Preface xxxv
About the Authors xlv
Chapter 1: Introduction 1
Graphics is a broad field; to understand it, you need information from perception, physics, mathematics, and engineering. Building a graphics application entails user-interface work, some amount of modeling (i.e., making a representation of a shape), and rendering (the making of pictures of shapes). Rendering is often done via a “pipeline” of operations; one can use this pipeline without understanding every detail to make many useful programs. But if we want to render things accurately, we need to start from a physical understanding of light. Knowing just a few properties of light prepares us to make a first approximate renderer.
1.1 An Introduction to Computer Graphics 1
1.2 A Brief History 7
1.3 An Illuminating Example 9
1.4 Goals, Resources, and Appropriate Abstractions 10
1.5 Some Numbers and Orders of Magnitude in Graphics 12
1.6 The Graphics Pipeline 14
1.7 Relationship of Graphics to Art, Design, and Perception 19
1.8 Basic Graphics Systems 20
1.9 Polygon Drawing As a Black Box 23
1.10 Interaction in Graphics Systems 23
1.11 Different Kinds of Graphics Applications 24
1.12 Different Kinds of Graphics Packages 25
1.13 Building Blocks for Realistic Rendering: A Brief Overview 26
1.14 Learning Computer Graphics 31
Chapter 2: Introduction to 2D Graphics Using WPF 35
A graphics platform acts as the intermediary between the application and the underlying graphics hardware, providing a layer of abstraction to shield the programmer from the details of driving the graphics processor. As CPUs and graphics peripherals have increased in speed and memory capabilities, the feature sets of graphics platforms have evolved to harness new hardware features and to shoulder more of the application development burden. After a brief overview of the evolution of 2D platforms, we explore a modern package (Windows Presentation Foundation), showing how to construct an animated 2D scene by creating and manipulating a simple hierarchical model. WPF’s declarative XML-based syntax, and the basic techniques of scene specification, will carry over to the presentation of WPF’s 3D support in Chapter 6.
2.1 Introduction 35
2.2 Overview of the 2D Graphics Pipeline 36
2.3 The Evolution of 2D Graphics Platforms 37
2.4 Specifying a 2D Scene Using WPF 41
2.5 Dynamics in 2D Graphics Using WPF 55
2.6 Supporting a Variety of Form Factors 58
2.7 Discussion and Further Reading 59
Chapter 3: An Ancient Renderer Made Modern 61
We describe a software implementation of an idea shown by Dürer. Doing so lets us create a perspective rendering of a cube, and introduces the notions of transforming meshes by transforming vertices, clipping, and multiple coordinate systems. We also encounter the need for visible surface determination and for lighting computations.
3.1 A Dürer Woodcut 61
3.2 Visibility 65
3.3 Implementation 65
3.4 The Program 72
3.5 Limitations 75
3.6 Discussion and Further Reading 76
3.7 Exercises 78
Chapter 4: A 2D Graphics Test Bed 81
We want you to rapidly test new ideas as you learn them. For most ideas in graphics, even 3D graphics, a simple 2D program suffices. We describe a test bed, a simple program that’s easy to modify to experiment with new ideas, and show how it can be used to study corner cutting on polygons. A similar 3D program is available on the book’s website.
4.1 Introduction 81
4.2 Details of the Test Bed 82
4.3 The C# Code 88
4.4 Animation 94
4.5 Interaction 95
4.6 An Application of the Test Bed 95
4.7 Discussion 98
4.8 Exercises 98
Chapter 5: An Introduction to Human Visual Perception 101
The human visual system is the ultimate “consumer” of most imagery produced by graphics. As such, it provides design constraints and goals for graphics systems. We introduce the visual system and some of its characteristics, and relate them to engineering decisions in graphics. The visual system is both tolerant of bad data (which is why the visual system can make sense of a child’s stick-figure drawing), and at the same time remarkably sensitive. Understanding both aspects helps us better design graphics algorithms and systems. We discuss basic visual processing, constancy, and continuation, and how different kinds of visual cues help our brains form hypotheses about the world. We discuss primarily static perception of shape, leaving discussion of the perception of motion to Chapter 35, and of the perception of color to Chapter 28.
5.1 Introduction 101
5.2 The Visual System 103
5.3 The Eye 106
5.4 Constancy and Its Influences 110
5.5 Continuation 111
5.6 Shadows 112
5.7 Discussion and Further Reading 113
5.8 Exercises 115
Chapter 6: Introduction to Fixed-Function 3D Graphics and Hierarchical Modeling 117
The process of constructing a 3D scene to be rendered using the classic fixed-function graphics pipeline is composed of distinct steps such as specifying the geometry of components, applying surface materials to components, combining components to form complex objects, and placing lights and cameras. WPF provides an environment suitable for learning about and experimenting with this classic pipeline. We first present the essentials of 3D scene construction, and then further extend the discussion to introduce hierarchical modeling.
6.1 Introduction 117
6.2 Introducing Mesh and Lighting Specification 120
6.3 Curved-Surface Representation and Rendering 128
6.4 Surface Texture in WPF 130
6.5 The WPF Reflectance Model 133
6.6 Hierarchical Modeling Using a Scene Graph 138
6.7 Discussion 147
Chapter 7: Essential Mathematics and the Geometry of 2-Space and 3-Space 149
We review basic facts about equations of lines and planes, areas, convexity, and parameterization. We discuss inside-outside testing for points in polygons. We describe barycentric coordinates, and present the notational conventions that are used throughout the book, including the notation for functions. We present a graphics-centric view of vectors, and introduce the notion of covectors.
7.1 Introduction 149
7.2 Notation 150
7.3 Sets 150
7.4 Functions 151
7.5 Coordinates 153
7.6 Operations on Coordinates 153
7.7 Intersections of Lines 165
7.8 Intersections, More Generally 167
7.9 Triangles 171
7.10 Polygons 175
7.11 Discussion 182
7.12 Exercises 182
Chapter 8: A Simple Way to Describe Shape in 2D and 3D 187
The triangle mesh is a fundamental structure in graphics, widely used for representing shape. We describe 1D meshes (polylines) in 2D and generalize to 2D meshes in 3D. We discuss several representations for triangle meshes, simple operations on meshes such as computing the boundary, and determining whether a mesh is oriented.
8.1 Introduction 187
8.2 “Meshes” in 2D: Polylines 189
8.3 Meshes in 3D 192
8.4 Discussion and Further Reading 198
8.5 Exercises 198
Chapter 9: Functions on Meshes 201
A real-valued function defined at the vertices of a mesh can be extended linearly across each face by barycentric interpolation to define a function on the entire mesh. Such extensions are used in texture mapping, for instance. By considering what happens when a single vertex value is 1, and all others are 0, we see that all our piecewise-linear extensions are combinations of certain basic piecewise linear mesh functions; replacing these basis functions with other, smoother functions can lead to smoother interpolation of values.
9.1 Introduction 201
9.2 Code for Barycentric Interpolation 203
9.3 Limitations of Piecewise Linear Extension 210
9.4 Smoother Extensions 211
9.5 Functions Multiply Defined at Vertices 213
9.6 Application: Texture Mapping 214
9.7 Discussion 217
9.8 Exercises 217
Chapter 10: Transformations in Two Dimensions 221
Linear and affine transformations are the building blocks of graphics. They occur in modeling, in rendering, in animation, and in just about every other context imaginable. They are the natural tools for transforming objects represented as meshes, because they preserve the mesh structure perfectly. We introduce linear and affine transformations in the plane, because most of the interesting phenomena are present there, the exception being the behavior of rotations in three dimensions, which we discuss in Chapter 11. We also discuss the relationship of transformations to matrices, the use of homogeneous coordinates, the uses of hierarchies of transformations in modeling, and the idea of coordinate “frames.”
10.1 Introduction 221
10.2 Five Examples 222
10.3 Important Facts about Transformations 224
10.4 Translation 233
10.5 Points and Vectors Again 234
10.6 Why Use 3 × 3 Matrices Instead of a Matrix and a Vector? 235
10.7 Windowing Transformations 236
10.8 Building 3D Transformations 237
10.9 Another Example of Building a 2D Transformation 238
10.10 Coordinate Frames 240
10.11 Application: Rendering from a Scene Graph 241
10.12 Transforming Vectors and Covectors 250
10.13 More General Transformations 254
10.14 Transformations versus Interpolation 259
10.15 Discussion and Further Reading 259
10.16 Exercises 260
Chapter 11: Transformations in Three Dimensions 263
Transformations in 3-space are analogous to those in the plane, except for rotations: In the plane, we can swap the order in which we perform two rotations about the origin without altering the result; in 3-space, we generally cannot. We discuss the group of rotations in 3-space, the use of quaternions to represent rotations, interpolating between quaternions, and a more general technique for interpolating among any sequence of transformations, provided they are “close enough” to one another. Some of these techniques are applied to user-interface designs in Chapter 21.
11.1 Introduction 263
11.2 Rotations 266
11.3 Comparing Representations 278
11.4 Rotations versus Rotation Specifications 279
11.5 Interpolating Matrix Transformations 280
11.6 Virtual Trackball and Arcball 280
11.7 Discussion and Further Reading 283
11.8 Exercises 284
Chapter 12: A 2D and 3D Transformation Library for Graphics 287
Because we represent so many things in graphics with arrays of three floating-point numbers (RGB colors, locations in 3-space, vectors in 3-space, covectors in 3-space, etc.) it’s very easy to make conceptual mistakes in code, performing operations (like adding the coordinates of two points) that don’t make sense.We present a sample mathematics library that you can use to avoid such problems. While such a library may have no place in high-performance graphics, where the overhead of type checking would be unreasonable, it can be very useful in the development of programs in their early stages.
12.1 Introduction 287
12.2 Points and Vectors 288
12.3 Transformations 288
12.4 Specification of Transformations. 290
12.5 Implementation 290
12.6 Three Dimensions 293
12.7 Associated Transformations 294
12.8 Other Structures 294
12.9 Other Approaches 295
12.10 Discussion 297
12.11 Exercises 297
Chapter 13: Camera Specifications and Transformations 299
To convert a model of a 3D scene to a 2D image seen from a particular point of view, we have to specify the view precisely. The rendering process turns out to be particularly simple if the camera is at the origin, looking along a coordinate axis, and if the field of view is 90 degrees in each direction. We therefore transform the general problem to the more specific one. We discuss how the virtual camera is specified, and how we transform any rendering problem to one in which the camera is in a standard position with standard characteristics. We also discuss the specification of parallel (as opposed to perspective) views.
13.1 Introduction 299
13.2 A 2D Example 300
13.3 Perspective Camera Specification 301
13.4 Building Transformations from a View Specification 303
13.5 Camera Transformations and the Rasterizing Renderer Pipeline 310
13.6 Perspective and z-values 313
13.7 Camera Transformations and the Modeling Hierarchy. 313
13.8 Orthographic Cameras 315
13.9 Discussion and Further Reading 317
13.10 Exercises 318
Chapter 14: Standard Approximations and Representations 321
The real world contains too much detail to simulate efficiently from first principles of physics and geometry. Models make graphics computationally tractable but introduce restrictions and errors. We explore some pervasive approximations and their limitations. In many cases, we have a choice between competing models with different properties.
14.1 Introduction 321
14.2 Evaluating Representations 322
14.3 Real Numbers 324
14.4 Building Blocks of Ray Optics 330
14.5 Large-Scale Object Geometry 337
14.6 Distant Objects 346
14.7 Volumetric Models 349
14.8 Scene Graphs 351
14.9 Material Models 353
14.10 Translucency and Blending 361
14.11 Luminaire Models 369
14.12 Discussion 384
14.13 Exercises 385
Chapter 15: Ray Casting and Rasterization 387
A 3D renderer identifies the surface that covers each pixel of an image, and then executes some shading routine to compute the value of the pixel. We introduce a set of coverage algorithms and some straw-man shading routines, and revisit the graphics pipeline abstraction. These are practical design points arising from general principles of geometry and processor architectures. For coverage, we derive the ray-casting and rasterization algorithms and then build the complete source code for a render on top of it. This requires graphics-specific debugging techniques such as visualizing intermediate results. Architecture-aware optimizations dramatically increase the performance of these programs, albeit by limiting abstraction. Alternatively, we can move abstractions above the pipeline to enable dedicated graphics hardware. APIs abstracting graphics processing units (GPUs) enable efficient rasterization implementations. We port our render to the programmable shading framework common to such APIs.
15.1 Introduction 387
15.2 High-Level Design Overview 388
15.3 Implementation Platform 393
15.4 A Ray-Casting Renderer 403
15.5 Intermezzo 417
15.6 Rasterization 418
15.7 Rendering with a Rasterization API 432
15.8 Performance and Optimization 444
15.9 Discussion 447
15.10 Exercises 449
Chapter 16: Survey of Real-Time 3D Graphics Platforms 451
There is great diversity in the feature sets and design goals among 3D graphics platforms. Some are thin layers that bring the application as close to the hardware as possible for optimum performance and control; others provide a thick layer of data structures for the storage and manipulation of complex scenes; and at the top of the power scale are the game-development environments that additionally provide advanced features like physics and joint/skin simulation. Platforms supporting games render with the highest possible speed to ensure interactivity, while those used by the special effects industry sacrifice speed for the utmost in image quality. We present a broad overview of modern 3D platforms with an emphasis on the design goals behind the variations.
16.1 Introduction 451
16.2 The Programmer’s Model: OpenGL Compatibility (Fixed-Function) Profile 454
16.3 The Programmer’s Model: OpenGL Programmable Pipeline 464
16.4 Architectures of Graphics Applications 466
16.5 3D on Other Platforms 478
16.6 Discussion 479
Chapter 17: Image Representation and Manipulation 481
Much of graphics produces images as output. We describe how images are stored, what information they can contain, and what they can represent, along with the importance of knowing the precise meaning of the pixels in an image file. We show how to composite images (i.e., blend, overlay, and otherwise merge them) using coverage maps, and how to simply represent images at multiple scales with MIP mapping.
17.1 Introduction 481
17.2 What Is an Image? 482
17.3 Image File Formats 483
17.4 Image Compositing 485
17.5 Other Image Types 490
17.6 MIP Maps 491
17.7 Discussion and Further Reading 492
17.8 Exercises 493
Chapter 18: Images and Signal Processing 495
The pattern of light arriving at a camera sensor can be thought of as a function defined on a 2D rectangle, the value at each point being the light energy density arriving there. The resultant image is an array of values, each one arrived at by some sort of averaging of the input function. The relationship between these two functions—one defined on a continuous 2D rectangle, the other defined on a rectangular grid of points—is a deep one. We study the relationship with the tools of Fourier analysis, which lets us understand what parts of the incoming signal can be accurately captured by the discrete signal. This understanding helps us avoid a wide range of image problems, including “jaggies” (ragged edges). It’s also the basis for understanding other phenomena in graphics, such as moiré patterns in textures.
18.1 Introduction 495
18.2 Historical Motivation 498
18.3 Convolution 500
18.4 Properties of Convolution 503
18.5 Convolution-like Computations 504
18.6 Reconstruction 505
18.7 Function Classes 505
18.8 Sampling 507
18.9 Mathematical Considerations 508
18.10 The Fourier Transform: Definitions 511
18.11 The Fourier Transform of a Function on an Interval 511
18.12 Generalizations to Larger Intervals and All of R 516
18.13 Examples of Fourier Transforms 516
18.14 An Approximation of Sampling 519
18.15 Examples Involving Limits 519
18.16 The Inverse Fourier Transform 520
18.17 Properties of the Fourier Transform 521
18.18 Applications 522
18.19 Reconstruction and Band Limiting 524
18.20 Aliasing Revisited 527
18.21 Discussion and Further Reading 529
18.22 Exercises 532
Chapter 19: Enlarging and Shrinking Images 533
We apply the ideas of the previous two chapters to a concrete example—enlarging and shrinking of images—to illustrate their use in practice. We see that when an image, conventionally represented, is shrunk, problems will arise unless certain high-frequency information is removed before the shrinking process.
19.1 Introduction 533
19.2 Enlarging an Image 534
19.3 Scaling Down an Image 537
19.4 Making the Algorithms Practical 538
19.5 Finite-Support Approximations 540
19.6 Other Image Operations and Efficiency 541
19.7 Discussion and Further Reading 544
19.8 Exercises 545
Chapter 20: Textures and Texture Mapping 547
Texturing, and its variants, add visual richness to models without introducing geometric complexity. We discuss basic texturing and its implementation in software, and some of its variants, like bump mapping and displacement mapping, and the use of 1D and 3D textures. We also discuss the creation of texture correspondences (assigning texture coordinates to points on a mesh) and of the texture images themselves, through techniques as varied as “painting the model” and probabilistic texture synthesis algorithms.
20.1 Introduction 547
20.2 Variations of Texturing 549
20.3 Building Tangent Vectors from a Parameterization 552
20.4 Codomains for Texture Maps 553
20.5 Assigning Texture Coordinates 555
20.6 Application Examples 557
20.7 Sampling, Aliasing, Filtering, and Reconstruction 557
20.8 Texture Synthesis 559
20.9 Data-Driven Texture Synthesis 562
20.10 Discussion and Further Reading 564
20.11 Exercises 565
Chapter 21: Interaction Techniques 567
Certain interaction techniques use a substantial amount of the mathematics of transformations, and therefore are more suitable for a book like ours than one that concentrates on the design of the interaction itself, and the human factors associated with that design. We illustrate these ideas with three 3D manipulators—the arcball, trackball, and Unicam—and with a a multitouch interface for manipulating images.
21.1 Introduction 567
21.2 User Interfaces and Computer Graphics 567
21.3 Multitouch Interaction for 2D Manipulation 574
21.4 Mouse-Based Object Manipulation in 3D 580
21.5 Mouse-Based Camera Manipulation: Unicam 584
21.6 Choosing the Best Interface 587
21.7 Some Interface Examples 588
21.8 Discussion and Further Reading 591
21.9 Exercises 593
Chapter 22: Splines and Subdivision Curves 595
Splines are, informally, curves that pass through or near a sequence of “control points.” They’re used to describe shapes, and to control the motion of objects in animations, among other things. Splines make sense not only in the plane, but also in 3-space and in 1-space, where they provide a means of interpolating a sequence of values with various degrees of continuity. Splines, as a modeling tool in graphics, have been in part supplanted by subdivision curves (which we saw in the form of corner cutting curves in Chapter 4) and subdivision surfaces. The two classes—splines and subdivision—are closely related. We demonstrate this for curves in this chapter; a similar approach works for surfaces.
22.1 Introduction 595
22.2 Basic Polynomial Curves 595
22.3 Fitting a Curve Segment between Two Curves: The Hermite Curve 595
22.4 Gluing Together Curves and the Catmull-Rom Spline 598
22.5 Cubic B-splines 602
22.6 Subdivision Curves 604
22.7 Discussion and Further Reading 605
22.8 Exercises 605
Chapter 23: Splines and Subdivision Surfaces 607
Spline surfaces and subdivision surfaces are natural generalizations of spline and subdivision curves. Surfaces are built from rectangular patches, and when these meet four at a vertex, the generalization is reasonably straightforward. At vertices where the degree is not four, certain challenges arise, and dealing with these “exceptional vertices” requires care. Just as in the case of curves, subdivision surfaces, away from exceptional vertices, turn out to be identical to spline surfaces. We discuss spline patches, Catmull-Clark subdivision, other subdivision approaches, and the problems of exceptional points.
23.1 Introduction 607
23.2 Bézier Patches 608
23.3 Catmull-Clark Subdivision Surfaces 610
23.4 Modeling with Subdivision Surfaces 613
23.5 Discussion and Further Reading 614
Chapter 24: Implicit Representations of Shape 615
Implicit curves are defined as the level set of some function on the plane; on a weather map, the isotherm lines constitute implicit curves. By choosing particular functions, we can make the shapes of these curves controllable. The same idea applies in space to define implicit surfaces. In each case, it’s not too difficult to convert an implicit representation to a mesh representation that approximates the surface. But the implicit representation itself has many advantages. Finding a ray-shape intersection with an implicit surface reduces to root finding, for instance, and it’s easy to combine implicit shapes with operators that result in new shapes without sharp corners.
24.1 Introduction 615
24.2 Implicit Curves 616
24.3 Implicit Surfaces 619
24.4 Representing Implicit Functions 621
24.5 Other Representations of Implicit Functions 624
24.6 Conversion to Polyhedral Meshes 625
24.7 Conversion from Polyhedral Meshes to Implicits 629
24.8 Texturing Implicit Models 629
24.9 Ray Tracing Implicit Surfaces 631
24.10 Implicit Shapes in Animation 631
24.11 Discussion and Further Reading 632
24.12 Exercises 633
Chapter 25: Meshes 635
Meshes are a dominant structure in today’s graphics. They serve as approximations to smooth curves and surfaces, and much mathematics from the smooth category can be transferred to work with meshes. Certain special classes of meshes—height field meshes, and very regular meshes—support fast algorithms particularly well. We discuss level of detail in the context of meshes, where practical algorithms abound, but also in a larger context. We conclude with some applications.
25.1 Introduction 635
25.2 Mesh Topology 637
25.3 Mesh Geometry 643
25.4 Level of Detail 645
25.5 Mesh Applications 1: Marching Cubes, Mesh Repair, and Mesh Improvement 652
25.6 Mesh Applications 2: Deformation Transfer and Triangle-Order Optimization 660
25.7 Discussion and Further Reading 667
25.8 Exercises 668
Chapter 26: Light 669
We discuss the basic physics of light, starting from blackbody radiation, and the relevance of this physics to computer graphics. In particular, we discuss both the wave and particle descriptions of light, polarization effects, and diffraction. We then discuss the measurement of light, including the various units of measure, and the continuum assumption implicit in these measurements. We focus on the radiance, from which all other radiometric terms can be derived through integration, and which is constant along rays in empty space. Because of the dependence on integration, we discuss solid angles and integration over these. Because the radiance field in most scenes is too complex to express in simple algebraic terms, integrals of radiance are almost always computed stochastically, and so we introduce stochastic integration. Finally, we discuss reflectance and transmission, their measurement, and the challenges of computing integrals in which the integrands have substantial variation (like the specular and nonspecular parts of the reflection from a glossy surface).
26.1 Introduction 669
26.2 The Physics of Light 669
26.3 The Microscopic View 670
26.4 The Wave Nature of Light 674
26.5 Fresnel’s Law and Polarization 681
26.6 Modeling Light as a Continuous Flow 683
26.7 Measuring Light 692
26.8 Other Measurements 700
26.9 The Derivative Approach 700
26.10 Reflectance 702
26.11 Discussion and Further Reading 707
26.12 Exercises 707
Chapter 27: Materials and Scattering 711
The appearance of an object made of some material is determined by the interaction of that material with the light in the scene. The interaction (for fairly homogeneous materials) is described by the reflection and transmission distribution functions, at least for at-the-surface scattering. We present several different models for these, ranging from the purely empirical to those incorporating various degrees of physical realism, and observe their limitations as well. We briefly discuss scattering from volumetric media like smoke and fog, and the kind of subsurface scattering that takes place in media like skin and milk. Anticipating our use of these material models in rendering, we also discuss the software interface a material model must support to be used effectively.
27.1 Introduction 711
27.2 Object-Level Scattering 711
27.3 Surface Scattering 712
27.4 Kinds of Scattering 714
27.5 Empirical and Phenomenological Models for Scattering 717
27.6 Measured Models 725
27.7 Physical Models for Specular and Diffuse Reflection 726
27.8 Physically Based Scattering Models 727
27.9 Representation Choices 734
27.10 Criteria for Evaluation 734
27.11 Variations across Surfaces 735
27.12 Suitability for Human Use 736
27.13 More Complex Scattering 737
27.14 Software Interface to Material Models 740
27.15 Discussion and Further Reading 741
27.16 Exercises 743
Chapter 28: Color 745
While color appears to be a physical property—that book is blue, that sun is yellow—it is, in fact, a perceptual phenomenon, one that’s closely related to the spectral distribution of light, but by no means completely determined by it. We describe the perception of color and its relationship to the physiology of the eye. We introduce various systems for naming, representing, and selecting colors. We also discuss the perception of brightness, which is nonlinear as a function of light energy, and the consequences of this for the efficient representation of varying brightness levels, leading to the notion of gamma, an exponent used in compressing brightness data. We also discuss the gamuts (range of colors) of various devices, and the problems of color interpolation.
28.1 Introduction 745
28.2 Spectral Distribution of Light 746
28.3 The Phenomenon of Color Perception and the Physiology of the Eye 748
28.4 The Perception of Color 750
28.5 Color Description 756
28.6 Conventional Color Wisdom 758
28.7 Color Perception Strengths andWeaknesses 761
28.8 Standard Description of Colors 761
28.9 Perceptual Color Spaces 767
28.10 Intermezzo 768
28.11 White 769
28.12 Encoding of Intensity, Exponents, and Gamma Correction 769
28.13 Describing Color 771
28.14 CMY and CMYK Color 774
28.15 The YIQ Color Model 775
28.16 Video Standards 775
28.17 HSV and HLS 776
28.18 Interpolating Color 777
28.19 Using Color in Computer Graphics 779
28.20 Discussion and Further Reading 780
28.21 Exercises 780
Chapter 29: Light Transport 783
Using the formal descriptions of radiance and scattering, we derive the rendering equation, an integral equation characterizing the radiance field, given a description of the illumination, geometry, and materials in the scene.
29.1 Introduction 783
29.2 Light Transport 783
29.3 A Peek Ahead 787
29.4 The Rendering Equation for General Scattering 789
29.5 Scattering, Revisited 792
29.6 AWorked Example 793
29.7 Solving the Rendering Equation 796
29.8 The Classification of Light-Transport Paths 796
29.9 Discussion 799
29.10 Exercise 799
Chapter 30: Probability and Monte Carlo Integration 801
Probabilistic methods are at the heart of modern rendering techniques, especially methods for estimating integrals, because solving the rendering equation involves computing an integral that’s impossible to evaluate exactly in any but the simplest scenes. We review basic discrete probability, generalize to continuum probability, and use this to derive the single-sample estimate for an integral and the importance-weighted single-sample estimate, which we’ll use in the next two chapters.
30.1 Introduction 801
30.2 Numerical Integration 801
30.3 Random Variables and Randomized Algorithms 802
30.4 Continuum Probability, Continued 815
30.5 Importance Sampling and Integration 818
30.6 Mixed Probabilities 820
30.7 Discussion and Further Reading 821
30.8 Exercises 821
Chapter 31: Computing Solutions to the Rendering Equation: Theoretical Approaches 825
The rendering equation can be approximately solved by many methods, including ray tracing (an approximation to the series solution), radiosity (an approximation arising from a finite-element approach), Metropolis light transport, and photon mapping, not to mention basic polygonal renderers using direct-lighting-plus-ambient approximations. Each method has strengths and weaknesses that can be analyzed by considering the nature of the materials in the scene, by examining different classes of light paths from luminaires to detectors, and by uncovering various kinds of approximation errors implicit in the methods.
31.1 Introduction 825
31.2 Approximate Solutions of Equations 825
31.3 Method 1: Approximating the Equation 826
31.4 Method 2: Restricting the Domain 827
31.5 Method 3: Using Statistical Estimators 827
31.6 Method 4: Bisection 830
31.7 Other Approaches 831
31.8 The Rendering Equation, Revisited 831
31.9 What Do We Need to Compute? 836
31.10 The Discretization Approach: Radiosity 838
31.11 Separation of Transport Paths 844
31.12 Series Solution of the Rendering Equation 844
31.13 Alternative Formulations of Light Transport 846
31.14 Approximations of the Series Solution 847
31.15 Approximating Scattering: Spherical Harmonics 848
31.16 Introduction to Monte Carlo Approaches 851
31.17 Tracing Paths 855
31.18 Path Tracing and Markov Chains 856
31.19 Photon Mapping 872
31.20 Discussion and Further Reading 876
31.21 Exercises 879
Chapter 32: Rendering in Practice 881
We describe the implementation of a path tracer, which exhibits many of the complexities associated with ray-tracing-like renderers that attempt to estimate radiance by estimating integrals associated to the rendering equations, and a photon mapper, which quickly converges to a biased but consistent and plausible result.
32.1 Introduction 881
32.2 Representations 881
32.3 Surface Representations and Representing BSDFs Locally 882
32.4 Representation of Light 887
32.5 A Basic Path Tracer 889
32.6 Photon Mapping 904
32.7 Generalizations 914
32.8 Rendering and Debugging 915
32.9 Discussion and Further Reading 919
32.10 Exercises 923
Chapter 33: Shaders 927
On modern graphics cards, we can execute small (and not-so-small) programs that operate on model data to produce pictures. In the simplest form, these are vertex shaders and fragment shaders, the first of which can do processing based on the geometry of the scene (typically the vertex coordinates), and the second of which can process fragments, which correspond to pieces of polygons that will appear in a single pixel. To illustrate the more basic use of shaders we describe how to implement basic Phong shading, environment mapping, and a simple nonphotorealistic renderer.
33.1 Introduction 927
33.2 The Graphics Pipeline in Several Forms 927
33.3 Historical Development 929
33.4 A Simple Graphics Program with Shaders 932
33.5 A Phong Shader 937
33.6 Environment Mapping 939
33.7 Two Versions of Toon Shading 940
33.8 Basic XToon Shading 942
33.9 Discussion and Further Reading 943
33.10 Exercises 943
Chapter 34: Expressive Rendering 945
Expressive rendering is the name we give to renderings that do not aim for photorealism, but rather aim to produce imagery that communicates with the viewer, conveying what the creator finds important, and suppressing what’s unimportant. We summarize the theoretical foundations of expressive rendering, particularly various kinds of abstraction, and discuss the relationship of the “message” of a rendering and its style. We illustrate with a few expressive rendering techniques.
34.1 Introduction 945
34.2 The Challenges of Expressive Rendering 949
34.3 Marks and Strokes 950
34.4 Perception and Salient Features 951
34.5 Geometric Curve Extraction 952
34.6 Abstraction 959
34.7 Discussion and Further Reading 961
Chapter 35: Motion 963
An animation is a sequence of rendered frames that gives the perception of smooth motion when displayed quickly. The algorithms to control the underlying 3D object motion generally interpolate between key poses using splines, or simulate the laws of physics by numerically integrating velocity and acceleration. Whereas rendering primarily is concerned with surfaces, animation algorithms require a model with additional properties like articulation and mass. Yet these models still simplify the real world, accepting limitations to achieve computational efficiency. The hardest problems in animation involve artificial intelligence for planning realistic character motion, which is beyond the scope of this chapter.
35.1 Introduction 963
35.2 Motivating Examples 966
35.3 Considerations for Rendering 975
35.4 Representations 987
35.5 Pose Interpolation 992
35.6 Dynamics 996
35.7 Remarks on Stability in Dynamics 1020
35.8 Discussion 1022
Chapter 36: Visibility Determination 1023
Efficient determination of the subset of a scene that affects the final image is critical to the performance of a renderer. The first approximation of this process is conservative determination of surfaces visible to the eye. This problem has been addressed by algorithms with radically different space, quality, and time bounds. The preferred algorithms vary over time with the cost and performance of hardware architectures. Because analogous problems arise in collision detection, selection, global illumination, and document layout, even visibility algorithms that are currently out of favor for primary rays may be preferred in other applications.
36.1 Introduction 1023
36.2 Ray Casting 1029
36.3 The Depth Buffer 1034
36.4 List-Priority Algorithms 1040
36.5 Frustum Culling and Clipping 1044
36.6 Backface Culling 1047
36.7 Hierarchical Occlusion Culling 1049
36.8 Sector-based Conservative Visibility 1050
36.9 Partial Coverage 1054
36.10 Discussion and Further Reading 1063
36.11 Exercise 1063
Chapter 37: Spatial Data Structures 1065
Spatial data structures like bounding volume hierarchies provide intersection queries and set operations on geometry embedded in a metric space. Intersection queries are necessary for light transport, interaction, and dynamics simulation. These structures are classic data structures like hash tables,trees, and graphs extended with the constraints of 3D geometry.
37.1 Introduction 1065
37.2 Programmatic Interfaces 1068
37.3 Characterizing Data Structures 1077
37.4 Overview of kd Structures 1080
37.5 List 1081
37.6 Trees 1083
37.7 Grid 1093
37.8 Discussion and Further Reading 1101
Chapter 38: Modern Graphics Hardware 1103
We describe the structure of modern graphics cards, their design, and some of the engineering tradeoffs that influence this design.
38.1 Introduction 1103
38.2 NVIDIA GeForce 9800 GTX 1105
38.3 Architecture and Implementation. 1107
38.4 Parallelism 1111
38.5 Programmability 1114
38.6 Texture, Memory, and Latency 1117
38.7 Locality 1127
38.8 Organizational Alternatives 1135
38.9 GPUs as Compute Engines 1142
38.10 Discussion and Further Reading 1143
38.11 Exercises 1143
List of Principles 1145
Bibliography 1149
Index 1183