Computed Tomography - From Photon Statistics to Modern Cone-Beam CT

Preface

Contents

1 Introduction

1.1 Computed Tomography – State of the Art

1.2 Inverse Problems

1.3 Historical Perspective

1.4 Some Examples

1.5 Structure of the Book

2 FundamentalsofX-rayPhysics

2.1 Introduction

2.2 X- ray Generation

2.3 Photon–Matter Interaction

2.4 Problems with Lambert–Beer’s Law

2.5 X- ray Detection

2.6 X- ray Photon Statistics

3 Milestones of Computed Tomography

3.1 Introduction

3.2 Tomosynthesis

3.3 Rotation– Translation of a Pencil Beam(First Generation)

3.4 Rotation– Translation of a Narrow Fan Beam (Second Generation)

3.5 Rotation of aWide Aperture Fan Beam ( Third Generation)

3.6 Rotation–Fix with Closed Detector Ring (Fourth Generation)

3.7 Electron BeamComputerized Tomography

3.8 Rotation in Spiral Path

3.9 Rotation in Cone-BeamGeometry

3.10 Micro- CT

3.11 PET- CT Combined Scanners

3.12 Optical Reconstruction Techniques

4 Fundamentals of Signal Processing

4.1 Introduction

4.2 Signals

4.3 Fundamental Signals

4.4 Systems

4.5 Signal Transmission

4.6 Dirac’s Delta Distribution

4.7 Dirac Comb

4.8 Impulse Response

4.9 Transfer Function

4.10 Fourier Transform

4.11 Convolution Theorem

4.12 Rayleigh’s Theorem

4.13 Power Theorem

4.14 Filtering in the Frequency Domain

4.15 Hankel Transform

4.16 Abel Transform

4.17 Hilbert Transform

4.18 Sampling Theoremand Nyquist Criterion

4.19 Wiener–Khintchine Theorem

4.20 Fourier Transform of Discrete Signals

4.21 Finite Discrete Fourier Transform

4.22 z- Transform

4.23 Chirp z- Transform

5 Two-Dimensional Fourier-Based ReconstructionMethods

5.1 Introduction

5.2 Radon Transformation

5.3 Inverse Radon Transformation and Fourier Slice Theorem

5.4 Implementation of the Direct Inverse Radon Transform

5.5 LinogramMethod

5.6 Simple Backprojection

5.7 Filtered Backprojection

5.8 Comparison Between Backprojection and Filtered Backprojection

5.9 Filtered Layergram: Deconvolution of the Simple Backprojection

5.10 Filtered Backprojection and Radon’s Solution

5.11 Cormack Transform

6 Algebraic and Statistical ReconstructionMethods

6.1 Introduction

6.2 Solution with Singular Value Decomposition

6.3 Iterative Reconstruction with ART

6.4 Pixel Basis Functions and Calculation of the SystemMatrix

6.5 Maximum Likelihood Method

7 Technical Implementation

7.1 Introduction

7.2 Reconstruction with Real Signals

7.3 Practical Implementation of the Filtered Backprojection

7.4 Minimum Number of Detector Elements

7.5 Minimum Number of Projections

7.6 Geometry of the Fan-Beam System

7.7 Image Reconstruction for Fan-Beam Geometry

7.8 Quarter-Detector Offset and Sampling Theorem

8 Three-Dimensional Fourier-Based ReconstructionMethods

8.1 Introduction

8.2 Secondary Reconstruction Based on 2D Stacks of Tomographic Slices

8.3 Spiral CT

8.4 Exact 3D Reconstruction in Parallel-Beam Geometry

8.5 Exact 3D Reconstruction in Cone-Beam Geometry

8.6 Approximate 3D Reconstructions in Cone-BeamGeometry

8.7 Helical Cone-Beam Reconstruction Methods

9 Image Quality and Artifacts

9.1 Introduction

9.2 Modulation Transfer Function of the Imaging Process

9.3 Modulation Transfer Function and Point Spread Function

9.4 Modulation Transfer Function in Computed Tomography

9.5 SNR, DQE, and ROC

9.6 2D Artifacts

9.7 3D Artifacts

9.8 Noise in Reconstructed Images

10 Practical Aspects of Computed Tomography

10.1 Introduction

10.2 Scan Planning

10.3 Data Representation

10.4 Some Applications in Medicine

11 Dose

11.1 Introduction

11.2 Energy Dose, Equivalent Dose, and Effective Dose

11.3 Definition of Specific CT Dose Measures

11.4 Device-Related Measures for Dose Reduction

11.5 User-Related Measures for Dose Reduction

References

Subject Index