Link¨oping Studies in Science and Technology Dissertation No. 672 Cone-Beam Reconstruction Using Filtered Backprojection Henrik Turbell Department of Electrical Engineering Link¨opings universitet, SE-581 83 Link¨oping, Sweden Link¨oping, February 2001 ISBN 91-7219-919-9 ISSN 0345-7524 PrintedinSwedenbyUniTryck,Link¨oping 2001 To my parents Abstract The art of medical computed tomography is constantly evolving and the last years have seen new ground breaking systems with multi-row detectors. These tomo- graphs are able to increase both scanning speed and image quality compared to the single-row systems more commonly found in hospitals today. This thesis deals with three-dimensional image reconstruction algorithms to be used in future gen- erations of tomographs with even more detector rows than found in current multi- row systems. The first practical algorithm for three-dimensional reconstruction from cone- beam projections acquired from a circular source trajectory is the FDK method. We present a novel version of this algorithm that produces images of higher quality. We also formulate a version of the FDK method that performs the backprojection in O(N 3 log N) steps instead of the O(N 4) steps traditionally required. An efficient way to acquire volumetric patient data is to use a helical source trajectory together with a multi-row detector. We present an overview of existing reconstruction algorithms for this geometry. We also present a new family of algo- rithms, the PI methods, which seem to surpass other proposals in simplicity while delivering images of high quality. The detector used in the PI methods is limited to a window that exactly fits the cylindrical section between two consecutive turns of the helical source path. A rebinning to oblique parallel beams yields a geometry with many attractive properties. The key property behind the simplicity of the PI methods is that each object point to be reconstructed is illuminated by the source during a rotation of exactly half a turn. This allows for fast and simple reconstruction. v Acknowledgements I would like to take this opportunity to thank all those who have contributed to this thesis, directly or indirectly. First and foremost, I would like to thank my supervisor Prof. Per-Erik Daniels- son who introduced me to the field and whose enthusiasm has been a daily source of inspiration. Much of the work in this thesis is based on his ideas. It has been a privilage to have a supervisor who always keeps the door open and is eagerly willing to discuss the work. Prof. Paul Edholm, Dr. Maria Magnusson-Seger, and Jan Eriksson, Tec. Lic., for letting me participate in their research on helical reconstruction and for many interesting discussions where their knowledge and experience always were of great help. Dr. Roland Proksa, Dr. Michael Grass, and Dr. Thomas K¨ohler at Philips Research Laboratories in Hamburg, for showing an early interest in the PI method and taking the time to explain many of the real-world objectives and restraints in medical imaging. The thesis shows several examples of the symbiotic relationship developed over the years, where the Link¨oping algorithms have been improved upon by the Hamburg lab and vice versa. Prof. Gabor Herman, Dr. Samuel Matej, Dr. Robert M. Lewitt, and Bruno Motta de Carvalho at the Medical Image Processing Group at Pennsylvania Uni- versity for their hospitality and for sharing their deep knowledge in algebraic reconstruction techniques during my stay in Philadelphia in the autumn of 1998. Although this month of research did not result in any conclusive results, it gave me an invaluable perspective on image reconstruction. vii viii Acknowledgements Prof. Michel Defrise not only for publishing mind-boggling algorithms with firm theoretical foundations but also for giving creative feedback on an early draft of the thesis. Jens, our system engineer, and his predecessors, for providing a reliable com- puter system. All past and present members of Image Processing Laboratory for a most en- joyable working environment. Qingfen for all her food, love, and understanding during the final stages of the work. The financial support from the Swedish Research Council for Engineering Sci- ences (281-95-470) and from Philips Medical Systems is gratefully acknowledged. Contents 1 Introduction 1 1.1ModernComputedTomography.................... 1 1.2Cone-BeamReconstruction....................... 5 1.3MainContributions............................ 7 1.4OutlineoftheThesis........................... 7 1.5Publications................................ 9 2 Two-Dimensional Reconstruction 11 2.1ProjectionsandtheFourierSliceTheorem............... 11 2.2 Filtered Backprojection . .................... 13 2.2.1 Parallel-BeamReconstruction.................. 14 2.2.2 Fan-BeamReconstruction.................... 17 2.2.3 Short-ScanReconstruction.................... 19 2.3FastBackprojection............................ 24 2.3.1 LinksandtheBasicStep..................... 24 2.3.2 ACompleteAlgorithm...................... 27 2.3.3 ComplexityAnalysis....................... 29 3 Circular Cone-Beam Reconstruction 31 3.1ExactMethods............................... 31 3.1.1 TheCone-BeamGeometry.................... 32 3.1.2 TheThree-DimensionalRadonTransform........... 33 3.1.3 ReconstructionAlgorithms................... 34 ix x Contents 3.2TheFDKMethod............................. 35 3.2.1 ReconstructionfromPlanarDetectorData.......... 35 3.2.2 Reconstruction from Cylindrical Detector Data . ..... 37 3.2.3 Properties............................. 38 3.3VariationsoftheFDKMethod...................... 41 3.3.1 TheP-FDKMethod........................ 41 3.3.2 TheT-FDK,HT-FDK,andS-FDKMethods.......... 45 3.3.3 TheFDK-SLANTMethod.................... 47 3.3.4 ExperimentalResults....................... 50 3.3.5 Discussion............................. 52 3.4TheFDK-FASTMethod......................... 57 3.4.1 LinksandtheBasicStep..................... 57 3.4.2 ComplexityAnalysis....................... 60 3.4.3 IterativeInterpolation...................... 66 3.4.4 HardwareImplementation................... 67 3.4.5 ExperimentalResults....................... 72 3.5Discussion................................. 73 4 Helical Cone-Beam Reconstruction 77 4.1ExactMethods............................... 77 4.1.1 TheHelixGeometry....................... 78 4.1.2 ReconstructionofShortObjects................. 79 4.1.3 ReconstructionofLongObjects................. 81 4.2ApproximateMethods.......................... 84 4.2.1 Two-DimensionalBackprojection................ 84 4.2.2 NutatingSurfaceReconstruction................ 90 4.2.3 Three-DimensionalBackprojection............... 93 4.2.4 MultirowFourierReconstruction................ 95 4.3ThePIMethods.............................. 99 4.3.1 ThePI-ORIGINALMethod................... 99 4.3.2 ThePI-SLANTMethod..................... 109 4.3.3 ThePI-2DMethod........................ 112 4.3.4 ThePI-FASTMethod....................... 113 4.3.5 The n-PIMethods......................... 121 4.3.6 NoiseHomogeneity....................... 124 4.3.7 ExperimentalResults....................... 128 4.4Discussion................................. 139 5 Forward Projection through Voxel Volumes 141 5.1Methods.................................. 141 5.1.1 Siddon’sMethod......................... 141 5.1.2 Joseph’sMethod......................... 142 Contents xi 5.1.3 ASimpleMethod......................... 143 5.1.4 K¨ohler’sMethod......................... 143 5.2Experimentalresults........................... 144 5.3Discussion................................. 146 6 Summary 149 A Preservation of Line Integrals 151 A.1ProjectionofaPoint............................ 151 A.2FDKReconstruction........................... 153 A.3 Slanted Filtering . ........................... 154 B Phantom Definitions 157 B.1TheSphereClockPhantom....................... 157 B.2TheThree-DimensionalShepp-LoganPhantom............ 159 B.3TheVoxelisedHeadPhantom...................... 159 C Notation 161 Author Index 165 Bibliography 169 1 Introduction Computed tomography (CT) is a technique for imaging cross-sections of an ob- ject using a series of X-ray measurements taken from different angles around the object. It has had a revolutionary impact in diagnostic medicine and has also been used successfully in industrial non-destructive testing applications. In 1972 Hounsfield patented the first CT scanner and he was awarded a Nobel Prize to- gether with Cormack for this invention in 1979. Ever since, new developments have led to faster scanning, better dose usage and improved image quality. An important part in this story of success has been the development of new efficient image reconstruction algorithms. Although the problem of image recon- struction in its purest mathematical form was solved by Johann Radon in 1917, the field is steadily evolving and gives rise to a seemingly never ceasing flow of new algorithms. 1.1 Modern Computed Tomography There exist many texts on the history of tomography. Webb (1990) gives a detailed survey of classical tomography, i.e. techniques used before the computerised ver- sion was invented, and also discusses the first CT-related patents. Kalender (2000) presents the main developments thereafter. In this section we will focus on the on the developments in CT during the last ten years. The so-called slip-ring technique was introduced in CT around 1990. Previ- ously, power support to the X-ray tube and connectors for tapping detector data 1 2 Chapter 1 Introduction Figure 1.1 A modern tomograph (by courtesy