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Manipulation of Object-Based 3D THE RECONSTRUCTION AND MANIPULATION OF OBJECT-BASED 3D X-RAY IMAGES by Simant Prakoonwit This thesisis submittedin partial fulfilment of the requirementsfor the Degree of Doctor of Philosophy (Ph. D. ) and the Diploma of Imperial College (D. I. C) Departmentof Electricaland Electronic Engineering hnperial Collegeof Science,Technology and Medicine University of London May 1995 ýy4y. AKSTRAICT Computersand graphic peripherals have come to play a significant part in scientific visualisation.In medical applications, many non-invasivetechniques have beendevelopedfor visualising the inner entities of the human body. A major interest has arisen in representationof the entities by digital objects that can be manipulated and displayedusing methods developed in ComputerGraphics. A novel methodin 3D reconstructionis presentedusing the assumptionthat all the entities of interest can be representedas a set of discrete objects. An object is characterisedby its commoncharacteristic. Each object is reconstructedfrom the projections of its surface curvesin about 10 conventional2D X-ray images taken at suitableprojection angles.Yhe method automatically generates an optimumnumber of the object's surface points that are appropriately distributed Yhe method also determines the object's closed surface ftom these surface points. Each solid reconstructedobject is thenrepresented by a Boundary-representation(B-rep) scheme, which is compatiblewith any standarddisplay and manipulationtechnique, and can be manipulatedseparately. Experimentswere performed on both computer-generatedobjects and physical objects.Results show that the methodcan be usedin a wide range of applications as an aMtional to existing techniques,and can allow a &amatic decreasein radiation exposureand econon;y in data manipulation. 2 plrlý- my mum, my, dad M.. S.ds. fer L ST Page ABSTRACT 2 DEDICATION 3 CONTENTS LIST 4 FIGURE LIST 10 TABLE LIST 20 GLOSSARY AND NOTATION 21 ACKNOWLEDGEMENTS 23 CHAPTER 1: INTRODUCTION 24 1.1 3D Representation 24 1.1.1Wire-Frames 25 1.1.2Constructive Solid Geometry(CSG) 26 1.1.3Volumetric Representation 26 1.1.4Boundary Representation (B-rep) 27 1.1.5Discussion 27 1.2 3D Reconstruction 28 1.2.1Penetrating imaging modalities 32 1.2.2 Non-penetratingimaging modalities 33 1.2.33D reconstructionfrom penetrating 33 - imagingmodalities 1.2.43D reconstructionfrom non-penetrating 35 imagingmodalities 1.3 X-ray basedimaging 37 4 Contentslist 1.4 Motivation for this research 40 1.4.1Discussion on the previousapproaches on 40 3D reconstruction 1.4.2Motivation 44 1.5 The scopeof this research 45 1.6 Organisationof this thesis 46 1.7 References 51 PARTLCONCEPT CHAPTER 2: CONCEPT OF OBJECT-BASED 3D X-RAY HVIAGING 60 2.1 Introduction 60 2.2 Topologicalproperties of an object 62 2.3 Geometricalproperties of an object 64 2.3.1 Smoothand non-smooth object 64 2.3.2 Convexand non-convex ob j ect 65 2.4 Projectionof an object 66 2.4.1 Projection 66 2.4.2 Curvesin X-ray imaging 67 2.4.3 Relationsbetween the contrastcurves in 2D 71 andthe surfacein 3D 2.5 Intuitive conceptof object-Based3D X-ray Imaging 73 2.5.1 Tangentcurves 73 2.5.2 Singularcurves 74 2.5.3 3D reconstruction 74 2.5.4 Multiple objectsand object identification 77 2.6 Descriptionsand definitions 78 2.7 References 95 5 Contentslist PART 2: IMPLEMENTATION CHAPTER 3: DATA ACQUISITION 97 3.1 Introduction 97 3.2 Projection system 98 3.2.1 Projection directions 98 3.2.2 Co-ordinate and projection systems 102 3.3 X-ray systems 105 3.3.1 Systemoverview 106 3.3.2 Systemused in the project 108 3.4 Phantoms 109 3.4.1 Computer-generatedobjects 109 3.4.2 Physical objects 118 3.5 References 124 CHAPTER 4: CURVE REPRESENTATION 125 4.1 Introduction 125 4.2 Contrastcurve representation 126 4.3 Contrastcurve determination 127 4.4 Occludingcontrast curve determination 131 4.5 References 135 CHAPTER 5: OBJECT SURFACE CURVE RECONSTRUCTION 137 5.1 Introduction 137 5.2 Common tangent plane 140 5.2.1 Parallel projection 141 5.2.2 Conical projection 143 5.3 Determination of tangent points 145 5.3.1 Parallel projection 146 5.3.2 Conical projection 152 6 Contentslist 5.4 Determinationof commontangent points 158 5.4.1 Tangentpoint matching 159 5.4.2 Determinationof commontangent points 160 5.5 Verification of a commontangent point 164 5.6 Distribution of tangentpoints 171 5.6.1 Uniformly distributedprojection directions 172 5.6.2Non uniformly distributedprojection directions 179 5.7 Reconstructionof tangentand singularcurves 179 5.7.1 Reconstructionof a tangentor singularcurve 179 5.7.2 Tangentor singularcurves linking 191 5.8 Multiple objectsand objectidentification 194 5.8.1 Objectidentification 194 5.8.2 Curveidentification 202 5.9 References 206 CHAPTER 6: OBJECT SURFACE RECONSTRUCTION 208 6.1 Introduction 208 6.2 General primary surface reconstruction 212 6.2.1 Background principles 212 6.3 Implementation 216 6.3.1 Main connectednetwork 217 6.3.2 Referenceplane 218 6.3.3 Definition of the first primary polyhedron's face 219 6.3.4 Definition of the rest of the primary faces 223 6.4 Bounding primary surface reconstruction 225 6.4.1 Definition of the first primary face 226 6.4.2 Definition of the rest of the primary faces 226 6.5 Face triangulation 227 6.6 References 233 7 Contentslist CHAPTER 7: MANIEPULATION AND VISUALISATION 236 7.1 Introduction 236 7.2 Objectmanipulation and visualisation 237 7.3 Someessential tools for manipulationand visualisation 239 7.3.1 Viewing functions 240 7.3.2 Surfaceand materials 240 7.3.3 Lighting and shading 241 7.3.4 Rendering 243 7.4 Manipulationand visualisation software 246 7.5 References 247 CHAPTER 8: RESULTS 249 8.1 Introduction 249 8.2 Singleobjects 249 8.2.1 Ellipsoid 250 8.2.2Dimple-shaped object 252 8.2.3 Bone-shapeobject 254 8.3 Multiple objects 257 CHAPTER 9: DISCUSSION AND CONCLUSIONS 266 9.1 Discussion 266 9.1.1 General discussion 266 9.1.2 Data acquisition 268 9.1.3 Tangent and singular curve reconstruction 270 9.1.4 Surface reconstruction 271 9.1.5 Incomplete objects 272 9.2 Conclusions 273 9.3 Further work 273 9.3.1 Data acquisition 273 9.3.2 Incomplete objects 274 8 Contentslist 9.3.3 Processautomation and systemintegration 275 9.3.4 Surface formation 276 APPENDICES APPENDIX A: TOPOLOGICAL PROPERTIES OF SURFACES 277 APPENDIX B: SURFACESAND CONTRAST CURVES 283 APPENDIX C: PROJECTION DIRECTIONS 286 APPENDIX D: GEOMETRICAL TOOLS 292 APPENDIX E: OVERVIEW ON COMPUTER SYSTEMS 300 9 FIG'(7R,W LIST Page Fig 1.2a Digital 3D imaging. 29 Fig 1.2b 2D-2D imageprocessing (low-level). 31 Fig 1.2c 2D-2D imageprocessing (high-level). 31 Fig 1.2d 3D reconstruction. 31 Fig 1.2e2D imagingmodalities. 31 Fig 1.5a The scopeof this research. 47 Fig 1.6a Flow of operationsin this thesis. so Relevantchapter numbers are in parentheses. Fig 2.2a Sphereand bone are topologicallyequivalent, 64 torus and coffeemug is topologicallyequivalent. Fig 2.3a Non-smoothobjects. 64 Fig 2.3b (1) Convexobject (2) non-convexob j ect. 65 Fig 2.4a Contrastcurve from a tangentcurve. 67 Fig 2.4b Contrastcurve from tangentcurve and discontinuity. 68 Fig 2.4c Contrastcurve and occludingcontrast curve. 70 Fig 2.4d Curveswith branchesand junction pointsA, B. 70 Fig 2.4e Curveswith swallowtailsand crossing point A. 71 Fig 2.4f Curve with a butterfly andcrossing point B and 71 junction points A, C. Fig 2.4g Singularpoints. 72 10 Figure list Fig 2.5a (1) Contrastcurves on a spherefrom two projectiondirections. 74 (2) Contrastcurves from more projectiondirections. (3) Contrastcurves as a wire-framerepresentation of the sphere. Fig 2.5b Contrastcurves A, D from tangentcurve B, C respectivelyand 75 points o andrn arethe projectionsof point n. Fig 2.5c Tangentplane. 76 Fig 2.5d Singularcurve and tangent plane. 77 Fig 2.5e Identifyingobject's contrastcurves by two tangentplanes. 78 Fig 2.6a 1) Cartesianco-ordinate system in E'. 79 2) Right-handedtriad Cartesianco-ordinate system in V. Fig 2.6b World co-ordinatesystem (xyz) in E' and 80 3D projectionplane co-ordinate system (xp, yp, z. ) in E',' and 1P 2D projectionplane co-ordinate system (xp, yp) in EP. P Fig 2.6c Projectioncone and projection cylinder, case the projectionplanes 81 are boundedby rectangularclosed curves. Fig 2.6d The set of projectionplanes in the example. 82 Fig 2.6e Geometricprojection in parallelprojection and conicalprojection. 83 Fig 2.6f Commontangent plane in parallelprojection. 84 Fig 2.6g Commontangent plane in conicalprojection. 84 Fig 2.6h (1) Onetangent curve, one singularcurve and two contrastcurves. 86 (2) Onetangent curve, one isolatedsingular point in E', one contrast 2 curve and one isolatedsingular point in E;'. (3) Onetangent curve with one singularcurve and contrastcurves. (4) Onetangent curve, one singularpoint in E' and one contrast curvewith one singularpoint in E2P Fig 2.6i (1) Onetangent point in one commontangent plane. 87 (2) Two tangentpoints in one commontangent plane. Fig 2.6j Commontangent point andtangent points. 88 Fig 2.6k Point P is outsidesphere C andits projections. 89 11 Figure list Fig 2.61Distribution of tangentand commontangent points. 92 Fig 2.6m Pair (xV 92 0,y 1). Fig 2.6n Pair (yo, yj). 92 Fig 2.6o Distribution of planecommon tangent points. 93 Fig Ma Flow of operations in the data acquisition. 98 Relevant section numbersare in parentheses. Fig 3.2a Five regular polyhedra: (1) cube (2) octahedron 100 (3) tetrahedron (4) icosahedron(5) dodecahedron. Fig 3.2b Tetrahedron and the four projection directions. 101 Fig 3.2c (1) World co-ordinate system ft, y, z). 103 (2) 3D projection plane co-ordinate system (x,,, y..., z,. ). (3) 2D projection plane co-ordinate system (x,,, ypd. Fig 3.3a Digital x-ray system overview. 106 Fig 3.3b Transfering data from the X-ray machine into a workstation.
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