Principles of computerized tomographic imaging

I Introduction References 3 2 Signal Processing Fundamentals 2.1 One-Dimensional Signal Processing 5 Continuous and Discrete One-Dimensional Functions l Linear Operations l Fourier Representation l Discrete Fourier Transform (DFT) l Finite Fourier Transform l Just How Much Data Is Needed? l Interpretation of the FFT Output l How to Increase the Display Resolution in the Frequency Domain l How to Deal with Data Defined for Negative Time l How to Increase Frequency Domain Display Resolution of Signals Defined for Negative Time l Data Truncation Effects 2.2 Image Processing 28 Point Sources and Delta Functions l Linear Shift Invariant Operations l Fourier Analysis l Properties of Fourier Transforms l The Two-Dimensional Finite Fourier Transform l Numerical Implementation of the Two-Dimensional FFT 2.3 References 47 3 Algorithms for Reconstruction with Nondiffracting Sources 3.1 Line Integrals and Projections 49 3.2 The Fourier Slice Theorem 56 3.3 Reconstruction Algorithms for Parallel Projections 60 The Idea l Theory l Computer Implementation of the Algorithm 3.4 Reconstruction from Fan Projections 75 Equiangular Rays l Equally Spaced Collinear Detectors l A Re-sorting Algorithm 3.5 Fan Beam Reconstruction from a Limited Number of Views 93 3.6 Three-Dimensional Reconstructions 99 Three-Dimensional Projections l Three-Dimensional Filtered Backprojection 5 49 3.7 Bibliographic Notes 107 3.8 References 110 4 Measurement of Projection Data- The Nondiffracting Case 113 4.1 X-Ray Tomography 114 Monochromatic X-Ray Projections l Measurement of Projection Data with Polychromatic Sources l Polychromaticity Artifacts in X-Ray CT l Scatter l Different Methods for Scanning l Applications 4.2 Emission Computed Tomography 134 Single Photon Emission Tomography l Attenuation Compensation for Single Photon Emission CT l Positron Emission Tomography l Attenuation Compensation for Positron Tomography 4.3 Ultrasonic Computed Tomography 147 FundamentalC onsiderationsl Ultrasonic Refractive Index Tomography l Ultrasonic Attenuation Tomography l Applications 4.4 Magnetic Resonance Imaging 158 4.5 Bibliographic Notes 168 4.6 References 169 5 Abasing Artifacts and Noise in CT Images 5.1 Aliasing Artifacts 177 What Does Aliasing Look Like? l Sampling in a Real System 5.2 Noise in Reconstructed Images 190 The Continuous Case l The Discrete Case 5.3 Bibliographic Notes 200 5.4 References 200 6 Tomographic Imaging with Diffracting Sources 6.1 Diffracted Projections 204 HomogeneousW ave Equation l InhomogeneousW ave Equation 6.2 Approximations to the Wave Equation 211 The First Born Approximation l The First Rytov Approximation 6.3 The Fourier Diffraction Theorem 218 Decomposing the Green’s Function l Fourier Transform Approach l Short Wavelength Limit of the Fourier Diffraction Theorem l The Data Collection Process 6.4 Interpolation and a Filtered Backpropagation Algorithm for Diffracting Sources 234 Frequency Domain Interpolation l Backpropagation Algorithms 177 203 6.5 Limitations 247 Mathematical Limitations l Evaluation of the Born Approximation l Evaluation of the Rytov Approximation l Comparison of the Born and Rytov Approximations 6.6 Evaluation of Reconstruction Algorithms 252 6.7 Experimental Limitations 261 Evanescent Waves l Sampling the Received Wave l The Effects of a Finite Receiver Length l Evaluation of the Experimental Effects l Optimization l Limited Views 6.8 Bibliographic Notes 268 6.9 References 270 7 Algebraic Reconstruction Algorithms 7.1 Image and Projection Representation 276 7.2 ART (Algebraic Reconstruction Techniques) 283 7.3 SIRT (Simultaneous Iterative Reconstructive Technique) 284 7.4 SART (Simultaneous Algebraic Reconstruction Technique) 285 Modeling the Forward Projection Process l Implementation of the Reconstruction Algorithm 7.5 Bibliographic Notes 292 7.6 References 295 8 Reflection Tomography 8.1 Introduction 297 8.2 B-Scan Imaging 298 8.3 Reflection Tomography 303 Plane Wave Reflection Transducers l Reflection Tomography vs. Diffraction Tomography l Reflection Tomography Limits 8.4 Reflection Tomography with Point Transmitter/Receivers 313 Reconstruction Algorithms l Experimental Results 8.5 Bibliographic Notes 321 8.6 References 321 275 297 Index About the Authors