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4.8 Nonlinear Mapping 185

4.9 Self-Organized Maps 186

4.10 Supervised Classification: Use of Templates 188

4.11 Inference from Two to Three Dimensions 189

CHAPTER 5 Three-Dimensional Reconstruction 193

1 Introduction 193

2 General Mathematical Principles 194

2.1 The Projection Theorem and Radon's Theorem 194

2.2 Object Boundedness, Shape Transform, and Resolution 196

2.3 Definition of Eulerian Angles, and Special

Projection Geometries: Single-Axis and Conical Tilting 198

3 The Rationales of Data Collection: Reconstruction Schemes 201

3.1 Introduction 201

3.2 Cylindrically Averaged Reconstruction 202

3.3 Compatibility of Projections 205

3.4 Relating Projections to One Another Using Common Lines 206

3.5 The Random-Conical Data Collection Method 210

3.6 Comparison of Common Lines Versus Random-Conical Methods 212

3.7 Reconstruction Schemes Based on Uniform Angular Coverage 213

4 Overview of Existing Reconstruction Techniques 213

4.1 Preliminaries 213

4.2 Weighted Back-Projection 214

4.3 Fourier Reconstruction Methods 219

4.4 Iterative Algebraic Reconstruction Methods 221

5 The Random-Conical Reconstruction in Practice 222

5.1 Overview 222

5.2 Optical Diffraction Screening 222

5.3 Interactive Tilted/Untilted Particle Selection 225

5.4 Optical Density Scaling 226

5.5 Processing of Untitled-Particle Images 227

5.6 Processing of Tilted-Particle Images 228

5.7 Carrying Out the Reconstruction 231

6 Common-Lines Methods (or "Angular Reconstitution'') in Practice 232

7 Reference-Based Methods and Refinement 232

7.1 Introduction 232

7.2 Three-Dimensional Projection Matching 236

7.3 Numerical Aspects 240

7.4 Three-Dimensional Radon Transform Method 242

7.5 The Size of Angular Deviations 243

7.6 Model Dependence of the Reconstruction 246

7.7 Consistency Check by Reprojection 247

8 Resolution Assessment 247

8.1 Theoretical Resolution of the 3D Reconstruction 247

8.2 Practically Achieved Resolution 248

8.3 Cross-Validation Using Excision of Fourier Data from the 3D Reference 253

9 Contrast Transfer Function and Fourier Amplitude Correction 255

9.1 Introduction 255

9.2 Contrast Transfer Function Correction 255

9.3 Fourier Amplitude Correction 259

10 Three-Dimensional Restoration 261

10.1 Introduction 261

10.2 Theory of Projection onto Convex Sets 262

10.3 Projection onto Convex Sets in Practice 264

11 Reconstructions from Heterogeneous Data Sets 266

11.1 Introduction 266

11.2 Separating Ligand-Bound from Ligand-Free Complexes 266

11.3 Separating Populations with Different Conformations 267

12 Merging and Averaging of Reconstructions 270

12.1 The Rationale for Merging 270

12.2 Negatively Stained Specimens: Complications due to Preparation-Induced Deformations 271

12.3 Alignment of Volumes 272

12.4 Merging of Reconstructions through Merging of Projection Sets into a Common Coordinate Frame 275

12.5 Classification of 3D Volumes 276

CHAPTER 6 Interpretation of Three-Dimensional Images of Macromolecules 277

1 Introduction 277

2 Assessment of Statistical Significance 279

2.1 Introduction 279

2.2 Three-Dimensional Variance Estimation from Projections 280

2.3 Use of the 3D Variance Estimate to Ascertain the Statistical Significance 284

3 Validation and Consistency 286

3.1 Internal Consistency 286

3.2 Reconstructions from the Same Data Set with Different Algorithms 286

3.3 Consistency with X-Ray Structures 287

3.4 Concluding Remarks 291

4 Visualization and Rendering 293

4.1 Surface Rendering 294

4.2 Definition of Boundaries 296

4.3 Volume Rendering 297

5 Segmentation of Volumes 298

5.1 Manual (Interactive) Segmentation 299

5.2 Segmentation Based on Density Alone 299

5.3 Knowledge-Based Segmentation, and Identification of Regions 303

6 Methods for Docking and Fitting 308

6.1 Manual Fitting 309

6.2 Quantitative Fitting 311

7 Classification of Volumes 316

Appendix 1 Some Important Definitions and Theorems 319

Appendix 2 Profiles, Point-Spread Functions, and Effects of Commonly Used Low-Pass Filters 327

Appendix 3 Bibliography of Methods 331

Appendix 4 Bibliography of Structures 337

Appendix 5 Special Journal Issues on Image Processing Techniques 343

References 345 Index 399

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