THE MATHEMATICS OF COMPUTERIZED TOMOGRAPHY
CONTENTS INTRODUCTION 1 1 TWO DIMENSIONAL X RAY PROPAGATION IMAGING 5 1 1 PROPAGATION OF X RAYS IN FREE SPACE 6 1 1 1 WAVE EQUATIONS IN FREE SPACE 6 1 1 2 ANGULAR SPECTRUM APPROACH 7 1 1 3 PARAXIAL HELMHOTZ EQUATION 9 1 1 4 FRESNEL DIRACTION 10 1 1 5 FRAUNHOFER DIRACTION 12 1 1 6 FRESNEL SCALING THEOREM 12 1 1 7 NUMERICAL IMPLEMENTATION OF FRESNEL DIRACTION 14 1 2 X RAY INTERACTION WITH MATTER 15 1 2 1 WAVE EQUATIONS IN THE PRESENCE OF MATTER 15 1 2 2 REFRACTIVE INDEX FOX X RAYS 16 1 2 3 THE PROJECTION APPROXIMATION 17 1 2 4 PHASE SHIFT AND ABSORPTION WITHIN OBJECTS 19 1 3 CONTRAST FORMATION AND IMAGING REGIMES 20 1 3 1 CONTRAST TRANSFER FUNCTION 21 1 3 2 TRANSPORT OF INTENSITY EQUATION 22 1 3 3 IMAGING REGIMES 23 1 4 EECTIVE NEAR ELD AND EMPTY BEAM DIVISION 25 1 5 COHERENCE 26 1 6 RADIATION DAMAGE AND DOSE 30 2 SOLVING THE PHASE PROBLEM 31 2 1 HOLOGRAPHIC RECONSTRUCTION 32 2 2 TRANSPORT OF INTENSITY EQUATION 35 2 2 1 PURE PHASE OBJECT 37 2 2 2 TREATMENT OF ABSORPTION 38 2 3 CONTRAST TRANSFER FUNCTION 41 2 3 1 PURE PHASE OBJECT 41 2 3 2 TREATMENT OF ABSORPTION 44 2 4 ITERATIVE ALGORITHMS 44 2 4 1 GERCHBERG SAXTON FIENUP TYPE ALGORITHMS 46 2 4 1 1 GERCHBERG SAXTON ALGORITHM 46 2 4 1 2 ERROR REDUCTION ALGORITHM 47 2 4 1 3 HYBRID INPUT OUTPUT ALGORITHM 48 2 4 2 NEWTON METHODS 50 2 5 HYBRID APPROACHES 51 3 THREE DIMENSIONAL X RAY PROPAGATION IMAGING 53 3 1 FOURIER SLICE THEOREM 55 3 2 FILTERED BACKPROJECTION 56 3 3 CONE BEAM TOMOGRAPHY 57 3 4 RECONSTRUCTION AND VISUALIZATION SOFTWARE 58 4 IMPLEMENTATION OF X RAY PROPAGATION IMAGING 59 4 1 X RAY DETECTORS 59 4 2 LABORATORY SETUP JULIA 60 4 3 GINIX AT BEAMLINE P10 OF PETRA III 62 4 3 1 COHERENCE BEAMLINE P10 62 4 3 2 GINIX 63 4 3 3 X RAY WAVEGUIDES 66 4 3 4 ORIGINAL WAVEGUIDE CONGURATION 69 4 3 5 OPTIMIZED WAVEGUIDE CONGURATION 70 4 3 6 KB CONGURATION 72 4 4 BEAMLINE ID22NI OF THE ESRF 76 4 5 DISTANCE CALIBRATION 79 4 5 1 DISTANCE CALIBRATION WITH PERIODIC STRUCTURE 79 4 5 2 DISTANCE CALIBRATION USING HOLOGRAPHIC RECONSTRUCTION 81 5 THREE DIMENSIONAL IMAGING OF RODENT COCHLEA 83 5 1 CHALLENGES OF COCHLEAR IMAGING 84 5 2 EXPERIMENT 85 5 2 1 EXPERIMENTAL SETUP 85 5 2 2 SAMPLE PREPARATION AND MOUNTING 86 5 2 3 ALIGNMENT OF ROTATION AXIS 89 5 3 DATA ANALYSIS 91 5 3 1 PHASE AND TOMOGRAPHIC RECONSTRUCTION 91 5 3 2 THRESHOLD BASED SEGMENTATION OF BONE 93 5 4 RESULTS 94 5 4 1 SOLID TARGET SOURCE 94 5 4 2 LIQUID METAL JET SOURCE 98 5 4 3 OPTIMIZING SPATIAL RESOLUTION 99 5 4 4 SPACE ESTIMATION FOR INTRA COCHLEAR IMPLANTS 105 5 5 SUMMARY 106