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Simultaneous Reconstruction and 3D Motion Estimation for Gated Myocardial Emission Tomography


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SIMULTANEOUS RECONSTRUCTION AND 3D MOTION ESTIMATION FOR GATED MYOCARDIAL EMISSION TOMOGRAPHY By ZIXIONG CAO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2003

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This dissertation is dedicated to my parents and my wife

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ACKNOWLEDGMENTS It is impossible to adequately thank my long-time advisor and mentor, David Gilland for his tremendous guidance, considerable patience and encouragement needed for me to proceed through the graduate study and complete this dissertation. His rich experience in the medical-imaging area, his thoughtful insight, his brilliant and creative ideas, and his great sense of humor have made my graduate study very rewarding. Bernard Mair deserves many thanks. His lecture on numerical analysis introduced me to this area and led to all of the computation work described in this dissertation. His constant guidance and supervision laid a smooth way for my studies and research work. I also thank Chris Batich and David Hintenlang for their discussions, suggestions, and encouragement during the development of this dissertation. It is a great honor to have them serve on my committee. I would like to thank Mu Chen and Karen Gilland for their helpful suggestion on my numerical observer study. I must express my sincere gratitude particularly to William Ditto for his support. I thank the Biomedical Engineering Department (especially April-Lane Derfinyak and Laura Studstill) for their help, kindness, and patience. I thank Rami, Uday, and Ping-Fang for making the lab such a great place to work. I also thank Shu and Jiangyong for their comments and suggestions. Finally, I thank my dear parents, my sister Xia, and my wife Xueshuang for their endless love, support, encouragement, and understanding in dealing with all of the challenges that I have faced. I owe a great debt to them. iii

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TABLE OF CONTENTS Page ACKNOWLEDGMENTS .................................................................................................iii LIST OF TABLES ............................................................................................................vii LIST OF FIGURES .........................................................................................................viii ABSTRACT .......................................................................................................................xi CHAPTER 1 INTRODUCTION......................................................................................................1 1.1 Introduction..........................................................................................................1 1.2 Significance..........................................................................................................3 2 BACKGROUND........................................................................................................5 2.1 Myocardial Emission Tomography......................................................................5 2.2 Projection and Image Reconstruction..................................................................6 2.2.1 Radon Transform.....................................................................................6 2.2.2 Maximum Likelihood and Image Reconstruction...................................9 2.3 Myocardium Motion Estimation........................................................................11 2.4 Image Observer Evaluation................................................................................13 2.4.1 Human Observer Study..........................................................................13 2.4.2 Numerical Observer: Background.........................................................15 2.4.3 Hotelling Observer and Channelized Hotelling Observer.....................16 3 FORMULATION OF SIMULTANEOUS ESTIMATION METHOD...................20 3.1 Objective Functions............................................................................................20 3.2 Euler-Lagrange Equations..................................................................................21 3.3 Optimization Algorithm: RM.............................................................................26 3.3.1 Scheme of Optimization Algorithm.......................................................26 3.3.2 Computation of R Step..........................................................................27 3.3.3 Computation of M Step.........................................................................29 iv

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4 IMPLEMENTATION AND EXPERIMENTAL RESULTS: GEOMETRIC PHANTOM..............................................................................................................31 4.1 Simulated Source Object and Projection Data...................................................31 4.1.1 Steps to Generate Geometric Phantom..................................................31 4.1.2 Dimensions of Geometric Phantom.......................................................32 4.1.3 Intensity and Defect...............................................................................33 4.1.4 Wall Motion Simulation........................................................................34 4.1.5 Forward Projection Simulation and Detector Response........................34 4.2 Bi-Value Lam Constants Model.......................................................................35 4.3 Convergence Properties of RM Algorithm........................................................38 4.4 Motion Estimation Results.................................................................................39 4.4.1 Figures of Merit.....................................................................................39 4.4.2 Lam Constants Study Results..............................................................41 4.4.3 Global Motion Evaluation.....................................................................42 4.4.4 Regional Motion Evaluation..................................................................44 4.5 Image Reconstruction Results............................................................................45 4.5.1 Figures of Merit.....................................................................................45 4.5.2 Global Image Evaluation.......................................................................46 4.5.3 Defect Contrast Evaluation....................................................................49 4.5.4 Regional Image Evaluation....................................................................49 5 IMPLEMENTATION AND EXPERIMENTAL RESULTS: NCAT PHANTOM.53 5.1 Simulated NCAT Phantom.................................................................................53 5.2 Motion Estimation Results.................................................................................54 5.2.1 Global Motion Evaluation.....................................................................54 5.2.2 Regional Motion Evaluation..................................................................57 5.3 Image Reconstruction Results............................................................................58 5.3.1 Global Image Evaluation.......................................................................58 5.3.2 Defect Contrast Evaluation....................................................................59 5.3.3 Regional Image Evaluation....................................................................62 6 OBSERVER EVALUATION OF IMAGE QUALITY...........................................64 6.1 Methods ..........................................................................................................64 6.1.1 Single-Slice CHO Model.......................................................................64 6.1.2 Multi-Frame Multi-Slice CHO-HO Model............................................66 6.2 Results of Geometric Phantom Study................................................................69 6.2.1 Single-Slice CHO Results......................................................................69 6.2.2 Multi-Frame Multi-Slice CHO-HO Results..........................................69 6.3 Results of NCAT Phantom Study......................................................................72 6.3.1 Single-Slice CHO Results......................................................................72 6.3.2 Multi-Frame Multi-Slice CHO-HO Results..........................................74 6.4 Multi-Frame Multi-Slice Multi-View CHO-HO Study.....................................74 v

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7 CONCLUSIONS AND FUTURE WORK..............................................................79 APPENDIX NUMERICAL ANALYSIS FOR MOTION ESTIMATION..................82 LIST OF REFERENCES...................................................................................................92 BIOGRAPHICAL SKETCH.............................................................................................98 vi

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LIST OF TABLES Table page 2-1. Conditional probabilities of a defect detection task...................................................14 4-1. Ellipsoid semi-axes lengths for source object............................................................33 4-2. Poisson ratio and Lam constants...............................................................................37 4-3. Global motion evaluation results of the geometric phantom......................................44 4-4. Regional motion evaluation results of the geometric phantom..................................45 5-1. Global motion evaluation results of the NCAT phantom...........................................56 5-2. Regional motion evaluation results of the NCAT phantom.......................................58 5-3. The contrast and noise level of the NCAT phantom..................................................62 5-4. Five sets of hyper-parameters for RM used for regional image evaluation...............62 6-1. SCHO detectability index results of the geometric phantom.....................................71 6-2. MMCHO detectability index results of the geometric phantom................................71 6-3. SCHO detectability index results of the NCAT phantom..........................................74 6-4. MMCHO detectability index results of the NCAT phantom.....................................74 6-5. MMMCHO detectability index results of the NCAT phantom..................................78 vii

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LIST OF FIGURES Figure page 2-1. Anatomic orientation of the heart.................................................................................6 2-2. Principle of data acquisition and geometric considerations for SPECT.......................7 2-3. Coordinate systems transformation..............................................................................8 2-4. Images of MLEM and PS-MLEM..............................................................................11 2-5. Motion vector representing the displacement of voxels.............................................12 2-6. Kleins iterative motion-estimation optimization method..........................................13 3-1. Scheme of the RM algorithm......................................................................................27 4-1. Source objects of the geometric phantom...................................................................32 4-2. Simulation steps of the geometric phantom...............................................................33 4-3. Ideal motion fields of the geometric phantom in 3 short-axis slices..........................35 4-4. Collapse high-resolution projection to low-resolution projection..............................36 4-5. Segmentation for bi-value Lam constants model.....................................................38 4-6. Convergence of the RM algorithm.............................................................................39 4-7. Convergence of R step................................................................................................40 4-8. Convergence of M step...............................................................................................40 4-9. Global motion error vs. for several Poisson ratios..................................................41 4-10. Ideal motion vector fields superimposed on true images.........................................42 4-11. Estimated motion by M step using true images........................................................43 4-12. Individually reconstructed images by PS-MLEM and estimated motion by M step.43 4-13. Estimated motion and reconstructed images by RM................................................43 viii

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4-14. Chosen region to calculate regional motion error for the geometric phantom.........45 4-15. RMS errors of the RM and PS-MLEM images........................................................47 4-16. Hyper-parameter effects on the RM images.............................................................48 4-17. RMS error of the RM images as a function of and .............................................48 4-18. Defect contrast of the PS-MLEM and RM images...................................................49 4-19. The trade-off between the contrast and noise level of the geometric phantom........51 4-20. Sub-regions for regional image evaluation for the geometric phantom...................51 4-21. The trade-off between regional intensity bias and variance of the geometric phantom....................................................................................................................52 5-1. Longand shortaxis views of the NCAT cardiac phantom......................................54 5-2. Ideal motion vector fields superimposed on true images...........................................55 5-3. Estimated motion by M step using true images..........................................................55 5-4. Individually reconstructed images by PS-MLEM and estimated motion by M step..56 5-5. Estimated motion and reconstructed images by RM..................................................56 5-6. Global motion errors of three methods.......................................................................57 5-7. Chosen region to calculate regional motion error for the NCAT phantom................58 5-8. RMS errors of the RM and PS-MLEM images..........................................................59 5-9. RMS error vs. iteration number of RM outer loop.....................................................59 5-10. Normal and defect regions in the NCAT phantom...................................................60 5-11. Contrast vs. cut-off frequency of PS-MLEM...........................................................61 5-12. The trade-off between the contrast and noise level of the NCAT phantom.............61 5-13. Sub-regions for regional image evaluation for the NCAT phantom........................63 5-14. The trade-off between regional intensity bias and variance of the NCAT phantom..63 6-1. The RSC-model channels used for channelization in single-slice CHO model.........65 6-2. Defect locations in the phantoms................................................................................65 6-3. The RSC-model channels represented as spatial-domain templates..........................67 ix

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6-4. Two-step procedure of the multi-frame multi-slice CHO-HO model........................68 6-5. SP images of the geometric phantom reconstructed by PS-MLEM..............................70 6-6. SP images of the geometric phantom reconstructed by RM......................................70 6-7. MMCHO detectability index results of the geometric phantom................................71 6-8. SP true images of the NCAT phantom.......................................................................72 6-9. SP images of the NCAT phantom reconstructed by PS-MLEM................................73 6-10. SP images of the NCAT phantom reconstructed by RM..........................................73 6-11. MMCHO detectability index results of the NCAT phantom...................................75 6-12. Multi-frame multi-slice multi-view SP images reconstructed by PS-MLEM..........76 6-13. Multi-frame multi-slice multi-view SP images reconstructed by RM.....................77 6-14. Multi-frame multi-slice multi-view SP images reconstructed by RM.....................77 6-15. MMMCHO detectability index results of the NCAT phantom................................78 A-1. Trilinear interpolation coordinates............................................................................90 x

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SIMULTANEOUS RECONSTRUCTION AND 3D MOTION ESTIMATION FOR GATED MYOCARDIAL EMISSION TOMOGRAPHY By Zixiong Cao December 2003 Chair: David Gilland Major Department: Biomedical Engineering A new method for simultaneous image reconstruction and wall-motion estimation in gated myocardial emission tomography (ET) is presented. We implemented the method using simulated phantoms and evaluated the performance of the method. To reduce the blur in the images induced by the heart motion, gated ET uses the patients electrocardiogram (ECG) to trigger and acquire a time sequence of image frames, each capturing a particular stage of the cardiac cycle. However, the individual gated image frames reconstructed by conventional image-reconstruction algorithms are very noisy due to the reduced counts of detected gamma rays. Non-rigid deformation of myocardium can be characterized by means of a vector field called the motion field, which describes the relative displacement of each voxel, and thus establishes a voxel-intensity correspondence between two image frames. Previous research incorporated a deformable elastic material model into a motion-estimation algorithm to more accurately describe the deformation of the myocardium. The estimated xi

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motion field was used to warp each frame to a common reference volume. All warped frames were then recombined to generate an improved composite image. Instead of using the motion to warp and recombine frames, our simultaneous estimation method uses the motion field to regularize the individual reconstructed image frames, whereas the projection data are also a constraint for motion estimation. Thus, the simultaneous method is expected to improve the motion-estimation accuracy and the image quality of individual frames, relative to the independent motion-estimation and image-reconstruction methods. We presented the mathematical formulation of the simultaneous estimation method, followed by an optimization algorithm. The algorithm was implemented on a simulated geometric phantom and a realistic non-uniform-rational-B-splines-based cardiac torso (NCAT) phantom. We evaluated the simultaneous method in terms of the motion accuracy and the image quality. A numerical observer model was used to assess the detectability provided by the simultaneous method for the defect detection task. Results showed that the simultaneous method produces better motion accuracy and image quality, relative to the independent motion estimation and an alternative image-reconstruction algorithm. xii

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CHAPTER 1 INTRODUCTION 1.1 Introduction Nuclear medicine provides images of diseases by administering small amounts of radioactive materials into the patients body and detecting the emitted gamma rays using special types of cameras. By processing the counts of gamma rays detected from various angles, a computer reconstructs images, which can provide information about the function of the body area being imaged. Nuclear medicine is unique in its ability to create functional images of blood flow or metabolic processes rather than the more conventional structural or anatomic images produced by X-ray examination, computed tomography (CT), and magnetic resonance imaging (MRI) [1]. Tomography achieves detailed organ studies by separating a three-dimensional (3D) object into a stack of two-dimensional (2D) cut sections or slices. Unlike the X-ray CT that detects transmitted X-ray photons, tomography in nuclear medicine detects emitted gamma rays and therefore is referred to as emission tomography, or ET in short [2]. Single Photon Emission Computed Tomography (SPECT) and Positron Emission Tomography (PET) are two widely used modalities in the clinical nuclear medicine. By measuring both the heart blood flow and the metabolic rate of the patients, physicians can find areas of decreased blood flow (such as that caused by blockages), and differentiate diseased from healthy muscle. This information is particularly important in diagnosing coronary artery diseases (CAD) [3, 4]. Myocardial ET is one way to measure blood flow and metabolism in the myocardium. SPECT imaging assesses the severity and 1

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2 extent of perfusion defects in patients with coronary stenosis. It accounts for over 90% of all myocardial perfusion imaging performed in the US today [5]. The motion of the patients heart causes blurring in myocardial ET images. This motion blurring can be reduced by gated myocardial ET, which uses the patients electrocardiogram (ECG) signal to trigger and acquire a time sequence (typically 8 or 16 frames) of acquisition over the cardiac cycle [6-9]. Data are acquired over many cardiac cycles to produce the final set of image frames. Each gated image frame records a particular stage of cardiac cycle. In addition to reducing motion blurring in each image frame, gated imaging has the advantage of allowing an estimation of cardiac wall motion and ejection fraction. However, the statistical quality of each gated image suffers as the total number of detected gamma rays is distributed over a series of time frames. Because of the low gamma ray counts, the images produced by conventional image-reconstruction algorithms usually are very noisy [4, 6]. Previous researchers [10-13] presented several reconstruction methods to improve the quality of the gated image frames. Lalush et al. [10] considered the time sequence of gated frames as a four-dimensional (4D) image and reconstructed it using a penalized maximum-likelihood (ML) technique with space-time Gibbs priors to ensure smoothness in individual image frames and between frames. Narayanan et al. [11], Wernick et al. [12] and Brankov et al. [13] used an alternative technique based on principle component analysis, in which the time sequence of gated datasets were Karhunen-Loeve (KL) transformed and then rapidly constructed frame-by-frame. The gated image frames were finally obtained by applying the inverse KL transform. Other previous studies [14-16] presented methods to obtain a vector field

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3 describing the motion of the myocardium between successive frames. Klein and Huesmans method [15, 16] views the myocardium as an isotropic elastic material whose deformations are governed by the equations of continuum mechanics. It was shown that deforming and combining the gated image frames using the estimated motion resulted in improved composite images. Based on these previous studies, we generated a new idea to simultaneously reconstruct two gated image frames and estimate the motion vector field between them. Our hypotheses are: constraints from the material property will reduce the noise level in individual gated image frames, and less-noise images will in turn produce more accurate motion estimation. 1.2 Significance Based on the 3D motion-estimation method developed by Klein et al. [15, 16] and the conventional maximum-likelihood expectation-maximization (MLEM) algorithm [17, 18], we proposed a new estimation method for gated myocardial ET: the simultaneous image-reconstruction and 3D motion-estimation method. From an image reconstruction viewpoint, the simultaneous estimation method uses the deformation of elastic material that enforces the consistency between frames as a penalty to perform penalized ML reconstruction. Unlike the usual penalized ML algorithm [10, 19], our penalty uses the deformation to regularize the reconstructed images, without any spatial smoothing. In their initial attempt, Gilland and Mair [20] used the Polak-Ribiere conjugate gradient optimization algorithm to solve the problem. Because of the complexity of the objective function, it is not clear that this algorithm converges and may produce negative image intensities. Our study developed a new two-step iterative algorithm (RM) that

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4 alternately updated the gated images and the motion vector field, and guaranteed non-negativity in the images. This RM algorithm and some results were partly reported by Cao et al. [21]. We evaluated the motion-estimation accuracy and image quality provided by the simultaneous method using a simulated geometric phantom and a realistic Non-uniform-rational-B-splines-based Cardiac Torso (NCAT) cardiac phantom. We also evaluated the image quality using a numerical observer model for both phantoms. The numerical observer results have been reported by Cao et al. in An observer model evaluation of simultaneous reconstruction and motion estimation for emission tomography submitted to 2004 IEEE International Symposium on Biomedical Imaging, Arlington, Virginia, US. Myocardial ET provides perfusion metabolic and wall-motion information in a single imaging procedure. Our simultaneous estimation method promises to improve motion-estimation accuracy and individual gated image quality simultaneously, which will improve the accuracy in diagnosing cardiac diseases.

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CHAPTER 2 BACKGROUND 2.1 Myocardial Emission Tomography While other cardiac imaging modalities (angiography, echocardiography, etc.) provide purely anatomical information, myocardial ET provides physiological information, such as that used to identify areas of relatively or absolutely reduced myocardial blood flow associated with ischemia or scar [1]. SPECT and PET are two main myocardial ET protocols used today. Although we presented and implemented our method on the basis of SPECT model because of its fundamental concept and simplicity, it can be expanded to estimate the image and motion for PET imaging. In SPECT myocardial perfusion imaging procedure, the patient is intravenously given a pharmaceutical bound with a radioisotope label (such as Tc-99m). The radioactive pharmaceutical is then taken up in the patients myocardium in proportion to perfusion or tissue metabolism. Single photons are emitted as the Tc-99m radioisotope decays, and a fraction of these photons are able to penetrate the surrounding body tissue and reach the SPECT gamma camera [22]. The gamma camera records the spatial distribution of the photons and forms a 2D projection of the 3D activity distribution. A 3D image of the organ is then reconstructed from the projection data collected from various angles. Projection and reconstruction procedures are discussed later. Patients with significant coronary artery stenosis usually have diminished radioactive-pharmaceutical concentration in the area of decreased perfusion, which will cause a defective area in the perfusion image. By comparing the images taken in two 5

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6 cases, when the tracer is administered during stress and rest, physicians can distinguish ischemia from scar tissue. If the defect in the image taken in stress is worse than that taken at rest, it is most likely due to ischemia. If the defect remains unchanged, the lesion is most likely from scar [1]. The human heart usually points downward, approximately 45 to the left and anterior, but individual variations are considerable. Cuts perpendicular to the long axis are called coronal or short axis cuts. Cuts perpendicular to the short axis are called sagittal (vertical long axis) and transaxial (horizontal long axis) cuts [4]. Figure 2-1 shows anatomic orientation of the heart. Figure 2-1. Anatomic orientation of the heart. 2.2 Projection and Image Reconstruction 2.2.1 Radon Transform Figure 2-2 shows the 2D data acquisition scheme in SPECT using a parallel collimator [22]. Rotating the detector allows one to observe the photon emission in the field of view from many angles. The number of scintillations detected at location s along

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7 the detector when the detector head is at an angle of is defined as g(s, ). We denote the position of a point in the 2D slice as (x, y) and the estimated number of photons emitted from this point as a function f(x, y). This estimated image function f(x, y) is assumed to be proportional to the tracer concentration of interest. Function g is the projection of f onto the detector as allowed by the parallel collimator. A sinogram is an image representative of g in which the horizontal axis represents the angular position of the detector and the vertical axis represents the count location on the detector. s g(s,) y x f(x,y) detector object collimator u Figure 2-2. Principle of data acquisition and geometric considerations for SPECT. Mathematically, the projection operator can be described by the Radon transform [23]. The Radon transform g(s, ) of the function f(x, y) is the line integral of the values of f(x, y) along the line parallel to the collimator at a distance s from the origin: uucossins,sinucossu)u,s(),u,s()s,(dfdyxfg (2-1) where u denotes the location of the points along the integral line. Equation 2-1 involved a

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8 coordinate system transformation, which is illustrated in Figure 2-3. Assuming the two coordinate systems share the origin, the transformation between them is ucossins and sinucoss yx (2-2) From an operators point of view, the Radon transform can be represented as a procedure of an operator product: rrrdfhHfg)()()s,()s,(s (2-3) where r = (x, y) is the coordinate vector, H is called a forward-projection operator, and is a weight function that represents the probability that a photon is emitted from r and detected at (s, ) )(srh y s x u Figure 2-3. Coordinate systems transformation. In numerical analysis by computers, (s, ) and (x, y) are all represented as discrete variables, and g(s, ) and f(x, y) are functions of these discrete variables. A discretization procedure for all the variables and the functions is inevitable. In our study, each element of the 3D object image is called a voxel, and each measurement on the detector at a particular projection angle is called a bin. In the discretized image and sinogram, the element is the j )(jf th voxel in the image and )(ig is the i th bin in the sinogram.

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9 In the discretized domain, the sinogram and the image can both be considered as vectors. The elements in the sinogram vector are the counts of all the bins. The elements in the image vector are the intensities of all the voxels. In our study, for convenience, we use the same notation for function and vector. For example, when we use g(s, ), g is a continuous function; when we use g is a vector with the element index of i. )(ig For the discretized image and sinogram, the forward-projection operator H becomes a matrix. Vector g is then the matrix product of H and vector f : g = Hf, or in the form of components, qjijjfhiHfig1)()()( i =1, 2,, p. (2-4) where H is now called a forward-projection matrix with elements of h ij q is the total number of voxels in the image slice, and p is the total number of bins in the sinogram. The entries of H can be carefully chosen to take into account the geometry of acquisition and, more precisely, the detector response, attenuation and scatter. 2.2.2 Maximum Likelihood and Image Reconstruction By comparing the projection calculated from the current image estimate to the measured projection data, an iterative image-reconstruction algorithm adjusts the image estimate iteratively, and eventually reconstructs an image [23]. The iterative algorithms differ in the comparison method and the correction applied to create the new estimate. Equation 2-4 is based on the assumption of the ideal noise-free situation. In experimental measurements, however, the counts are subject to randomly change under the Poisson statistics of the radioactive disintegrations. In emission tomography, the well-known log-likelihood function can be written as

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10 piiHfiHfigfgL1)]())(log()([),( (2-5) which measures the likelihood that the image f produces the sinogram g. Maximizing the log-likelihood function leads to finding the most likely image f to produce the sinogram g. Maximum-likelihood expectation-maximization (MLEM) is an iterative algorithm that reconstructs this most likely image. Each MLEM iteration is divided into two steps [18, 24]: Expectation step (E step) in which a formula is formed to express the likelihood of any image given the measured data and current estimate f (k) and Maximization step (M step) in which a new image f (k+1) that has the greatest likelihood is found. The iterative image-updating formula of the MLEM algorithm is given as [18, 24] piijqjkijpiijkkhjfhighjfjf11')('1)()1()'()()()( j = 1, 2,, q (2-6) MLEM shows its advantage of monotonicity of the likelihood function versus iteration [17]. The convergence, however, is very slow and the images appear unacceptably noisy before it converges [25]. To smooth the noisy image reconstructed by MLEM, a digital post-processing filter is usually applied to the image reconstructed after many iterations of MLEM. This Post-Smoothed MLEM (PS-MLEM) algorithm has demonstrated excellent results in previous studies [26, 27]. The most important parameter in the PS-MLEM algorithm is the cut-off frequency of the filter, which determines the smoothness of resulting images. In our study, we use 100 iterations MLEM and low-pass Hann filter because of its simplicity. Figure 2-4 (A) shows a sample image reconstructed by MLEM with 100 iterations. Figure 2-4 (B) demonstrates that the

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11 smoothness of the image reconstructed by PS-MLEM is influenced significantly by the amount of the post-smoothing filter. Nyquist frequency (0.5 cycle/pixel) is usually used as a feasible cut-off. Because of this, when we compare our algorithm to PS-MLEM algorithm, we usually use a range of cut-off frequencies (from 0.1 to 1.0 cycle/pixel) for PS-MLEM. A B Figure 2-4. Images of MLEM and PS-MLEM. A) 100-iteration MLEM image B) PS-MLEM image using Hann filter with various cut-off frequencies. From 1 to 10, the cut-off frequency was 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0 cycle/pixel. 2.3 Myocardium Motion Estimation A variety of methods have been proposed recently to estimate and represent the cardiac motion [14-16, 28-32]. Based on a deformable elastic model and the assumption that voxels corresponding to the same tissue in two frames conserve their intensity values, Klein and Huesman [15, 16, 32] presented a voxel-based method to estimate the cardiac

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12 motion vector field. This motion field is compatible for use by the simultaneous method since the images are also represented as voxel-based intensity scalar field. Kleins motion-estimation method [32] involves minimizing a two-component objective function. The first component is an image matching objective function serves as a driving force to push deformed image intensities of two frames into correspondence. The second component is a regularization of strain energy function that prevents deformations that are physically unlikely or meaningless. We denote the frame 1 and 2 as and, respectively, and the motion vector field as m(r) = (u(r), v(r), w(r)), where r = (x, y, z) are the voxel coordinates in the spatial domain. As shown in Figure 2-5, the motion vector m(r) presents the displacement from a voxel in Frame 1 to its correspondence that represents the same tissue in Frame 2. [32] )(1rf )(2rf m(r) Frame1: f1(r) Frame2: f2(r+m(r)) rr+m(r) x y Figure 2-5. Motion vector representing the displacement of voxels The image matching objective function is formulated as rrmrrmdffffEI22121)()(,, (2-7) and the strain energy E s (m) is

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13 rrrmdwvwuvudwvudwvuEyzxzxyzyxzyxS222222222)( (2-8) where are elastic weighting parameters called Lam constants that reflect the degree of compressibility of the object, x uux is the derivative of the motion component u with respect to x. The entire objective function is grouped as E = E I + E S where is a hyper-parameter that control the balance between the two terms. In the 3D discrete numerical domain, E can be expressed as a function of all the variables and their derivates, where j =1, 2, q denotes the voxel index in the image. ),,(jjjwvu qjjyjzjxjzjxjyqjjzjyjxqjjzjyjxqjjjjwvwuvuwvuwvuffE1222122212122122)()(mrr (2-9) Kleins iterative optimization method [32] to minimize the entire objective function E consists of steps shown in Figure 2-6. Approximate E to a quadratic form (E~)around the current estimate m~(k) Find the minimum point of E~, denoted as m Use mas the new estimatem~( k +1) Figure 2-6. Kleins iterative motion-estimation optimization method 2.4 Image Observer Evaluation 2.4.1 Human Observer Study Besides the quantitative assessment such as the signal-to-noise ratio and defect contrast, the image quality and the performance of a reconstruction algorithm can also be

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14 assessed by observer studies, in which the observers evaluate the images from the perspective of a clinically specific task. Defect detection in myocardial ET imaging is a specific two-state classification task that requires the observer to diagnose by viewing the image and classifying it as either normal or some particular disease state. The accuracy of the diagnosis can also be viewed as an observer measurement of the image quality. Psychophysical study with human observers (usually the physicians) is the standard means of observer evaluation of detection task performance in medical imaging [33]. For a defect detection task, there are two possible true states of the object: signal present (SP) and signal absent (SA). There are also two possible responses: positive and negative, corresponding to the observers classification of SP and SA, respectively. Table 2-1 shows the conditional probabilities of a certain response given the occurrence of a certain stimulus [34]. Table 2-1. Conditional probabilities of a defect detection task Stimulus = SP Stimulus = SA Response = Positive True positive P(TP)=P( Positive | SP ) False positive P(FP)= P( Positive | SA ) Response = Negative False negative P(FN)=P( Negative | SP ) True negative P(TN)= P( Negative | SA ) The probabilities in the Table 2-1 are not independent of each other. The relations between them are: P(TP)+P(FN)=1 and P(FP)+P(TN)=1. Thus we have only two independent probabilities in the defect detection problem. In an observer study to evaluate the performance of an imaging system, the human observers are shown a large number of images and asked to rate their confidence as to whether a defect is present or not [33]. The resulting P(TP) and P(FP) represent the probability of correctly distinguishing, or the detectability of the system. However, these two probabilities are obviously determined by the decision criteria the observer used. To

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15 remove the dependence on the decision criteria, a Receiver Operating Characteristics (ROC) curve [34] is plotted with P(TP) versus P(FP) as the decision criteria change. The area under the ROC curve is a common figure of merit to represent the detectability of the imaging system. The ROC curve can also be used to optimize the parameters to produce better images [35-37]. 2.4.2 Numerical Observer: Background The human observer tests are very time-consuming and difficult to perform in practice. Therefore, there is considerable interest in the use of numerical observers of which the performance indices can be calculated rather than measured. A numerical observer correlating well with the human observers would be extremely valuable for optimizing and evaluating imaging systems. The numerical observer procedure for the defect detection task is as follows: compute a scalar feature )(f where f is the image to be classified, then compare it to a threshold C. Choose positive if >C (or C). The scalar feature is also called a test statistic or decision variable. A common figure of merit for the numerical observer is the detectability index d a (also known as d ) [38], defined by )|var()|var()|()|(222SASPSAESPEda (2-10) where )|(kE and )|var(k are the conditional mean and variance of )(f respectively, given that f was produced by an object in state k (k = SP or SA). An ideal observer is defined as one who has full statistical knowledge of the task and who makes best use of the knowledge to minimize a suitably defined risk. The test

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16 statistic of the ideal observer is defined by [39] )|()|(logidealSPfPSAfP (2-11) where is the conditional probability of f given that f was produced by an object in state k (k = SP or SA). The right side of Equation 2-11 is called log-likelihood ratio. (Note: log-likelihood ratio is unrelated to log-likelihood function discussed in 2.2.2). )|(kfP Ideal observer maximizes the area under the ROC curve and sets an upper limit to the performance obtainable by any other observers including the human, though it is only applicable for simple situations. In practice the log-likelihood ratio is rarely calculable, except for the detection of an exactly known signal superimposed on an exactly known background (SKE-BKE) situation. In this case the likelihood ratio can be calculated by simple linear filtering process. If the noise is stationary, white and Gaussian, the likelihood ratio is the output of a matched filter. If the noise is stationary and Gaussian but not white, a prewhitening procedure is required before the matched filter [40]. As the performance of the human observer was found to be dramatically degraded by correlated noise [41, 42], and some spatial-frequency channels were found to exist in the human visual system [43, 44], Myers et al. [45] added a frequency-selective channel mechanism to the ideal observer. The channel model has demonstrated the ability to improve the correlation between numerical and human observers for detection tasks by other researchers [46-50]. 2.4.3 Hotelling Observer and Channelized Hotelling Observer In practice, a linear observer model is usually more applicable than the ideal observer. A linear observer is defined by

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17 21T)()(Ljjjfufuf (2-12) where u is called a discriminant vector, and T denotes the transpose. The j th component value of u, denoted as u j can be viewed as the importance assigned to the j )(jf th pixel value of the image vector f. Before define Hotelling observer, we need to introduce two scatter matrices that are used to describe the firstand secondorder statistics of f [39]. The interclass scatter matrix S 1 measures how far the state means deviate from the grand mean f and is defined by KkTkkkffffP11))((S (2-13) where K is the number of states, P k is the probability of occurrence of state k, kf is the state mean for the k th state. The grand mean is given by KkkkfPf1 (2-14) The intraclass scatter matrix S 2 is defined as the average covariance matrix across all states: KkkkP12KS (2-15) where the k th state covariance matrix is given by kK kTkkkffff))((K (2-16) The angular brackets in Equation 2-16 have the same meaning as the overbar, that is, a full ensemble average over all objects f in state k.

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18 Based on the covariance matrices S 1 and S 2 a criterion measuring the separability of the states was proposed by Hotelling in his classic paper [51], and is now often referred to as the Hotelling trace: )(112SStrJ (2-17) where tr denotes the trace (sum of the diagonal elements) of the matrix. The Hotelling trace is a common measure of classification performance in pattern recognition [39]. It increases if the variability in the image (due to noise of other factors) is decreased, since that corresponds to reducing the covariance terms that go into S 2 and hence increase It also increases if the system is modified in such a way that the two state means become more widely separated, since that increases the norm of the vectors that constitute S 12S 1 Previous research [40, 52, 53] presented that Hotelling Observer (HO) as a linear observer optimizing the Hotelling trace. The test statistic of HO is obtained by ffffuTSASPTHOHO12)(S (2-18) Implementing HO begins with creating a training set of realizations of objects in two states (SP and SA), and estimating the scatter matrix S 2 from the training set. If there are N pixels in the image, then S 2 is an NN matrix with the rank of N The number of images in the training set should be larger than N 2 Otherwise, the inverse of the estimated S 2 will not exist [54, 55]. The Channelized Hotelling Observer (CHO) incorporates the channel mechanism into HO [46]. Using a bank of M band-pass frequency filters, CHO transform each image to an M1 vector. The values in the resulting vector represent the frequency information of the image, so this channelization procedure can be viewed as a feature extraction procedure. Mathematically, it can be described by

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19 Uf (2-19) where U is an NM matrix, which represents the channelization procedure and is determined by the channel model. For an SKE task, the m th row vector in the matrix U describes the m th channels impulse response centered on the defect location. CHO calculates the test statistic of the resulting vector by 12)(STSASPTCHOCHOu (2-20) where SP and SA are the state means of for SP and SA, respectively, and is the intraclass scatter matrix of 2S CHO model has demonstrated to predict human observer performance for a variety of SKE-BKE detection tasks. Gifford et al. [47] investigated the application of CHO to detect hepatic lesions using various channel models, eye filters and internal noise models. Burgess et al. [56] showed the correlation between the human observer and CHO for well-defined two-component noise fields (Poisson noise correlated with a Gaussian function) with various channel models.

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CHAPTER 3 FORMULATION OF SIMULTANEOUS ESTIMATION METHOD 3.1 Objective Functions Given two gated projection datasets )1( ~ g and )2( ~ g our method simultaneously reconstructs the image frames and estimates the motion vector field that describes the motion from to Here and m(r) are presented as functions of a continuous variable in the spatial domain, where x, y, z are coordinates in three dimensions. )(),(21rrff )(),(),()(rrrrmwvu )(1rf )(2rf )(),(21rrff zyx,,r We define an energy function as )(),,(),(),,(212121mmmSIEffEffLffE (3-1) The first term is a sum of the negative log-likelihood functions of two frames 211)(21)(log)(~)(),(mpimmmiHfigiHfffL (3-2) where m = 1, 2 denotes the frame index, i=1, 2, p denotes the element index in the datasets, and H is the forward-projection matrix that gives qjmijmjfhiHf1)()( (3-3) The second term in Equation 3-1 is the image matching term that measures the agreement of two frames under the assumed motion m: drffffEI22121)()(,,rmrrm (3-4) The third term is the strain energy term of deforming elastic material: 20

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21 rrrrrrmdwvwuvudwvudwvuEyzxzxyzyxzyxS2222222)(21)()(21)( (3-5) where the Lam constants and determine the material compressibility. Due to the different compressibility of the blood pool and the myocardium, we allow the values of and to change between regions by representing them as the functions (r), (r). The values of the hyper-parameters and reflect the influence that the projection data and the strain energy have on the estimates obtained by this method. In our study, we chose these values by experience. The reconstructed image frames and estimated motion vector field m )(*),(21rrf*f are obtained by minimizing the function that is, ),,(21mffE ),,(minarg)*,*,(21,0,2121mmmffEf*fff (3-6) 3.2 Euler-Lagrange Equations We use the Frechet derivatives of the function E to obtain the Euler-Lagrange equations. As the first step, we compute the Gateaux differentials [57] of E with increment t. Let D m E(f 1 f 2 m; t), D u E(f 1 f 2 m; t), D v E(f 1 f 2 m; t), D w E(f 1 f 2 m; t) denote the Gateaux differentials of E with increment t relative to f m u, v, w, respectively. The Gateaux differentials are computed by using the well-known formula as 021211),,();,,(sfstfEdtdtffEDmm (3-7) Applying similar formulas to the functions E I E S L, which determine E, for differentials relative to all the variables, we obtain the Gateaux differentials

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22 rrrmrrmdtfftffEDI)())](()([2);,,(21211 (3-8) rrmrrmrrmdtfftffEDI))(())](()([2);,,(21212 (3-9) 2 1, )],(/)()(~)([);,(1)(21miHfiHtigiHttffLDpimmm (3-10) rrrmrrmrrmdtxffftffEDIu)())(())](()([2);,,(22121 (3-11) rrrmrrmrrmdtyffftffEDIv)())(())](()([2);,,(22121 (3-12) rrrmrrmrrmdtzffftffEDIw)())(())](()([2);,,(22121 (3-13) rmmdtwutvututEDzxzyxyxxSu])()()2[();( (3-14) rmmdtwvtvutvtEDzyzxxyyySv])()()2[();( (3-15) rmmdtwvtwutwtEDyyzxxzzzSw])()()2[();( (3-16) Now, let denote the set of all infinitely differentiable functions with compact support on Then the Euler-Lagrange equations are )(cC 0);,,(21 tffEDmm , 0);,,(21tffEDum 0);,,(21 tffEDvm and for m = 1, 2 and all All these equations involve integrals containing the function t. To remove the integral and the dependence on t from these equations, note that if for all then g = 0. To use this fact, each of the Gateaux differentials of E needs to be expressed in the form for some g. Such a function g is called a Frechet derivative. The Frechet derivatives of E with respect to f 0);,,(21tffEDwm cCt 0)()(rrrdtg cCt rrrdt)()(g m u, v, w are denoted by D m E, D u E, D v E, D w E respectively, and are weighted sums of the Frechet derivatives of E I E S

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23 and L. Some Frechet derivatives of E I are immediately obtained from Equations 3-8, 3-11, 3-12, and 3-13 by easy calculation: ))](()([2),,(21211rmrrm ffffEDI (3-17) ))(())](()([2),,(22121rmrrmrrm xfffffEDIu (3-18) ))(())](()([2),,(22121rmrrmrrm yfffffEDIv (3-19) ))(())](()([2),,(22121rmrrmrrm zfffffEDIw (3-20) To obtain D 2 E I (f 1 f 2 m), we need to express the integral in Equation 3-9 as Consider the coordinate transformation rrrdt)()(g ))(()( ~ 3rmrmrr I where I 3 is the identity 33 matrix. Then, the Jacobian is given by 1133)))]~()(('[det(~)~(rmmrrrIIJ (3-21) where (3-22) zyxzyxzyxwwwvvvuuu)('rm So, if ~ is the image of under the operator (I 3 +m), from Equation 3-9 we obtain ~2131212~~)~()]~())~()(([2);,,(rrrrrrmmdtfIftffEDI (3-23) Now, the motion changes the location of pixels within the ROI but it leaves the entire ROI unaltered, hence ~ So, from Equations 3-21, 3-23 we obtain |))('det(|/)]()([2),,(321212rmrrm IffffEDI (3-24) where )()(13rmrI

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24 The Frechet derivatives of L are given by )()()( ~ )(),(1)(21pimimmiHfhighffLDrr (3-25) where So, from Equations 3-17, 3-24, 3-25 and the fact that E piihh1)()(rr S does not depend on f m we obtain the Frechet derivative of E with respect to f 1 and f 2 piiiHfhighffffED11)1(21211)()()( ~ )())](()([2),,(rrrmrrm (3-26) piiiHfhighIffffED12)2(321212)()()( ~ )())('det()]()([2),,(rrrmrrm (3-27) We now determine the Frechet derivatives of E S with respect to u, v, w. The Gateaux differentials in Equations 3-14, 3-15, 3-16 are all of the form for some vector field F. By using the divergence theorem, for all rFtd )(cCt rFrFFrFtddtttd)())()(( (3-28) so the Frechet derivatives are F From Equations 3-14, 3-15, 3-16, we see that the vector field F for , and are );(tEDSum );(tEDSvm );(tEDSwm ))(),(,2)((xzxyxzyxwuvuuwvu (3-29) ))(,2)(),((yzyzyxxywvvwvuvu (3-30) )2)(),(),((zzyxyzxzwwvuwvwu (3-31) respectively. In each case we obtain F as

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25 ))(()()2(xzxyzzyyxxwvuuu (3-32) yyzzxxyzxyvvvwu)2()())(( (3-33) zzyyxxyzxzwwwvu)2()())(( (3-34) Hence we obtain the Frechet derivatives of E S ))(()()2()(xzxyzzyyxxSuwvuuuED m (3-35) yyzzxxyzxySvvvvwuED)2()())(()( m (3-36) zzyyxxyzxzSwwwwvuED)2()())(()( m (3-37) Since L is independent of m, we obtain the Frechet derivatives of E relative to u, v, w as ))(())](()([2))])(()()2[(),,(22121rmrrmrrm xfffwvuuuffEDxzxyzzyyxxu (3-38) ))(())](()([2])2()())([(),,(22121rmrrmrrm yfffvvvwuffEDyyzzxxyzxyv (3-39) ))(())](()([2])2()())([(),,(22121rmrrmrrm zfffwwwvuffEDzzyyxxyzxzw (3-40) Finally, the Euler-Lagrange equations for E, satisfied by are )*,*,(21mf*f 0)(/)(~)()())(()(211)1(21piiiHfighhffrrrmrr (3-41) 0)(/)(~)()())('det()()(212)2(321piiiHfighhIffrrrmrr (3-42) 0))(()()2()()()(2221xzxyzzyyxxwvuuuxfffmrmrr (3-43)

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26 0))(()()2()()()(2221yzxyzzxxyywuvvuyfffmrmrr (3-44) 0))(()()2()()()(2221yzxzyyxxzzvuwwwzfffmrmrr (3-45) The Equations 3-43, 3-44, 3-45 are valid on regions with uniform Lam constants and So we assume throughout our study that the frames are segmented into two regions with uniform values for the Lam constants on each. 3.3 Optimization Algorithm: RM 3.3.1 Scheme of Optimization Algorithm We estimate the minimizer by a two-step iterative procedure, which updates the estimates iteratively to in the k )*,*,(21mf*f )()(2)(1,,kkkffm )1()1(2)1(1,,kkkffm th iteration. Figure 3-1 shows the scheme of this optimization algorithm. In R step (R for reconstruction), we assume the motion is reasonable and fixed, and minimize the sum of the image matching and likelihood functions to generate the image updates, that is, (3-46) ),,(),(minarg),,(minarg),()(21210,)(210,)1(2)1(12121kIffkffkkffEffLffEffmm In M step (M for motion), we assume the images are reasonable and fixed, and minimize the sum of the image matching and strain energy functions to generate a new motion estimate, that is, )(),,(minarg),,(minarg)1(2)1(1)1(2)1(1)1(mmmmmmSkkIkkkEffEffE (3-47)

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27 Hence this iterative optimization algorithm is called RM algorithm. The image frames and motion vector field are simultaneously updated after one repetition of these two steps, called an outer loop of the RM algorithm. The minimization problems in R and M steps are also optimized using iterative algorithms, and therefore, they are referred to as inner loops. Outer Loop M Step: Minimize EI + ES over m R Step: MinimizeL + EI over f1, f2 0)1()1()1(kkkffm21,)()()(kkkffm21, )1()1(kkff21,)(km Figure 3-1. Scheme of the RM algorithm. 3.3.2 Computation of R Step Equation 3-46 can be regarded as a penalized maximum-likelihood approach of image reconstruction in which the penalty term enforces natural constraints due to the limitedness of material deformations, rather than forcing (possibly unnatural) prior smoothing constraints. Greens one-step-late (OSL) algorithm [58] was proposed for the penalized maximum-likelihood estimation of a single frame using a general class of priors. It was used [10] for reconstructing gated frames, and convergence was obtained for a modified version [59]. We used a modified OSL for our R-step optimization. Since E I and L are convex in f 1 f 2 the Kuhn-Tucker optimality conditions are necessary and sufficient for the minimizer. That is 0),,()(21kmffEDm (3-48) and for m =1,2. (3-49) 0),,()(21kmmffEDfm

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28 From Equations 3-26, 3-27, we define )()(2)()()(~)(),,(2111)1(211mrrrrmffhiHfighffpii (3-50) |))('det(|)()(2)()()(~)(),,(32112)2(212rIffhiHfighffpiimrrrrm (3-51) Hence the necessary and sufficient conditions for Equation 3-46 are 2,1,1),,()(21mffkmm (3-52) 2,1,),,()(21mffffmkmmm (3-53) So, it seems natural to estimate by the fixed-point iterations given by )1(kmf ,...2,1),, ~ ~ ( ~ ~ )(,2,1,1,lffffkllmlmlmm (3-54) and )(,)1( ~ lim ~ klmlkmff If the image matching term E I is not present in the objective function in Equation 3-46, then Equation 3-54 reduces to MLEM algorithm. If E I is replaced by a Gibbs prior, this approach yields Greens OSL algorithm. However, the multiplier is not guaranteed to be nonnegative. We use the following modification of the iteration in Equation 3-54 to generate nonnegative approximations , of m )(0,kmf )(1,kmf )(2,kmf )1(kmf R Step Algorithm 1. Initialization: .)()(0,kmkmff 2. Given compute the values of for )(,klmf ),,,()()()(,2)(,1)(,kklklmklmffmr r 3. Determine constant satisfying 0lt

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29 (3-55) 1}2,1,:}0),(1max{max{)(,jtklmlrr 4. Then is the update of )(,)(,)(,)(1,)1(klmklmlklmklmftff )(,klmf 5. Go back to 2. To apply this algorithm we need to compute the variable for each which occurs in the quantities However, the mathematical form of the transformation is completely unknown. Only numerical values of are available, so we choose to use a fast, direct method of obtaining an approximation of by using a linear approximation of )()(13rmrI r )()(,rklm )()(rmk )()(rmk r )()(rmk By definition, ))((')()(rrrmrmrrmrr hence, which gives the approximation as )))(('()(3rrrmrmI )())('(13rmrmrrI (3-56) This approximation is valid when is not an eigenvalue of )('rm 3.3.3 Computation of M Step M Step is to compute the motion update after the image frames have been updated to This has been reduced to the motion-estimation problem considered by Klein and Huesman [15]. In the iterates of M step, the update of is obtained by minimizing a simpler, quadratic approximation of the original objective function, using Taylor approximation )1(km )1(2)1(1,kkff )(1klm )(klm ))(())()(())(())()()(())(()()(2)()()(2)()()(2)(2rmrrmrmrmrrmrmrmrrmrklkklklkklklkkffff (3-57) Hence, approximately we need only to compute

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30 )()(minarg,)1(mmmmSlkkEQ (3-58) where rrmrrmrmrmrrmdfffQklkklklkklk2)()(2)()()(2)(1,))(())()(())(()()( (3-59) From the Frechet derivatives of and E lkQ, S the Euler-Lagrange equations for are )(1klmm 0))(()(2)()()()()(2)()(2)()(2)()()(2)(1xzxyzzyyxxklkklkklklkkwvuuuxffffmrmrmmmrr (3-60) 0))(()(2)()()()()(2)()(2)()(2)()()(2)(1yzxyzzxxyyklkklkklklkkwuvvvyffffmrmrmmmrr (3-61) 0))(()(2)()()()()(2)()(2)()(2)()()(2)(1yzxzyyxxzzklkklkklklkkvuwwwzffffmrmrmmmrr (3-62) Using finite difference approximations for the derivatives, these partial differential equations (PDE) can be changed to an algebraic linear system for the motion components at all voxels, that is, (u j v j w j j = 1,2,q). The algebraic linear equations are solved using the regular conjugate gradient algorithm. The estimates ,,, are then iteratively obtained by solving Equations 3-60, 3-61, and 3-62 with the initialization Appendix gives details of the numerical analysis for motion estimation. )(0km )(1km )(2km )()(0kkmm

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CHAPTER 4 IMPLEMENTATION AND EXPERIMENTAL RESULTS: GEOMETRIC PHANTOM 4.1 Simulated Source Object and Projection Data Since it is difficult to obtain a standard data set that presents the truth volume, using real medical imaging data to test an algorithm is usually not applicable. To demonstrate some of the basic characteristics of the RM algorithm, this chapter implements the algorithm with a simulated 3D geometric phantom. Based on a hollow semi-ellipsoid model, the source object of this phantom is designed to simulate the contracting left ventricle in a gated myocardial ET study. The inner and outer boundaries of the myocardium are defined by 2 concentric semi-ellipsoid surfaces with dimensions of short and long axes chosen to mimic the shape and size of a regular left ventricle. Three orthogonal views of the source object are shown in Figure 4-1. 4.1.1 Steps to Generate Geometric Phantom The simulated geometric phantom is represented as a 303030 voxel density volume. In order to create a more realistic source distribution in which object boundaries have some degree of smoothness, we generated the 303030 source object from a high-resolution (120120120) object with the steps illustrated in Figure 4-2. Two simulated data files are generated for each frame: a noise-added projection data file and a true source object image file. The projection data file will be used to reconstruct the image by the RM algorithm and the PS-MLEM algorithm, and the true image will be used to evaluate the reconstructed images. 31

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32 A B Figure 4-1. Source objects of the geometric phantom. A) Frame 1. B) Frame 2. For both frames, the 3 orthogonal planes are in 2 long axis slices (two left columns) and a short axis slice (right column). 4.1.2 Dimensions of Geometric Phantom We generated 2 frames of the gated ET image (Frame 1 and Frame 2) representing the end-systolic and mid-systolic phases of the heart cycle, respectively. The source objects of the 2 frames had different lengths of semi-axes in a manner that achieved a constant myocardium volume across the 2 frames and an anatomically realistic myocardial-to-chamber volume ratio. Assuming a nominal voxel linear dimension of 3.5 mm, the myocardial volume was 120 ml, and the chamber volumes of Frame 1 and Frame 2 were 43.3 ml and 74.4 ml, respectively. These volumes were chosen to represent the normal human heart at these contraction phases [60]. The lengths in mm of the ellipsoid semi-axes for the 2 frames are shown in Table 4-1. Two of the three ellipsoid semi-axes were equal in length and represented the circular, short-axis slices.

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33 High-Resolution Source Object (120120120) Smoothed High-Resolution Source Object (120120120) 3D Hann filter w/ Nyquist cut-off High-Resolution Projection (12060120) Collapse and Scale True Source Object (303030) Noise-free Projection (306030) Project w/ detector response Noise-added Projection(306030) Add Poisson noise Collapse Figure 4-2. Simulation steps of the geometric phantom Table 4-1. Ellipsoid semi-axes lengths for source object. (Unit: mm) Frame 1 Frame 1 Frame 2 Frame 2 Axis Inner boundary Outer boundary Inner boundary Outer boundary Short axis 21.0 38.5 26.2 40.6 Long axis 70.0 78.8 77.0 84.0 4.1.3 Intensity and Defect A uniform intensity was assigned to the myocardium region between the half ellipsoids, and zero intensity was assigned to the region in the inner chamber and external

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34 to the myocardium. For the contrast study purpose, two myocardial defects were simulated with 50% decreased intensity and located at basal and mid-ventricular points along the long axis, as indicated by the arrows in Figure 4-1. The total counts of the projection for each frame was scaled to 99,000, which was chosen to match that of a clinical, gated myocardial perfusion study using Tc-99m sestamibi (70 y.o. male, 10 mCi injected activity). 4.1.4 Wall Motion Simulation To mimic more realistic movement of heart, a wringing motion was simulated in addition to the contraction. This motion was found in contrast ventriculography [61] and tagged MR images of the human heart [28]. We characterized this wringing motion by a rotation of the basal and apical myocardium about the heart long axis in opposite directions. In our study, the extreme basal and apical points on the heart long axis rotated 5 degrees in opposite directions, and points between rotated according to a linear gradient connecting these extremes. We calculated the motion geometrically and referred to it as the ideal motion. To illustrate the wringing motion visually, the ideal motion fields in 3 short-axis slices (#5, #15 and #20) are shown in Figure 4-3. 4.1.5 Forward Projection Simulation and Detector Response Collimator-detector response was simulated assuming a low-energy high-resolution parallel-hole collimator. The projection data were filtered using a Gaussian kernel with FWHM=1.9 pixels. With the nominal 3.5 mm pixel size, this kernel has an equivalent FWHM equal to 6.7 mm. Forward projection was performed at 60 angles over 180 degrees. Effects of attenuation, scatter, and randoms were not considered.

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35 A B C Figure 4-3. Ideal motion fields of the geometric phantom in 3 short-axis slices. A) Slice #5. B) Slice #15. C) Slice #20. The high-resolution projection data were then collapsed to a low-resolution projection (306030) by a procedure described in Figure 4-4 (only a 2D slice is shown). While the final projection dimensions (306030) are small relative to those used by large field-of-view imaging systems, the sampling rate used here, relative to the size of the heart, is comparable to these systems. For example, the thickness of the myocardium in humans at end-systole is typically close to 14 mm; in our heart model this thickness is approximately 4 pixels, which represents a nominal pixel size of 3.5 mm. Finally, Poisson noise was added to this scaled, collapsed projection data. 4.2 Bi-Value Lam Constants Model The compressibility degree of the myocardium, the blood pool in the chamber, and other outside adjacent tissue differs from each other. Thus the Lam constants and should have different values for these regions. In this chapter, due to the assumption of constant myocardium volumes across the 2 frames of the geometric phantom, and the fact that the region of the blood pool is not conserved during contraction, we propose a bi-value Lam constants model: a fairly incompressible model for myocardium and a compressible model for the other regions.

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36 Figure 4-4. Collapse high-resolution projection to low-resolution projection In practice, there are two more often used constants presenting the compressibility of material: E, called Youngs elasticity modulus and called Poisson ratio, which are related to Lam constants as )()23( E and )(2 (4-1) Youngs elasticity modulus E relates the tension of the object to its stretch in the same direction. Poisson ratio is the ratio between lateral contraction and axial extension [62]. The term in the constraint equation penalizes non-zero divergence and the term penalizes sharp discontinuities in the flow field [32]. For highly incompressible material, the Poisson ratio approaches 0.5, which yields a approaching infinity. At the other extreme situation of highly compressible material, Bin Projection Angle Projection Angle Bin High-resolution 12060 Projection 3060 Projection 1 2 1 2 1 2 60 1 30 60 2 120

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37 and approach 0. Table 4-2 gives the Lam constants we used in our study for some specific Poisson ratios. Table 4-2. Poisson ratio and Lam constants Poisson ratio ( ) 0 0.1 0.2 0.3 0.4 ~0.5 Lam constants (, ) =0; =1; =0.25; =1; =0.67; =1; =1.5; =1; =4; =1; =100; =1; To implement the bi-value Lam constants model, a preprocessing segmentation of the left ventricle from the background is required. Since the accurate automatic segmentation of the myocardium boundaries may be a formidable task itself, in our study we used a simple automatic threshold-based procedure. Firstly, we reconstructed the image from the noisy data using the PS-MLEM algorithm with Nyquist cut-off frequency. Then we compared each voxel intensity of the resulting image to a pre-chosen threshold and generated a segmentation bitmap. A feasible threshold was reassured by examining the segmentation bitmap visually. Because the segmentation is only to model elastic characteristics of different tissue types, the RM algorithm is expected not to be sensitive to slight errors of a few voxels along the myocardium boundaries. This insensitivity has also been demonstrated by the insignificant difference of the images reconstructed from using various thresholds for segmentation. The segmentation procedure and results are illustrated in Figure 4-5. In our study, the threshold was chosen to be 1/8 of the mean intensity of myocardium voxels. The segmentation bitmaps generated from the images with and without defect were similar to each other.

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38 A B Figure 4-5. Segmentation for bi-value Lam constants model. A) With defect. B) Without defect. (Note: Left column are the images to be segmented with threshold. Right column are resulting segmentation bitmap where white region represents the myocardium and black region represents the background.) 4.3 Convergence Properties of RM Algorithm Since the RM algorithm is to minimize the objective function iteratively, its convergence property can be illustrated by the plot of the objective function value versus iteration number. Similarly, the convergence of the objective functions considered in the individual R and M steps illustrate the individual convergence properties. The convergence results for the RM algorithm are shown in Figure 4-6. The total objective function E and the negative log-likelihood function L decrease with iteration

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39 and are approximately constant after 20 iterations. Due to the initial zero motion estimate and initial uniform image estimates, the E I and E S functions present low initial value and increase with iteration before flattening out. Figure 4-7 shows the convergence of the individual R step. R step using either the true or the estimated motion largely converges by 25 iterations. The M step also has converged by 25 iterations as indicated in Figure 4-8. The objective functional value in M step achieved with the PS-MLEM images is larger than that with true images. This reflects the inconsistencies between the estimated frames by PS-MLEM due to the statistical noise. Figure 4-6. Convergence of the RM algorithm 4.4 Motion Estimation Results 4.4.1 Figures of Merit The motion estimation is evaluated by comparing the estimated motion to ideal motion globally and regionally. Based on the image matching function, we define a global motion error that measures the agreement of two true image frames under the estimated motion. It represents the overall accuracy of the motion estimation.

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40 Figure 4-7. Convergence of R step Figure 4-8. Convergence of M step rrmrr221))(()(ff (4-2) where and are the true images, and is the estimated motion vector field. Regional motion evaluation is performed at some selected sub-regions. We define magnitude error as the absolute value of the magnitude difference between the estimated motion and the ideal motion, and angle error as the angle between the two vectors. 1f 2f m

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41 4.4.2 Lam Constants Study Results As we discussed in 4.2, motion estimation involves the assignment of Lam constants for myocardium and background regions. To investigate how the Lam constants influences motion estimation, we performed M step using true images with various Poisson ratio v (equivalent to various Lam constants, see Table 4-2) and hyper-parameter We used six values (0, 0.1, 0.2, 0.3, 0.4, and 0.5) of v for myocardium, and v =0 for the background. The results are shown in Figure 4-9. The global motion error using v = 0.5 for myocardium was much more sensitive to than those using other v s. The minimum global motion error was achieved with v = 0.4 and = 0.02. Our results agree with Kleins claim [32], that is, for the fairly incompressible deformation, the motion-estimation algorithm performs best with a chosen Poisson ratio value near 0.4, instead of the value of 0.5 for the ideally incompressible material. Thus in our following geometric phantom study, we kept the Poisson ratio at 0.4 for the myocardium and 0 for the background. Figure 4-9. Global motion error vs. for several Poisson ratios.

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42 4.4.3 Global Motion Evaluation To compare the motion-estimation performance of RM and the individual motion-estimation algorithm, we implemented the following 3 methods: Method 1: M step using true images with change of Method 2: Individual method. M step using PS-MLEM images with change of cut-off frequency and ; Method 3: RM algorithm with change of hyper-parameter and Figure 4-10 shows the ideal motion superimposed on the true Frame 1 images of the geometric phantom, in both longand shortaxis views. Figure 4-11 shows the motion estimated by Method 1, superimposed on the true images. Figure 4-12 shows the images and motion individually estimated by Method 2. Figure 4-13 shows the images and motion simultaneously estimated by RM. The minimum global motion error each method achieved is given in Table 4-3. The optimum conditions to produce the minimum errors for each method are also given. The numbers in parentheses are the standard deviations of the global motion error calculated across a 20-image ensemble. Method 1 and 2 used 40 iterations for M step, and Method 3 used 40 outer-loop iterations, each containing 1 M-step iteration and 1 R-step iteration. Figure 4-10. Ideal motion vector fields superimposed on true images.

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43 Figure 4-11. Estimated motion by M step using true images. ( = 0.02). Figure 4-12. Individually reconstructed images by PS-MLEM and estimated motion by M step. (cut-off = 0.3 cycle/pixel; = 0.01). Figure 4-13. Estimated motion and reconstructed images by RM. (=0.02; = 0.01).

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44 Table 4-3. Global motion evaluation results of the geometric phantom Method Optimum Conditions Minimum global motion error 1 = 0.02. 21.4 2 cut-off = 0.3 cycle/pixel; = 0.01. 46.4(1.2) 3 =0.02; = 0.01. 37.2(1.4) The results in Table 4-3 show the smallest motion error was achieved by M step using the true images, as we expected. Of the methods those were operating on noisy data, the RM algorithm outperformed the individual method in terms of the global motion error. This result reflects the increased noise control of the RM algorithm. 4.4.4 Regional Motion Evaluation Three sub-regions within the myocardium of Frame 1 of the geometric phantom were selected for the regional motion evaluation: (1) basal and in normal region, (2) basal and in defect region, (3) mid-ventricle and in normal region. Each sub-region was a 222 voxel volume. Their locations are indicated by the squares in a slice view in Figure 4-14. To obtain the magnitude and angle errors of motion estimate, the mean magnitude and angle of the estimated motion in the sub-regions were computed and compared with the mean magnitude and angle of the ideal motion in those sub-regions accordingly. These errors were computed for a 20-image ensemble, and the mean and standard deviation of the errors across the ensemble were obtained. The regional motion evaluation was performed for both the RM algorithm and the individual method. The results of the regional motion evaluation are shown in Table 4-4. The mean magnitude errors are in units of pixels, and mean angle errors are in units of degrees. Standard deviations are given in parentheses. Overall the table indicates the improved motion-estimation accuracy for RM compared with the individual method.

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45 Figure 4-14. Chosen region to calculate regional motion error for the geometric phantom. Table 4-4. Regional motion evaluation results of the geometric phantom Individual method RM Sub-region Ideal mag. Mag. error Angle error Mag. error Angle error 1 1.52 -0.18(0.17) 31.7(14.1) -0.18(0.12) 28.7(11.5) 2 1.52 -0.49(0.28) 30.1(14.8) -0.45(0.22) 25.1(11.1) 3 0.66 0.24(0.37) 61.0(16.8) 0.02(0.22) 43.3(17.3) 4.5 Image Reconstruction Results 4.5.1 Figures of Merit The RM images were evaluated both globally and regionally by comparing to the PS-MLEM images. The evaluation methods include global image evaluation, regional image evaluation, and defect contrast evaluation. Since the cut-off frequency of the post-filtering used in the PS-MLEM algorithm can greatly affect the quality of the resulting images, a range of cut-off frequencies was investigated. The figure-of-merit for the global image evaluation was the Root Mean Squared (RMS) image error defined as rrr2)()(1RMSffN (4-3) where is the true image, is the reconstructed image, and N is the total )(rf )(rf

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46 number of voxels in the image. To get an overall image evaluation of the 2-frame system, the mean RMS error was computed across the 2 frames. The regional image evaluation also considered the 3 sub-regions used for the regional motion evaluation. For each sub-region, the mean bias and variance of the intensity of the estimated image were computed across a 20-image ensemble. For the PS-MLEM algorithm, a set of bias/variance points was computed across a range of cut-off frequencies from 0.1 to 1.0 cycle/pixel. The bias/variance curves were plotted for each sub-region to show the trade-off between the absolute value of the bias and the variance, as the cut-off frequency changed. Defect contrast is defined as normaldefectnormalContrastIII (4-4) where and are mean image intensities within the normal and defect regions, respectively. The normal and defect regions were defined from the known source object (true image) and included the entire region volume except for boundary voxels. Contrast was measured for each image out of the 20-image ensemble, and then the mean contrast was computed. Like the regional image evaluation, a range of cut-off frequencies was considered for the PS-MLEM algorithm. normalI defectI 4.5.2 Global Image Evaluation The results of the global image evaluation for RM and PS-MLEM are given in Figure 4-15. The RMS error of the RM algorithm decreases with iteration until approximately iteration 20, then increases very slightly. This demonstrates the stability of the RM images with iteration, unlike MLEM. The PS-MLEM algorithm shows a minimum RMS error with a cut-off frequency of approximately 0.4 cycles/pixel. This

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47 minimum, however, is still substantially larger than the stable RMS error of the RM algorithm (0.045 compared with 0.03). The hyper-parameter effects on the RM images are illustrated in Figure 4-16. The figure shows the image of Frame 2 as and are varied. As is reduced, there is less reliance on the likelihood function (i.e., the measured projection data), which causes the image of the myocardium to broaden and approach a more uniform intensity image. As increases, the algorithm becomes more like MLEM, and a moderate increase in noise is evident. The sensitivity of the RM image to change in is much less. Further illustration of the hyper-parameter effect is shown in Figure 4-17. These surface plots show the change in the global motion and image errors as a 2D function of and The minimum RMS error is located at =0.02 and =0.04. When was smaller than 0.02, the RMS error increased dramatically and became irregularly sensitive to which corresponded the image shown in Figure 4-16 (B). A B Figure 4-15. RMS errors of the RM and PS-MLEM images. A) RM. B) PS-MLEM.

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48 Figure 4-16. Hyper-parameter effects on the RM images. A) = 0.03, = 0.01. B) unchanged, smaller (0.015). C) unchanged, larger (0.06). D) unchanged, smaller (0.002). E). unchanged, larger (0.1). Figure 4-17. RMS error of the RM images as a function of and

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49 4.5.3 Defect Contrast Evaluation The results of the contrast measurements are shown in Figure 4-18. With the PS-MLEM algorithm for both defects, the defect contrast increases as the cut-off frequency increases and, at the highest cut-off frequency, is slightly less than that of the RM image. Although at this cut-off frequency the 2 methods have similar defect contrast, the quality of the images is substantially different, as shown in Figure 4-19. A B Figure 4-18. Defect contrast of the PS-MLEM and RM images. A) Defect 1. B) Defect 2. Figure 4-19 (A) and (B) show the images of Frame 1 for RM (40 iteration) and PS-MLEM (1.4 cycles/pixel cut-off). It is evident that while the images have similar contrast, the RM image has substantially lower noise level. Also included in Figure 4-19 (C) are the PS-MLEM images with Nyquist cut-off frequency. While these images have a noise level closer that of the RM images, the defect contrast is smaller. This is particularly evident in the short axis images in the right column. 4.5.4 Regional Image Evaluation The bias and variance of the voxel intensity within the 3 sub-regions (indicated by the squares in Figure 4-20) were calculated in both 2 frames. Figure 4-21 shows the plots of the absolute bias versus variance for the PS-MLEM and RM images. In each plot, the RM bias/variance point is downward and left to the PS-MLEM bias/variance curve, indicating that RM produced smaller bias for equal variance, or smaller variance for

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50 equal bias. This demonstrates that the RM algorithm achieved better regional image estimation, relative to PS-MLEM. In summary, in this chapter we implemented the RM algorithm using a simulated geometric phantom that mimics the human left ventricle. We evaluated the motion accuracy by comparing the motion estimated by 3 methods with the ideal motion. The motion estimated by RM using noisy projection data was superior to that estimated individually by applying the motion estimation method to PS-MLEM reconstructed images. Also we evaluated the image reconstruction by comparing the RM images to the PS-MLEM images in terms of the global image error, the regional intensity bias/variance, and the defect contrast. All the results showed that the RM algorithm produces better image quality. In Chapter 6 we will assess the image quality by observer study.

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51 A B C Figure 4-19. The trade-off between the contrast and noise level of the geometric phantom. A) RM images. B) PS-MLEM images at similar defect contrast but higher noise level. C) PS-MLEM images at similar noise level but lower contrast. A B Figure 4-20. Sub-regions for regional image evaluation for the geometric phantom. A) Frame 1. B) Frame 2.

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52 A B C D E F Figure 4-21. The trade-off between regional intensity bias and variance of the geometric phantom. A) Sub-region in basal normal region of Frame 1. B) Sub-region in basal normal region of Frame 2. C) Sub-region in basal defect region of Frame 1. D) Sub-region in basal defect region of Frame 2. E) Sub-region in mid-ventricle normal region of Frame 1. F) Sub-region in mid-ventricle normal region of Frame 2.

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CHAPTER 5 IMPLEMENTATION AND EXPERIMENTAL RESULTS: NCAT PHANTOM 5.1 Simulated NCAT Phantom Besides the geometric phantom, we also implemented the RM algorithm on a more realistic, complex physical phantom: 4D NURBS-Based Cardiac Torso (NCAT) cardiac phantom. The NCAT phantom simulation software was developed at the Johns Hopkins University and fully described by Segars et al. [63, 64]. It uses Non-Uniform Rational B-Splines (NURBS) method to define the geometric conditions of a human heart and produce the source object of heart at a series of phrases over the cardiac cycle. The NCAT software also generates the myocardial motion vector field for the gated image frames from the tagged MRI data of normal human patients. We used the NCAT software to generate 8 gated image frames of the cardiac phantom, each represented as a 323219 volume with a voxel linear dimension of 4.0 mm. The volume of the left myocardium was approximately 85 ml. We selected the 2 frames with maximal deformation to implement the RM algorithm: end-diastole and end-systole, which we referred to as Frame 1 and Frame 2, respectively. A defect region with 50% decreased voxel intensity was initially imposed on one side of the left ventricle in Frame 1. By deforming the defect in Frame 1 with the ideal motion field, we could impose the defect region onto Frame 2. The volume of the defect was approximately 5 ml. Similarly to the geometric phantom simulation, we scaled the total counts of each frame to 99,000 and added Poisson noise. The detector response was simulated with a FWHM of 7.5 mm. Figure 5-1 shows the long-axis and short-axis slice views of these 2 frames. 53

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54 A B Figure 5-1. Longand shortaxis views of the NCAT cardiac phantom. A) Frame 1: end-diastole. B) Frame 2: end-systole. 5.2 Motion Estimation Results 5.2.1 Global Motion Evaluation As indicated in Kleins study of the physical phantom [32], the best motion estimation is achieved with a Poisson ratio near 0.45 for the myocardium. Thus in this chapter we use 0.45 for the Poisson ratio of myocardium and the Lam constants are 1,9 for voxels within the myocardium; 1,0 for voxels of the background. Similarly to the geometric phantom study, motion estimation for the NCAT phantom was also evaluated in terms of the global motion error and the regional motion error. We also compared the global motion error of the following 3 methods: Method 1: M step using true images with change of

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55 Method 2: Individual method. M step using PS-MLEM images with change of cut-off frequency and ; Method 3: RM algorithm with change of hyper-parameter and Figure 5-2 shows the ideal motion superimposed on the true frame 1 images of the NCAT phantom, in both longand shortaxis views. Figure 5-3 shows estimated motion by M step using true images, superimposed on the true images. Figure 5-4 shows the images and motion individually estimated by Method 2. Figure 5-5 shows the images and motion simultaneously estimated by RM. Figure 5-2. Ideal motion vector fields superimposed on true images Figure 5-3. Estimated motion by M step using true images. ( = 0.10).

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56 Figure 5-4. Individually reconstructed images by PS-MLEM and estimated motion by M step. (cut-off = 0.4 cycle/pixel; = 0.05). Figure 5-5. Estimated motion and reconstructed images by RM. (=0.10; = 0.05). The global motion evaluation results are shown in Figure 5-6. The minimum global motion errors of the 3 methods are shown in Table 5-1. The numbers in the parentheses are the standard deviations of global motion error calculated from a 20-image ensemble. Each method implemented 40 iterations. Table 5-1. Global motion evaluation results of the NCAT phantom Method Optimum conditions Minimum global motion error 1 = 0.10. 267.7 2 cut-off = 0.4 cycle/pixel; = 0.05. 304.1 (11.3) 3 =0.10; = 0.05. 288.6 (5.8)

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57 A B C Figure 5-6. Global motion errors of three methods. A) global motion error vs. for Method 1. B) global motion error vs. cut-off frequency and for Method 2. C) global motion error vs. (, ) for Method 3. 5.2.2 Regional Motion Evaluation A 22 sub-region within the defect region in slice 7 (indicated by the square in Figure 5-7) was chosen for the regional motion evaluation. The results are shown in Table 5-2. The mean magnitude errors are in units of pixels, and mean angle errors are in units of degrees. The numbers in the parentheses are standard deviations of the errors. Overall the table indicates the improved motion-estimation accuracy for the RM algorithm compared with the individual method.

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58 Figure 5-7. Chosen region to calculate regional motion error for the NCAT phantom Table 5-2. Regional motion evaluation results of the NCAT phantom Individual method RM Ideal mag. Mag. error Angle error Mag. error Angle error 1.56 -0.84 (0.28) 20.2 (8.5) -0.52 (0.21) 18.6 (5.6) 5.3 Image Reconstruction Results To evaluate the image-reconstruction performance of the RM algorithm for the NCAT phantom study, we also compare the global image error (RMS), the regional image error (bias/variance), and the defect contrast of the RM images to those of the PS-MLEM images over a range of cut-off frequencies. 5.3.1 Global Image Evaluation For PS-MLEM algorithm, the mean RMS error of 2 frames changed with the cut-off frequency, as indicated in Figure 5-8 (A). The minimum RMS error (0.284) was achieved at approximately 0.6 cycles/pixel cut-off. For RM algorithm, a surface representing the change of the mean RMS error with the hyper-parameters (, ) is shown in Figure 5-8 (B). The minimum RMS error of RM (0.268) was achieved when =0.13 and =0.07. Figure 5-9 shows the mean RMS error of the 2 frames versus the iteration number of RM outer loop. The mean RMS error decreased with iteration until approximately 40,

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59 then stayed flat up to iteration 60. This demonstrates again the stability of the RM image with the outer-loop iteration. A B Figure 5-8. RMS errors of the RM and PS-MLEM images. A) RMS error vs. cut-off frequency for PS-MLEM. B) RMS error vs. (, ) for RM. Figure 5-9. RMS error vs. iteration number of RM outer loop. 5.3.2 Defect Contrast Evaluation To compute the defect contrast, we need first determine the normal and defect regions. The defect regions of 2 frames were both chosen during the simulation of the phantom. We generated the normal region by comparing the voxel intensity of the true

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60 images to a selected threshold. Figure 5-10 shows the normal and defect regions for 2 frames. White pixels present normal and gray pixels present defect. Most voxels in left myocardium were included into the normal or defect region. A B Figure 5-10. Normal and defect regions in the NCAT phantom. A) Frame 1. B) Frame2. As indicated in Figure 5-11, PS-MLEM using a higher cut-off frequency produced a higher contrast. The highest contrast of PS-MLEM images (0.53) was achieved with the cut-off of 1.4 cycle/pixel. RM images had a lower contrast (0.5) compared with the highest contrast achieved by PS-MLEM. However, RM images presented better trade-off between the contrast and noise than PS-MLEM. That is, RM had lower noise level while their contrasts were similar, and higher contrast while their noise levels were similar. Table 5-3 illustrates this trade-off between the contrast and noise of RM and PS-MLEM. We used the standard deviation of the voxel intensity in the normal myocardium region to represent the noise level. For RM, this standard deviation was 0.15. The PS-MLEM image had comparable noise level at the cut-off of 0.4 cycle/pixel, but the contrast was substantially reduced to 0.43. At a higher cut-off of 0.7 cycle/pixel in which the contrast was comparable, the standard deviation was substantially higher (0.25). The images of the 3 cases are shown in Figure 5-12.

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61 Figure 5-11. Contrast vs. cut-off frequency of PS-MLEM A B C Figure 5-12. The trade-off between the contrast and noise level of the NCAT phantom. A) RM images. B) PS-MLEM images with similar contrasts but higher noise levels. C) PS-MLEM images with similar noise levels but lower contrasts.

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62 Table 5-3. The contrast and noise level of the NCAT phantom Method Contrast Std. Dev. RM 0.50 0.15 PS-MLEM (cut-off=0.7 cycle/pixel) 0.50 0.25 PS-MLEM (cut-off=0.4 cycle/pixel) 0.43 0.15 5.3.3 Regional Image Evaluation Figure 5-13 shows the 2 sub-regions of interest in Frame 1 and Frame 2: one in the normal region and the other in the defect region. Figure 5-14 shows the trade-off between intensity bias and variance at these sub-regions in Frame 1 and Frame 2. We investigated 5 sets of hyper-parameters (see Table 5-4) for the RM algorithm. Set #1 (=0.13; =0.07) is the one that produces minimum RMS error. Set #2 increases and keeps unchanged, to investigate the influence of a larger value of Set #3, #4, #5 are to investigate the influence of a smaller a smaller and a larger respectively. These 5 sets of hyper-parameters presented different bias/variance property, as indicated in Figure 5-14. For the sub-region in the defect of Frame 1, all of the five RM images presented close or slightly worse trade-off between intensity bias and variance than PS-MLEM. For the sub-region in the defect of Frame 2 or the normal of Frame 1 and 2, all of the five RM images presented better trade-off between intensity bias and variance than PS-MLEM. The one out of these 5 sets that achieved the best bias/variance trade-off was the set #5, which grouped a medium (0.13) and a larger (0.15). It seems that the regional bias/variance property was benefited by the larger which imposed more material property constraints and produced smaller and smoother motion field. Table 5-4. Five sets of hyper-parameters for RM used for regional image evaluation Set #1 Set #2 Set #3 Set #4 Set #5 0.13 0.25 0.03 0.13 0.13 0.07 0.07 0.07 0.01 0.15

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63 A B Figure 5-13. Sub-regions for regional image evaluation for the NCAT phantom. A) Frame 1. B) Frame 2. A B C D Figure 5-14. The trade-off between regional intensity bias and variance of the NCAT phantom. A) Sub-region in the defect region of Frame 1. B) Sub-region in the defect region of Frame 2. C) Sub-region in the normal region of Frame 1. D) Sub-region in the normal region of Frame 2.

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CHAPTER 6 OBSERVER EVALUATION OF IMAGE QUALITY 6.1 Methods 6.1.1 Single-Slice CHO Model Our study of observer evaluation of image quality for the RM algorithm started with a single-slice CHO (SCHO) model. A number of channel models have been presented since Myers et al. [45] added a channel mechanism to the ideal observer. Previous research [45, 49] revealed that the performance of the CHO model was rather insensitive to the details of the channel model, such as the band-pass cut-off frequencies and the geometric properties of the channels. Since Radially Symmetric Channel (RSC) model demonstrated excellent performance to predict human performance [47, 49, 56], we used this relatively simple but efficient channel model in our study. Our RSC model consisted of rotationally symmetric, non-overlapping channels with cut-off frequencies of (0.03125, 0.0625), (0.0625, 0.125), (0.125, 0.25), (0.25, 0.5) cycles/pixel. Choice of cut-off frequencies was somewhat arbitrary. Figure 6-1 shows the RSC-model channels in the frequency domain. The channelization procedure was described in Equation 2-19 as Uf To compute the vector from the image f for a location-known defect, we Fourier transformed each frequency-domain channel to a spatial-domain template centered at the defect location. The i th component of the vector was then calculated by integration of the pixel-by-pixel product of the image and the i th spatial-domain template. This is to avoid excessive Fourier transformation for the large number of images. 64

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65 Figure 6-1. The RSC-model channels used for channelization in single-slice CHO model. (Note: Images are shown in the frequency domain.) Figure 6-2 shows the 2 defects in the geometric phantom and the defect in the NCAT phantom. Figure 6-3 shows the spatial-domain templates of the 4 RSC-model channels used in our study. Three rows indicate the different defect locations in the two phantoms. A B Figure 6-2. Defect locations in the phantoms. A) Geometric phantom. B) NCAT phantom. Before implementing CHO to compute the test statistic CHO we need to create a training set of objects in two states (SP and SA), and estimate the scatter matrix S 2 and the mean vectors SP and SA The size of the training set is usually determined by the channel number. As previous studies [54, 55] presented, for an N-channel model, the size of the training set for each state (SP or SA) should be larger than N 2 Otherwise the inverse of S 2 will not exist. In our study, we used 4-channel RSC model, so the size of the training set should at least be 16. We used 40 to ensure the existence of S 2

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66 To compare RM with PS-MLEM, we generated 280 images (140 SP and 140 SA) for each algorithm. We used 80 out of them (40 SP and 40 SA) as the training set. After generated the scatter matrix S 2 and the mean vector SP and SA we applied the CHO discriminant vector on the other 200 images (100 SP and 100 SA) to compute the SCHO detectability index (SCHO-d a ) To acquire the standard deviation of the detectability index, we divided the 280 images arbitrarily to 7 groups, each containing 40 images (20 SP and 20 SA). Each time we selected 2 groups of images to be used as the training set, then applied the resulting CHO discriminant vector to the remaining 5 groups to estimate an SCHO-d a By randomly selecting 2 out of the 7 groups as the training set, we estimated 21 different values of SCHO-d a s. Thus we are able to compute the mean and standard deviation of SCHO-d a 6.1.2 Multi-Frame Multi-Slice CHO-HO Model Since a defect in a 3D object may persist from slice to slice, in practice the physicians generally make their diagnostic decisions by observing multiple slices of images, and even multiple frames in the case of gated emission tomography imaging. Chen et al. [65] presented a multi-slice multi-view CHO-HO model for ungated SPECT myocardial perfusion imaging. We modified it to a multi-frame multi-slice CHO-HO (MMCHO) model for gated myocardial perfusion imaging. The MMCHO model involves 2 steps. In the first CHO step we apply a CHO model to each slice of all frames to assess the probability that the defect is present in that slice. The result of the first step for each multi-frame multi-slice image is a vector of test statistics of which the element is an individual test statistic where i=1, 2 ,, I (I is the number of slices) is the slice index and j=1, 2, J (J is the number of frames) is the ijCHO

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67 frame index. In the second HO step, we apply an HO model to the ensemble of vectors of (SP and SA) to compute a final MMCHO detectability index (MMCHO-d ijCHO a ). Figure 6-4 shows the scheme for this 2-step procedure. A B C Figure 6-3. The RSC-model channels represented as spatial-domain templates. The channels are Fourier transformed and shifted to the location centers of the defects. A) Channels for Defect 1 in the geometric phantom. B) Channels for Defect 2 in the geometric phantom. C) Channels for the defect in the NCAT phantom. In our MMCHO study, we considered 2 frames, each containing 3 chosen slices. The first CHO step applied the SCHO model to all the 3 slices of the 2 frames, and generated vectors containing 6 test statistics, (i=1, 2; j=1, 2, 3). The second HO step used 60 test-statistic vectors as the training set to determine the HO discriminant vector, and then applied the discriminant to the remaining test-statistic vectors to compute the MMCHO-d ijCHO a The mean and standard deviation of MMCHO-d a were computed by the same procedure used in the SCHO model.

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68 SA ensemble SA ensemble SP ensemble 2 frames CHO channels SP ensemble 3 slices 21CHOSP11CHOSP12CHOSP13CHOSP22CHOSP23CHOSP Step 1: CHO Step 2: HO HOCHOSA 21CHOSA11CHOSA12CHOSA13CHOSA22CHOSA23CHOSA 3 slices HOCHOSP Multi-frame multi-slice CHO-HO detectability index (MMCHO-da) Figure 6-4. Two-step procedure of the multi-frame multi-slice CHO-HO model.

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69 6.2 Results of Geometric Phantom Study Since the 2 defects in the geometric phantom were both centered in slice #15 of the long axis, we picked that slice and its 2 adjacent long axis slices, #14 and #16, to implement the SCHO and MMCHO study. The 3 images slices of 2 frames reconstructed by PS-MLEM and RM are shown in Figure 6-5 and Figure 6-6, respectively. In each figure, the image slices for each frame are grouped in the order of #14, #15, and #16. The cut-off frequency for PS-MLEM was 0.3 cycle/pixel. The RM algorithm used the hyper-parameters (=0.02, =0.04), which produced minimum RMS image error. The PS-MLEM images present more smoothness than the RM images, particularly at the myocardium boundaries. 6.2.1 Single-Slice CHO Results We performed SCHO model to the 2 defects in the 3 long-axis slices of the 2 frames. For PS-MLEM, we used a range of cut-off frequency from 0.1 to 1.0 cycle/pixel and found the minimum SCHO-d a The results of PS-MLEM and RM are shown in Table 6-1. Standard deviations of the SCHO-d a are given in parentheses. The RM images had significantly improved SCHO-d a in all of the cases except for the defect 1 in slice #15 of Frame 2. The improvement of RM for one slice was evidently different from another, while the SCHO performance of the slices differed, too. This also motivated us to develop a multi-frame multi-slice observer model that produces a single detectability index for a series of gated myocardial ET images. 6.2.2 Multi-Frame Multi-Slice CHO-HO Results We applied the MMCHO model to the 2 frames and 3 slices shown in Figure 6-5 and Figure 6-6. The results are shown in Table 6-2. The numbers in parentheses are the

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70 A B Figure 6-5. SP images of the geometric phantom reconstructed by PS-MLEM. A) Frame 1. B) Frame 2. A B Figure 6-6. SP images of the geometric phantom reconstructed by RM. A) Frame 1. B) Frame 2.

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71 standard deviations of MMCHO-d a The results show that the RM images presented significantly higher MMCHO detectability than the PS-MLEM images. Figure 6-7 shows how the MMCHO-d a of the PS-MLEM images changed with the cut-off frequency. For both defects, the maximum detectability indices of PS-MLEM were achieved with the cut-off of 0.4 cycle/pixel. The MMCHO-d a s of the RM images are also plotted. Table 6-1. SCHO detectability index results of the geometric phantom SCHO-d a for Defect 1 SCHO-d a for Defect 2 Slice Algorithm Frame 1 Frame 2 Frame 1 Frame 2 PS-MLEM 6.98 (0.22) 6.92 (0.21) 7.53 (0.13) 8.89 (0.24) 14 RM 9.26 (0.28) 8.85 (0.42) 11.52 (0.63) 11.93 (0.46) PS-MLEM 8.38 (0.25) 9.06 (0.23) 8.27 (0.19) 9.46 (0.28) 15 RM 8.95 (0.32) 8.03 (0.16) 10.69 (0.30) 9.89 (0.23) PS-MLEM 9.07 (0.35) 8.58 (0.36) 9.28 (0.35) 8.85 (0.35) 16 RM 12.89 (0.51) 11.87 (0.23) 13.24 (0.49) 12.62 (0.41) A B Figure 6-7. MMCHO detectability index results of the geometric phantom. A) Defect 1. B) Defect 2. (Note: For the purpose of comparison, RM result is also represented as a point with error bar in each plot disregarding the cut-off frequency.) Table 6-2. MMCHO detectability index results of the geometric phantom MMCHO-d a for Defect 1 MMCHO-d a for Defect 2 PS-MLEM 14.23 (0.25) 15.71 (0.23) RM 19.25 (1.13) 20.81 (0.91)

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72 6.3 Results of NCAT Phantom Study For the NCAT phantom, we selected long axis slice #7, #8 and #9. The defect is centered in slice #7 and #8. Figure 6-8 shows the true image slices of two frames. Figure 6-9 shows the images reconstructed by the PS-MLEM algorithm using the cut-off frequency of 0.3 cycle/pixel. Figure 6-10 shows the images reconstructed by the RM algorithm using the hyper-parameters of ( = 0.03, = 0.07). In each figure, the image slices for each frame are grouped in the order of #7, #8, and #9. 6.3.1 Single-Slice CHO Results The results of single-slice CHO performance for the PS-MLEM and RM images are shown in Table 6-3. Again, the minimum SCHO-d a for PS-MLEM was found by using a range of cut-off frequencies from 0.1 to 1.0 cycle/pixel. A B Figure 6-8. SP true images of the NCAT phantom. A) Frame 1. B) Frame 2.

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73 A B Figure 6-9. SP images of the NCAT phantom reconstructed by PS-MLEM. A) Frame 1. B) Frame 2 A B Figure 6-10. SP images of the NCAT phantom reconstructed by RM. A) Frame 1. B) Frame 2

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74 Table 6-3. SCHO detectability index results of the NCAT phantom Slice Algorithm SCHO-d a for Frame 1 SCHO-d a for Frame 2 PS-MLEM 11.85(0.23) 12.81(0.64) 7 RM 20.35(0.80) 19.80(1.11) PS-MLEM 10.71(0.30) 13.01(0.81) 8 RM 18.94(0.69) 18.45(0.45) PS-MLEM 8.01(0.26) 9.05(0.46) 9 RM 8.32(0.35) 16.66(0.49) 6.3.2 Multi-Frame Multi-Slice CHO-HO Results To investigate the algorithm performance for a lower defect contrast, we generated another simulated NCAT phantom which had a defect with 25% decreased intensity. We performed the MMCHO observer to both of the two phantoms (50% defect and 25% defect). The results are all shown in Table 6-4. It is evident that for both 50% and 25% defect phantoms, the RM images presented significantly higher MMCHO detectability than the PS-MLEM images. Figure 6-11 shows how the MMCHO-d a of the PS-MLEM images changed with the cut-off frequency. Table 6-4. MMCHO detectability index results of the NCAT phantom MMCHO-d a for 50% defect MMCHO-d a for 25% defect PS-MLEM 16.98(0.77) 8.14 (0.36) RM 33.41(0.66) 14.59 (0.52) 6.4 Multi-Frame Multi-Slice Multi-View CHO-HO Study Since a 3D defect in practice may present different shape, size, and probably different detectability in different axis views, it is meaningful and feasible to expand the multi-frame multi-slice CHO-HO model to a multi-frame multi-slice multi-view CHO-HO (MMMCHO) model. There are also 2 steps involved in the MMMCHO model. In the first CHO step, the SCHO model is applied to each slice of all views and all frames. The first step produces a test-statistic vector with elements as which is from the i ikjCHO th

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75 slice in the k th view of the j th frame. The second HO step takes the resulting test-statistic vectors to compute a final MMMCHO detectability index (MMMCHOd a ). A B Figure 6-11. MMCHO detectability index results of the NCAT phantom. A) 50% defect. B) 25% defect. In our MMMCHO study, we considered 2 frames, 2 views (a long axis and a short axis), and 4 slices. The 16 images of a SP realization are showed in Figure 6-12, 6-13 and 6-14, presenting the PS-MLEM algorithm (cut-off = 0.4 cycle/pixel), and the RM algorithm ( = 0.03 and 0.13), respectively. In each figure, the 1 st row is a long axis view of Frame 1, the 2 nd row is a short axis view of Frame 1, the 3 rd row is a long axis view of Frame 2 and the 4 th row is a short axis view of Frame 2. Because our previous studies showed that the RM images were more sensitive to than to we implemented the algorithm using 2 different values: 0.03 and 0.13 for while was kept at 0.07. Since the test-statistic vector produced in the first CHO step has 16 elements, the size of the training set for each state (SP or SA) in the second HO step should be larger than 16 2 = 256. In our study, we used 350 images of each state as the training set. Thus, we generated 800 images (400 SP and 400 SA) for each algorithm. The first CHO step

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76 used 80 images (40 SP and 40 SA) as the training set. The second HO step used 700 sets of test-statistic vectors (350 SP and 350 SA) produced in the first step as the training set for the HO model. The HO discriminant vector was then applied to the remaining 100 test-statistic vectors (50 SP and 50 SA) to estimate the MMMCHO detectability index (MMMCHO-d a ). The MMMCHO-d a versus cut-off frequency of PS-MLEM is shown in Figure 6-15. The highest MMMCHO-d a of PS-MLEM was achieved with the cut-off of 0.4 cycle/pixel. The MMMCHO-d a s and their standard deviations of the RM images and the PS-MLEM images are shown in Table 6-5. The detectability of RM algorithm was affected by the value of A smaller value of produced smoother images and achieved higher defect detectability. Future studies will investigate the MMMCHO-d a as a function of and Figure 6-12. Multi-frame multi-slice multi-view SP images reconstructed by PS-MLEM.

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77 Figure 6-13. Multi-frame multi-slice multi-view SP images reconstructed by RM (= 0.03). Figure 6-14. Multi-frame multi-slice multi-view SP images reconstructed by RM (= 0.13).

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78 Figure 6-15. MMMCHO detectability index results of the NCAT phantom. Table 6-5. MMMCHO detectability index results of the NCAT phantom. MMMCHO-d a PS-MLEM 9.93 (0.78) RM (= 0.03) 16.55 (0.76) RM (= 0.13) 13.58 (0.83) In summary, we presented a single-slice CHO model and two multi-image CHO-HO models for observer evaluation of the gated myocardial ET images. Results of all the 3 numerical observers demonstrated that the RM algorithm produced images with higher detectability, relative to the PS-MLEM algorithm.

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CHAPTER 7 CONCLUSIONS AND FUTURE WORK The objective of this study was to implement a new method that simultaneously reconstructed gated images and estimated cardiac motion for myocardial ET, and evaluate the performance of this method on a simulated geometric phantom and an NCAT cardiac phantom. The theoretical derivation of the optimization RM algorithm dealing with the non-rigid motion vector field and the emission images in two time frames of gated myocardial ET has been described. The RM algorithm may be viewed as a penalized version of Kleins motion-estimation algorithm in which the data likelihood forms an additional constraint. It may also be viewed as penalized likelihood reconstruction in which image smoothing is achieved by assuming the motion is a smooth deformation between frames, and that the intensity of the material points changes little between frames. The reconstructed images were compared with an independent reconstruction method consisting of high iterations of the MLEM algorithm applied to each projection dataset followed by smoothing with the Hann filter, an algorithm often called PS-MLEM. Our results showed that the RM images presented lower global image error and regional image error, relative to PS-MLEM. For defect contrast study, the geometric phantom and the NCAT phantom were simulated one or more defective regions with 50% decrease in voxel intensity. Our results demonstrated that the RM images achieved substantial improvement in the contrast-noise trade-off over the PS-MLEM images. The RM images 79

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80 had a substantially higher defect contrast compared to the PS-MLEM images except for the highest cut-off frequency PS-MLEM images. However, at this high cut-off frequency, the PS-MLEM images had substantially higher noise than the RM images. The RM estimated motion vector field was compared with that obtained from applying Kleins method to the PS-MLEM images. Our method outperformed that method both in how well the resulting deformed image matched the reference image and in regional comparisons between the resulting motion fields and the ideal motion field. For the observer evaluation of image quality on specific detection task, we used the CHO model because it had shown good correlation to human observer. Based on the fundamental single-slice CHO model, we proposed a multi-image CHO-HO model for gated myocardial ET. Results with both the single-slice CHO model and the multi-image CHO-HO model demonstrated that the RM images had significantly improved detectability than the PS-MLEM images. Our future work includes 1. Multi-frame RM algorithm: Current RM algorithm deals with two time frames due to the restriction of huge amounts of computation. With the help of more powerful computers, multi-frame processing will expand the current two-frame algorithm and is expected to present even higher quality gated images. 2. Human observer study: Though numerical observer is an efficient path for observer assessment of image quality, currently it is not a substitute of human observer. Human observer study is still necessary for further observer assessment study. 3. Spatial resolution study of RM: The regularization of MLEM by combing the likelihood with a penalty often results in position and image dependent spatial resolution and bias. More study about the spatial resolution and noise property of the RM algorithm is required. 4. More concerns such as attenuation, scatter, and randoms: Additional study is required to evaluate how these issues affect the performance of the RM algorithm. It plays an important role to ensure the advantage of RM in clinical application.

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81 In conclusion, the new RM algorithm has demonstrated the potential to improve both image quality and motion-estimation accuracy for gated myocardial ET, relative to the independent motion-estimation and image-reconstruction algorithms. More studies are required to investigate the performance of the RM algorithm in clinical application.

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APPENDIX A NUMERICAL ANALYSIS FOR MOTION ESTIMATION Numerical analysis is the mathematical method that uses numerical approximations to obtain numerical answers to the problems that do not have analytical solutions. The foundation of differential and integral calculus is based upon limits of approximations. The definition of the first-order derivative is a difference quotient: xxuxxudxdux)()(lim0 (A-1) Therefore, there are various approximations for the derivative. Each of them has its own benefits and disadvantages. Some examples of the approximations are Forward difference: xxuxxudxdu )()( (A-2) Backward difference: xxxuxudxdu )()( (A-3) Central difference: xxxuxxudxdu)21()21( or xxxuxxu 2)()( (A-4) and the central difference approximation for the second-order derivative: 222)()()(2)(xxxuxuxxudxud (A-5) The central difference requires information in front of, and behind the location being approximated, whereas the forward and backward differences use the information 82

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83 on only one side of the location being approximated. In our study, we use the central difference because of its better accuracy. To obtain the numerical solution of the motion-estimation problem, we rewrite the Equations 3-60, 3-61, 3-62 as )())()()((2)()(2))(()(2)()(2)()(2)()()(2)(1)()(2)()(2klkklkklklkkklkklkxzxyzzyyxxxffffxffwvuuumrmrmmrrmrmrm (A-6) )()())()(2)()(2))(()(2)()(2)()(2)()()(2)(1)()(2)()(2klkklkklklkkklkklkyzxyzzxxyyyffffyffwuvvvmrmrmmrrmrmrm (A-7) )()()()(2)()(2))(()(2)()(2)()(2)()()(2)(1)()(2)()(2klkklkklklkkklkklkyzxzyyxxzzzffffzffvuwwwmrmrmmrrmrmrm (A-8) where k denotes the k th outer loop, l denotes the l th M-step inner loop, and is current motion estimation. The solution m of Equations A-6, A-7, A-8 is the motion estimation for the (l+1) )(klm th M-step inner loop. Equations A-6, A-7, A-8 are linear to m and can be written in the matrix form as BMA Assuming the 3D image has dimensions of IJK, the total voxel number is N=IJK. Then, is a 3N matrix, is the 3Nmotion vector field to estimate, and is a 3N )3)(3(2)3(1)3()3(22221)3(11211,...,,:,...,,,...,,NNNNNNaaaaaaaaaA TNNNwwwvvvuuu,...,,,...,,,,...,,212121M TNNNNNNNbbbbbbbbb3221222121,...,,,,...,,,,...,,B

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84 motion estimation. In numerical approach, the voxel coordinate components of r, (x, y, z) are represented as the discrete indices (i, j, k), respectively. Using the central difference approximation, the firstand seconddifferential of the motion component u at the location (i, j, k) can be expressed by the linear combination of the motion at its adjacent voxel locations. Substituting u with v and w gives the first and second differential of v and w. 2,,1,,1kjikjixuuu 2,1,,1,kjikjiyuuu 21,,1,, kjikjizuuu (A-9) kjikjikjikjikjikjikjixxuuuuuuuu,,1,,,,1,,1,,,,,,12)()( (A-10) kjikjikjikjikjikjikjiyyuuuuuuuu,1,,,,1,,1,,,,,,1,2)()( (A-11) 1,,,,1,,1,,,,,,1,,2)()( kjikjikjikjikjikjikjizzuuuuuuuu (A-12) 4)()(,1,1,1,1,1,1,1,1kjikjikjikjixyuuuuu (A-13) 4)()(1,1,1,1,1,1,1,1, kjikjikjikjiyzuuuuu (A-14) 4)()(1,,11,,11,,11,,1 kjikjikjikjixzuuuuu (A-15) For each voxel (i, j, k), Equations A-6, A-7, A-8 generate 3 rows of entries in the matrix A. The coefficients of () for voxel (i, j, k) from Equation A-6 can be filled into a chart as follows, where i-1, i, i+1, j-1, j, j+1, k-1, k, k+1 are voxel location indices in 3 dimensions, and the values in the chart are the coefficients for the motion component u at the voxel located by the indices. For example, the coefficients for is N21,...,,uuu kjiu,,1 )2(

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85 j-1 j j+1 j-1 j j+1 j-1 J j+1 i-1 0 0 0 0 )2( 0 0 0 0 i 0 0 2)(2)(2)82(xfk 0 0 i+1 0 0 0 0 )2( 0 0 0 0 k-1 k k+1 Similarly, the coefficients of () in Equation A-6 for (i, j, k) can be expressed using the following chart: N21,...,,vvv j-1 j j+1 j-1 j j+1 j-1 j j+1 i-1 0 0 0 4 0 4 0 0 0 i 0 0 0 0 ))((2)(2)(2yfxfkk 0 0 0 0 i+1 0 0 0 4 0 4 0 0 0 k-1 k k+1 The coefficients of () in Equation A-6 for voxel (i, j, k) are N21,...,,www j-1 j j+1 j-1 j j+1 j-1 j j+1 i-1 0 4 0 0 0 0 0 4 0 i 0 0 0 0 ))((2)(2)(2zfxfkk 0 0 0 0 i+1 0 4 0 0 0 0 0 4 0 k-1 k k+1 The coefficients of () for Equation A-7 for voxel (i, j, k) are N21,...,,uuu

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86 j-1 j j+1 j-1 j j+1 j-1 j j+1 i-1 0 0 0 4 0 4 0 0 0 i 0 0 0 0 ))((2)(2)(2yfxfkk 0 0 0 0 i+1 0 0 0 4 0 4 0 0 0 k-1 k k+1 The coefficients of () for Equation A-7 for voxel (i, j, k) are N21,...,,vvv j-1 j j+1 j-1 j j+1 j-1 j j+1 i-1 0 0 0 0 0 0 0 0 i 0 0 )2( 2)(2)(2)82(yfk )2( 0 0 i+1 0 0 0 0 0 0 0 0 k-1 k k+1 The coefficients of () for Equation A-7 for voxel (i, j, k) are N21,...,,www j-1 j j+1 j-1 j j+1 j-1 j j+1 i-1 0 0 0 0 0 0 0 0 0 i 4 0 4 0 )((2)(2)(2zfxfkk 0 4 0 4 i+1 0 0 0 0 0 0 0 0 0 k-1 k k+1 The coefficients of () for Equation A-8 for voxel (i, j, k) are N21,...,,uuu

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87 j-1 j j+1 j-1 j j+1 j-1 j j+1 i-1 0 4 0 0 0 0 0 4 0 i 0 0 0 0 ))((2)(2)(2zf x fkk 0 0 0 0 i+1 0 4 0 0 0 0 0 4 0 k-1 k k+1 The coefficients of () for Equation A-8 for voxel (i, j, k) are N21,...,,vvv j-1 j j+1 j-1 j j+1 j-1 j j+1 i-1 0 0 0 0 0 0 0 0 0 i 4 0 4 0 ))((2)(2)(2zfyfkk 0 4 0 4 i+1 0 0 0 0 0 0 0 0 0 k-1 k k+1 The coefficients of () for Equation A-8 for voxel (i, j, k) are N21,...,,www j-1 j j+1 j-1 j j+1 j-1 j j+1 i-1 0 0 0 0 0 0 0 0 i 0 )2( 0 2)(2)(2)82(zfk 0 )2( 0 i+1 0 0 0 0 0 0 0 0 k-1 k k+1 A numerical image in practice has certain boundaries. Numerical analysis methods have to consider the physical processes in the boundaries. Different boundary conditions may cause very different results. Because of the ideally compressible model for the space outside myocardium, it is reasonable to assume the motion at the image

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88 boundaries to be approximately zero. Thus in our study, we used fixed boundary conditions, in which m(x b y b z b ) = 0 where (x b y b z b ) is a boundary voxel. The calculation of and its derivatives is not straightforward since is a floating number while r is integer indices (i, j, k). Two approaches to approximate and its derivatives are described and compared here. For notation convenience, we write as )()()(2klkfmr )(klm )()()(2klkfmr )()()(2klkfmr )(2mr f in following text. 1. Taylor approximation approach Let mmm where m = ),,(wvu and ),,(wvu are the closest integer approximation to ; ),,(wvu mmm is the difference vector. Then from Taylor approximation we obtain )()()()(2222mrmmrmmrmr ffff (A-16) )()()()()(222222222mrmrmrmrmrzxfwyxfvxfuxfxf (A-17) )()()()()(222222222mrmrmrmrmrzyfwyfvyxfuyfyf (A-18) )()()()()(222222222mrmrmrmrmrzfwzyfvzxfuzfzf (A-19) The differential of )(2mr f can be approximated using central difference. 2),,1(),,1()(222wkvjuifwkvjuifxfmr (A-20)

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89 2),1,(),1,()(222wkvjuifwkvjuifyfmr (A-21) 2)1,,()1,,()(222wkvjuifwkvjuifzfmr (A-22) ),,1(),,(2),,1()(222222wkvjuifwkvjuifwkvjuifxfmr (A-23) ),1,(),,(2),1,()(222222wkvjuifwkvjuifwkvjuifyfmr (A-24) )1,,(),,(2)1,,()(222222wkvjuifwkvjuifwkvjuifzfmr (A-25) 4)),1,1(),1,1(()),1,1(),1,1(()(222222wkvjuifwkvjuifwkvjuifwkvjuifyxfmr (A-26) 4))1,1,()1,1,(())1,1,()1,1,(()(222222wkvjuifwkvjuifwkvjuifwkvjuifzyfmr (A-27) 4))1,,1()1,,1(())1,,1()1,,1(()(222222wkvjuifwkvjuifwkvjuifwkvjuifxzfmr (A-28) 2. Trilinear interpolation method

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90 Figure A-1. Trilinear interpolation coordinates Trilinear interpolation is to linearly interpolate a point within a box (3D) given values at the vertices of the box. Consider a unit cube as shown in Figure A-1. Denote the values at each vertex as , ..., then the value at position (x, y, z) within the cube is given by 000V 100V 010V 111V xyzV xyzV-z)xy(V-x)yz(V-y)zx(V-y)z-x)((V-z)-x)y( (V-z)-y)(x(V-z)-y)(-x)((V Vxyz111110011101001010100000111111111111 (A-29) Let where are the largest integers approximation of so mmm ),,(wvu ),,(wvu ),,(),,(wwvvuuwvu m are the positive difference. Hence wvuwkvjuifwvuwkvjuifwvuwkvjuifwvuwkvjuifwvuwkvjuifwvuwkvjuifwvuwkvjuifwvuwkvjuifff )1,1,1()1(),1,1()1)(1,1,()1()1,,1()1)(1)(1,,()1()1)(,1,()1)(1(),,1()1)(1)(1)(,,()()(2222222222mmrmr (A-30) The derivative of with respect to x is given by )(2mrf

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91 wvuwkvjuixfwvuwkvjuixfwvuwkvjuixfwvuwkvjuixfwvuwkvjuixfwvuwkvjuixfwvuwkvjuixfwvuwkvjuixfxfxf )1,1,1()1(),1,1()1)(1,1,()1()1,,1()1)(1)(1,,()1()1)(,1,()1)(1(),,1()1)(1)(1)(,,()()(2222222222mmrmr (A-31) The other derivatives can be obtained similarly. For the motion estimation using noise-free images, no significant difference was observed between the Taylor approximation approach and the trilinear interpolation approach. For noisy images, trilinear interpolation approach showed more stable and robust to the noise than Taylor approximation approach did.

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LIST OF REFERENCES 1. P.J. Early and D.B. Sodee, Principles and practice of nuclear medicine (2nd edition). Mosby, St. Louis, Missouri, 1995. 2. R. J. Jaszczak, D. Huard, P. Murphy, and J. Burdine, Radionuclide emission computed tomography with a scintillation camera. Journal of Nuclear Medicine, 1976. 17: p. 551. 3. E.G. DePuey, D.S. Berman, and E.V. Garcia, Cardiac SPECT imaging. Raven, New York, 1995. 4. G. Germano, Clinical gated cardiac SPECT. Futura, Armonk, New York, 1999. 5. G. Germano, Technical aspects of myocardial SPECT imaging. Journal of Nuclear Medicine, 2001. 42: p. 1499-1507. 6. Y. Najm, A. Timmis, M. Maisey, S. Ellam, R. Mistry, P. Curry, and E. Sowton, The evaluation of ventricular function using gated myocardial imaging with tc-99m mibi. European Heart Journal, 1989. 10(2): p. 142-148. 7. T.L. Faber, M.S. Akers, R.M. Peshock, and J.R. Corbett, Three-dimensional motion and perfusion quantification in gated single-photon emission computed tomograms. Journal of Nuclear Medicine, 1991. 32(12): p. 2311-2317. 8. F. Mannting and M.G. Morgan-Mannting, Gated SPECT with technetium-99m-sestamibi for assessment of myocardial perfusion abnormalities. Journal of Nuclear Medicine, 1993. 34: p. 601-608. 9. T. Chua, H. Kiat, G. Germano, G. Maurer, K. van Train, J. Friedman, and D. Berman, Gated technetium-99m sestamibi for simultaneous assessment of stress myocardial perfusion, postexercise regional ventricular function and myocardial viability. Correlation with echocardiography and rest thallium-201 scintigraphy. Journal of the American College of Cardiology, 1994. 23(5): p. 1107-1114. 10. D.S. Lalush and B.M.W. Tsui, Block-iterative techniques for fast 4d reconstruction using a priori motion models in gated cardiac SPECT. Physics in Medicine and Biology, 1998. 43(4): p. 875-886. 11. M.V. Narayanan, M.A. King, E.J. Soares, C.L. Byrne, P.H. Pretorius, and M.N. 92

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BIOGRAPHICAL SKETCH Zixiong Cao was born in Anhui, China on November 1, 1975. He earned a B.S. in physics from the University of Science and Technology of China; and an M.S. in physics from Peking University, China. He studied physics at the Texas A&M University before joining the Biomedical Engineering Department at the University of Florida. He is glad to be an engineer now. He loves physics, he loves engineering, and he loves the University of Florida. 98


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Permanent Link: http://ufdc.ufl.edu/UFE0002464/00001

Material Information

Title: Simultaneous Reconstruction and 3D Motion Estimation for Gated Myocardial Emission Tomography
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0002464:00001

Permanent Link: http://ufdc.ufl.edu/UFE0002464/00001

Material Information

Title: Simultaneous Reconstruction and 3D Motion Estimation for Gated Myocardial Emission Tomography
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0002464:00001


This item has the following downloads:


Full Text











SIMULTANEOUS RECONSTRUCTION AND 3D MOTION
ESTIMATION FOR GATED MYOCARDIAL EMISSION TOMOGRAPHY
















By

ZIXIONG CAO


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2003



























This dissertation is dedicated to my parents and my wife
















ACKNOWLEDGMENTS

It is impossible to adequately thank my long-time advisor and mentor, David

Gilland for his tremendous guidance, considerable patience and encouragement needed

for me to proceed through the graduate study and complete this dissertation. His rich

experience in the medical-imaging area, his thoughtful insight, his brilliant and creative

ideas, and his great sense of humor have made my graduate study very rewarding.

Bernard Mair deserves many thanks. His lecture on numerical analysis introduced

me to this area and led to all of the computation work described in this dissertation. His

constant guidance and supervision laid a smooth way for my studies and research work. I

also thank Chris Batich and David Hintenlang for their discussions, suggestions, and

encouragement during the development of this dissertation. It is a great honor to have

them serve on my committee.

I would like to thank Mu Chen and Karen Gilland for their helpful suggestion on

my numerical observer study. I must express my sincere gratitude particularly to William

Ditto for his support. I thank the Biomedical Engineering Department (especially

April-Lane Derfinyak and Laura Studstill) for their help, kindness, and patience. I thank

Rami, Uday, and Ping-Fang for making the lab such a great place to work. I also thank

Shu and Jiangyong for their comments and suggestions.

Finally, I thank my dear parents, my sister Xia, and my wife Xueshuang for their

endless love, support, encouragement, and understanding in dealing with all of the

challenges that I have faced. I owe a great debt to them.


















TABLE OF CONTENTS
Page

ACKNOWLEDGMENT S ................. ................. iii........ ....


LIST OF TABLES ................ ..............vii .......... ....


LI ST OF FIGURE S ................. ................. viii............


AB STRAC T ................ .............. xi


CHAPTER


1 INT RODUC TION ................. ...............1............ ....


1.1 Introduction ............ ...... ._ .............. 1...
1.2 Si gnificance ............ ..... ._ ...............3....


2 BACKGROUND ............ ..... ._ ...............5....


2. 1 Myocardial Emission Tomography ................. ...............5............ ...
2.2 Proj section and Image Reconstruction ................. ...............6...............
2.2. 1 Radon Transform .................. ........... .......... ... .. ......6
2.2.2 Maximum Likelihood and Image Reconstruction ................ ................9
2.3 Myocardium Motion Estimation ................ ...............11................
2.4 Image Observer Evaluation ................. ...............13................
2.4.1 Human Observer Shtdy................ ...............13.
2.4.2 Numerical Observer: Background .................. ............ ..................15
2.4.3 Hotelling Observer and Channelized Hotelling Observer...................16


3 FORMULATION OF SIMULTANEOUS ESTIMATION METHOD..................20


3.1 Objective Functions............... ...............2
3.2 Euler-Lagrange Equations ............ ......_ .. ...............21
3.3 Optimization Algorithm: RM ................. ........... ............... 26.....
3.3.1 Scheme of Optimization Algorithm............... ...............2
3.3.2 Computation of R Step .............. ...............27....
3.3.3 Computation of M Step .............. ...............29....













4 IMPLEMENTATION AND EXPERIMENTAL RESULTS: GEOMETRIC
PHANTOM ................. ...............3.. 1..............


4. 1 Simulated Source Obj ect and Proj section Data ................. ........................3 1
4. 1.1 Steps to Generate Geometric Phantom ................ ................ ...._.31
4. 1.2 Dimensions of Geometric Phantom ................. ......... ................32
4. 1.3 Intensity and Defect ................. ...............33..............
4. 1.4 W all M otion Simulation .............. ............. ... ............3
4. 1.5 Forward Proj section Simulation and Detector Response ................... .....34
4.2 Bi-Value Lame Constants Model ................. ...............35...............
4.3 Convergence Properties of RM Algorithm .............. ...............38....
4.4 Motion Estimation Results ................. ...............39........... ...
4.4. 1 Figures of M erit ................. ........... ...............39.....
4.4.2 Lame Constants Study Results .............. ...............41....
4.4.3 Global Motion Evaluation .............. ...............42....
4.4.4 Regional Motion Evaluation............... ...............4
4.5 Image Reconstruction Results ................. ...............45................
4.5.1 Figures of M erit ................. ...............45..............
4.5.2 Global Image Evaluation ................ ...............46...............
4.5.3 Defect Contrast Evaluation ................. ...............49...............
4.5.4 Regional Image Evaluation............... ...............4


5 IMPLEMENTATION AND EXPERIMENTAL RESULTS: NCAT PHANTOM.53


5.1 Simulated NCAT Phantom............... ...............53
5.2 Motion Estimation Results ................. ...............54........... ...
5.2.1 Global Motion Evaluation .............. ...............54....
5.2.2 Regional Motion Evaluation............... ...............5
5.3 Image Reconstruction Results ................. ...............58........... ...
5.3.1 Global Image Evaluation ................ ...............58..............
5.3.2 Defect Contrast Evaluation ................. ...............59........... ..
5.3.3 Regional Image Evaluation............... ...............6


6 OB SERVER EVALUATION OF IMAGE QUALITY ................. ............... .....64


6.1 M ethods ........... ......... ............ ..........6
6. 1.1 Single-Slice CHO Model ............... ... .............. ....................6
6. 1.2 Multi-Frame Multi-Slice CHO-HO Model ................. .....................66
6.2 Results of Geometric Phantom Study .............. ...............69....
6.2.1 Single-Slice CHO Results.................. ...... ...........6
6.2.2 Multi-Frame Multi-Slice CHO-HO Results .............. .....................6
6.3 Results of NCAT Phantom Study .............. ...............72....
6.3.1 Single-Slice CHO Results.................. ...... ...........7
6.3.2 Multi-Frame Multi-Slice CHO-HO Results .............. .....................7
6.4 Multi-Frame Multi-Slice Multi-View CHO-HO Study .............. ...................74













7 CONCLUSIONS AND FUTURE WORK .............. ...............79....


APPENDIX NUMERICAL ANALYSIS FOR MOTION ESTIMATION .................. 82


LIST OF REFERENCES ................. ...............92................


BIOGRAPHICAL SKETCH .............. ...............98....


















LIST OF TABLES


Table pg

2-1. Conditional probabilities of a defect detection task .............. .....................14

4-1. Ellipsoid semi-axes lengths for source obj ect ................. ...............33.............

4-2. Poisson ratio and Lame constants ................. ...............37...............

4-3. Global motion evaluation results of the geometric phantom ................. ................ .44

4-4. Regional motion evaluation results of the geometric phantom ........._..... ...............45

5-1. Global motion evaluation results of the NCAT phantom ................. ............... .....56

5-2. Regional motion evaluation results of the NCAT phantom ................ ................. .58

5-3. The contrast and noise level of the NCAT phantom ................. ................ ...._.62

5-4. Five sets of hyper-parameters for RM used for regional image evaluation ...............62

6-1. SCHO detectability index results of the geometric phantom .............. ...................71

6-2. MMCHO detectability index results of the geometric phantom .............. .... .........._.71

6-3. SCHO detectability index results of the NCAT phantom ................. .............. ....74

6-4. MMCHO detectability index results of the NCAT phantom ................ ................. 74

6-5. MMMCHO detectability index results of the NCAT phantom ................. ...............78


















LIST OF FIGURES


Figure pg

2-1. Anatomic orientation of the heart. ............ .....___ ...............6

2-2. Principle of data acquisition and geometric considerations for SPECT.............._._.....7

2-3. Coordinate systems transformation. ........._._. ......_ ....__ ......._.........8

2-4. Images of MLEM and PS-MLEM ................. ...............11...............

2-5. Motion vector representing the displacement of voxels ................. ............. .......12

2-6. Klein's iterative motion-estimation optimization method............. .. ........._ ....13

3-1. Scheme of the RM al gorithm. ................ ........... ........ ......... ...........27

4-1. Source objects of the geometric phantom............... ...............32

4-2. Simulation steps of the geometric phantom .............. ...............33....

4-3. Ideal motion fields of the geometric phantom in 3 short-axis slices. ................... ......3 5

4-4. Collapse high-resolution proj section to low-resolution proj section ................... ...........36

4-5. Segmentation for bi-value Lame constants model. ............. .....................3

4-6. Convergence of the RM algorithm ................ ...............39...............

4-7. Convergence of R step............... ...............40..

4-8. Convergence of M step............... ...............40..

4-9. Global motion error vs. fl for several Poisson ratios. ............. .....................4

4-10. Ideal motion vector fields superimposed on true images. ............. .....................42

4-11. Estimated motion by M step using true images ................. ......... ................43

4-12. Individually reconstructed images by PS-MLEM and estimated motion by M step.43

4-13. Estimated motion and reconstructed images by RM...................... ...............4











4-14. Chosen region to calculate regional motion error for the geometric phantom.........45

4-15. RMS errors of the RM and PS-MLEM images. .......... ................ ................47

4-16. Hyper-parameter effects on the RM images ................. ............... ......... ...48

4-17. RMS error of the RM images as a function of a and B............_.._. .....................48

4-18. Defect contrast of the PS-MLEM and RM images ................. .......................49

4-19. The trade-off between the contrast and noise level of the geometric phantom........5 1

4-20. Sub-regions for regional image evaluation for the geometric phantom. ..................51

4-21. The trade-off between regional intensity bias and variance of the geometric
phantom ..........._._ ....._. ._ ...............52.....

5-1. Long- and short- axis views of the NCAT cardiac phantom. ................ ................ .54

5-2. Ideal motion vector fields superimposed on true images .............. .....................5

5-3. Estimated motion by M step using true images............... ...............55.

5-4. Individually reconstructed images by PS-MLEM and estimated motion by M step..56

5-5. Estimated motion and reconstructed images by RM .................... ...............5

5-6. Global motion errors of three methods............... ...............57

5-7. Chosen region to calculate regional motion error for the NCAT phantom ................58

5-8. RMS errors of the RM and PS-MLEM images. ................ ................. ..........59

5-9. RMS error vs. iteration number of RM outer loop. .................. ................5

5-10. Normal and defect regions in the NCAT phantom ................. ................ ...._.60

5-1 1. Contrast vs. cut-off frequency of PS-MLEM ................ ...............61.............

5-12. The trade-off between the contrast and noise level of the NCAT phantom ............61

5-13. Sub-regions for regional image evaluation for the NCAT phantom. .......................63

5-14. The trade-off between regional intensity bias and variance of the NCAT phantom..63

6-1. The RSC-model channels used for channelization in single-slice CHO model.........65

6-2. Defect locations in the phantoms. ...._._._.. .... ..__.. ...............65..

6-3. The RSC-model channels represented as spatial-domain templates.. ......................67











6-4. Two-step procedure of the multi-frame multi-slice CHO-HO model. .................. .....68

6-5. SP images of the geometric phantom reconstructed by PS-MLEM. .............................70

6-6. SP images of the geometric phantom reconstructed by RM. ............. ...................70

6-7. MMCHO detectability index results of the geometric phantom. ............. ..... ........._.71

6-8. SP true images of the NCAT phantom. ................ ...............72..............

6-9. SP images of the NCAT phantom reconstructed by PS-MLEM ........._..... ..............73

6-10. SP images of the NCAT phantom reconstructed by RM ................. ................ ...73

6-11i. MMCHO detectability index results of the NCAT phantom. ................ ...............75

6-12. Multi-frame multi-slice multi-view SP images reconstructed by PS-MLEM.._......76

6-13. Multi-frame multi-slice multi-view SP images reconstructed by RM .....................77

6-14. Multi-frame multi-slice multi-view SP images reconstructed by RM .....................77

6-15. MMMCHO detectability index results of the NCAT phantom. .............. ..... ........._.78

A-1. Trilinear interpolation coordinates .............. ...............90....
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

SIMULTANEOUS RECONSTRUCTION AND 3D MOTION
ESTIMATION FOR GATED MYOCARDIAL EMIS SION TOMOGRAPHY

By

Zixiong Cao

December 2003

Chair: David Gilland
Maj or Department: Biomedical Engineering

A new method for simultaneous image reconstruction and wall-motion estimation

in gated myocardial emission tomography (ET) is presented. We implemented the

method using simulated phantoms and evaluated the performance of the method.

To reduce the blur in the images induced by the heart motion, gated ET uses the

patient' s electrocardiogram (ECG) to trigger and acquire a time sequence of image

frames, each capturing a particular stage of the cardiac cycle. However, the individual

gated image frames reconstructed by conventional image-reconstruction algorithms are

very noisy due to the reduced counts of detected gamma rays.

Non-rigid deformation of myocardium can be characterized by means of a vector

field called the motion field, which describes the relative displacement of each voxel, and

thus establishes a voxel-intensity correspondence between two image frames. Previous

research incorporated a deformable elastic material model into a motion-estimation

algorithm to more accurately describe the deformation of the myocardium. The estimated









motion field was used to warp each frame to a common reference volume. All warped

frames were then recombined to generate an improved composite image.

Instead of using the motion to warp and recombine frames, our simultaneous

estimation method uses the motion field to regularize the individual reconstructed image

frames, whereas the proj section data are also a constraint for motion estimation. Thus, the

simultaneous method is expected to improve the motion-estimation accuracy and the

image quality of individual frames, relative to the independent motion-estimation and

image-reconstruction methods.

We presented the mathematical formulation of the simultaneous estimation method,

followed by an optimization algorithm. The algorithm was implemented on a simulated

geometric phantom and a realistic non-uniform-rational-B-splines-based cardiac torso

(NCAT) phantom. We evaluated the simultaneous method in terms of the motion

accuracy and the image quality. A numerical observer model was used to assess the

detectability provided by the simultaneous method for the defect detection task. Results

showed that the simultaneous method produces better motion accuracy and image quality,

relative to the independent motion estimation and an alternative image-reconstruction

algorithm.















CHAPTER 1
INTTRODUCTION

1.1 Introduction

Nuclear medicine provides images of diseases by administering small amounts of

radioactive materials into the patient's body and detecting the emitted gamma rays using

special types of cameras. By processing the counts of gamma rays detected from various

angles, a computer reconstructs images, which can provide information about the

function of the body area being imaged. Nuclear medicine is unique in its ability to create

functional images of blood flow or metabolic processes rather than the more conventional

structural or anatomic images produced by X-ray examination, computed tomography

(CT), and magnetic resonance imaging (MRI) [1].

Tomography achieves detailed organ studies by separating a three-dimensional (3D)

obj ect into a stack of two-dimensional (2D) cut sections or slices. Unlike the X-ray CT

that detects transmitted X-ray photons, tomography in nuclear medicine detects emitted

gamma rays and therefore is referred to as emission tomography, or ET in short [2].

Single Photon Emission Computed Tomography (SPECT) and Positron Emission

Tomography (PET) are two widely used modalities in the clinical nuclear medicine.

By measuring both the heart blood flow and the metabolic rate of the patients,

physicians can find areas of decreased blood flow (such as that caused by blockages), and

differentiate diseased from healthy muscle. This information is particularly important in

diagnosing coronary artery diseases (CAD) [3, 4]. Myocardial ET is one way to measure

blood flow and metabolism in the myocardium. SPECT imaging assesses the severity and









extent of perfusion defects in patients with coronary stenosis. It accounts for over 90% of

all myocardial perfusion imaging performed in the US today [5].

The motion of the patient' s heart causes blurring in myocardial ET images. This

motion blurring can be reduced by gated myocardial ET, which uses the patient' s

electrocardiogram (ECG) signal to trigger and acquire a time sequence (typically 8 or 16

frames) of acquisition over the cardiac cycle [6-9]. Data are acquired over many cardiac

cycles to produce the final set of image frames. Each gated image frame records a

particular stage of cardiac cycle. In addition to reducing motion blurring in each image

frame, gated imaging has the advantage of allowing an estimation of cardiac wall motion

and ej section fraction. However, the statistical quality of each gated image suffers as the

total number of detected gamma rays is distributed over a series of time frames. Because

of the low gamma ray counts, the images produced by conventional image-reconstruction

algorithms usually are very noisy [4, 6].

Previous researchers [10-13] presented several reconstruction methods to improve

the quality of the gated image frames. Lalush et al. [10] considered the time sequence of

gated frames as a four-dimensional (4D) image and reconstructed it using a penalized

maximum-likelihood (ML) technique with space-time Gibbs priors to ensure smoothness

in individual image frames and between frames. Narayanan et al. [1l], Wernick et al. [12]

and Brankov et al. [13] used an alternative technique based on principle component

analysis, in which the time sequence of gated datasets were Karhunen-Loeve (KL)

transformed and then rapidly constructed frame-by-frame. The gated image frames were

finally obtained by applying the inverse KL transform.

Other previous studies [14-16] presented methods to obtain a vector field









describing the motion of the myocardium between successive frames. Klein and

Huesman's method [15, 16] views the myocardium as an isotropic elastic material whose

deformations are governed by the equations of continuum mechanics. It was shown that

deforming and combining the gated image frames using the estimated motion resulted in

improved composite images.

Based on these previous studies, we generated a new idea to simultaneously

reconstruct two gated image frames and estimate the motion vector field between them.

Our hypotheses are: constraints from the material property will reduce the noise level in

individual gated image frames, and less-noise images will in turn produce more accurate

motion estimation.

1.2 Significance

Based on the 3D motion-estimation method developed by Klein et al. [15, 16] and

the conventional maximum-likelihood expectation-maximization (MLEM) algorithm [17,

18], we proposed a new estimation method for gated myocardial ET: the simultaneous

image-reconstruction and 3D motion-estimation method.

From an image reconstruction viewpoint, the simultaneous estimation method uses

the deformation of elastic material that enforces the consistency between frames as a

penalty to perform penalized ML reconstruction. Unlike the usual penalized ML

algorithm [10, 19], our penalty uses the deformation to regularize the reconstructed

images, without any spatial smoothing.

In their initial attempt, Gilland and Mair [20] used the Polak-Ribiere conjugate

gradient optimization algorithm to solve the problem. Because of the complexity of the

obj ective function, it is not clear that this algorithm converges and may produce negative

image intensities. Our study developed a new two-step iterative algorithm (RM) that









alternately updated the gated images and the motion vector field, and guaranteed

non-negativity in the images. This RM algorithm and some results were partly reported

by Cao et al. [21].

We evaluated the motion-estimation accuracy and image quality provided by the

simultaneous method using a simulated geometric phantom and a realistic

Non-uniform-rational-B-splines-based Cardiac Torso (NCAT) cardiac phantom. We also

evaluated the image quality using a numerical observer model for both phantoms. The

numerical observer results have been reported by Cao et al. in "An observer model

evaluation of simultaneous reconstruction and motion estimation for emission

tomography" submitted to 2004 IEEE International Symposium on Biomedical Imaging,

Arlington, Virginia, US.

Myocardial ET provides perfusion metabolic and wall-motion information in a

single imaging procedure. Our simultaneous estimation method promises to improve

motion-estimation accuracy and individual gated image quality simultaneously, which

will improve the accuracy in diagnosing cardiac diseases.















CHAPTER 2
BACKGROUND

2.1 Myocardial Emission Tomography

While other cardiac imaging modalities (angiography, echocardiography, etc.)

provide purely anatomical information, myocardial ET provides physiological

information, such as that used to identify areas of relatively or absolutely reduced

myocardial blood flow associated with ischemia or scar [1]. SPECT and PET are two

main myocardial ET protocols used today. Although we presented and implemented our

method on the basis of SPECT model because of its fundamental concept and simplicity,

it can be expanded to estimate the image and motion for PET imaging.

In SPECT myocardial perfusion imaging procedure, the patient is intravenously

given a pharmaceutical bound with a radioisotope label (such as Tc-99m). The

radioactive pharmaceutical is then taken up in the patient's myocardium in proportion to

perfusion or tissue metabolism. Single photons are emitted as the Tc-99m radioisotope

decays, and a fraction of these photons are able to penetrate the surrounding body tissue

and reach the SPECT gamma camera [22]. The gamma camera records the spatial

distribution of the photons and forms a 2D proj section of the 3D activity distribution. A

3D image of the organ is then reconstructed from the proj section data collected from

various angles. Proj section and reconstruction procedures are discussed later.

Patients with significant coronary artery stenosis usually have diminished

radioactive-pharmaceutical concentration in the area of decreased perfusion, which will

cause a defective area in the perfusion image. By comparing the images taken in two










cases, when the tracer is administered during stress and rest, physicians can distinguish

ischemia from scar tissue. If the defect in the image taken in stress is worse than that

taken at rest, it is most likely due to ischemia. If the defect remains unchanged, the lesion

is most likely from scar [1].

The human heart usually points downward, approximately 450 to the left and

anterior, but individual variations are considerable. Cuts perpendicular to the long axis

are called coronal or short axis cuts. Cuts perpendicular to the short axis are called

sagittal (vertical long axis) and transaxial (horizontal long axis) cuts [4]. Figure 2-1

shows anatomic orientation of the heart.



Cozonal=
-Short Axis

















Figure 2-1. Anatomic orientation of the heart.

2.2 Projection and Image Reconstruction

2.2.1 Radon Transform

Figure 2-2 shows the 2D data acquisition scheme in SPECT using a parallel

collimator [22]. Rotating the detector allows one to observe the photon emission in the

field of view from many angles. The number of scintillations detected at location s along









the detector when the detector head is at an angle of 6 is defined as g(s, 6). We denote the

position of a point in the 2D slice as (x, y) and the estimated number of photons emitted

from this point as a function f(x, y). This estimated image function f(x, y) is assumed to be

proportional to the tracer concentration of interest. Function g is the proj section of onto

the detector as allowed by the parallel collimator. A sinogram is an image representative

of g in which the horizontal axis represents the angular position of the detector and the

vertical axis represents the count location on the detector.


g(s,0)
detector

collimator






0 x
f(x,y)

object


Figure 2-2. Principle of data acquisition and geometric considerations for SPECT.

Mathematically, the proj section operator can be described by the Radon transform

[23]. The Radon transform g(s, 6) of the function f(x, y) is the line integral of the values

of f(x, y) along the line parallel to the collimator at a distance s from the origin:

g(s, 6) = f (x(s,u), y(s, u))du
(2- 1)
= f (s cos 6 u sin 6, s sin 6 + ucos 6)d~u

where u denotes the location of the points along the integral line. Equation 2-1 involved a









coordinate system transformation, which is illustrated in Figure 2-3. Assuming the two

coordinate systems share the origin, the transformation between them is

x = s cos 6 u sin 6 and y = s sin 6 + ucos 6 (2 -2)

From an operator' s point of view, the Radon transform can be represented as a

procedure of an operator product:

g(s, 6) = Hf(s, 6) = hs (r) f(r)dr (2 -3 )

where r = (x, y) is the coordinate vector, H is called a forward-projection operator, and

h ()is a weight function that represents the probability that a photon is emitted from r

and detected at (s, 6)















Figure 2-3. Coordinate systems transformation.

In numerical analysis by computers, (s, 6) and (x, y) are all represented as discrete

variables, and g(s, 6) and f(x, y) are functions of these discrete variables. A discretization

procedure for all the variables and the functions is inevitable. In our study, each element

of the 3D obj ect image is called a voxel, and each measurement on the detector at a

particular proj section angle is called a bin. In the discretized image and sinogram, the

element f (j) is the jth VOxel in the image and g(i) is the ith bin in the sinogram.









In the discretized domain, the sinogram and the image can both be considered as

vectors. The elements in the sinogram vector are the counts of all the bins. The elements

in the image vector are the intensities of all the voxels. In our study, for convenience, we

use the same notation for function and vector. For example, when we use g(s, 6), g is a

continuous function; when we use g(i), g is a vector with the element index of i.

For the discretized image and sinogram, the forward-projection operator H

becomes a matrix. Vector g is then the matrix product of H and vector f : g = HJ; or in the

form of components,


g () = Hf (i) = h, f( j) i= 1, 2,.. .,p. (2 -4 )


where H is now called a forward-proj section matrix with elements of h,,, q is the total

number of voxels in the image slice, and p is the total number of bins in the sinogram.

The entries of H can be carefully chosen to take into account the geometry of acquisition

and, more precisely, the detector response, attenuation and scatter.

2.2.2 Maximum Likelihood and Image Reconstruction

By comparing the proj section calculated from the current image estimate to the

measured proj section data, an iterative image-reconstruction algorithm adjusts the image

estimate iteratively, and eventually reconstructs an image [23]. The iterative algorithms

differ in the comparison method and the correction applied to create the new estimate.

Equation 2-4 is based on the assumption of the ideal noise-free situation. In

experimental measurements, however, the counts are subj ect to randomly change under

the Poisson statistics of the radioactive disintegrations. In emission tomography, the

well-known log-likelihood function can be written as










L(gi, f ) = [gig(i)lgHf(i)) Hf(i)] (2-5)


which measures the likelihood that the image produces the sinogram g.

Maximizing the log-likelihood function leads to finding the most likely image fto

produce the sinogram g. Maximum-likelihood expectation-maximization (MLEM) is an

iterative algorithm that reconstructs this most likely image. Each MLEM iteration is

divided into two steps [18, 24]: Expectation step (E step) in which a formula is formed to

express the likelihood of any image given the measured data and current estimate f(k),

and Maximization step (M step) in which a new image f(k+1) that has the greatest

likelihood is found. The iterative image-updating formula of the MLEM algorithm is

given as [18, 24]

f ~ fk (k+1 ( )=hg ,.,q



MLEM shows its advantage of monotonicity of the likelihood function versus

iteration [17]. The convergence, however, is very slow and the images appear

unacceptably noisy before it converges [25]. To smooth the noisy image reconstructed by

MLEM, a digital post-processing filter is usually applied to the image reconstructed after

many iterations of MLEM. This Post-Smoothed MLEM (PS-MLEM) algorithm has

demonstrated excellent results in previous studies [26, 27]. The most important parameter

in the PS-MLEM algorithm is the cut-off frequency of the filter, which determines the

smoothness of resulting images. In our study, we use 100 iterations MLEM and

low-pass Hann filter because of its simplicity. Figure 2-4 (A) shows a sample image

reconstructed by MLEM with 100 iterations. Figure 2-4 (B) demonstrates that the










smoothness of the image reconstructed by PS-MLEM is influenced significantly by the

amount of the post-smoothing filter. Nyquist frequency (0.5 cycle/pixel) is usually used

as a feasible cut-off. Because of this, when we compare our algorithm to PS-MLEM

algorithm, we usually use a range of cut-off frequencies (from 0. 1 to 1.0 cycle/pixel) for

PS-MLEM.


























Figure 2-4. Images of MLEM and PS-MLEM. A) 100-iteration MLEM image B)
PS-MLEM image using Hann filter with various cut-off frequencies. From 1
to 10, the cut-off frequency was 0. 1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0
cycle/pixel.

2.3 Myocardium Motion Estimation

A variety of methods have been proposed recently to estimate and represent the

cardiac motion [14-16, 28-32]. Based on a deformable elastic model and the assumption

that voxels corresponding to the same tissue in two frames conserve their intensity values,

Klein and Huesman [15, 16, 32] presented a voxel-based method to estimate the cardiac









motion vector Hield. This motion Hield is compatible for use by the simultaneous method

since the images are also represented as voxel-based intensity scalar Hield.

Klein's motion-estimation method [32] involves minimizing a two-component

obj ective function. The first component is an image matching obj ective function serves as

a driving force to push deformed image intensities of two frames into correspondence.

The second component is a regularization of strain energy function that prevents

deformations that are physically unlikely or meaningless.

We denote the frame 1 and 2 as f,(r) and f,(r) respectively, and the motion

vector Hield as m(r) = (u(r), v(r), w(r)), where r = (x, y, z) are the voxel coordinates in the

spatial domain. As shown in Figure 2-5, the motion vector m(r) presents the

displacement from a voxel in Frame 1 to its correspondence that represents the same

tissue in Frame 2. [32]

Framel ,- Frame2:



r r> mr







Figure 2-5. Motion vector representing the displacement of voxels

The image matching obj ective function is formulated as

E,(J f;,m)= Cf(r)- f(r +m(r))(dr (2-7)

and the strain energy Es(m) is










Es (m) = fIx +", Vy +W X dr+pus+
(2-8)



where ii, pu are elastic weighting parameters called Lame constants that reflect the degree

du
of compressibility of the object, ux = -is the derivative of the motion component u


with respect to x. The entire obj ective function is grouped as E = EI + BEs, where a is a

hyper-parameter that control the balance between the two terms. In the 3D discrete

numerical domain, E can be expressed as a function of all the variables (u,, v,, w, ) and

their derivates, where j=1, 2, ... q denotes the voxel index in the image.


E= [f (r~,)-f( r +m ) +1; ap, it, ",3 l)i
(2-9)
ipp C u, + ", + .,,) u,, .L +v, i~+ ul +w, /",= l+ 3,-N~
]=1 L =1

Klein' s iterative optimization method [32] to minimize the entire obj ective function

E consists of steps shown in Figure 2-6.

Approximate E to a Find the minimum
quadratic form (E) around I a point of E
the current estimate m (k) denoted as m


Use li as the new estimate m~kl

Figure 2-6. Klein's iterative motion-estimation optimization method

2.4 Image Observer Evaluation

2.4.1 Human Observer Study

Besides the quantitative assessment such as the signal-to-noise ratio and defect

contrast, the image quality and the performance of a reconstruction algorithm can also be









assessed by observer studies, in which the observers evaluate the images from the

perspective of a clinically specific task. Defect detection in myocardial ET imaging is a

specific two-state classification task that requires the observer to diagnose by viewing the

image and classifying it as either normal or some particular disease state. The accuracy of

the diagnosis can also be viewed as an observer measurement of the image quality.

Psychophysical study with human observers (usually the physicians) is the standard

means of observer evaluation of detection task performance in medical imaging [33].

For a defect detection task, there are two possible true states of the obj ect: signal

present (SP) and signal absent (SA). There are also two possible responses: positive and

negative, corresponding to the observer' s classification of SP and SA, respectively. Table

2-1 shows the conditional probabilities of a certain response given the occurrence of a

certain stimulus [34].

Table 2-1. Conditional probabilities of a defect detection task
Stimulus = SP Stimulus = SA
Response True positive False positive
= Positive P(TP)=P( Positive | SP ) P(FP)= P( Positive | SA)
Response False negative True negative
= Negative P(FN)=P( Negative | SP ) P(TN)= P( Negative | SA )

The probabilities in the Table 2-1 are not independent of each other. The relations

between them are: P(TP)+P(FN)=1 and P(FP)+P(TN)=1. Thus we have only two

independent probabilities in the defect detection problem.

In an observer study to evaluate the performance of an imaging system, the human

observers are shown a large number of images and asked to rate their confidence as to

whether a defect is present or not [33]. The resulting P(TP) and P(FP) represent the

probability of correctly distinguishing, or the detectability of the system. However, these

two probabilities are obviously determined by the decision criteria the observer used. To









remove the dependence on the decision criteria, a Receiver Operating Characteristics

(ROC) curve [34] is plotted with P(TP) versus P(FP) as the decision criteria change. The

area under the ROC curve is a common figure of merit to represent the detectability of

the imaging system. The ROC curve can also be used to optimize the parameters to

produce better images [35-37].

2.4.2 Numerical Observer: Background

The human observer tests are very time-consuming and difficult to perform in

practice. Therefore, there is considerable interest in the use of numerical observers of

which the performance indices can be calculated rather than measured. A numerical

observer correlating well with the human observers would be extremely valuable for

optimizing and evaluating imaging systems.

The numerical observer procedure for the defect detection task is as follows:

compute a scalar feature Z( f), where fis the image to be classified, then compare it to a

threshold C. Choose positive if ii >C (or ii C). The

scalar feature ii is also called a test statistic or decision variable.

A common figure of merit for the numerical observer is the detectability index da

(also known as d) [38], defined by

22[E'(A | SP) (A | SA)]2
d = (2-10)
Svar(il | SP) + var(il | SA)

where E(AZ | k) and var(A2 | k) are the conditional mean and variance of Z( f),

respectively, given that fwas produced by an obj ect in state k (k = SP or SA).

An ideal observer is defined as one who has full statistical knowledge of the task

and who makes best use of the knowledge to minimize a suitably defined risk. The test









statistic of the ideal observer is defined by [39]


jldeal =IP( f | SP)P S (-1


where P( f | k) is the conditional probability of given that fwas produced by an object

in state k (k = SP or SA). The right side of Equation 2-11 is called log-likelihood ratio.

(Note: log-likelihood ratio is unrelated to log-likelihood function discussed in 2.2.2).

Ideal observer maximizes the area under the ROC curve and sets an upper limit to

the performance obtainable by any other observers including the human, though it is only

applicable for simple situations. In practice the log-likelihood ratio is rarely calculable,

except for the detection of an exactly known signal superimposed on an exactly known

background (SKE-BKE) situation. In this case the likelihood ratio can be calculated by

simple linear filtering process. If the noise is stationary, white and Gaussian, the

likelihood ratio is the output of a matched filter. If the noise is stationary and Gaussian

but not white, a prewhitening procedure is required before the matched filter [40].

As the performance of the human observer was found to be dramatically degraded

by correlated noise [41, 42], and some spatial-frequency channels were found to exist in

the human visual system [43, 44], Myers et al. [45] added a frequency-selective channel

mechanism to the ideal observer. The channel model has demonstrated the ability to

improve the correlation between numerical and human observers for detection tasks by

other researchers [46-50].

2.4.3 Hotelling Observer and Channelized Hotelling Observer

In practice, a linear observer model is usually more applicable than the ideal

observer. A linear observer is defined by










n( f ) = u f = Cu, f( j) (2-12)


where u is called a discriminant vector, and T denotes the transpose. The fh COmponent

value of u, denoted as u,, can be viewed as the importance assigned to f( j), the fh piXel

value of the image vector f

Before define Hotelling observer, we need to introduce two scatter matrices that are

used to describe the first- and second- order statistics off[39]. The interclass scatter

matrix S1 measures how far the state means deviate from the grand mean f and is

defined by


S, _C pf Pk( k k)T (2-13)


where K is the number of states, Pk is the probability of occurrence of state k, fk is the

state mean for the kth state. The grand mean is given by


faf Pk k (2 14)
k=1

The intraclass scatter matrix S2 is defined as the average covariance matrix across all

states:


S2 a PkKk (2-15)


where the kth state covariance matrix Kk is given by


Kk=(f kX/-7k k,' (2-16)

The angular brackets in Equation 2-16 have the same meaning as the overbar, that is, a

full ensemble average over all obj ects fin state k.









Based on the covariance matrices S1 and S2, a criterion measuring the separability

of the states was proposed by Hotelling in his classic paper [5 1], and is now often

referred to as the Hotelling trace:

J = tr(S 'S,) (2-17)

where tr denotes the trace (sum of the diagonal elements) of the matrix. The Hotelling

trace is a common measure of classification performance in pattern recognition [39]. It

increases if the variability in the image (due to noise of other factors) is decreased, since

that corrnrnrespnd to reduci;ng, the covriance te~rmsn that go\ into S2 and hence increase S.

It also increases if the system is modified in such a way that the two state means become

more widely separated, since that increases the norm of the vectors that constitute S1.

Previous research [40, 52, 53] presented that Hotelling Observer (HO) as a linear

observer optimizing the Hotelling trace. The test statistic of HO is obtained by

Arr = u of = (fe f)"S' f, (2-1 8)

Implementing HO begins with creating a training set of realizations of obj ects in

two states (SP and SA), and estimating the scatter matrix S2 %Om the training set. If there

are N pixels in the image, then S2 is an NX Nmatrix with the rank of N The number of

images in the training set should be larger than N2. Otherwise, the inverse of the

estimated S2 Will HOt exist [54, 55].

The Channelized Hotelling Observer (CHO) incorporates the channel mechanism

into HO [46]. Using a bank of2~band-pass frequency filters, CHO transform each image

to an M~X 1 vector. The values in the resulting vector represent the frequency information

of the image, so this channelization procedure can be viewed as a feature extraction

procedure. Mathematically, it can be described by









v =Uf (2-19)

where U is an NX M matrix, which represents the channelization procedure and is

determined by the channel model. For an SKE task, the mth row vector in the matrix U

describes the mth channel's impulse response centered on the defect location.

CHO calculates the test statistic of the resulting vector v by

ACHO HO~ SP SA) TS V (2-20)

where vSP and vSA are the state means of v for SP and SA, respectively, and Sv is

the intraclass scatter matrix of v.

CHO model has demonstrated to predict human observer performance for a variety

of SKE-BKE detection tasks. Gifford et al. [47] investigated the application of CHO to

detect hepatic lesions using various channel models, eye filters and internal noise models.

Burgess et al. [56] showed the correlation between the human observer and CHO for

well-defined two-component noise fields (Poisson noise correlated with a Gaussian

function) with various channel models.














CHAPTER 3
FORMULATION OF SIMULTANEOUS ESTIMATION METHOD

3.1 Objective Functions

Given two gated projection datasets g "' and g, 2, our method simultaneously

reconstructs the image frames f, (r), f, (r) and estimates the motion vector field

m(r) = (r(r), v(r), w(r)) that describes the motion from f, (r) to f, (r) .Here

f,(r, fr) nd m(r) are presented as functions of a continuous variable in the spatial

domain, r = (x, y, z-), where x, y_, z are coordinates in three dimensions.

We define an energy function as

Elf,, f,, m)= ai(f,, f,)+ Eil is/, fm)+ PEs(m) (3 -1 )

The first term is a sum of the negative log-likelihood functions of two frames


L (f f, ) = H f ,, (i) J"' (i) lo gHf,, (i) (3 -2 )


where m = 1, 2 denotes the frame index, i=1, 2, ..., p denotes the element index in the

datasets, and H is the forward-proj section matrix that gives


Hf ,c i)= [hes J, j ( 3-3 )


The second term in Equation 3-1 is the image matching term that measures the agreement

of two frames under the assumed motion m:

El 71,72,m)= Cf;(r)-fl(r+m(r))]dr (3-4)

The third term is the strain energy term of deforming elastic material:










Es(m)= IAlrux y zI 2d+ lyu2 2 z2
(3-5)
+ pl(r) lu, v


where the Lame constants 31 and p determine the material compressibility. Due to the

different compressibility of the blood pool and the myocardium, we allow the values of 31

and p to change between regions by representing them as the functions 31(r), p(r). The

values of the hyper-parameters a and fl reflect the influence that the proj section data and

the strain energy have on the estimates obtained by this method. In our study, we chose

these values by experience.

The reconstructed image frames f, (r), f2 (r) and estimated motion vector field

m are obtained by minimizing the function E(fy, f2,m), that is,


(J; f2 ,m )= argminE(;f f,m). (3-6)


3.2 Euler-Lagrange Equations

We use the Frechet derivatives of the function E to obtain the Euler-Lagrange

equations. As the first step, we compute the Gateaux differentials [57] of E with

increment t. Let DmEV;fif, m; t), DuEV;fif, m; t), DEfifi,f2 m; t), DwEV;fi, fm; t)

denote the Gateaux differentials of E with increment t relative to fm, u, v, w, respectively.

The Gateaux differentials are computed by using the well-known formula as


DE(.fif2,m;t)= ~E~f, +srt, f2 0~s (3-7)


Applying similar formulas to the functions EI, Es, L, which determine E, for differentials

relative to all the variables, we obtain the Gateaux differentials









D,E, if2,, ,m; t)= 25f [(r)- ~f2(r + m(r))]t(r)dr (3 -8)

D2E,(f,, f2,m; t)= 25 [f, (r)- f2(r + m(r))]t(r + m(r))dr (3 -9)


DmL~,, 2;t= [H~i) ge)(i)t~i/Hf~i)] m=1,2(3-10)


Dz,:(I,72m~t= 2 [,(r- f 2(r + m(r))t(r)dr (3-11)




DE(II72,~t) -2[/()- 2 2(r + m(r))t(r)dr (3-12)



DzE,E(m; t) = [(jlV m + 2puxl)tr + pl(u~ + vx)tv + pr(uz, x )tz ]dr (3-14)

DE,E(m; t) = [(lV -m +2pny,)ty + p(ul + vr)tz*l 1Vz + y z]dr (3-15)

D_,E (m; t) = [(AV m + 2pez)tz + pr(uz + x)tr #r(' y y~t ]dr (3-16)

Now, let Cc (R) denote the set of all infinitely differentiable functions with

compact support on s Then the Euler-Lagrange equations are DmE( fi, f2, m; t) = 0 ,

Dz,E( f,, f2, m; t) = 0, D,E( f,, f2, m; t) = 0, and D,E( f,, f2, m; t) = 0 for m = 1, 2 and

all t E Cfm All these equations involve integrals containing the function t. To remove the

integral and the dependence on t from these equations, note that if Sg(r)t(r)dr = 0 for

all t E Cfm then g = 0. To use this fact, each of the Gateaux differentials of E needs to be

expressed in the form Sg(r)t(r)dr for some g. Such a function g is called a Frechet

derivative. The Frechet derivatives of E with respect to f,,, u, v, w are denoted by D~,,E

Dz,E, DE, DwE respectively, and are weighted sums of the Frechet derivatives of EI, Es,









and L. Some Frechet derivatives of E are immediately obtained from Equations 3-8, 3-11,

3-12, and 3-13 by easy calculation:

DE,(fi,, 2,m) = 2[f,(r)- f2(r + m(r))] (3-17)


D E, (f,,f2, m) = -2[ f, (r)- f2( 2 ~r) (r + m(r)) (3-18)


DE E(II,2, fm) = -2[J;(r)- f2( 2 ~r) (r + m(r)) (3-19)


DE E(II,2, fm) = -2[f, (r) f2( 2 ~r) (r + m(r)) (3-20)


To obtain D2Ef;(, f2, m), We need to express the integral in Equation 3-9 as g (r)t(r)dr .

Consider the coordinate transformation r = r + m(r) = (I3 + m)(r) Where I3 is the

identity 3 X 3 matrix. Then, the Jacobian is given by


dr
JWxr) [det(I 3 (3-21)


where m'(r) =~ vx vy (-2



So, if a is the image of R under the operator (I3+m), fTOm Equation 3-9 we obtain


D2E:(fi72, ; t = 2 [ (IS F) (F)t(F dr(3-23)


Now, the motion changes the location of pixels within the ROI R, but it leaves the

entire ROI unaltered, hence R = 0 So, from Equations 3-21, 3-23 we obtain

D2E, (ft,, 2,m) = -2[f, (r)- f2(r)]/|I det(3 + m' (r))|1 (3-24)


where r = (I; -1)(r






24


The Frechet derivatives of L are given by



Hf, (i)


(3-25)


where h(r) [ ha (r) So, from Equations 3-17, 3-24, 3-25 and the fact that E does not
=1-


depend onf,,, we obtain the Frechet derivative of E with respect to fi and f2



g~'Hf, (i)(ih r


(3-26)


D2E( ,f,, fm )


We now determine the Frechet derivatives of Es with respect to u, v, w. The

Gateaux differentials in Equations 3-14, 3-15, 3-16 are all of the form SF Vtdlr for

some vector field F. By using the divergence theorem, for all t e Cf (R)


SF Vtdr = (V -o (tF) (V F)t)dr = -5 (V F)tdr (3

so the Frechet derivatives are V -F.

From Equations 3-14, 3-15, 3-16, we see that the vector field F for De,Es(m; t)

D,Es (m; t), and D,Es (m; t) are

(AZ(ux + v, + w,) + 2puux, p(u, + vx), pu(u, *x)) (3

(pU(u, + vx), A(ux + vy+ wZ) + 2pUy ,p(vZ y )) (3

(pU(u, *), pU(v W), *y x y z,)+ 2puw,) (3

respectively. In each case we obtain V F as


-28)


,


-29)

-30)

-3 1)


2[ f (r 2 ()]/detI3 m'(r) + ~ r)- ()H (i)(ih, (r)1


(3-27)









(AZ + 2pu)uxx + pu(u, + uz)+ (/ + Pu)(xy xzw) (3-32)

(AZ + pu)(uxy+ yz)+ xxv, z) + (AZ + 2pu)v,, (3-33)

(AZ+ pu)(uxz + vyz)+ Pu(w + yy)+ (AZ+ 2pu)waz (3-34)

Hence we obtain the Frechet derivatives of Es

Dz,Es (m) = -(AZ + 2pu)uxx pu(u, + uzz) (/ + Pu)(xy xzw) (3-35)

DV,Es(m) = -(AZ + pu)(uxy + yz) uvxx zz) (A + 2pu)v,, (3-36)

Dw,Es(m) = -(AZ + pu)(uxz + yz) u wx yyw) (AZ + 2pu)waz (3 -3 7)

Since L is independent of m, we obtain the Frechet derivatives ofE relative to u, v, w as

Dz,E(f;, f2,m) = -P[(il+ 2p)uxx pu(u, + uz) (i #~)(xy xz,
(3f (3-3 8)
2[ f (r) f2( 2 ~)l ( ~)


DE(f,, f2,m -[i u(xy yz) uvxx zz) (il+ 2U)vw]
afi (3-39)
2[J; (r) f2( 2 ~)l ( ~)


DwE(f,, f2,Il -[i ~ u(xz vyz uxx yyW) (jl+ 2p)waz i
dfi (3-40)
2[f, (r) f2( 2 ~)l ( ~)


Finally, the Euler-Lagrange equations for E, satisfied by (f, f2 ,m ), are


2[ zr)-f2(~m~r)]+ h~)- h(r~ *()/Hf(i)= 0(3-41)



-2[ z(r)- f2(r)]/ det(I3 + m'(r) + hj(r)- h(r)rj()i/H2i (3-42)


2[ (r) -f2 2ml~f
Dx (3-43)
B(A+2pU)ux + p(ul~ +uzz) (~ )(V xy xz1?) = 0










dy (3-44)
Pi(A + 2p)ul, + p(v, + v ) + I~(A + p)(u +w = 0


2[f, (r) -f,(r + m) r +m)
8: ( 3- 45 )
B(A+2r)w_ +pI(w +Mw )+(Al+p)(ux= f's) =0

The Equations 3-43, 3-44, 3-45 are valid on regions with uniform Lame constants 32

and pu. So we assume throughout our study that the frames are segmented into two regions

with uniform values for the Lame constants on each.

3.3 Optimization Algorithm: RM

3.3.1 Scheme of Optimization Algorithm

We estimate the minimizer ( f, f m ) by a two-step iterative procedure,

which updates the estimates ~f,(k),>2(k) m(k) iteratively to ~f,(k+1)> 2(,k+1) 'k+1) in

the kth iteration. Figure 3-1 shows the scheme of this optimization algorithm. In R step (R

for reconstruction), we assume the motion is reasonable and fixed, and minimize the sum

of the image matching and likelihood functions to generate the image updates, that is,

(fi (k+1), f(k+1)) = arg min Elf,, f,,m (k))
(3-46)
= arg minaol(f,, f2) +E~ (i i /2,m(k) )

In M step (M for motion), we assume the images are reasonable and fixed, and minimize

the sum of the image matching and strain energy functions to generate a new motion

estimate, that is,

ml(k+1) = arg min E( f,(k+1), 2(k+1),m

= arg min E (fi"k+1) 2(k+1),m) Ss(m)] (3-47)









Hence this iterative optimization algorithm is called "RM" algorithm. The image

frames and motion vector field are simultaneously updated after one repetition of these

two steps, called an "outer loop" of the RM algorithm. The minimization problems in R

and M steps are also optimized using iterative algorithms, and therefore, they are referred

to as "inner loops".


J;(k), 2(k) R Step: Minimize f,'k+1) 2(k+1) M Step: i(k+1) f2(k+1)
(k) i,f f2 > EI + BEs over m

Outer Loop


Figure 3-1. Scheme of the RM algorithm.

3.3.2 Computation of R Step

Equation 3-46 can be regarded as a penalized maximum-likelihood approach of

image reconstruction in which the penalty term enforces natural constraints due to the

limitedness of material deformations, rather than forcing (possibly unnatural) prior

smoothing constraints.

Green's one-step-late (OSL) algorithm [58] was proposed for the penalized

maximum-likelihood estimation of a single frame using a general class of priors. It was

used [10] for reconstructing gated frames, and convergence was obtained for a modified

version [59]. We used a modified OSL for our R-step optimization.

Since EI and L are convex infi, f2, the Kuhn-Tucker optimality conditions are

necessary and sufficient for the minimizer. That is

Dm E(fi, f2," (k) ) > 0 (3-48)

and fmDmE( fi,f2, m(k)) = 0 for m =1,2. (3-49)









From Equations 3-26, 3-27, we define


af ha(r) g (i)Hf; (i)
Az (f,, f27M m> =1 (3-50)
ah7(r) + 2[ (r) f2( ~


afha(r)gl?(2) Hf2
A2 17, 2, _> 1=1 (3-51)
ah(r) 2[ (r)- f2 (r)] | det(I3+m'(^)

Hence the necessary and sufficient conditions for Equation 3-46 are

Am1 2mk) :, m =1,2 (3-52)

fAm(11,2,m~) mr''>=, m =1,2 (3-53)

So, it seems natural to estimate f,(ki+1) by the fixed-point iterations given by

my I 1+ = fml,t~l $1(f; 2f,17 (k)), l= 1,2,... (3-54)

and f,( k+) p ~kl)

If the image matching term EI is not present in the obj ective function in Equation

3-46, then Equation 3-54 reduces to MLEM algorithm. If E is replaced by a Gibbs prior,

this approach yields Green's OSL algorithm. However, the multiplier Asm is not

guaranteed to be nonnegative. We use the following modification of the iteration in

Eqluation 3-54 to generate nonnegative approximations / 0~, J~ ~fk) ... off kll+1)

R Step Alg~orithm

1. Initialization: f, o)_ = (k)


2. Given Jt compute the values of At (r) =' As '( f ,f fmk)FO


3. Determine constant tl > 0 satisfying









t, -mamaxax-~-l-A' zr) ,0):rf e ,, j=; 1,2)< 1 (3-55)

4. he ff',= f '+ tz (A' -) is the update of ~f l.

5. Go back to 2.

To apply this algorithm we need to compute the variable r = (I, + m) (r) for

each re E which occurs in the quantities A','t(r) f u.. Hwee, the mathemaical form


of the transformation m(k)(r) is completely unknown. Only numerical values of

mk()are available, so we choose to use a fast, direct method of obtaining an

approximation of i' by using a linear approximation of mk(k).

By definition, r = r; + mtr) = r + m(r) + m' (r)(r r), hence,

m(r) = -(r, + m'(r))(r r), which gives the approximation as

r r-I,+m(r))m~r)(3-56)

This approximation is valid when -1 is not an eigenvalue of m'(r) .

3.3.3 Computation of M Step

M Step is to compute the motion update m(k+1) after the image frames have been

updated to ~f,(k+1) 2(k+1)1 This has been reduced to the motion-estimation problem

considered by Klein and Huesman [15]. In the iterates of M step, the update mk) of

m k) is obtained by minimizing a simpler, quadratic approximation of the original

obj ective function, using Taylor approximation


(3-57)


Hence, approximately we need only to compute









m'k+1) arg min Qk,/,m) + PEs (m)] (3-58)


where Qk,l ( k)=5 i')(r _i 2(k (k _~ic, _mr (kx)cr, (kI)c mi(k)~

(3-59)

From the Frechet derivatives of Qk,l and Es, the Euler-Lagrange equations for

m = mI are

2 i(k) 2i(k) (k) _I--I (k) )7 2 (k)( (k 2I ~ nd(k)(k
Dx (3-60)
+ P(t/+ 2pU)ux + iuO,.+> uzz (;3 xyU),' xz,) = 0

2 (k,() 2() ( k m ") _m (k) 2(k (k) (k):(k
Sy (3-61)
+ P + 2pl)v,, + pl(v, zzv)+(i + pl)(uxy ~ y)] = 0


2 () 2() k) (k 2(k) () ] 2(k) (k))
8z (3-62)
+ P(2 + 29UM~ zz yyU' xz yz~+~ + ~(* ),r) = 0

Using finite difference approximations for the derivatives, these partial differential

equations (PDE) can be changed to an algebraic linear system for the motion components

at all voxels, that is, (u,, v,, w,, j= 1,2,... q). The algebraic linear equations are solved

using the regular conjugate gradient algorithm. The estimates m k) (k) (k)

then iteratively obtained by solving Equations 3-60, 3-61, and 3-62 with the initialization

m k) (k) Appendix gives details of the numerical analysis for motion estimation.















CHAPTER 4
IMPLEMENTATION AND EXPERIMENTAL RESULTS: GEOMETRIC PHANTOM

4.1 Simulated Source Object and Projection Data

Since it is difficult to obtain a standard data set that presents the "truth" volume,

using real medical imaging data to test an algorithm is usually not applicable. To

demonstrate some of the basic characteristics of the RM algorithm, this chapter

implements the algorithm with a simulated 3D geometric phantom. Based on a hollow

semi-ellipsoid model, the source obj ect of this phantom is designed to simulate the

contracting left ventricle in a gated myocardial ET study. The inner and outer boundaries

of the myocardium are defined by 2 concentric semi-ellipsoid surfaces with dimensions

of short and long axes chosen to mimic the shape and size of a regular left ventricle.

Three orthogonal views of the source obj ect are shown in Figure 4-1.

4.1.1 Steps to Generate Geometric Phantom

The simulated geometric phantom is represented as a 30 X 30 X 30 voxel density

volume. In order to create a more realistic source distribution in which obj ect boundaries

have some degree of smoothness, we generated the 30 X 30 X 30 source obj ect from a

high-resolution (120 X 120 X 120) obj ect with the steps illustrated in Figure 4-2.

Two simulated data files are generated for each frame: a noise-added proj section

data file and a true source obj ect image file. The proj section data file will be used to

reconstruct the image by the RM algorithm and the PS-MLEM algorithm, and the true

image will be used to evaluate the reconstructed images.




















A














Figure 4-1. Source objects of the geometric phantom. A) Frame 1. B) Frame 2. For both
frames, the 3 orthogonal planes are in 2 long axis slices (two left columns)
and a short axis slice (right column).

4.1.2 Dimensions of Geometric Phantom

We generated 2 frames of the gated ET image (Frame 1 and Frame 2) representing

the end-systolic and mid-systolic phases of the heart cycle, respectively. The source

obj ects of the 2 frames had different lengths of semi-axes in a manner that achieved a

constant myocardium volume across the 2 frames and an anatomically realistic

myocardial-to-chamb er volume ratio. Assuming a nominal voxel linear dimension of 3.5

mm, the myocardial volume was 120 ml, and the chamber volumes of Frame 1 and

Frame 2 were 43.3 ml and 74.4 ml, respectively. These volumes were chosen to represent

the normal human heart at these contraction phases [60]. The lengths in mm of the

ellipsoid semi-axes for the 2 frames are shown in Table 4-1. Two of the three ellipsoid

semi-axes were equal in length and represented the circular, short-axis slices.

















Nyquist cut-off


response


Figure 4-2. Simulation steps of the geometric phantom

Table 4-1. Ellipsoid semi-axes lengths for source object. (Unit: mm)
Frame 1 Frame 1 Frame 2 Frame 2
Axis Inner boundary Outer boundary Inner boundary Outer boundary
Short axis 21.0 38.5 26.2 40.6
Long axis 70.0 78.8 77.0 84.0


4.1.3 Intensity and Defect

A uniform intensity was assigned to the myocardium region between the half

ellipsoids, and zero intensity was assigned to the region in the inner chamber and external









to the myocardium. For the contrast study purpose, two myocardial defects were

simulated with 50% decreased intensity and located at basal and mid-ventricular points

along the long axis, as indicated by the arrows in Figure 4-1.

The total counts of the proj section for each frame was scaled to 99,000, which was

chosen to match that of a clinical, gated myocardial perfusion study using Tc-99m

sestamibi (70 y.o. male, 10 mCi injected activity).

4.1.4 Wall Motion Simulation

To mimic more realistic movement of heart, a "wringing" motion was simulated in

addition to the contraction. This motion was found in contrast ventriculography [61] and

tagged MR images of the human heart [28]. We characterized this wringing motion by a

rotation of the basal and apical myocardium about the heart long axis in opposite

directions. In our study, the extreme basal and apical points on the heart long axis rotated

5 degrees in opposite directions, and points between rotated according to a linear gradient

connecting these extremes.

We calculated the motion geometrically and referred to it as the "ideal motion". To

illustrate the wringing motion visually, the ideal motion fields in 3 short-axis slices (#5,

#15 and #20) are shown in Figure 4-3.

4.1.5 Forward Projection Simulation and Detector Response

Collimator-detector response was simulated assuming a low-energy high-resolution

parallel-hole collimator. The proj section data were filtered using a Gaussian kernel with

FWHM=1.9 pixels. With the nominal 3.5 mm pixel size, this kernel has an equivalent

FWHM equal to 6.7 mm. Forward proj section was performed at 60 angles over 180

degrees. Effects of attenuation, scatter, and randoms were not considered.





















Figure 4-3. Ideal motion fields of the geometric phantom in 3 short-axis slices.
A) Slice #5. B) Slice #15. C) Slice #20.

The high-resolution proj section data were then collapsed to a low-resolution

proj section (30 X 60 X 30) by a procedure described in Figure 4-4 (only a 2D slice is

shown). While the final projection dimensions (30 X 60 X 30) are small relative to those

used by large field-of-view imaging systems, the sampling rate used here, relative to the

size of the heart, is comparable to these systems. For example, the thickness of the

myocardium in humans at end-systole is typically close to 14 mm; in our heart model this

thickness is approximately 4 pixels, which represents a nominal pixel size of 3.5 mm.

Finally, Poisson noise was added to this scaled, collapsed proj section data.

4.2 Bi-Value LamC Constants Model

The compressibility degree of the myocardium, the blood pool in the chamber, and

other outside adj acent tissue differs from each other. Thus the Lame constants 32 and pu

should have different values for these regions. In this chapter, due to the assumption of

constant myocardium volumes across the 2 frames of the geometric phantom, and the fact

that the region of the blood pool is not conserved during contraction, we propose a

bi-value Lame constants model: a fairly incompressible model for myocardium and a

compressible model for the other regions.














Bin


1


1
2
1


.
Bm
I
I


1le 60

Projectionn Angle


High-resolution 120 X 60 Proj section



Figure 4-4. Collapse high-resolution proj section to low-resolution proj section
In practice, there are two more often used constants presenting the compressibility

of material: E, called Young' s elasticity modulus and v called Poisson ratio, which are

related to Lame constants as

pu(3il+ 2p) ii
E=~ and v=~ (4-1)
(a + p)> 2(il + pu)

Young' s elasticity modulus E relates the tension of the object to its stretch in the same
direction. Poisson ratio v is the ratio between lateral contraction and axial extension

[62]. The ii term in the constraint equation penalizes non-zero divergence and the pu

term penalizes sharp discontinuities in the flow field [32].

For highly incompressible material, the Poisson ratio approaches 0.5, which yields

a 3 approaching infinity. At the other extreme situation of highly compressible material, 3A










and v approach 0. Table 4-2 gives the Lame constants we used in our study for some

specific Poisson ratios.

Table 4-2. Poisson ratio and Lame constants
Poisson ratio (v) 0 0.1 0.2 0.3 0.4 ~0.5
Lame constants 3A=0; 3A=0.25; 3A=0.67; 3A=1.5; 3A=4; 3A=100;
(n, p)> iu= ; pu= ; pu= ; pu= ; pu= ; u= 1;

To implement the bi-value Lame constants model, a preprocessing segmentation of

the left ventricle from the background is required. Since the accurate automatic

segmentation of the myocardium boundaries may be a formidable task itself, in our study

we used a simple automatic threshold-based procedure. Firstly, we reconstructed the

image from the noisy data using the PS-MLEM algorithm with Nyquist cut-off frequency.

Then we compared each voxel intensity of the resulting image to a pre-chosen threshold

and generated a segmentation bitmap. A feasible threshold was reassured by examining

the segmentation bitmap visually.

Because the segmentation is only to model elastic characteristics of different tissue

types, the RM algorithm is expected not to be sensitive to slight errors of a few voxels

along the myocardium boundaries. This insensitivity has also been demonstrated by the

insignificant difference of the images reconstructed from using various thresholds for

segmentation.

The segmentation procedure and results are illustrated in Figure 4-5. In our study,

the threshold was chosen to be 1/8 of the mean intensity of myocardium voxels. The

segmentation bitmaps generated from the images with and without defect were similar to

each other.























A

















Figure 4-5. Segmentation for bi-value Lame constants model. A) With defect. B) Without
defect. (Note: Left column are the images to be segmented with threshold.
Right column are resulting segmentation bitmap where white region
represents the myocardium and black region represents the background.)

4.3 Convergence Properties of RM Algorithm

Since the RM algorithm is to minimize the obj ective function iteratively, its

convergence property can be illustrated by the plot of the obj ective function value versus

iteration number. Similarly, the convergence of the obj ective functions considered in the

individual R and M steps illustrate the individual convergence properties.

The convergence results for the RM algorithm are shown in Figure 4-6. The total

obj ective function E and the negative log-likelihood function L decrease with iteration










and are approximately constant after 20 iterations. Due to the initial zero motion estimate

and initial uniform image estimates, the EI and Es functions present low initial value and

increase with iteration before flattening out. Figure 4-7 shows the convergence of the

individual R step. R step using either the true or the estimated motion largely converges

by 25 iterations. The M step also has converged by 25 iterations as indicated in Figure

4-8. The obj ective functional value in M step achieved with the PS-MLEM images is

larger than that with true images. This reflects the inconsistencies between the estimated

frames by PS-MLEM due to the statistical noise.

















4.4.1~~ Fiue oft Meri
Th moio esiato iseautdb oprn h siae oint da
moin lbal adreinll.Bae o heiag achn fntin w efn

glba moinerrta esrsteareeto w reiaefae ne h
esiae mto.It rerset th vrl cuayo temto siain








40




2.4 x 105

--2.5

--2.6
Using ideal emotion
-2.7- U~sing: estimated motion

--2.8








-32 10 20 30 40 50r
Iteration number



Figure 4-7. Convergence of R step


11000
S- Using ideal images
10000 ."- Using PS MlLEM images

9000 -

8000

7000


5000

4(000


3000
0 10 20 30 40 50
Iteration Number



Figure 4-8. Convergence of M step



[,(r)f2F IT))2(4-2)



where f, and f2 are the true images, and in is the estimated motion vector field.


Regional motion evaluation is performed at some selected sub-regions. We define


magnitude error as the absolute value of the magnitude difference between the estimated


motion and the ideal motion, and angle error as the angle between the two vectors.










4.4.2 LamC Constants Study Results

As we discussed in 4.2, motion estimation involves the assignment of Lame

constants for myocardium and background regions. To investigate how the Lame

constants influences motion estimation, we performed M step using true images with

various Poisson ratio v (equivalent to various Lame constants, see Table 4-2) and

hyper-parameter B. We used six values (0, 0.1i, 0.2, 0.3, 0.4, and 0.5) of v for myocardium,

and v =0 for the background. The results are shown in Figure 4-9. The global motion

error using v = 0.5 for myocardium was much more sensitive to a than those using other

v 's. The minimum global motion error was achieved with v = 0.4 and B = 0.02.

Our results agree with Klein's claim [32], that is, for the fairly incompressible

deformation, the motion-estimation algorithm performs best with a chosen Poisson ratio

value near 0.4, instead of the value of 0.5 for the ideally incompressible material. Thus in

our following geometric phantom study, we kept the Poisson ratio at 0.4 for the

myocardium and 0 for the background.

80


7 0


60 v=0.4
O '


30


20* *
O 0.02 0.04 0.06 0.OB 0.1
Hyper-pararneter in M step: P

Figure 4-9. Global motion error vs. B for several Poisson ratios.









4.4.3 Global Motion Evaluation

To compare the motion-estimation performance of RM and the individual

motion-estimation algorithm, we implemented the following 3 methods:

Method 1: M step using true images with change ofa.

Method 2: Individual method. M step using PS-MLEM images with change of
cut-off frequency and B;

Method 3: RM algorithm with change of hyper-parameter a and B.

Figure 4-10 shows the ideal motion superimposed on the true Frame 1 images of

the geometric phantom, in both long- and short- axis views. Figure 4-11 shows the

motion estimated by Method 1, superimposed on the true images. Figure 4-12 shows the

images and motion individually estimated by Method 2. Figure 4-13 shows the images

and motion simultaneously estimated by RM.

The minimum global motion error each method achieved is given in Table 4-3. The

optimum conditions to produce the minimum errors for each method are also given. The

numbers in parentheses are the standard deviations of the global motion error calculated

across a 20-image ensemble. Method 1 and 2 used 40 iterations for M step, and Method 3

used 40 outer-loop iterations, each containing 1 M-step iteration and 1 R-step iteration.


Figure 4-10. Ideal motion vector fields superimposed on true images.
























Figure 4-11. Estimated motion by M step using true images. (B = 0.02).
















Figure 4-12. Individually reconstructed images by PS-MLEM and estimated motion by M
step. (cut-off = 0.3 cycle/pixel; B = 0.01).


Figure 4-13. Estimated motion and reconstructed images by RM. (a=0.02; J


0.01).










Table 4-3. Global motion evaluation results of the geometric phantom
Method Optimum Conditions Minimum global motion error
1 B = 0.02. 21.4
2 cut-off = 0.3 cycle/pixel; 46.4(1.2)
B = 0.01.
3 a=0.02; 37.2(1.4)
B = 0.01.

The results in Table 4-3 show the smallest motion error was achieved by M step

using the true images, as we expected. Of the methods those were operating on noisy data,

the RM algorithm outperformed the individual method in terms of the global motion error.

This result reflects the increased noise control of the RM algorithm.

4.4.4 Regional Motion Evaluation

Three sub-regions within the myocardium of Frame 1 of the geometric phantom

were selected for the regional motion evaluation: (1) basal and in normal region, (2) basal

and in defect region, (3) mid-ventricle and in normal region. Each sub-region was a

2 X 2 X 2 voxel volume. Their locations are indicated by the squares in a slice view in

Figure 4-14. To obtain the magnitude and angle errors of motion estimate, the mean

magnitude and angle of the estimated motion in the sub-regions were computed and

compared with the mean magnitude and angle of the ideal motion in those sub-regions

accordingly. These errors were computed for a 20-image ensemble, and the mean and

standard deviation of the errors across the ensemble were obtained. The regional motion

evaluation was performed for both the RM algorithm and the individual method.

The results of the regional motion evaluation are shown in Table 4-4. The mean

magnitude errors are in units of pixels, and mean angle errors are in units of degrees.

Standard deviations are given in parentheses. Overall the table indicates the improved

motion-estimation accuracy for RM compared with the individual method.
























Figure 4-14. Chosen region to calculate regional motion error for the geometric phantom.

Table 4-4. Regional motion evaluation results of the geometric phantom
Individual method RM
Sub-region Ideal mag. Mag. error Angle error Mag. error Angle error
1 1.52 -0.18(0.17) 31.7(14.1) -0.18(0.12) 28.7(11.5)
2 1.52 -0.49(0.28) 30.1(14.8) -0.45(0.22) 25.1(11.1)
3 0.66 0.24(0.37) 61.0(16.8) 0.02(0.22) 43.3(17.3)


4.5 Image Reconstruction Results

4.5.1 Figures of Merit

The RM images were evaluated both globally and regionally by comparing to the

PS-MLEM images. The evaluation methods include global image evaluation, regional

image evaluation, and defect contrast evaluation. Since the cut-off frequency of the

post-filtering used in the PS-MLEM algorithm can greatly affect the quality of the

resulting images, a range of cut-off frequencies was investigated.

The figure-of-merit for the global image evaluation was the Root Mean Squared

(RMS) image error defined as


RMS = fr-fr43


where f (r) is the true image, f (r) is the reconstructed image, and Nis the total










number of voxels in the image. To get an overall image evaluation of the 2-frame system,

the mean RMS error was computed across the 2 frames.

The regional image evaluation also considered the 3 sub-regions used for the

regional motion evaluation. For each sub-region, the mean bias and variance of the

intensity of the estimated image were computed across a 20-image ensemble. For the

PS-MLEM algorithm, a set of bias/variance points was computed across a range of

cut-off frequencies from 0. 1 to 1.0 cycle/pixel. The bias/variance curves were plotted for

each sub-region to show the trade-off between the absolute value of the bias and the

variance, as the cut-off frequency changed.

Defect contrast is defined as

I -I
Contrast = normal defect (4 -4)
normal

where Inormal and Idefect are mean image intensities within the normal and defect regions,

respectively. The normal and defect regions were defined from the known source obj ect

(true image) and included the entire region volume except for boundary voxels. Contrast

was measured for each image out of the 20-image ensemble, and then the mean contrast

was computed. Like the regional image evaluation, a range of cut-off frequencies was

considered for the PS-MLEM algorithm.

4.5.2 Global Image Evaluation

The results of the global image evaluation for RM and PS-MLEM are given in

Figure 4-15. The RMS error of the RM algorithm decreases with iteration until

approximately iteration 20, then increases very slightly. This demonstrates the stability of

the RM images with iteration, unlike MLEM. The PS-MLEM algorithm shows a

minimum RMS error with a cut-off frequency of approximately 0.4 cycles/pixel. This







47


minimum, however, is still substantially larger than the stable RMS error of the RM

algorithm (0.045 compared with 0.03).

The hyper-parameter effects on the RM images are illustrated in Figure 4-16. The

figure shows the image of Frame 2 as a and a are varied. As a is reduced, there is less

reliance on the likelihood function (i.e., the measured proj section data), which causes the

image of the myocardium to broaden and approach a more uniform intensity image. As a

increases, the algorithm becomes more like MLEM, and a moderate increase in noise is

evident. The sensitivity of the RM image to change in B is much less.

Further illustration of the hyper-parameter effect is shown in Figure 4-17. These

surface plots show the change in the global motion and image errors as a 2D function of a

and B. The minimum RMS error is located at a=0.02 and 8=0.04. When a was smaller

than 0.02, the RMS error increased dramatically and became irregularly sensitive to B,

which corresponded the image shown in Figure 4-16 (B).










It r t o n m e r r R 1 m r o o s A a t n s .I . .1 o l

Figre4-1. MS rrrsof heRM ndPS-LE iage. ) R. ) P-MEM







48















A B C












D


Figure 4-16. Hyper-parameter effects on the RM images. A) a = 0.03, B = 0.01. B)B
unchanged, smaller a (0.015). C)B unchanged, larger a (0.06). D) a
unchanged, smaller B (0.002). E). a unchanged, larger B (0.1i).





0.07-

0.065-

~006-

S0.055-

2 005-
.0l 0.045

3 0 04-

0.035,
0.1
0.080.


Figure 4-17. RMS error of the RM images as a function of a and B










4.5.3 Defect Contrast Evaluation

The results of the contrast measurements are shown in Figure 4-18. With the

PS-MLEM algorithm for both defects, the defect contrast increases as the cut-off

frequency increases and, at the highest cut-off frequency, is slightly less than that of the

RM image. Although at this cut-off frequency the 2 methods have similar defect contrast,

the quality of the images is substantially different, as shown in Figure 4-19.






S0.36 RM: 0.41 0.2t RM: 0.29



0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2
PS-MLEM cut-off frequency (cyclelpixel) A PS-MLEM cut-off frecluency (cyclelpixel) B

Figure 4-18. Defect contrast of the PS-MLEM and RM images. A) Defect 1. B) Defect 2.

Figure 4-19 (A) and (B) show the images of Frame 1 for RM (40 iteration) and

PS-MLEM (1.4 cycles/pixel cut-off). It is evident that while the images have similar

contrast, the RM image has substantially lower noise level. Also included in Figure 4-19

(C) are the PS-MLEM images with Nyquist cut-off frequency. While these images have a

noise level closer that of the RM images, the defect contrast is smaller. This is

particularly evident in the short axis images in the right column.

4.5.4 Regional Image Evaluation

The bias and variance of the voxel intensity within the 3 sub-regions (indicated by

the squares in Figure 4-20) were calculated in both 2 frames. Figure 4-21 shows the plots

of the absolute bias versus variance for the PS-MLEM and RM images. In each plot, the

RM bias/variance point is downward and left to the PS-MLEM bias/variance curve,

indicating that RM produced smaller bias for equal variance, or smaller variance for










equal bias. This demonstrates that the RM algorithm achieved better regional image

estimation, relative to PS-MLEM.

In summary, in this chapter we implemented the RM algorithm using a simulated

geometric phantom that mimics the human left ventricle. We evaluated the motion

accuracy by comparing the motion estimated by 3 methods with the ideal motion. The

motion estimated by RM using noisy proj section data was superior to that estimated

individually by applying the motion estimation method to PS-MLEM reconstructed

images. Also we evaluated the image reconstruction by comparing the RM images to the

PS-MLEM images in terms of the global image error, the regional intensity bias/variance,

and the defect contrast. All the results showed that the RM algorithm produces better

image quality. In Chapter 6 we will assess the image quality by observer study.













































Figure 4-19. The trade-off between the contrast and noise level of the geometric phantom.
A) RM images. B) PS-MLEM images at similar defect contrast but higher
noise level. C) PS-MLEM images at similar noise level but lower contrast.












Figure 4-20. Sub-regions for regional image evaluation for the geometric phantom.
A) Frame 1. B) Frame 2.












SPS-MLEM
\+ RM











O 0.05 0.1
Variance


0 0.05 0.1
Variance


+ PS-MLEM
- + RM


0.05~


U.055


0.05


CO 0 045




0.035


0.03


n ncE


SPS-MLEM
+ RM


m 0.08

." 0.06

n1 0.04

0.02






0.12

0.1

.m 0.OB

S0.06

.0 0.04

0.02


0.05
Variance


0.1
D


Variance


+ PS-MLEM
+ RM


S0.03


S0.02


0.01


U


O 0.05 0.1
Variance


0.15


0.05 0.1
Variance


Figure 4-21. The trade-off between regional intensity bias and variance of the geometric
phantom. A) Sub-region in basal normal region of Frame 1. B) Sub-region in
basal normal region of Frame 2. C) Sub-region in basal defect region of Frame
1. D) Sub-region in basal defect region of Frame 2. E) Sub-region in
mid-ventricle normal region of Frame 1. F) Sub-region in mid-ventricle
normal region of Frame 2.














CHAPTER 5
IMPLEMENTATION AND EXPERIMENTAL RESULTS: NCAT PHANTOM

5.1 Simulated NCAT Phantom

Besides the geometric phantom, we also implemented the RM algorithm on a more

realistic, complex physical phantom: 4D N~URBS-Based Cardiac Torso (NCAT) cardiac

phantom. The NCAT phantom simulation software was developed at the Johns Hopkins

University and fully described by Segars et al. [63, 64]. It uses Non-Uniform Rational

B-Splines (NURB S) method to define the geometric conditions of a human heart and

produce the source obj ect of heart at a series of phrases over the cardiac cycle. The

NCAT software also generates the myocardial motion vector field for the gated image

frames from the tagged MRI data of normal human patients.

We used the NCAT software to generate 8 gated image frames of the cardiac

phantom, each represented as a 32 X 32 X 19 volume with a voxel linear dimension of 4.0

mm. The volume of the left myocardium was approximately 85 ml. We selected the 2

frames with maximal deformation to implement the RM algorithm: end-diastole and

end-systole, which we referred to as Frame 1 and Frame 2, respectively. A defect region

with 50% decreased voxel intensity was initially imposed on one side of the left ventricle

in Frame 1. By deforming the defect in Frame I with the ideal motion field, we could

impose the defect region onto Frame 2. The volume of the defect was approximately 5 ml.

Similarly to the geometric phantom simulation, we scaled the total counts of each frame

to 99,000 and added Poisson noise. The detector response was simulated with a FWHM

of 7.5 mm. Figure 5-1 shows the long-axis and short-axis slice views of these 2 frames.


























T B







Figure 5-1. Long- and short- axis views of the NCAT cardiac phantom. A) Frame 1:
end-diastole. B) Frame 2: end-systole.

5.2 Motion Estimation Results

5.2.1 Global Motion Evaluation

As indicated in Klein' s study of the physical phantom [32], the best motion

estimation is achieved with a Poisson ratio near 0.45 for the myocardium. Thus in this

chapter we use 0.45 for the Poisson ratio of myocardium and the Lame constants are

ii = 9, pu = 1 for voxels within the myocardium;

Ai = 0, pu = 1 for voxels of the background.

Similarly to the geometric phantom study, motion estimation for the NCAT

phantom was also evaluated in terms of the global motion error and the regional motion

error. We also compared the global motion error of the following 3 methods:

*Method 1: M step using true images with change ofa.









Method 2: Individual method. M step using PS-MLEM images with change of
cut-off frequency and B;

Method 3: RM algorithm with change of hyper-parameter a and B.

Figure 5-2 shows the ideal motion superimposed on the true frame 1 images of the

NCAT phantom, in both long- and short- axis views. Figure 5-3 shows estimated motion

by M step using true images, superimposed on the true images. Figure 5-4 shows the

images and motion individually estimated by Method 2. Figure 5-5 shows the images and

motion simultaneously estimated by RM.


Figure 5-2. Ideal motion vector fields superimposed on true images


Figure 5-3. Estimated motion by M step using true images. (B = 0.10).

























Figure 5-4. Individually reconstructed images by PS-MLEM and estimated motion by M
step. (cut-off = 0.4 cycle/pixel; B = 0.05).
















Figure 5-5. Estimated motion and reconstructed images by RM. (a=0.10; B = 0.05).

The global motion evaluation results are shown in Figure 5-6. The minimum global

motion errors of the 3 methods are shown in Table 5-1. The numbers in the parentheses

are the standard deviations of global motion error calculated from a 20-image ensemble.

Each method implemented 40 iterations.

Table 5-1. Global motion evaluation results of the NCAT phantom
Method Optimum conditions Minimum global motion error
1 B = 0.10. 267.7
2 cut-off = 0.4 cycle/pixel; 304.1 (11.3)
B = 0.05.
3 a=0.10; 288.6 (5.8)
B = 0.05.







57



340

330

S320
O 310

S300

S290

280


O 02 O 04 0.06 O0 OBO1 0 12 0.14 O 16
P A




S500
700



S500
r~~ : 300

a 03 200

0.12 1 1

P 0.04 0.4 OO0
0.02 0.2 cut-off
B C

Figure 5-6. Global motion errors of three methods. A) global motion error vs. a for
Method 1. B) global motion error vs. cut-off frequency and B for Method 2. C)
global motion error vs. (a, B) for Method 3.

5.2.2 Regional Motion Evaluation

A 2 X 2 sub-region within the defect region in slice 7 (indicated by the square in


Figure 5-7) was chosen for the regional motion evaluation. The results are shown in

Table 5-2. The mean magnitude errors are in units of pixels, and mean angle errors are in

units of degrees. The numbers in the parentheses are standard deviations of the errors.

Overall the table indicates the improved motion-estimation accuracy for the RM

algorithm compared with the individual method.






















Figure 5-7. Chosen region to calculate regional motion error for the NCAT phantom

Table 5-2. Regional motion evaluation results of the NCAT phantom
Individual method RM
Ideal mag. Mag. error Angle error Mag. error Angle error
1.56 -0.84 (0.28) 20.2 (8.5) -0.52 (0.21) 18.6 (5.6)


5.3 Image Reconstruction Results

To evaluate the image-reconstruction performance of the RM algorithm for the

NCAT phantom study, we also compare the global image error (RMS), the regional

image error (bias/variance), and the defect contrast of the RM images to those of the

PS-MLEM images over a range of cut-off frequencies.

5.3.1 Global Image Evaluation

For PS-MLEM algorithm, the mean RMS error of 2 frames changed with the

cut-off frequency, as indicated in Figure 5-8 (A). The minimum RMS error (0.284) was

achieved at approximately 0.6 cycles/pixel cut-off. For RM algorithm, a surface

representing the change of the mean RMS error with the hyper-parameters (a, a) is shown

in Figure 5-8 (B). The minimum RMS error of RM (0.268) was achieved when a=0.13

and 8=0.07.

Figure 5-9 shows the mean RMS error of the 2 frames versus the iteration number

of RM outer loop. The mean RMS error decreased with iteration until approximately 40,







59


then stayed flat up to iteration 60. This demonstrates again the stability of the RM image

with the outer-loop iteration.




0.3




0.26.
2 0.3 .1

0. e 01 .0.
O 0. .015 .




~~f0. 0. .2








0.2 *
0 ~ ~ 20 400 6050
Itertio nurnberc of RMM oute loo


Figure 5-9. RMS errorvs.o ithe rato anume ofRM outger. loop. rrr s ct-

532DfrqetCntras EvaPS-luation err s (,B)fr

Tocmpt tedfetcotatw ne irtdeemnete oml n dfc


regions.~ Th eetrgosof2fae eebthcoe uigte iuaino h


phato. W gneate te ora reinb oprngtevxlitnst ftetu










images to a selected threshold. Figure 5-10 shows the normal and defect regions for 2

frames. White pixels present normal and gray pixels present defect. Most voxels in left

myocardium were included into the normal or defect region.














Figure 5-10. Normal and defect regions in the NCAT phantom. A) Frame 1. B) Frame2.

As indicated in Figure 5-11i, PS-MLEM using a higher cut-off frequency produced

a higher contrast. The highest contrast of PS-MLEM images (0.53) was achieved with the

cut-off of 1.4 cycle/pixel. RM images had a lower contrast (0.5) compared with the

highest contrast achieved by PS-MLEM. However, RM images presented better trade-off

between the contrast and noise than PS-MLEM. That is, RM had lower noise level while

their contrasts were similar, and higher contrast while their noise levels were similar.

Table 5-3 illustrates this trade-off between the contrast and noise of RM and PS-MLEM.

We used the standard deviation of the voxel intensity in the normal myocardium region to

represent the noise level. For RM, this standard deviation was 0.15. The PS-MLEM

image had comparable noise level at the cut-off of 0.4 cycle/pixel, but the contrast was

substantially reduced to 0.43. At a higher cut-off of 0.7 cycle/pixel in which the

contrast was comparable, the standard deviation was substantially higher (0.25). The

images of the 3 cases are shown in Figure 5-12.




















0.5 1
cut-off frequency


Figure 5-11i. Contrast vs. cut-off frequency of PS-MLEM


Figure 5-12. The trade-off between the contrast and noise level of the NCAT phantom.
A) RM images. B) PS-MLEM images with similar contrasts but higher noise
levels. C) PS-MLEM images with similar noise levels but lower contrasts.










Table 5-3. The contrast and noise level of the NCAT phantom
Method Contrast Std. Dev.
RM 0.50 0.15
PS-MLEM 0.50 0.25
(cut-off=0.7 cycle/pixel)
PS-MLEM 0.43 0.15
(cut-off=0.4 cycle/pixel)

5.3.3 Regional Image Evaluation

Figure 5-13 shows the 2 sub-regions of interest in Frame 1 and Frame 2: one in the

normal region and the other in the defect region. Figure 5-14 shows the trade-off between

intensity bias and variance at these sub-regions in Frame 1 and Frame 2. We investigated

5 sets of hyper-parameters (see Table 5-4) for the RM algorithm. Set #1 (a=0. 13; 8=0.07)

is the one that produces minimum RMS error. Set #2 increases a and keeps B unchanged,

to investigate the influence of a larger value of a. Set #3, #4, #5 are to investigate the

influence of a smaller a, a smaller B, and a larger B, respectively.

These 5 sets of hyper-parameters presented different bias/variance property, as

indicated in Figure 5-14. For the sub-region in the defect of Frame 1, all of the Hyve RM

images presented close or slightly worse trade-off between intensity bias and variance

than PS-MLEM. For the sub-region in the defect of Frame 2 or the normal of Frame 1

and 2, all of the Hyve RM images presented better trade-off between intensity bias and

variance than PS-MLEM. The one out of these 5 sets that achieved the best bias/variance

trade-off was the set #5, which grouped a medium a (0. 13) and a larger B(0. 15). It seems

that the regional bias/variance property was benefited by the larger a which imposed

more material property constraints and produced smaller and smoother motion Hield.

Table 5-4. Five sets of hyper-parameters for RM used for regional image evaluation
Set #1 Set #2 Set #3 Set #4 Set #5
a 0.13 0.25 0.03 0.13 0.13
fl 0.07 0.07 0.07 0.01 0.15

































PS-MLEM
\+ RM: set 1
+ RM: set 2
RM: set 3
\x RM: set 4
RM: set 5

+





63




















Figure 5-13. Sub-regions for regional image evaluation for the NCAT phantom.
A) Frame 1. B) Frame 2.


\ PS-MLEM
\+ RM: set 1
\+ RM: set 2
\RM: set 3
x RM: set 4
i~RM: set 5


ii+

0.05 0.1 0.15
Variance


0.3


0.2


0.05 0.1 0.16
Variance


0.2

B


1.5






-r
0

O o~


O 0.1 0.2 0.3 0.4
Variance


0.05 0.1 0.15 0.2
Variance


Figure 5-14. The trade-off between regional intensity bias and variance of the NCAT
phantom. A) Sub-region in the defect region of Frame 1. B) Sub-region in the
defect region of Frame 2. C) Sub-region in the normal region of Frame 1. D)
Sub-region in the normal region of Frame 2.














CHAPTER 6
OBSERVER EVALUATION OF IMAGE QUALITY

6.1 Methods

6.1.1 Single-Slice CHO Model

Our study of observer evaluation of image quality for the RM algorithm started

with a single-slice CHO (SCHO) model. A number of channel models have been

presented since Myers et al. [45] added a channel mechanism to the ideal observer.

Previous research [45, 49] revealed that the performance of the CHO model was rather

insensitive to the details of the channel model, such as the band-pass cut-off frequencies

and the geometric properties of the channels. Since Radially Symmetric Channel (RSC)

model demonstrated excellent performance to predict human performance [47, 49, 56],

we used this relatively simple but efficient channel model in our study.

Our RSC model consisted of rotationally symmetric, non-overlapping channels

with cut-off frequencies of (0.03 125, 0.0625), (0.0625, 0. 125), (0. 125, 0.25), (0.25, 0.5)

cycles/pixel. Choice of cut-off frequencies was somewhat arbitrary. Figure 6-1 shows the

RSC-model channels in the frequency domain.

The channelization procedure was described in Equation 2-19 as v = Uf To

compute the vector v from the image ffor a location-known defect, we Fourier

transformed each frequency-domain channel to a spatial-domain template centered at the

defect location. The ith COmponent of the vector v was then calculated by integration of

the pixel-by-pixel product of the image and the ith spatial-domain template. This is to

avoid excessive Fourier transformation for the large number of images.


















Figure 6-1. The RSC-model channels used for channelization in single-slice CHO model.
(Note: Images are shown in the frequency domain.)

Figure 6-2 shows the 2 defects in the geometric phantom and the defect in the NCAT

phantom. Figure 6-3 shows the spatial-domain templates of the 4 RSC-model channels

used in our study. Three rows indicate the different defect locations in the two phantoms.













A B

Figure 6-2. Defect locations in the phantoms. A) Geometric phantom. B) NCAT
phantom .

Before implementing CHO to compute the test statistic ACHO We need to create a

training set of obj ects in two states (SP and SA), and estimate the scatter matrix S2 and

the mean vectors v, and v The size of the training set is usually determined by the

channel number. As previous studies [54, 55] presented, for an N-channel model, the size

of the training set for each state (SP or SA) should be larger than N2. Otherwise the

inverse of S2 Will HOt exist. In our study, we used 4-channel RSC model, so the size of the

training set should at least be 16. We used 40 to ensure the existence of S2.









To compare RM with PS-MLEM, we generated 280 images (140 SP and 140 SA)

for each algorithm. We used 80 out of them (40 SP and 40 SA) as the training set. After

generated the scatter matrix S2 and the mean vector v, and va, we applied the CHO

discriminant vector on the other 200 images (100 SP and 100 SA) to compute the SCHO

detectability index (SCHO-d,) To acquire the standard deviation of the detectability

index, we divided the 280 images arbitrarily to 7 groups, each containing 40 images (20

SP and 20 SA). Each time we selected 2 groups of images to be used as the training set,

then applied the resulting CHO discriminant vector to the remaining 5 groups to estimate

an SCHO-d,. By randomly selecting 2 out of the 7 groups as the training set, we

estimated 21 different values of SCHO-,' s. Thus we are able to compute the mean and

standard deviation of SCHO-d,.

6.1.2 Multi-Frame Multi-Slice CHO-HO Model

Since a defect in a 3D obj ect may persist from slice to slice, in practice the

physicians generally make their diagnostic decisions by observing multiple slices of

images, and even multiple frames in the case of gated emission tomography imaging.

Chen et al. [65] presented a multi-slice multi-view CHO-HO model for ungated SPECT

myocardial perfusion imaging. We modified it to a multi-frame multi-slice CHO-HO

(MMCHO) model for gated myocardial perfusion imaging.

The MMCHO model involves 2 steps. In the first CHO step we apply a CHO

model to each slice of all frames to assess the probability that the defect is present in that

slice. The result of the first step for each multi-frame multi-slice image is a vector of test

statistics of which the element is an individual test statistic ASHO, Where i=1, 2 ,..., I (I is

the number of slices) is the slice index and j=1, 2, ..., J (J is the number of frames) is the









frame index. In the second HO step, we apply an HO model to the ensemble of vectors of

A O(SP and SA) to compute a final MMCHO detectability index (MMCHO-d,).

Figure 6-4 shows the scheme for this 2-step procedure.

























Figure 6-3. The RSC-model channels represented as spatial-domain templates. The
channels are Fourier transformed and shifted to the location centers of the
defects. A) Channels for Defect 1 in the geometric phantom. B) Channels for
Defect 2 in the geometric phantom. C) Channels for the defect in the NCAT
phantom .

In our MMCHO study, we considered 2 frames, each containing 3 chosen slices.

The first CHO step applied the SCHO model to all the 3 slices of the 2 frames, and

generated vectors containing 6 test statistics, A HO (i=1, 2; j=1, 2, 3). The second HO

step used 60 test-statistic vectors as the training set to determine the HO discriminant

vector, and then applied the discriminant to the remaining test-statistic vectors to

compute the MMCHO-d,. The mean and standard deviation of MMCHO-d, were

computed by the same procedure used in the SCHO model.



















fr-ames


/SP
ensemble


SA \
ensemble


3 slices 3 slices


/ SP
ensemble


ensemble


S~A- CHO HO O SP~- CHO -HO


Multi-frame multi-slice CHO-HO detectability index

(MMCHO-der)

Figure 6-4. Two-step procedure of the multi-frame multi-slice CHO-HO model.


CHO channels


Step 1: CHO


Step 2: HO









6.2 Results of Geometric Phantom Study

Since the 2 defects in the geometric phantom were both centered in slice #15 of the

long axis, we picked that slice and its 2 adj acent long axis slices, #14 and #16, to

implement the SCHO and MMCHO study. The 3 images slices of 2 frames reconstructed

by PS-MLEM and RM are shown in Figure 6-5 and Figure 6-6, respectively. In each

figure, the image slices for each frame are grouped in the order of #14, #15, and #16. The

cut-off frequency for PS-MLEM was 0.3 cycle/pixel. The RM algorithm used the

hyper-parameters (a=0.02, 8=0.04), which produced minimum RMS image error. The

PS-MLEM images present more smoothness than the RM images, particularly at the

myocardium boundaries.

6.2.1 Single-Slice CHO Results

We performed SCHO model to the 2 defects in the 3 long-axis slices of the 2

frames. For PS-MLEM, we used a range of cut-off frequency from 0. 1 to 1.0 cycle/pixel

and found the minimum SCHO-d,. The results of PS-MLEM and RM are shown in Table

6-1. Standard deviations of the SCHO-d, are given in parentheses.

The RM images had significantly improved SCHO-d, in all of the cases except for

the defect 1 in slice #15 of Frame 2. The improvement of RM for one slice was evidently

different from another, while the SCHO performance of the slices differed, too. This also

motivated us to develop a multi-frame multi-slice observer model that produces a single

detectability index for a series of gated myocardial ET images.

6.2.2 Multi-Frame Multi-Slice CHO-HO Results

We applied the MMCHO model to the 2 frames and 3 slices shown in Figure 6-5

and Figure 6-6. The results are shown in Table 6-2. The numbers in parentheses are the

































Figure 6-5. SP images of the geometric phantom reconstructed by PS-MLEM.
A) Frame 1. B) Frame 2.


Figure 6-6. SP images of the geometric phantom reconstructed by RM.
A) Frame 1. B) Frame 2.









































~---PS-MLEM


standard deviations of MMCHO-d,. The results show that the RM images presented

significantly higher MMCHO detectability than the PS-MLEM images. Figure 6-7 shows

how the MMCHO-d, of the PS-MLEM images changed with the cut-off frequency. For

both defects, the maximum detectability indices of PS-MLEM were achieved with the

cut-off of 0.4 cycle/pixel. The MMCHO-,' s of the RM images are also plotted.


Table 6-1. SCHO detectability index results of the geometric phantom


SCHO-d, for Defect 1
Frame 1 Frame 2
6.98 (0.22) 6.92 (0.21)
9.26 (0.28) 8.85 (0.42)
8.38 (0.25) 9.06 (0.23)
8.95 (0.32) 8.03 (0.16)
9.07 (0.35) 8.58 (0.36)
12.89 (0.51) 11.87 (0.23)









25


SCHO-d, for Defect 2
Frame 1 Frame 2
7.53 (0.13) 8.89 (0.24)
11.52 (0.63) 11.93 (0.46)
8.27 (0.19) 9.46 (0.28)
10.69 (0.30) 9.89 (0.23)
9.28 (0.35) 8.85 (0.35)
13.24 (0.49) 12.62 (0.41)


Slice Algorithm
14 PS-MLEM
RM
15 PS-MLEM
RM
16 PS-MLEM
RM


25
x
a,
ri 20


-i 15
n
rd
~ io
a,
0 5
I
y
L.


r-


0 0. 2 0. 4 0. 6 0. 8 1 1. 2
cut-off frequency (cycle,'pixel)


0 0.2 0.4 0.6 0.8 1 1.2
cut-off frequency (cycle,'pixel)


Figure 6-7. MMCHO detectability index results of the geometric phantom.
A) Defect 1. B) Defect 2. (Note: For the purpose of comparison, RM
result is also represented as a point with error bar in each plot disregarding the
cut-off frequency.)

Table 6-2. MMCHO detectability index results of the geometric phantom
MMCHO-d, for Defect 1 MMCHO-d, for Defect 2
PS-MLEM 14.23 (0.25) 15.71 (0.23)
RM 19.25 (1.13) 20.81 (0.91)









6.3 Results of NCAT Phantom Study

For the NCAT phantom, we selected long axis slice #7, #8 and #9. The defect is

centered in slice #7 and #8. Figure 6-8 shows the true image slices of two frames. Figure

6-9 shows the images reconstructed by the PS-MLEM algorithm using the cut-off

frequency of 0.3 cycle/pixel. Figure 6-10 shows the images reconstructed by the RM

algorithm using the hyper-parameters of (a = 0.03, a = 0.07). In each figure, the image

slices for each frame are grouped in the order of #7, #8, and #9.

6.3.1 Single-Slice CHO Results

The results of single-slice CHO performance for the PS-MLEM and RM images

are shown in Table 6-3. Again, the minimum SCHO-da for PS-MLEM was found by

using a range of cut-off frequencies from 0. 1 to 1.0 cycle/pixel.


Figure 6-8. SP true images of the NCAT phantom.
A) Frame 1. B) Frame 2.

































Figure 6-9. SP images of the NCAT phantom reconstructed by PS-MLEM.
A) Frame 1. B) Frame 2
























Figure 6-10. SP images of the NCAT phantom reconstructed by RM.
A) Frame 1. B) Frame 2









Table 6-3. SCHO detectability index results of the NCAT phantom
Slice Algorithm SCHO-d, for Frame 1 SCHO-d, for Frame 2
7 PS-MLEM 11.85(0.23) 12.81(0.64)
RM 20.35(0.80) 19.80(1.11)
8 PS-MLEM 10.71(0.30) 13.01(0.81)
RM 18.94(0.69) 18.45(0.45)
9 PS-MLEM 8.01(0.26) 9.05(0.46)
RM 8.32(0.35) 16.66(0.49)


6.3.2 Multi-Frame Multi-Slice CHO-HO Results

To investigate the algorithm performance for a lower defect contrast, we generated

another simulated NCAT phantom which had a defect with 25% decreased intensity. We

performed the MMCHO observer to both of the two phantoms (50% defect and 25%

defect). The results are all shown in Table 6-4. It is evident that for both 50% and 25%

defect phantoms, the RM images presented significantly higher MMCHO detectability

than the PS-MLEM images. Figure 6-11 shows how the MMCHO-d, of the PS-MLEM

images changed with the cut-off frequency.

Table 6-4. MMCHO detectability index results of the NCAT phantom
MMCHO-d, for 50% defect MMCHO-d, for 25% defect
PS-MLEM 16.98(0.77) 8.14 (0.36)
RM 33.41(0.66) 14.59 (0.52)

6.4 Multi-Frame Multi-Slice Multi-View CHO-HO Study

Since a 3D defect in practice may present different shape, size, and probably

different detectability in different axis views, it is meaningful and feasible to expand the

multi-frame multi-slice CHO-HO model to a multi-frame multi-slice multi-view

CHO-HO (MMMCHO) model. There are also 2 steps involved in the MMMCHO model.

In the first CHO step, the SCHO model is applied to each slice of all views and all frames.

The first step produces a test-statistic vector with elements as AfHO,, Which is from the ith










slice in the kth VieW Of the jth frame. The second HO step takes the resulting test-statistic

vectors to compute a Einal MMMCHO detectability index (MMMCHO- da).


40 16
35 ~- -PS-MLEM __-A-PS-MLEM
RM HR
3012


25 10-


10 -



0 0.2 0.4 0.6 0.8 1 20 0. 2 0. 4 0. 6 0. 8 1 1. 2
cut-off frequency (cycle/pixel) Acut-off frequency (cycle/pixel) B


Figure 6-1 1. MMCHO detectability index results of the NCAT phantom.
A) 50% defect. B) 25% defect.

In our MMMCHO study, we considered 2 frames, 2 views (a long axis and a short

axis), and 4 slices. The 16 images of a SP realization are showed in Figure 6-12, 6-13 and

6-14, presenting the PS-MLEM algorithm (cut-off = 0.4 cycle/pixel), and the RM

algorithm (a = 0.03 and 0.13), respectively. In each Eigure, the 1st row is a long axis view

of Frame 1, the 2nd TOW iS a short axis view of Frame 1, the 3rd TOW iS a, long axis view of

Frame 2 and the 4th TOW iS a short axis view of Frame 2. Because our previous studies

showed that the RM images were more sensitive to a than to a, we implemented the

algorithm using 2 different values: 0.03 and 0. 13 for a, while B was kept at 0.07.

Since the test-statistic vector produced in the first CHO step has 16 elements, the

size of the training set for each state (SP or SA) in the second HO step should be larger

than 162 = 256. In our study, we used 350 images of each state as the training set. Thus,

we generated 800 images (400 SP and 400 SA) for each algorithm. The first CHO step









used 80 images (40 SP and 40 SA) as the training set. The second HO step used 700 sets

of test-statistic vectors (3 50 SP and 3 50 SA) produced in the first step as the training set

for the HO model. The HO discriminant vector was then applied to the remaining 100

test-statistic vectors (50 SP and 50 SA) to estimate the MMMCHO detectability index

(MMMCHO-d,).

The MMMCHO-d, versus cut-off frequency of PS-MLEM is shown in Figure 6-15.

The highest MMMCHO-d, of PS-MLEM was achieved with the cut-off of 0.4 cycle/pixel.

The MMMCHO-,' s and their standard deviations of the RM images and the PS-MLEM

images are shown in Table 6-5.

The detectability of RM algorithm was affected by the value of a. A smaller value

of a produced smoother images and achieved higher defect detectability. Future studies

will investigate the MMMCHO-d, as a function of a and B.


Figure 6-12. Multi-frame multi-slice multi-view SP images reconstructed by PS-MLEM.


































Figure 6-13. Multi-frame multi-slice multi-view SP images reconstructed by RM
(a= 0.03).


Figure 6-14. Multi-frame multi-slice multi-view SP images reconstructed by RM
(a= 0.13).







78



20
-A- PS-MLEM
x 18
.~16 -r -4-RM 2











0 0. 2 0. 4 0. 6 0. 8 1 1. 2
cut-of ffrequency (cycle/pixel)


Figure 6-15. MMMCHO detectability index results of the NCAT phantom.

Table 6-5. MMMCHO detectability index results of the NCAT phantom.
MMMCHO-da
PS-MLEM 9.93 (0.78)
RM (a= 0.03) 16.55 (0.76)
RM (a= 0.13) 13.58 (0.83)

In summary, we presented a single-slice CHO model and two multi-image

CHO-HO models for observer evaluation of the gated myocardial ET images. Results of

all the 3 numerical observers demonstrated that the RM algorithm produced images with

higher detectability, relative to the PS-MLEM algorithm.














CHAPTER 7
CONCLUSIONS AND FUTURE WORK

The obj ective of this study was to implement a new method that simultaneously

reconstructed gated images and estimated cardiac motion for myocardial ET, and

evaluate the performance of this method on a simulated geometric phantom and an

NCAT cardiac phantom.

The theoretical derivation of the optimization RM algorithm dealing with the

non-rigid motion vector field and the emission images in two time frames of gated

myocardial ET has been described. The RM algorithm may be viewed as a penalized

version of Klein' s motion-estimation algorithm in which the data likelihood forms an

additional constraint. It may also be viewed as penalized likelihood reconstruction in

which image smoothing is achieved by assuming the motion is a smooth deformation

between frames, and that the intensity of the material points changes little between

frames.

The reconstructed images were compared with an independent reconstruction

method consisting of high iterations of the MLEM algorithm applied to each proj section

dataset followed by smoothing with the Hann filter, an algorithm often called PS-MLEM.

Our results showed that the RM images presented lower global image error and regional

image error, relative to PS-MLEM. For defect contrast study, the geometric phantom and

the NCAT phantom were simulated one or more defective regions with 50% decrease in

voxel intensity. Our results demonstrated that the RM images achieved substantial

improvement in the contrast-noise trade-off over the PS-MLEM images. The RM images









had a substantially higher defect contrast compared to the PS-MLEM images except for

the highest cut-off frequency PS-MLEM images. However, at this high cut-off frequency,

the PS-MLEM images had substantially higher noise than the RM images.

The RM estimated motion vector field was compared with that obtained from

applying Klein's method to the PS-MLEM images. Our method outperformed that

method both in how well the resulting deformed image matched the reference image and

in regional comparisons between the resulting motion fields and the ideal motion field.

For the observer evaluation of image quality on specific detection task, we used the

CHO model because it had shown good correlation to human observer. Based on the

fundamental single-slice CHO model, we proposed a multi-image CHO-HO model for

gated myocardial ET. Results with both the single-slice CHO model and the multi-image

CHO-HO model demonstrated that the RM images had significantly improved

detectability than the PS-MLEM images.

Our future work includes

1. Multi-frame RM algorithm: Current RM algorithm deals with two time frames due
to the restriction of huge amounts of computation. With the help of more powerful
computers, multi-frame processing will expand the current two-frame algorithm and
is expected to present even higher quality gated images.

2. Human observer study: Though numerical observer is an efficient path for observer
assessment of image quality, currently it is not a substitute of human observer.
Human observer study is still necessary for further observer assessment study.

3. Spatial resolution study of RM: The regularization of MLEM by combing the
likelihood with a penalty often results in position and image dependent spatial
resolution and bias. More study about the spatial resolution and noise property of the
RM algorithm is required.

4. More concerns such as attenuation, scatter, and randoms: Additional study is
required to evaluate how these issues affect the performance of the RM algorithm. It
plays an important role to ensure the advantage of RM in clinical application.









In conclusion, the new RM algorithm has demonstrated the potential to improve

both image quality and motion-estimation accuracy for gated myocardial ET, relative to

the independent motion-estimation and image-reconstruction algorithms. More studies

are required to investigate the performance of the RM algorithm in clinical application.















APPENDIX A
NUMERICAL ANALYSIS FOR MOTION ESTIMATION

Numerical analysis is the mathematical method that uses numerical approximations

to obtain numerical answers to the problems that do not have analytical solutions. The

foundation of differential and integral calculus is based upon limits of approximations.

The definition of the first-order derivative is a difference quotient:

du .u(x + x) u(x)
= hm (A-1)


Therefore, there are various approximations for the derivative. Each of them has its own

benefits and disadvantages. Some examples of the approximations are

du u(x + &x) u(x)
Forward difference: 4 ( A -2)
dx~ hx

du u(x) u(x &x)
Backward difference: 4 (A-3)
dx~ bx

1 1

Central difference: 22



or (A-4)
2hx

and the central difference approximation for the second-order derivative:

d2u u(x + &) 2(x)+ u(x h)
(A-5)
dx~C ( )2

The central difference requires information in front of, and behind the location

being approximated, whereas the forward and backward differences use the information









on only one side of the location being approximated. In our study, we use the central

difference because of its better accuracy.

To obtain the numerical solution of the motion-estimation problem, we rewrite the

Equations 3-60, 3-61, 3-62 as


p[(A + 2~)u_ + p~(uY + uzz) + (I C P)(xy xzW ) 2m Vf2(k)( k) 2(,k) k
Dx (A-6)
= 2(fi(k) ( f2(k) k) k) 2(k1k) ) 2(k) k)



P$(A + 2pu)vy, + pu(v, + v, ) +(Al + pu)(uxy+ yz) -. 2m1 Vf2(k) k) k)1 ''
(A-7)
= -2 fi*(k) 2(k) + ik)) Ilk)) V2(k) ( lik)) k)+II
ay


$(A+ 2~w,+ pw +w )+ ( + )(u yz m -Vf2k) (k)) 2(k) +(k))
8z (A-8)
= f(k 2() (k) k) 2(k (k)1 2(k) (k)I
8z

where k denotes the kth outer loop, I denotes the lth M-step inner loop, and m k) is

current motion estimation. The solution m of Equations A-6, A-7, A-8 is the motion

estimation for the (Il+)th M-step inner loop. Equations A-6, A-7, A-8 are linear to m and

can be written in the matrix form as A M = B .

Assuming the 3D image has dimensions ofIX JX K, the total voxel number is

11, 127 1(3N)

N=IX JXK. Then, A= a21, 227 2(3N) iS a 3NX 3Nmatrix,

a(3N)1, (3N)2, > (3N)(3N)


M = [u1, u,.,27"' N> 1 2 N>,11 1'7 W2,l' NT iS the 3NX 1 motion vector field to estimate,


and B = b,, b2,..., bN b ,bN+2,.,bN 2+,bN2.. 3 Sa3X1vco










determined by image frames and current motion estimation.

In numerical approach, the voxel coordinate components of r, (x, y, z) are

represented as the discrete indices (i, j, k), respectively. Using the central difference

approximation, the first- and second- differential of the motion component u at the

location (i, j, k) can be expressed by the linear combination of the motion at its adjacent

voxel locations. Substituting u with v and w gives the first and second differential of v

and w.


u u+1,],k -1Ik, +1,k 1, ,],k+1 ,,- A 9
x2 2 a2


uxx = (uz+,l,,k ],k ,, 2-1], l+,,,k 2ur], 2-1] (A-10)


uW = (u",J+1,k U,,,k (U,,,k 2,J-1,k Ui,J+1,k 2uI,,,k + 2,J-1,k (A-11)


uzz = (ul,],k+1 U, ,k (U,,,k 2,,,k-1 Ul,],k+1 2uI,,,k + 2,,,k-1 (A-12)


u (u+1,+1, 2-,J+,k +,J-,k 1,J1,k(A-13)
v 4


u ( rJ1k1 2J1k+ J1k1 ,-,- (A-14)



u,,~ (A-15)


For each voxel (i, j, k), Equations A-6, A-7, A-8 generate 3 rows of entries in the

matrix A. The coefficients of (uz, u27"'>, N) for voxel (i, j, k) from Equation A-6 can be

filled into a chart as follows, where i-1, i, i+1, j-1,j +, k-1, k, k+1 are voxel

location indices in 3 dimensions, and the values in the chart are the coefficients for

the motion component u at the voxel located by the indices. For example, the coefficients

for uz+1,],k is (il+2u)/7.











i-1 0 0 0 0 (a +2pu)P 0 0 0
(2l+ 8pu)P
i pf 0 upf df2 (k) 2 pf 0 up
2( )2
8x
i+1 0 O2)


l+pu i+pu 0
i-1 0 0 0 P 0 p 0 0
4 4
df(k) df(k) 0
i 0 0 00 2(2 2 )00 0

i+1 l+pu i+pu 0
0 00 P 0 p 0
4 4


.- O p o 0

f2(i 2(k) 2(k


i+1 l+pu i+pu 0
0 p 0 0 0 0 0 P
4 4


j-1 j j+1


j-1 J j+1


Similarly, the coefficients of (v,, v27"'>, N) in Equation A-6 for (i, j, k) can be expressed

using the following chart:


j-1 j j+1


j-1 j j+1


The coefficients of (w,,w27"' N,) in Equation A-6 for voxel


(i, j, k)


j-1 j j+1


j-1 j j+1


The coefficients of (uz, u27"'>, N) for Equation A-7 for voxel


(i, j, k) are












4 4
(3/ (k) (f(k) 0 0
i 0 0 0 0 -2( 2 O
8x S
i+ o -p o~ 0 0
4 4
k-1 k k+1

The coefficients of (v, ,v27"'>v N) for Equation A-7 for voxel (i, j, k) are

j-1 j j+1 j-1 j j+1 j-1 j j+1
i-1 0 0 0 0 ##p 0 0 0 0

(2l+ 8pu)P
pup (a + 2 p) f(k (a + 2 p) f
i 0 0 of 2o 0 #
2( )2

i+1

k-1 k k+1

The coefficients of (w ,w27"' N,) for Equation A-7 for voxel (i, j, k) are

j-1 j j+1 j-1 j j+1 j-1 j j+1

i-1 0 0 0 0 0 0 0 0

il+u p A+ p af2 (k) d2(k) lu
c az2( )(/ 4


i+1 0


j-1 j j+1


j-1 j j+1


The coefficients of (u, ,u27"'>u N) for Equation A-8 for voxel (i, j, k) are












i- Op o o o -p0
4 4
8f~:k) ~S(k) 0
i 0 0 0 0 ~- 2(2 2) 0 0 0

i+1 l+pu i+pu 0
4 4
k-1 k k+1

The coefficients of (v,, v27"'>, N) for Equation A-8 for voxel (i, j, k) are

j-1 j +1 j-1 j +1 j-1 j +1

i-1 0 0 0 0 0 0 0 0 0

i +u p f, (k) df(k) lu
2+#-2( 2
4 y 8 4 0 4

i+1

k-1 k k+1

The coefficients of (w,,w27"' N,) for Equation A-8 for voxel (i, j, k) are

j-1 j +1 j-1 j +1 j-1 j +1

i-1 0 0 0 0 ##p 0 0 0 0

(2il + 8pu)P

2( )2
8z
i+1 #


j-1 j


A numerical image in practice has certain boundaries. Numerical analysis

methods have to consider the physical processes in the boundaries. Different boundary

conditions may cause very different results. Because of the ideally compressible model

for the space outside myocardium, it is reasonable to assume the motion at the image


j+1 j-1









boundaries to be approximately zero. Thus in our study, we used "fixed boundary

conditions", in which m(xb, yb, Zb) = 0 where (xb, yb, Zb) is a boundary voxel.

The calculation of f 'k(k) (kIl)) and its derivatives is not straightforward since

m k) iS a floating number while r is integer indices (i, j, k). Two approaches to


approximate f,(k)( ( mk)) and its derivatives are described and compared here. For

notation convenience, we write f,(k) (~mk)) as f (r +m) in following text.

1. Taylor approximation approach

Let m = m +Sm where m =(u, v,w) and (u, v, w) are the closest integer

approximation to (u, v, w); 6m = m -m is the difference vector. Then from Taylor

approximation we obtain




(A-17)


(rrm)= (r )m)+& (r +m1)+ Sv (r +m1)+ &v (r +m1)


(A-18)


(r m =(r + m1) + At (r + m1) + Sv (r + m1) + Av (r + m1)


(A-19)




The differential of f (r + m1) can be approximated using central difference.

d, f,(i + u+1, j +v, k +w)- f,(i + u-1, j +v, k +w)
(r +m) =(A-20)
Dx 2