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Contents List of Tables ................................... v List of Figures .................................. vi List of Abbreviations and Symbols ..................... ix Acknowledgments ................................ xi Abstract ...................................... xii 1 Introduction .................................. 1 1.1 SPECTImaging .............................. 1 1.2 MR Imaging ................................ 2 2 Background .................................. 5 2.1 The Registration Problem ........................ 5 2.1.1 Imaging Geometry ........................ .5 2.1.3 What is Registration ....................... 7 2.2 TheAbsoluteOrientationProblem ................... 10 2.3 Descriptions of Rigid Rotation ..................... 15 3 Theory and Implementation of the New Method ........... 18 3.1 Non-iterative Direct Method ....................... 18 3.2 Revisiting Methods Based on the SVD ................. 23 3.3 Reference System ............................. 26 3.4 Preliminary Studies ............................ 27 3.5 Reference System Orientation ...................... 28 3.5.1 Image Segmentation ..................... 30 3.5.2 Valid versus Invalid Images ................... 32 ... 111 3.6 Determination of Scale .......................... 33 3.6.1 Volume Rendering ........................ 34 3.7 Determination of Translation and Rotation ............... 35 3.7.1 Fittingastraight Linetoa3-D9C1oud" ............ 35 3.7.2 Translational Part ........................ 39 3.7.3 Rotational Part .......................... 39 3.8 Applying the Registration Transformation ................ 43 3.8.1 Mu1tilinea.r Interpolation ..................... 15 3.9 Implementation .............................. 50 4 Experirnental Results ............................ 52 4.1 Experimental Setups and Data Sets ................... 52 4.2 Strategy for Accuracy Estimation .................... 53 4.3 SPECT-teSPECT Registration ..................... 54 4.4 SPECT-bMRRegistration ....................... 57 4.5 MR-to-MR Registration ......................... 55 5 Conclusions .................................. 61 1 Discussion ................................. 61 5.1.1 Reference System Design ..................... 62 5.1.2 Interpretation of Error Measurements .............. 62 5.2 Possible Applications ........................... 63 5.3 Future Development ........................... 64 References ..................................... 65 Appendix A Quaternion Mathematics ................... 68 A.l Conjugate and Norm of Quaternions .................. 69 A.2 Product of Quaternions .......................... 69 A.3 Division of Quaternions .......................... 70 Appendix B Program Listings ........................ 71 B.1 First ModuIe ............................... 71 B.2 Second Module .............................. 76 List of Tables 3.1 Example of coordinates (in pixels) of reference points for an MRI data set ..................................... 37 4.1 Example of a registration enor table for an MRI data set ....... 55 4.2 Angles of rotation for the SPECT-teSPECT registration ........ 56 1.3 Angles of rotation for the SPECT-teMR registration .......... 58 4.4 Statistics on the MR-teMR registration ................. 59 4.5 Angles of rotation for the MR-t*MR registration ............ 60 A .1 Quaternion multiplication table ...................... 69 List of Figures 1.1 SPECT imagingscheme .......................... 1.2 A transaxid SPECT Image........................ 1.3 A transaxial MR Image .......................... 2.1 Imaging geometry............................. 2.2 Imaging orientations . S-sagittal, C.coronal, T.transaxia.l ....... 2.3 SPECT Imaging . (a) Placement of the imaging planes . (b.c) Two adjacent slices ............................... 2.4 MR Imaging . (a)Placement of the imaging planes . (b.c) Two adjacent slices.................................... 2.5 Orientations and spacings of the section planes in different machines . 2.6 Steps for registration ........................... 2.7 Corresponding MRI and SPECT slices.................. 2.8 Problem: match triangles ......................... 2.9 Reference points. known positions .................... 3-10 Reference points. unknown positions ................... 2.11 Tree diagram illustrating the classification of the computat iond met h- ods for solving the absolute orientation problem ............ 3.1 Structure of triangles ........................... 3.2 Unique position of a rigid body ...................... 3.3 Left to right: initial position, rotated position. flipped position .... 3.4 Solid angle of plausible rotations ..................... 3.5 Acquired data............................... 3.6 Difficulty in using of vertices as reference................ 3.7 Reference system prototype ........................ 3.8 Problems encountered in preliminary studies. (a) Marginal slice is "empty* . (b) Vertex lies between slices. (c) Vertex does not belong to the set . (d) Tubing curvature imposes a problem ............ 3.9 Marginal markers added to the reference system ............ 3.10 Location of reference points in images is a function of proximity to the triangle vertex ............................... 3.11 (a) Original monochrome image . (b) Blurred image thresholded at 10% of the maximum intensity. (c) Original image thresholded at 10% of the maximum intensity.......................... 3.12 (a) Mask image. (b) Result of masking . (c) Masked image thresholded at 30% of the maximum intensity .................... 3.13 Close-up of the tubing crossed by a slice ................. 3.14 Reduction of the 3-D line-fitting problem to 2-D cases in planes YZ andXZ.................................. 3.15 Vectors representing sides......................... 3-16 Rotation in terrns of quaternions ..................... 3.17 Finding the rotational part of the registration transformation as a com- bination of two rotations . (a) First rotation . (b) Second rotation ... 3.18 Voxel analysis in an MR image ...................... 3.19 Voxel analysis in a SPECT image .................... 3.20 Pixelation effect and "edgy" transitions ................. 3.21 Explanation for the pixelation effect and "edgy" transitions ...... 3.22 One-dimensional cross.section ...................... 3.23 Won-interpolated data........................... 3.24 Goal of interpolation ........................... 3.25 Neighbouring supervoxels ......................... 3.26 Influence of neighbours .......................... 3.27 Computation of the grey scale value of a new voxel . (a) In a "level" .(b) Between the " levels" ........................... 3.25 Results of interpolation . (a)SPECT image, no interpolation . (b)SPECT image. interpolated ............................ 4.1 SPECT-to-SPECT registration . (a) Benchmark image. (b) Registered image . (c) Result of subtraction ..................... vii 4.2 SPECT-to-SPECT registration error surfaces. (a) " Idle triangle". (b) " Reference triangle". ........................... 57 4.3 SPECT-twMR registration. (a) Benchmark image. (b) Registered image. (c) Result of subtraction. .................... 57 4.4 SPECT-to-MR registration error surfaces. (a) "Idle triangle". (b) " Reference trianglen. ........................... 58 4.5 MR-t-MR registration. (a) Benchmark image. (b) Registered image. (c) Result of subtraction. ........................ 59 1.6 MR-to-MR registration error surfaces. (a) " Idle triangle". ( b) " Refer- ence trianglen. .............................. 60 List of Abbreviations and Symbols MRI Magnetic Resonance Imaging ................................... 1 SPECT Single Photon Emission Cornputed Tomography ................ 1 PET Positron Emission Tomography ................................. 4 PSF Point Spread Function ......................................... 1 SNR Signal to Noise Ratio .......................................... 2 Radio Frequency ............................................... 3 SVD Singular Value Decomposi t ion .................................. 16 DVR Direct Volume Rendering ...................................... 31 Coordinate system of the scanner ............................... 9 Coordinate system of the body (head)........................... 9 Sagittal, Coronal . and Transaxial irnaging orientations .......... 5 ICP Iterative Closest Point Algorithm ..............................