SURFACE RECONSTRUCTION FROM POINT CLOUD

Bo GaoMaster’s ThesisDecember, 2007

Thesis Committee: Professor Harriet FellProfessor Robert Futrelle

College of Computer ScienceNortheastern University

3D Scanner

A 3D scanner is a device that analyzes a real-world object or environment to collect data on its shape and possibly its appearance (i.e. color).

Technology Contact

CMM (Coordinate Measuring Machine) Non-Contact

Laser Structured Light

3D Scanner (Pictures)

Brontes Technologies, A 3M Company

Point Cloud

A point cloud is a set of 3D points describing the outlines or surface features of an object

Stanford 3D scanning repository Brontes’ 3D scanning data Others

Surface Reconstruction

Given a set of sample points X assumed to lie on or near an unknown surface F, create a surface model S approximating F.

Automatic, accurate, concise, piecewise, etc.

CAD designs, medical imaging, visualization, entertainment, reverse engineering, etc.

Related Work

Computer Graphics Hoppe et al. (Zero-set Algorithm) Curless and Levoy (Laser scan)

Computational Geometry Edelsbrunner et al. (α-shape Algorithm) Amenta et al. (Crust & Power Crust

Algorithms)

Computer Graphics Based Approach

Marching Cubes Algorithm Isovalue Normal Normal Consistency Zero-set Algorithm Demo

Computer Graphics Based Approach – Marching Cubes Algorithm

Marching Cubes Algorithm Use a series of logical

cubes, determine how the surface intersects these cubes, and generate a triangle mesh to represent the surface

Isovalue – a scalar value in comparing to the Isosurface

2D example 3D patterns

R3

Computer Graphics Based Approach – Marching Cubes Algorithm (2D)

Computer Graphics Based Approach – Marching Cubes Algorithm (Cont.)

Computer Graphics Based Approach – Marching Cubes Algorithm (Cont.)

Computer Graphics Based Approach – Isovalue

Density Low accuracy Two layers

Signed distance

One layer Accurate

dist p= p−o i⋅n

Computer Graphics Based Approach – Normal

Scanned Normal Data sets from Brontes Technologies

Computed Normal All other data sets online Eigenvector

Create covariance matrix of as follows:

Get eigenvalues , and related eigenvectors , choose as normal

321iii

321iii vvv 33 ii vorv

CV = ∑p ∈ Nbhd x i

p− o i ° p− o i

ni

Nbhd x i

Computer Graphics Based Approach – Normal Consistency

Inconsistent normal problem

Reason

Computer Graphics Based Approach – Normal Consistency (Cont.)

Solution Create a Graph G(V, E) V are all sample points For each edge ,

Use Kruskal’s algorithm to find the MST (Minimum Spanning Tree)

Traverse the MST in DFS order

Evu ),(u ∈ Nbhd v ∨ v ∈ Nhbd u

Computer Graphics Based Approach – Zero-set Algorithm

Separate points into mxmxm blocks according to their coordinates

For each point , compute its ,average center , normal , and make normal consistent

Compute isovalue at each cube vertex

Do Marching Cube process Write triangles into mesh file

x iNbhd x i o i

n i

Computer Graphics Based Approach – Marching Cubes Algorithm (Cont.)

Marching Cubes Trick

Computer Graphics Based Approach – Demo

DEMO

Computational Geometry Based Approach

Voronoi Diagram and Delaunay Triangulation

Pole and Medial Axis Crust Algorithm Demo

Computational Geometry Based Approach – Voronoi Diagram & Delaunay Triangulation

Voronoi Diagram is a subdivision of space

Delaunay triangulation is the dual graph of the Voronoi diagram

Voronoi/Delaunay Applet

Computational Geometry Based Approach – Pole & Medial Axis

Computational Geometry Based Approach – Crust Algorithm

2D Crust Algorithm

Computational Geometry Based Approach – Crust Algorithm (Cont.)

Compute the 8 vertices Z of the surrounding box of the point set S.

Compute the Voronoi diagram of , and get the Voronoi vertices V.

Compute poles for each sample point Let P denote all poles. Compute the

Delaunay triangulation of . Keep only those triangles in which all

three vertices are sample points. Write triangles into final mesh file.

S∪Z

S∪P

Computational Geometry Based Approach – Crust Algorithm (Cont.)

Improvements 8 vertices of the surrounding box

hull – by Ken Clarkson, an open-source convext hull program

qhull – quick hull, 4 times faster than hull, http:www.qhull.org

Computational Geometry Based Approach – Demo

DEMO

Voronoi Based Zero-set Approach

Introduction Pole cube Algorithm Future Work Demo

Voronoi Based Zero-set Approach – Introduction

Use Pole as normal Use Delaunay triangles to get Nghb Lower time complexity than Zero-set

Algorithm

Lower space complexity than Crust Algorithmqhull needs more than 2GB memory for the second round computation of “Buddha” (543k points).

)lg( VS )lg'lg( 323 mnnnnOmnnnvnO

Voronoi Based Zero-set Approach – Pole cube Algorithm

Compute the 8 vertices Z of the surrounding box of the point set S.

Compute the Voronoi diagram of , and get the Voronoi vertices V and Delaunay triangles DT.

Separate points according to their coordinates into blocks.

Compute pole for each sample point, and set the normal as pole.

Compute Nghb from DT. Make normals consistent. Compute Isovalue at each cube vertex. Do Marching Cubes process. Write triangles into mesh file.

S∪Z

m×m×m

Voronoi Based Zero-set Approach – Future Work

Label each pole inside or outside of sample S.

Make normal consistent without finding k-Nbhd

Using “power diagram” or “eigenvector” to separate poles described in two recent papers.

Voronoi Based Zero-set Approach – Demo

DEMO

Comparisons

Scanned normal VS. Computed normal

Performance (speed) Surface Quality

Comparisons – Scanned Normal VS. Computed Normal

D of Zero-set D of Pole Cube

Point Cloud k = 6 k = 8 k = 6 k = 8

Horse 0.00722519 0.00531292 0.00793274 0.00793274

Die 0.00842918 0.0053838 0.00640121 0.00640121

Face(121) 0.121609 0.0733952 0.0661861 0.0661861

Face(124) 0.0270493 0.0129876 0.0171255 0.0171255

Wheel(125) 0.0379457 0.0249135 0.0307598 0.0307598

Weapon(126) 0.0503735 0.0396337 0.0516142 0.0516142

Weapon(127) 0.112135 0.0527464 0.0591031 0.0591031

D=∑ 1− C⋅S × C⋅S / n , where C is the computed normal ,

S is the scanned normal , n is the mumber of sample points.

Comparisons – Performance (speed)

Point Cloud Points Crust Zero-set Pole CubeBasic K=8, 400 K=8, 400

Triangles Time Triangles Time Triangles TimeBunny 35947 232082 21.536s 1111425 1m20.083s 1150689 1m21.278sFace 52338 319840 31.735s 457513 1m14.905 697204 1m25.954Wheel 61785 385450 47.050s 459843 1m20.034s 621637 1m30.269Horse 101358 630082 1m0.823s 310644 2m10.432s 319433 2m5.349sDie 104038 610074 57.957s 501109 2m6.128s 493507 1m57.465sWeapon 162504 1024956 1m36.994s 577345 3m45.271 603768 3m10.488sDragon 437645 N/A N/A 842249 17m49.813s 853484 14m54.365sBuddha 543652 N/A N/A 668717 29m25.840s 752417 20m29.280s

Comparisons – Surface QualityCrust Zero-set (k=8, 3400 ) Pole Cube (k=8, 3400 )

Comparisons – Surface Quality (Cont.)

Comparisons – Surface Quality (Cont.)

Comparisons – Surface Quality (Cont.)

Surface Viewer

The “Surface Viewer” is for viewing and analyzing generated surfaces. This tool was written in C++ with OpenGL as the display API. Basic functions of the tool are: rotation, scale, filled/wired polygon view mode, surface/point view mode, pole view mode (for “Crust” algorithm), and normal comparison mode (for “Zero-set” and “Pole Cube” algorithms).

Demo

Questions

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