COMPUTER VISION

Preface									xiii
Symbols									xv

1	Introduction							1
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 in Shape Reconstruction	7
1.1.3	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	Introductory Example: Static Stereo Image Analysis		28
1.4.1	Coplanar Stereo Image Geometry					29
1.4.2	Shirai Algorithm						32
1.5 	References							37
1.6	Exercises							38

2	Image Acquisition						41
2.1	Geometric Camera Model						41
2.1.1	Central and Parallel Projection					42
2.1.2	A Simple Camera Model with Central Projection			46
2.1.3	Calibration by Direct Linear Transformation			50
2.1.4	Camera Model with Radial Lens Distortion			53
2.1.5	Tsai Calibration						58
2.2	Sensor Model							64
2.2.1	Cameras for Color Image Acquisition				65
2.2.2 	Photometric Sensor Model					67
2.2.3 	Attenuation, Clipping and Blooming				68
2.3	Photometric and Colormetric Calibration				71
2.3.1	Non-linear Camera Behavior					71
2.3.2	Measurement of Camera Linearity					73
2.3.3	White Balance and Black Value Calculation			75
2.3.4	Additive Color Mixing and Standard Color Values			76
2.3.5	Calibration into the Standard Color Space			80
2.4	References							84
2.5 	Exercises							85

3	Geometry of Object Surfaces					87
3.1	Functional Representations					87
3.1.1	Patches and Differentiable Functions				87
3.1.2	Normals and Gradients						92
3.1.3	Taylor Row Expansion						94
3.1.4	Sphere and Spatial Angles					96
3.2	Projection und Reconstruction					100
3.2.1	Depth Map, Height Map and Gradient Map				100
3.2.2	Backprojection							104
3.2.3	Visualization of Gradient Maps					105
3.3	Depth Maps from Gradients					108
3.3.1	Propagation Methods						108
3.3.2	Frankot-Chellappa Algorithm					112
3.4	Gradient Space							119
3.4.1	Three Coordinate Systems					119
3.4.2	Properties of the Gradient Space				123
3.5	References							126
3.6	Exercises							127

4	Static Stereo Analysis						131
4.1	Geometry of Stereo Image Acquisition Systems			132
4.2	Constraints							136
4.2.1	Epipolar Geometry						137
4.2.2	Uniqueness, Photometric Compatibility and Similarity		139
4.2.3	Continuity of Disparities					140
4.2.4	Compatibility of Features					141
4.2.5	Disparity Limit and Disparity Gradient Limit			143
4.2.6	Ordering of Projected Points					144
4.3	Intensity-based Correspondence Analysis				145
4.3.1	Block Matching Method						146
4.3.2	Block Matching Method for Color Stereo Analysis			151
4.4	Feature-based Correspondence Analysis				156
4.4.1	A Method for Histogram-based Stereo Analysis			157
4.4.2	Feature-based Color Stereo Analysis				161
4.5	Stereo Analysis with more than Two Cameras			165
4.5.1	Assignment Strategies						166
4.5.2	A Geometrical Approach						166
4.6	References							169
4.7	Exercises							171

5	Dynamic Stereo Analysis						173
5.1	Displacement Vectors  and  Reconstruction			173
5.1.1	Local Displacement Vectors					174
5.1.2	Object Motion and Local Displacement				177
5.1.3	Object Motion and Gradients					178
5.1.4	Local Displacements and Gradients				180
5.1.5	Camera Rotation around the Projection Center			185
5.2	Optical Flow							187
5.2.1	Computational Strategies					187
5.2.2	Horn-Schunck Method						189
5.2.3	Evaluation of Results						197
5.3	Object Rotation and Reconstruction				203
5.3.1	World Coordinates from Corresponding Points			203
5.3.2	Restricted Search Domain for Correspondence Analysis		207
5.3.3	Discussion							211
5.4	References							213
5.5	Exercises							214

6	Surface Reflection						217
6.1	Parameters and Laws in Radiation Physics			218
6.1.1	Space-angle Independent Quantities				219
6.1.2	Space-angle Dependent Quantities				220
6.1.3	Photometric Fundamental Law					221
6.1.4	Photometric Law of Distances					222
6.2	General Reflection Function					223
6.2.1	Definition of BRDF						223
6.2.2	BRDF of a perfectly diffuse Surface				225
6.2.3	Lambertian Cosine Law						226
6.2.4	Reflection Constant						226
6.2.5	Measurement of BRDF						227
6.3	Reflectance Maps						228
6.3.1	Definition and Representation					229
6.3.2	Linear Reflectance Map						230
6.3.3	Lambertian Reflectance Map					231
6.3.4	Reflectance Map Computation					237
6.4	Reflection Components						239
6.4.1	Diffuse Reflection						240
6.4.2	Specular Reflection						242
6.4.3	Dichromatic Reflection Model					245
6.4.4	Interreflections						248
6.5	Image Irradiance Equation					250
6.5.1	Image Formation							250
6.5.2	General Equation						251
6.6	References							252
6.7	Exercises							254

7	Shape from Shading						255
7.1	Introduction							255
7.1.1	Conditional SFS Problem						256
7.1.2	Classification							259
7.1.3	Direct 3D Interpretation of Image Irradiances			260
7.2	Propagation Methods						263
7.2.1	Linear Reflectance Map						264
7.2.2	Rotation-symmetric Reflectance Map				266
7.2.3	General Reflectance Maps					267
7.2.4	More Robust Methods						268
7.3	Global Minimization Schemes					271
7.3.1	Constraint Formulation						271
7.3.2	Constraint Combination						275
7.3.3	SFS as Variation Problem					276
7.4	Local Shape from Shading					284
7.4.1	Spherical Approximation and Tilt Calculation			284
7.4.2	Slant Calculation						286
7.5	References							289
7.6	Exercises							291

8	Photometric Stereo Analysis					293
8.1	Limitations  of SFS Methods					294
8.2	Analysis of Irradiance Tuples					299
8.2.1	Linear Reflectance Maps						301
8.2.2	Albedo-dependent Analysis					302
8.2.3	Uniqueness by Integrability					310
8.2.4	Albedo-independent Analysis					318
8.2.5	Uniqueness by Spherical Approximation				321
8.3	Analysis of Irradiance Triples					323
8.3.1	Albedo-dependent Analysis					324
8.3.2	Albedo-independent Analysis					329
8.3.3	Calculation of Light Source Direction				334
8.4	References							335
8.5	Exercises							337

9	Structured Illumination						339
9.1	Projection of Simple Geometric Patterns				339
9.1.1	Light Dot Technique						340
9.1.2	Light Dot Stereo Analysis					344
9.1.3	Light Plane Technique						346
9.1.4	Simultaneous Projection of several Light Planes			352
9.2	Projection of Encoded Patterns					354
9.2.1	Binary Codes and Light Plane Technique				354
9.2.2	Utilization of Moire-Patterns					356
9.2.3	Color Codes and Light Plane Technique				360
9.2.4	Active Color Stereo Analysis					362
9.3	References							368
9.4	Exercises							370

Index									373
List of Algorithms							379
Appendix: Color Images							381

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