1.1 Shape Reconstruction 2 1.1.1 Tasks and Tools 2 1.1.2 Formal Specification of the Basic Task 6 1.1.3 Three Straightforward Limitations for Shape Reconstruction 7 1.1.4 Utilization of Context Knowledge 10 1.2 Gray Value and Color Images 11 1.2.1 Image Parameters and Two Color Models 12 1.2.2 Conversion Between These Color Models 15 1.3 Edge Detection 19 1.3.1 Edges in Gray Value Images 20 1.3.2 Laplacian-of-Gaussian Edge Detector 24 1.4 An Introductory Example - Static Stereo Image Analysis 29 1.4.1 Coplanar Stereo Image Geometry 29 1.4.2 Shirai Algorithm 33 1.5 References 38 1.6 Exercises 40
2.1 Geometric Camera Model 43 2.1.1 Central and Parallel Projection 44 2.1.2 A Camera Model for Central Projection 48 2.1.3 Calibration by Direct Linear Transformation 53 2.1.4 A Camera Model with Radial Lens Distortion 57 2.1.5 Tsai's Calibration Method 62 2.2 Sensor Model 68 2.2.1 Camera Hardware for Color Image Acquisition 69 2.2.2 Photometric Sensor Model 71 2.2.3 Pre-kneeing, Clipping, and Blooming 73 2.3 Photometric Calibration 75 2.3.1 Gamma Re-Correction 75 2.3.2 Black Level and White Balance 77 2.4 References 78 2.5 Exercises 79
3 Geometry of Object Surfaces 81 3.1 Functional Representations 81 3.1.1 Facets or Differentiable Functions 81 3.1.2 Normals and Gradients 86 3.1.3 Taylor Expansion 90 3.1.4 Sphere and Solid Angles 91 3.2 Projection and Reconstruction 95 3.2.1 Range Image, Depth Map, Height Map, and Gradient Map 95 3.2.2 Backprojection 99 3.2.3 Visualization of Gradient Maps 102 3.3 Depth Maps from Gradient Maps 105 3.3.1 Local Propagation Methods 105 3.3.2 Frankot-Chellappa Algorithm 109 3.4 Gradient Space 116 3.4.1 Three Coordinate Systems 117 3.4.2 Properties of the Gradient Space 121 3.5 References 124 3.6 Exercises 125
4.1 Geometry of Static Stereo 130 4.2 Assumptions and Constraints 135 4.2.1 Epipolar Line Constraint 136 4.2.2 Uniqueness, Compatibility, and Similarity 138 4.2.3 Continuity of Disparities 139 4.2.4 Compatibility of Features 140 4.2.5 Disparity Limit and Disparity Gradient Limit 142 4.2.6 Ordering of Projected Points in the Image Plane 144 4.3 Intensity Based Correspondence Analysis 145 4.3.1 Block-Matching Method 146 4.3.2 Matching of Epipolar Lines using Dynamic Programming 151 4.3.3 Block-Matching Method for Color Image Stereo Analysis 155 4.4 Feature Based Correspondence Analysis 159 4.4.1 Stereo Analysis based on Zero-crossing Vectors 160 4.4.2 Feature Based Color Stereo Analysis 165 4.5 Stereo Analysis with more than Two Cameras 168 4.5.1 Assignment Strategies 169 4.5.2 A Geometric Method 170 4.6 References 172 4.7 Exercises 174
5.1 Displacement Vectors and Reconstruction 177 5.1.1 Local Displacement Vectors 178 5.1.2 Object Motion and Local Displacement 181 5.1.3 Object Motion and Gradients 182 5.1.4 Local Displacement and Gradients 184 5.1.5 Camera Rotation around the Projection Center 189 5.2 Optical Flow 190 5.2.1 Solution Strategies 190 5.2.2 Horn-Schunck Method 193 5.2.3 Discussion 201 5.3 Object Rotation and Reconstruction 207 5.3.1 World Coordinates from Point Correspondence 207 5.3.2 Constrained Search Space for Correspondence Analysis 212 5.3.3 Discussion 216 5.3.4 3D Models from Occluding Boundaries 218 5.4 References 222 5.5 Exercises 223
6.1 Radiometric Quantities and Laws 228 6.1.1 Quantities Independent from Solid Angles 228 6.1.2 Quantities Dependent on Solid Angles 229 6.1.3 A Fundamental Relationship 230 6.1.4 Inverse Square Law 231 6.2 Reflectance-Distribution Function 232 6.2.1 Definition of BRDF 233 6.2.2 BRDF of a Perfectly Diffuse Surface 234 6.2.3 Lambert's Cosine Law 235 6.2.4 Albedo 236 6.2.5 BRDF Measurement 236 6.3 Reflectance Maps 237 6.3.1 Definition and Representation 238 6.3.2 Linear Reflectance Maps 238 6.3.3 Lambertian Reflectance Maps 240 6.3.4 Generation of Reflectance Maps 244 6.4 Reflection Components 248 6.4.1 Diffuse Reflection 248 6.4.2 Specular Reflection 249 6.4.3 Dichromatic Reflection Model 252 6.4.4 Interreflections 256 6.5 Image Irradiance Equation 257 6.5.1 Image Formation 257 6.5.2 General Equation 258 6.6 References 259 6.7 Exercises 261
7.1 Introduction 263 7.1.1 SFS Constraints 264 7.1.2 Classification of SFS Methods 267 7.1.3 Direct Interpretation of Image Irradiances 268 7.2 Propagation Methods 271 7.2.1 Linear Reflectance Maps 271 7.2.2 Rotationally Symmetric and General Reflectance Maps 275 7.2.3 More Robust Methods 277 7.3 Global Minimization Approaches 280 7.3.1 Formulation of Constraints 280 7.3.2 Combination of Constraints 284 7.3.3 SFS as a Variational Problem 276 7.4 Local Shape from Shading 293 7.4.1 Spherical Approximation and Calculation of Tilt 293 7.4.2 Calculation of Slant 295 7.5 References 298 7.6 Exercises 300
8.1 Limitations of SFS 302 8.2 Analysis of Irradiance Pairs 307 8.2.1 Linear Reflectance Maps 308 8.2.2 Albedo Dependent Analysis 310 8.2.3 Uniqueness by Integrability 317 8.2.4 Albedo Independent Analysis 326 8.2.5 Uniqueness by Spherical Approximation 328 8.3 Analysis of Irradiance Triplets 331 8.3.1 Albedo Dependent Analysis 331 8.3.2 Albedo Independent Analysis 335 8.3.3 Calculation of Illumination Direction 340 8.4 References 341 8.5 Exercises 343
9.1 Projection of Simple Geometric Patterns 349 9.1.1 Light Spot Projection 349 9.1.2 Single Spot Stereo Analysis 353 9.1.3 Light Stripe Projection 355 9.1.4 Static Light Pattern Projection 362 9.2 Projection of Encoded Patterns 363 9.2.1 Binary Encoded Light Stripes and Phase Shifting 363 9.2.2 Color Encoded Light Stripe Projection 367 9.2.3 Active Color Stereo Analysis 369 9.3 References 373 9.4 Exercises 374