n p n c 2n variables( )i

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

0

00εK w dε


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

RBF Basis FunctionsRBF Basis FunctionsRBF Basis FunctionsRBF Basis Functions

• Tri‐harmonic basis functions

– Globally supported– Leads to dense symmetric linear system– C2 smoothness– Provably smooth

– Works well for highly irregular samplingWorks well for highly irregular sampling

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

• Multiquadratic

• Gaussian

• B‐Spline (compact support)

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

Without normal length lid ti

With normal length lid tivalidation validation

ExamplesExamplesExamplesExamples

ComparisonComparisonComparisonComparison

Distance Compact RBF Global RBFDistanceto plane

Compact RBF Global RBFTriharmonic

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

Extracting the SurfaceExtracting the SurfaceExtracting the SurfaceExtracting the Surface

F(x) = 0 surface

ology.com

F(x) < 0 inside

e from

: www.fa

rfieldtechno

F(x) > 0 outside

Image

Sample the SDFSample the SDFSample the SDFSample the SDF

2D: Marching Squaresg q

?

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

Marching CubesMarching CubesMarching CubesMarching Cubes

• Cell classification:Cell classification:– Inside

Outside– Outside

– Intersecting

Marching CubesMarching Cubes256 cases 15 cases

InversionRotationMarching CubesMarching Cubes

Marching CubesMarching CubesMarching CubesMarching Cubes

Marching Cubes ProblemsMarching Cubes ProblemsMarching Cubes ProblemsMarching Cubes Problems

• AmbiguityH l– Holes

• Generates HUGE meshes– Millions of polygons

AmbiguityAmbiguityAmbiguityAmbiguity

Separate pink

Separate blue

The Inversion ProblemThe Inversion ProblemThe Inversion ProblemThe Inversion Problem

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

The Inversion ProblemThe Inversion ProblemThe Inversion ProblemThe Inversion Problem

Inversion

InversionMismatch

Ambiguity SolutionAmbiguity SolutionAmbiguity SolutionAmbiguity Solution• 256 cases 23 cases

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

• 8 new cases

Without InversionWithout InversionWithout InversionWithout Inversion

Match

Ambiguity

Nodemo Ambiguity

Ambiguity No Ambiguity

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.

Beyond RBFBeyond RBFBeyond RBFBeyond RBF

• Poisson reconstructionPoisson reconstruction[Kazhdan 06]

Beyond RBFBeyond RBFBeyond RBFBeyond RBF

• Interactive approachInteractive approach[Sharf, 2007]

Surface Reconstruction: Part II - Computer...

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