COMPUTED TOMOGRAPHY FROM IMAGERY GENERATED BY FLUOROSCOPY ALONG AN ARBITRARY PATH by Christopher L. Baker A thesis submitted to the Faculty and Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Master of Science (Engineering Systems) Golden Colorado Date ___________________ Signed: _____________________ Christopher L. Baker Approved:___________________ Dr. Christian Debrunner Thesis Advisor Approved: ___________________ Dr. Mohamed Mahfouz Thesis Advisor Golden Colorado Date ___________________ ________________________ Dr. David Munoz Professor and Head Department of Engineering ii ABSTRACT An accurate geometrical three-dimensional (3D) model of human bone is required in many medical procedures including computer assisted surgery. Computed Tomography (CT) and Magnetic Resonance Imaging (MRI) are commonly used to obtain these models by reconstructing a 3D dataset from image projections. However, these methods are expensive and time consuming, and the instruments are not available in most operating rooms. An alternative to CT and MRI that is available in most operating rooms is fluoroscopy. If the viewing geometry is known, fluoroscopic imagery collected over a range of viewpoints can be processed using cone-beam tomographic reconstruction methods to produce a 3D volumetric image similar to that obtained from CT. Although there are fluoroscopy machines available that will process the projections in this manner, they must be carefully instrumented and constrained to gather images along a specific pre-defined path. We propose an alternative based on tomographic processing of fluoroscopic imagery collected along an arbitrary path, which will allow use of most of the fluoroscopy machines that are currently available. We use metallic markers (beads) attached to the object being scanned and 3D model matching techniques to recover the actual path of the source/detector from the image data. Given this path, we can use cone-beam tomographic reconstruction methods to reconstruct the 3D volume. With our method, careful control and instrumentation of the trajectory is not necessary, and the required motion can be obtained by moving the fluoroscopic sensor or the patient. This capability would provide 3D volumetric imaging capabilities to most facilities with fluoroscopes. We have performed various experiments to develop and demonstrate our approach. We first generated a set of synthetic phantom data to test the reconstruction algorithm along multiple paths. Next we used a fluoroscope to collect imagery of a physical phantom. We generated two sets of physical phantom data, the first of which we used to verify our pose estimation method. The next set was used to reconstruct the physical phantom from imagery iv collected along an arbitrary path. Finally we imaged a cadaver along an arbitrary path and reconstructed the 3D volume. v TABLE OF CONTENTS Abstract........................................................................................................................................................ iv Table of Contents...................................................................................................................................... vi List of Figures.............................................................................................................................................. x List of Tables............................................................................................................................................ xvi Acknowledgments .................................................................................................................................xviii Dedication.................................................................................................................................................. xx Chapter 1......................................................................................................................................................xx Chapter 2........................................................................................................................................................7 2.1 Pose Estimation .......................................................................................................................7 2.2 Cone-Beam Tomographic Reconstruction.........................................................................8 2.2.1 Feldkamp’s method ............................................................................................................9 2.2.2 Grangeat’s method............................................................................................................11 2.2.3 Defrise and Clack method...............................................................................................12 2.2.4 LINCON method.............................................................................................................13 2.3 Reconstruction Algorithm Speed Increase .......................................................................15 Chapter 3......................................................................................................................................................17 3.1 Data Collection ......................................................................................................................19 3.2 Sensor Calibration..................................................................................................................19 vi 3.2.1 Geometric distortion........................................................................................................ 20 3.2.2 Radiometric distortion..................................................................................................... 21 3.3 Pose Estimation..................................................................................................................... 23 3.3.1 Image feature matching to find the pose ..................................................................... 23 3.3.1.1 Finding the beads .................................................................................................... 23 3.3.1.2 Marker tracking correspondence.......................................................................... 24 3.3.1.3 Pose computation with least squares iterative technique................................. 26 3.4 Image Correction................................................................................................................... 27 3.4.1 Finding the optimal origin .............................................................................................. 27 3.4.2 Re-orientation of images ................................................................................................. 31 3.5 Tomographic Reconstruction ............................................................................................. 34 3.5.1 Reconstruction algorithm geometry.............................................................................. 34 3.5.2 Reconstruction algorithm................................................................................................ 36 3.5.2.1 Grangeat’s development ........................................................................................ 36 3.5.2.2 First derivative of the radon transform from the x-ray projections .............. 40 3.5.2.3 Resample to spherical coordinates....................................................................... 41 3.5.2.3.1 Backward mapping scheme with the source searching method ............... 42 3.5.2.3.2 Forward mapping scheme with the volume minimization method......... 46 3.5.2.4 Radon inversion by consecutive backprojections............................................. 49 3.6 Summary ................................................................................................................................. 52 Chapter 4...................................................................................................................................................... 53 vii 4.1 Experimental Setup...............................................................................................................53 4.1.1 Synthetic images of four cylinders.................................................................................54 4.1.2 Synthetic images of ellipsoids .........................................................................................55 4.1.3 Physical phantom description.........................................................................................56 4.1.4 Cadaver image....................................................................................................................57 4.1.5 Calibration accuracy experiments ..................................................................................59 4.1.5.1 Geometric calibration.............................................................................................59 4.1.5.2 Radiometric calibration ..........................................................................................60 4.1.6 Pose estimation accuracy experiments..........................................................................62 4.1.6.1 Ground truth from a rotational stage..................................................................63 4.1.6.2 Ground truth from the Optotrak.........................................................................65 4.2 Tomographic Reconstruction Results ...............................................................................70