# Geometric Modeling and Processing

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1 Geometric Modeling and Processing Tutorial of 3DIM&PVT 2011 (Hangzhou, China) May 16, 2011

2 About Presenter Prof. Dr. Ligang Liu Zhejiang University Research Interests Geometric modeling and processing Computer graphics Image processing Homepage (google Ligang Liu )

3 Goal of This Tutorial Present the geometry processing pipeline in computer graphics (and computer vision) Fundamental concepts Recent developments Show interesting connection between computer graphics and computer vision No big gap between these two fields Merging into one field

4 Introduction Topics Data acquisition Shape representations Geometric registration Surface smoothing Shape simplification Surface editing Surface segmentation Other topics Conclusion and wrap up Outline

5 1. Introduction

6 Digital Multimedia Sound Image Video Geometry The fourth wave of digital multimedia

7 Digitized 3D objects Digital Media

8 Digitized 3D Objects

9 Geometry Modeling and Processing 3D Geometry is based data for CAD/CAM, Engineering Visualization and simulation - medical, physics, etc... Graphics, multimedia Geometry processing Computerized modeling of 3D geometry

10 Digital Geometry Processing (DGP) Processing of discrete models Polygonal mesh (Typically triangular) Why discrete? Simplicity - ease of description Based data for rendering software/hardware Input to most simulation/analysis tools Output of most acquisition tools laser scanner, CT, MRI, etc...

11 Applications Medical Engineering Topography Simulation Game Movies E-commerce Art history

12 2. Data Acquisition

13 Getting Meshes from Real Objects

14 Getting Meshes from Real Objects Many models used in Graphics are obtained from real objects Well known Stanford bunny model?

15 Reverse Engineering Real Object CAD/Graphics Model Build new object

16 Different Acquisition Systems Volumetric scanning Photogrammetry Range scanning

17 2.1 Volume Scanning Build voxel structure by scanning slices

18 Volume Scanning Build voxel structure by scanning slices

19 2.2 Photogrammetry Reconstruction from photographs

20 Photogrammetry Reconstruction from a series of photos (video)

21 2.3 Range Scanning Reconstruction from point cloud Physical real model Acquired point cloud Reconstructed model

22 Range Scanning Systems Passive: stereo matching

23 Range Scanning Systems Active: structured light acquisition

24 Range Scanning Systems Active: laser scanning

25 Examples of Scan Systems

26 Range Scanning Active systems are superior Accurate calibration is crucial Multiple scans required for complex objects scan path planning scan registration Scans are incomplete and noisy model repair, hole filling smoothing for noise removal

27 Range Scanning: Reconstruction Set of raw scans Reconstructed model

28 2.4 Acquisition of Dynamic Objects

29 3D Animation Scanner New technology Record 3D video Active research area Ultimate goal 3D movie making New creative perspectives

30 What is Depth Image (DI)? A pair of aligned maps a texture map I: gives the color of all visible points a depth map D: gives the distance to each visible point

31 Time-of-flight (ToF) Camera A camera system that creates distance data with help of the time-of-flight (TOF) principle Light pulses Can measure depth scans at video rate Relatively new devices Become more popular (for everyday users)

32 Range image with height measurements Provide 2.5D structure of the scene

33 Advantages A single depth image Provide 2.5D structure of the scene A set of depth images Might provide hole-free 3D scene

34 Set of Depth Images 3D modeling of scene

35 Problems with Depth Images Pros Not only 2D image (2.5D) Easy acquisition Cons Substantial sensor level of random noise low quality data Non-trivial systematic bias

36 New Trend: Microsoft Kinect Capture depth images in video rates Based on PrimeSense of Israel 3D depth sensors RGB camera

37 Kinect Sensor for Xbox 360 Cheap home oriented

38 Kinect Data Low resolution 640x480 Very noisy Human skeleton Being a hot topic!

39 Questions?

40 3. Shape Representations

41 How to represent shape? Explicit representation Image of parameterization Implicit representation Kernel of distance function

42 How to represent shape? Explicit representation Image of parameterization Easy enumeration Implicit representation Kernel of distance function Easy in/out/distance test

43 How to represent shape? Explicit representation Image of parameterization Easy enumeration NURBS, triangular mesh Implicit representation Kernel of distance function Easy in/out/distance test Scalar-valued 3D grid

44 Analytic Surfaces: Spline Surfaces Piecewise polynomial approximation

45 Spline Surfaces Tensor product surfaces ( curves of curves ) Rectangular grid of control points Regular surface patch

46 Spline Surfaces Tensor product surfaces ( curves of curves ) Rectangular grid of control points Regular surface patch Problems: Many patches for complex models Smoothness across patch boundaries Trimming for non rectangular patches

47 Subdivision Surfaces Generalization of spline curves / surfaces Arbitrary control meshes Successive refinement (subdivision) Converges to smooth limit surface Connection between splines and meshes

48 Subdivision Surfaces Generalization of spline curves / surfaces Arbitrary control meshes Successive refinement (subdivision) Converges to smooth limit surface Connection between splines and meshes

49 Spline & Subdivision Surfaces Basis functions are smooth bumps Fixed support Fixed control grid Bound to control points Initial patch layout is crucial Requires experts!

50 Discrete Surfaces: Points & Meshes Triangular meshes piecewise linear approximation to surfaces Flexible for topology and adaptive refinement Suitable for highly detailed scanned data

51 Graph Embedding

52 Planar Graphs and Meshes

53 Mesh Data Structures

54 Mesh Data Structure How to store geometry and connectivity? Geometry queries What are the vertices of face #k? Are vertices #i and #j adjacent? Which faces are adjacent face #k? Geometry operations Remove/add a vertex/face Mesh simplification Vertex split, edge collapse

55 How good is a data structure? Space complexity Time Time to construct - preprocessing Time to answer a query Time to perform an operation (update the data structure) Trade-off between time and space Redundancy

56 Define a mesh Geometry Vertex coordinates Connectivity How do vertices connected?

57 Face Set (STL) Face: 3 positions

58 Shared Vertex (OBJ, OFF) Vertex position Face vertex indices

59 Half-edge based Connectivity vertex position 1 halfedge halfedge 1 vertex 1 face 1, 2, or 3 halfedges face 1 halfedge

60 Wavefront OBJ File Format Vertices Start with char v (x,y,z) coordinates Faces Start with char f Indices of its vertices in the file Other properties Normal, texture coordinates, material, etc. v v v v f f f f 1 3 4

61 Half-edge Based Libraries CGAL Computational geometry Free for non-commercial use OpenMesh Mesh processing Free, LGPL licence

62 Questions?

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