Geometric Modeling Summer Semester 2010 Introduction

Transcription

1 Geometric Modeling Summer Semester 2010 Introduction Motivation Topics Basic Modeling Techniques

2 Today... Topics: Formalities & Organization Introduction: Geometric Modeling Mathematical Tools (1) Geometric Modeling Summer Semester 2010 Introduction 2 / 93

3 Today... Topics: Formalities & Organization Introduction: Geometric Modeling Motivation Overview: Topics Basic modeling techniques Mathematical Tools (1) Geometric Modeling Summer Semester 2010 Introduction 3 / 93

4 Motivation

5 Motivation This lecture covers two related areas: Classic geometric modeling Geometry processing Common techniques (math, models, terminology), but different goals Geometric Modeling Summer Semester 2010 Introduction 5 / 93

6 Geometric Modeling Geometric Modeling: You start with a blank screen, design a geometric model Typical techniques: Triangle meshes Constructive Solid Geometry (CSG) Spline curves & surfaces Subdivision surfaces Goal is interactive modeling Mathematical tools are designed with the user in mind Geometric Modeling Summer Semester 2010 Introduction 6 / 93

7 Geometry Processing Geometry Processing You already have a geometric model Typically from a 3D range scanner (read: not nice) You need to process & edit the geometry The original model has not been build with the user in mind (stupid range scanner) Typical techniques: Noise removal, filtering Surface reconstruction Registration Freeform deformation modeling Statistical analysis (features, symmetry, hole-filling etc...) Geometric Modeling Summer Semester 2010 Introduction 7 / 93

8 Our Perspective The perspective of this lecture: The basic mathematical tools for handling geometry are the same Different usage, adaptation, specific algorithms We will discuss The basic concepts and tools (mathematical foundation, representations, basic algorithms)...and applications in both areas. Geometric Modeling Summer Semester 2010 Introduction 8 / 93

9 Examples: Geometric Modeling

10 The Modern World... designed on a computer (the building) designed on a computer as well (the cars) fortunately, not (yet) designed on a computer (the trees) (c.f. Danny Hillis, Siggraph 2001 keynote) Geometric Modeling Summer Semester 2010 Introduction 10 / 93

11 Impact of Geometric Modeling We live in a world designed using CAD Almost any man-made structure is nowadays planed and designed using computers Architecture Commodities: Chairs, furniture, your microwave & toaster Your car (in case you have one, but probably the bike as well) spline curves have actually been invented in the automotive industry Typesetting <advertising> Our abilities in geometric modeling shapes the world we live in each day. </advertising> Geometric Modeling Summer Semester 2010 Introduction 11 / 93

12 Different Modeling Tasks CAD / CAM Precision Guarantees Handle geometric constraints exactly (e.g. exact circles) Modeling guided by rules and constraints Geometric Modeling Summer Semester 2010 Introduction 12 / 93

13 Different Modeling Tasks Photorealistic Rendering Has to look good Ad-hoc techniques are ok Using textures & shaders to fake details More complexity, but less rigorous Just two examples, lots of stuff in between... Geometric Modeling Summer Semester 2010 Introduction 13 / 93

14 Examples: Geometry Processing

15 Geometry Processing A rather new area Motivation: 3D scanning You (your company) can buy devices that scan real world 3D objects You get (typically) clouds of measurement points Many other sources of geometry as well: Science (CT, [F]MRI, ET, Cryo-EM,...) 3D movie making The design department of your company has dozens of TByte of polygon soup... Crawl the internet Need to process the geometry further Geometric Modeling Summer Semester 2010 Introduction 15 / 93

16 Photoshopping Geometry Geometry Processing: Cleanup: Remove inconsistencies Make watertight (well defined inside/outside, for 3D printers) Simplify keep only the main structure Remove noise, small holes, etc... Touch-up /Edit: Texturing, painting, carving Deformation Stitch together pieces Lots of other stuff similar to image processing Geometric Modeling Summer Semester 2010 Introduction 16 / 93

17 Example Example: The Stanford Digital Michelangelo Project [Levoy et al.: The Digital Michelangelo Project, Siggraph 2000] Geometric Modeling Summer Semester 2010 Introduction 17 / 93

18 Scan Registration [data set: Stanford 3D Scanning Repository] Geometric Modeling Summer Semester 2010 Introduction 18 / 93

19 Feature Tracking Fully Automatic: [Implementation: Martin Bokeloh (Diploma thesis)] Geometric Modeling Summer Semester 2010 Introduction 19 / 93

20 Scanning the World... Example: The Wägele Laser scanners (2D sheets of distance measurments) [Biber et al. 2005] A pull-through measurement device can acquire complete buildings in a few hours Geometric Modeling Summer Semester 2010 Introduction 20 / 93

21 This is what you get... Corridor CS Building University of Tübingen (6.5 GB) CS Building, Outside (nicer colors...)...lots of artifacts (the scanner does not really like windows) Geometric Modeling Summer Semester 2010 Introduction 21 / 93

22 Automatic Processing Example: Automatic Outlier Removal Geometric Modeling Summer Semester 2010 Introduction 22 / 93

23 Think Big More Problems: Occluded areas, shiny / transparent objects holes (lots of holes, actually) Huge amounts of data (really huge) City Scanning There are big companies trying to scan large areas Think Google Earth in full resolution How about a virtual online walk through New York, Tokyo, Saarbrücken? Lots of open research problems to get there Geometric Modeling Summer Semester 2010 Introduction 23 / 93

24 [data set: Institute for Cartography, Leibnitz University Hannover]

25 HUGE Data Sets The Largest Data Set I have On My Hard-Drive... Data set: Outdoor Scan (structure from video) of a part of the UNC campus ( pts / 63.5 GB), courtesy of J.-M. Frahm, University of North Carolina Geometric Modeling Summer Semester 2010 Introduction 25 / 93

26 Hole Filling Wei-Levoy Texture Synthesis Algorithm: [Implementation: Alexander Berner (Diploma thesis)] Geometric Modeling Summer Semester 2010 Introduction 26 / 93

27 Filling Holes [implementation: Alexander Berner (Diploma thesis)] Geometric Modeling Summer Semester 2010 Introduction 27 / 93

28 Filling Holes [implementation: Alexander Berner (Diploma thesis)] Geometric Modeling Summer Semester 2010 Introduction 28 / 93

29 Symmetry Detection [data set: M. Wacker, HTW Dresden] Geometric Modeling Summer Semester 2010 Introduction 29 / 93

30 Results Geometric Modeling Summer Semester 2010 Introduction 30 / 93

31 Results Geometric Modeling Summer Semester 2010 Introduction 31 / 93

32 Results Geometric Modeling Summer Semester 2010 Introduction 32 / 93

33 Line Feature Matching [data sets: Kartographisches Institut, Universität Hannover / M. Wacker, HTW Dresden] Geometric Modeling Summer Semester 2010 Introduction 33 / 93

34 Reconstruction by Symmetry overlay of 16 parts [data sets: Kartographisches Institut, Universität Hannover] Geometric Modeling Summer Semester 2010 Introduction 34 / 93

35 Inverse Procedural Modeling Geometric Modeling Summer Semester 2010 Introduction 35 / 93

36 Overview Our approach Take existing model Analyse shape structure Derive shape modification rules Output Input Geometric Modeling Summer Semester 2010 Introduction 36 / 93

37 Technique Overview Conceptual Steps: Symmetry detection Finding docking sites and dockers Combine into a shape grammar Input Symmetry Docking site Result Geometric Modeling Summer Semester 2010 Introduction 37 / 93

38 Results Geometric Modeling Summer Semester 2010 Introduction 38 / 93

39 Results ~ triangles Geometric Modeling Summer Semester 2010 Introduction 39 / 93

40 Results Geometric Modeling Summer Semester 2010 Introduction 40 / 93

41 Results Geometric Modeling Summer Semester 2010 Introduction 41 / 93

42 Deformable Shape Matching [data set: Stanford 3D Scanning Repository] Geometric Modeling Summer Semester 2010 Introduction 42 / 93

43 Problem Statement Deformable Matching Two shapes: original, deformed How to establish correspondences? Looking for global optimum Arbitrary pose Assumption Approximately isometric deformation [data set: S. König, TU Dresden] Geometric Modeling Summer Semester 2010 Introduction 43 / 93

44 Results [data sets: Stanford 3D Scanning Repository / Carsten Stoll] Geometric Modeling Summer Semester 2010 Introduction 44 / 93

45 Animation Reconstruction [data set: P. Phong, Stanford University] Geometric Modeling Summer Semester 2010 Introduction 45 / 93

46 Scanning Moving Geometry Real-time 3D scanners: Acquire geometry at video rates Capture 3D movies: performance capture Technique still immature, but very interesting applications, in particular special effects for movies [Davis et al. 2003] Geometric Modeling Summer Semester 2010 Introduction 46 / 93

47 Real-Time 3D Scanners space-time stereo courtesy of James Davis University of California at Santa Cruz color-coded structured light courtesy of Phil Fong Stanford University high-speed structured light courtesy of Stefan Gumhold TU Dresden Geometric Modeling Summer Semester 2010 Introduction 47 / 93 47

48 Reconstruction Dynamic geometry reconstruction Hole filling Remove noise and outliers Establish correspondences Need to know where every point on the object goes to over time Simplifies further editing Geometric Modeling Summer Semester 2010 Introduction 48 / 93

49 Animation Reconstruction Remove noise, outliers Fill-in holes (from all frames) Dense correspondences Geometric Modeling Summer Semester 2010 Introduction 49 / 93

50 Factorization t = 0 t = 1 t = 2 f f data points d t,i f (x, t) deformation field S x point on urshape S [data set courtesy of P. Phong, Stanford University] Geometric Modeling Summer Semester 2010 Introduction 50 / 93

51 Geometric Modeling Summer Semester 2010 Introduction 51 / 93

52 Geometric Modeling Summer Semester 2010 Introduction 52 / 93

53 Overview Topics

54 Overview: Geometric Modeling 2010 Mathematical Background (Recap) Linear algebra: vector spaces, function spaces, quadrics Analysis: multi-dim. calculus, differential geometry Numerics: quadratic and non-linear optimization Geometric Modeling Smooth curves: polynomial interpolation & approximation, Bezier curves, B-Splines, NURBS Smooth surfaces: spline surfaces, implicit functions, variational modeling Meshes: meshes, multi-resolution, subdivision Geometric Modeling Summer Semester 2010 Introduction 54 / 93

55 Overview: Geometric Modeling 2010 Geometry Processing 3D Scanning: Overview Registration: ICP, NDT Surface Reconstruction: smoothing, topology reconstruction, moving least-squares Editing: free-form deformation Preliminary List: Topics might change Not presented strictly in this order Geometric Modeling Summer Semester 2010 Introduction 55 / 93

56 Topics Overview Current List of Topics (subject to changes): Math Background: Linear Algebra, Analysis, Differential Geometry, Numerics, Topology Interpolation and Approximation Spline Curves Blossoming and Polar Forms Rational Splines Spline Surfaces Subdivision Surfaces Implicit Functions Variational Modeling Point Based Representations Multi Resolution Representations Surface Parametrization Geometric Modeling Summer Semester 2010 Introduction 56 / 93

57 Overview Modeling Techniques

58 Geometric Modeling What do we want to do? d empty space (typically 3 ) B geometric object B 3 Geometric Modeling Summer Semester 2010 Introduction 58 / 93

59 Fundamental Problem The Problem: d B infinite number of points my computer: 8GB of memory We need to encode a continuous model with a finite amount of information Geometric Modeling Summer Semester 2010 Introduction 59 / 93

60 Modeling Approaches Two Basic Approaches Discrete representations Fixed discrete bins Continuous representations Mathematical description Evaluate continuously Geometric Modeling Summer Semester 2010 Introduction 60 / 93

61 Discrete Representations You know this... Fixed Grid of values: (i 1,..., i d s ) d s (x 1,..., x d t ) d t Typical scenarios: d s = 2, d t = 3: Bitmap images d s = 3, d t = 1: Volume data (scalar fields) d s = 2, d t = 1: Depth maps (range images) PDEs: Finite Differences models Geometric Modeling Summer Semester 2010 Introduction 61 / 93

62 Modeling Approaches Two Basic Approaches Discrete representations Fixed discrete bins Continuous representations Mathematical description Evaluate continuously Geometric Modeling Summer Semester 2010 Introduction 62 / 93

63 Continuous Models Basic Principle: Procedural Modeling Query Parameters (a finite set of numbers from a continuous set) Algorithm(s) determines the class of objects that can be represented finite set of Shape Parameters determines the object shape Answer Geometric Modeling Summer Semester 2010 Introduction 63 / 93

64 Example: Continuous Model Example: Sphere Shape Parameters: center, radius (4 numbers) Algorithms: Ray Intersection (e.g. for display) Input: Ray (angle, position: 5 numbers) Output: {true, false} Inside/outside test (e.g. for rasterization) Input: Position (3 numbers) Output: {true, false} Parametrization (e.g. for display) Input: longitude, latitude (, ) Output: position (3 numbers) Geometric Modeling Summer Semester 2010 Introduction 64 / 93

65 So Many Questions... Several algorithms for the same representation: Parametrization compute surface points according to continuous parameters (Signed) distance computation distance to surface of points in space, inside/outside test Intersection with rays (rendering), other objects (collision detection) Conversion into other representations. Many more... In addition, we also need algorithms to construct and alter the models. Geometric Modeling Summer Semester 2010 Introduction 65 / 93

66 Continuous, Procedural Models Continuous representations An algorithm describes the shape The shape is determined by a finite number of continuous parameters The shape can be queried with a finite number of continuous parameters More involved (have to ask for information) But potentially infinite resolution (continuous model) Structural model complexity still limited by algorithm and parameters This lecture examines these representations and the corresponding algorithms Geometric Modeling Summer Semester 2010 Introduction 66 / 93

67 Classes of Models (Main) classes of models in this lecture: Primitive meshes Parametric models Implicit models Particle / point-based models Remarks Most models are hybrid (combine several of these) Representations can be converted (may be approximate) Some questions are much easier to answer for certain representations Geometric Modeling Summer Semester 2010 Introduction 67 / 93

68 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 68 / 93

69 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 69 / 93

70 Parametric Models v d s S d t f f (u, v) (u, v) Parametric Models u Function f maps from parameter domain to target space Evaluation of f gives one point on the model Geometric Modeling Summer Semester 2010 Introduction 70 / 93

71 input: 3D input: 2D input: 1D output: 1D output: 2D output: 3D t f(t) t y t y z u x x function graph plane curve space curve u y u y v v z x x plane warp surface w u v y z x space warp

72 input: 3D input: 2D input: 1D output: 1D output: 2D output: 3D t f(t) t y t y z u x x function graph plane curve space curve u y u y v v z x x plane warp surface w u v y z x space warp

73 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 73 / 93

74 Primitive Meshes Primitive Meshes Collection of geometric primitives Triangles Quadrilaterals More general primitives (spline patches) Typically, the primitives are parametric surfaces Composite model: Mesh encodes topology, rough shape Primitive parameter encode local geometry Triangle meshes rule the world ( triangle soup ) Geometric Modeling Summer Semester 2010 Introduction 74 / 93

75 Primitive Meshes Complex Topology for Parametric Models Mesh of parameter domains attached in a mesh Domain can have complex shape ( trimmed patches ) Separate mapping function f for each part (typically of the same class) Geometric Modeling Summer Semester 2010 Introduction 75 / 93

76 Meshes are Great Advantages of mesh-based modeling: Compact representation (usually) Can represent arbitrary topology Using the right parametric surfaces as parts, many important geometric objects can be represented exactly (e.g. NURBS: circles, cylinders, spheres CAD/CAM) Geometric Modeling Summer Semester 2010 Introduction 76 / 93

77 Meshes are not so great Problem with Meshes: Need to specify a mesh first, then edit geometry Problems for larger changes Mesh structure and shape need to be adjusted Mesh encodes object topology Changing object topology is painful Difficult to use for many applications (such as surface reconstruction) Rule of thumb: If the topology or the coarse scale shape changes drastically and frequently during computations, meshes are hard to use Drastic example: Fluid simulation (surface of splashing water) Geometric Modeling Summer Semester 2010 Introduction 77 / 93

78 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 78 / 93

79 Implicit Modeling General Formulation: Curve / Surface S = {x f(x) = 0} x d (d = 2,3), f(x) S is (usually) a d-1 dimensional object This means...: The surface is obtained implicitly as the set of points for which some given function vanishes (f(x) = 0) Alternative notation: S = f -1 (0) ( inverse yields a set) Geometric Modeling Summer Semester 2010 Introduction 79 / 93

80 Implicit Modeling Example: Circle: x 2 + y 2 = r 2 f r (x,y) = x 2 + y 2 - r 2 = 0 Sphere: x 2 + y 2 + z 2 = r 2 r 2 x 2 y 2 Special Case: Signed distance field Function value is signed distance to surface f ( x, y) sign( x 2 y 2 r 2 ) x 2 y 2 r 2 Negative means inside, positive means outside Geometric Modeling Summer Semester 2010 Introduction 80 / 93

81 Implicit Modeling Example: Circle: x 2 + y 2 = r 2 f r (x,y) = x 2 + y 2 - r 2 = 0 Sphere: x 2 + y 2 + z 2 = r 2 Signed squared distance field r 2 (has some useful properties, y 2 e.g. from a statistical point x 2 of view) Special Case: Signed distance field Function value is signed distance to surface f ( x, y) sign( x 2 y 2 r 2 ) x 2 y 2 r 2 Negative means inside, positive means outside Geometric Modeling Summer Semester 2010 Introduction 81 / 93

82 Implicit Modeling: Pros & Cons Advantages: More general than parametric techniques Topology can be changed easily (depends on how f is specified, though) Implicit representations are the standard technique for simulations with free boundaries. Also known as level-set methods. Typical example: Fluid simulation (evolving water-air interface) Geometric modeling: Surface reconstruction, blobby surfaces Geometric Modeling Summer Semester 2010 Introduction 82 / 93

83 Implicit Modeling: Pros & Cons Disadvantages: Need to solve inversion problem S = f -1 (0) Algorithms for display, conversion etc. tend to be more difficult and more expensive (inside/outside test is easy though for signed distance fields) Representing objects takes more memory (we will discuss standard representations later) Geometric Modeling Summer Semester 2010 Introduction 83 / 93

84 Modeling Zoo Parametric Models Primitive Meshes Implicit Models Particle Models Geometric Modeling Summer Semester 2010 Introduction 84 / 93

85 Particle Representations Particle / Point-based Representations Geometry is represented as a set of points / particles The particles form a (typically irregular) sample of the geometric object Need additional information to deal with the empty space around the particles additional assumptions Geometric Modeling Summer Semester 2010 Introduction 85 / 93

86 Particle Representations Helpful Information Each particle may carries a set of attributes Must have: Its position Additional geometric information: Particle density (sample spacing), surface normals Additionally: Color, physical quantities (mass, pressure, temperature),... This information can be used to improve the reconstruction of the geometric object described by the particles Geometric Modeling Summer Semester 2010 Introduction 86 / 93

87 The Wrath of Khan Why Star Trek is at fault... Particle methods first used in computer graphics to represent fuzzy phenomena (fire, clouds, smoke) Particle Systems a Technique for Modeling a Class of Fuzzy Objects *Reeves Probably most well-known example: Genesis sequence Geometric Modeling Summer Semester 2010 Introduction 87 / 93

88 Genesis Sequence [Reeves 1983] Geometric Modeling Summer Semester 2010 Introduction 88 / 93

89 Non-Fire Objects Particle Traces for Modeling Plants (also from [Reeves 1983]) Geometric Modeling Summer Semester 2010 Introduction 89 / 93

90 Geometric Modeling How became the geometric modeling crowd interested in this? 3D Scanners 3D scanning devices yield point clouds (often: measure distance to points in space, one at a time) Then you have to deal with the problem anyway Need algorithms to directly work on point clouds (this is the geometry name for particle system) Geometric Modeling Summer Semester 2010 Introduction 90 / 93

91 Geometric Modeling How became the geometric modeling crowd interested in this? Other Reasons: Similar advantages as implicit techniques Topology does not matter (for the good and for the bad) Topology is easy to change Multi-scale representations are easy to do (more details on multi-resolution techniques later) Often easier to use than implicit or parametric techniques Geometric Modeling Summer Semester 2010 Introduction 91 / 93

92 Multi-Scale Geometry w/points Geometric Modeling Summer Semester 2010 Introduction 92 / 93

93 Summary Summary Lots of different representations No silver bullet In theory, everything always works, but might be just too complicated/expensive Best choice depends on the application We will look on all of this... Focus on parametric techniques though Most common approach Geometric Modeling Summer Semester 2010 Introduction 93 / 93