Motion Correction in PET Image. Reconstruction

Transcription

1 Motion Correction in PET Image Reconstruction Wenjia Bai Wolfson College Supervisors: Professor Sir Michael Brady FRS FREng Dr David Schottlander D.Phil. Transfer Report Michaelmas 2007

2 Abstract Positron Emission Tomography (PET) is an in vivo and non-invasive imaging method which provides important functional information about biological systems. It is widely used clinically to diagnose and characterise tumours, as well as brain diseases and cardiac disorders. The quality of reconstructed PET images has a great impact on the information we can derive and clinical decisions we will subsequently make. However, during PET scans, patient movements, such as head movement, respiratory and cardiac motions, are prevalent and inevitable due to the long duration of the scan (30-60 minutes). Such motions degrade PET images and will thus affect subsequent analysis. The motivation of this project is to develop computational techniques to address this problem. The aim is to correct for motion in PET imaging, improving the quality of reconstructed images. This D.Phil. transfer report introduces the physical principles of PET and the degrading impact of motion on PET image formation. It gives a literature review on PET data correction, image reconstruction, and motion correction methods. My work to date is then exhibited, which includes work on PET data simulation, data correction and image reconstruction. The experimental results are given and different image reconstruction algorithms are compared. Finally this report gives a plan for the remainder of the research work in my D.Phil, including simulation of PET data with motion, development and validation of motion correction methods.

3 Contents Contents i 1 Introduction Positron Emission Tomography Types of Coincidence Events Attenuation Image Reconstruction The Motion Blur Problem Literature Review Data Correction Attenuation Correction Correction for Random Coincidences Scatter Correction Normalisation Dead time Correction Image Reconstruction Analytic Reconstruction Iterative Reconstruction Motion Correction Motion Detection Motion Compensation i

4 3 Work to Date PET Data Generation Data Correction Precorrection for Random Coincidences Attenuation Correction Other Corrections Reconstruction Algorithms Analytic Reconstruction Iterative Reconstruction Moving phantom Future Work Data Generation Motion Correction Validation Other Work Proposed Timetable A Meetings and Seminars Attended 45 References 46 ii

5 Chapter 1 Introduction Positron Emission Tomography (PET) is an in vivo and non-invasive imaging method which provides important functional information about biological systems. It is widely used clinically to diagnose and characterise tumours, as well as brain diseases and cardiac disorders [55]. The quality of images reconstructed from PET data has a great impact on the information we can derive and clinical decisions we will subsequently make. However, image quality is often degraded by motion of the subject prevalent during the PET scan. The motivation of this project is to develop computational techniques to address the motion blur problem. The aim is to correct for motion in PET imaging, improving the quality of reconstructed images. The project first entails understanding the physical principles underlying PET image formation and computational techniques for reconstruction of PET images, then analysing the sources of motion blur, and finally developing and implementing motion correction algorithms to improve the quality of reconstruction. In this D.Phil. transfer report, Chapter 1 introduces the physical principles of PET and the degrading impact of motion on PET image formation. Chapter 2 reviews the computational techniques for PET data correction, image reconstruction, and motion correction proposed in recent years. Chapter 3 exhibits my work to date, including PET data simulation and development of data correction and image reconstruction algorithms. Chapter 4 gives a plan for the remainder of the research work in my D.Phil. 1

6 Figure 1.1: The process of positron emission and subsequent positron-electron annihilation which results in two 511 kev photons. 1.1 Positron Emission Tomography The physics underlying positron emission tomography (PET) is that a positron emitted by a certain kind of radionuclide annihilates with an electron to produce a pair of high energy photons (511 kev) which can be recorded by suitable detectors. Figure 1.1 shows the process of positron emission and subsequent positron-electron annihilation. The radionuclide is attached to a chemical compound of interest to form a biological radiotracer, which is injected into the subject under study. Using the annihilation photon pairs recorded by detectors, the radiotracer distribution in the subject can be determined. There are many radionuclides which decay by positron emission. Table 1.1 presents a selection of the radionuclides commonly used in PET imaging [55]. Among these radionuclides, 18 F is by far the most widely used [55]. The decay of 18 F is as follows, 18 F 18 O + e + + v (1.1) where e + is a positron, and v is a neutrino. Once produced, the positron leaves the decay site and rapidly loses its kinetic energy in interactions with atomic electrons in the surrounding tissue [44]. After most of its energy is lost, the positron eventually annihilates with a nearby electron, e + + e γ + γ (1.2) 2

7 where γ is a photon. The mass of the positron and electron is converted into energy. In order to preserve the energy and near-zero momentum, both photons have an energy of 511 kev and leave the annihilation site in opposite directions. Table 1.1: A selection of the radionuclides commonly used in PET imaging [55] Radionuclide Half-life Fluorine 18 ( 18 F) 110 min Carbon 11 ( 11 C) 20 min Nitrogen 13 ( 13 N) 10 min Oxygen 15 ( 15 O) 122 sec By tagging different chemical compounds with radionuclides, different radiotracers can be formed and provide a wide range of biochemical probes. A commonly used radiotracer is the 18 F-tagged glucose compound 18 F-fluoro-deoxy-glucose ( 18 F-FDG). It binds where glucose consumption is high, such as in tumours. This makes it a very useful radiotracer for glucose metabolism studies in cancer. The annihilation photon pair travels through the subject s body and is recorded as a coincidence event by a pair of detectors coupled to a timing circuit. Figure 1.2 shows the process of coincidence detection [41]. The position of annihilation is located along the line between the pair of detectors. This line is called the line of response (LOR). The total number of coincidence events detected by the detector pair is proportional to the total number of emissions along the LOR. Many detectors are coupled with each other to form a detector ring and many detector rings form a PET scanner. After measuring the coincidences along LORs at different angles through the subject, we can use mathematical algorithms to calculate images which reflect the distribution of radiotracer concentration in the subject. There are two ways to save the set of coincidences in a data file, respectively the sinogram and list mode. In a sinogram data, the total number of events detected in each LOR is recorded. In list mode, each coincidence event is recorded individually, with information of the LOR and the time of occurrence. Sinogram data are more efficient in memory, except when the number of LORs is very large and thus the average number of events per LOR < 1. List mode data occupy more memory but provide information of the time. They can be histogramed to form sinogram data. 3

8 Figure 1.2: The annihilation photon pair is recorded as a coincidence event by a pair of detectors coupled to a timing circuit. This figure is taken from [41]. Figure 1.3: Axial section of the PET scanner in 2D and 3D modes. This figure is taken from [41]. According to whether or not the rings are separated by septa between adjacent rings, the PET scanner can work either in 2D or 3D mode. In 2D mode, photons travelling between rings are stopped by the septa, while in 3D mode such photons are detected. A schematic view of the two modes is shown in Figure 1.3 [41]. According to the data acquisition protocol, the PET scanner can work either in static or dynamic acquisition mode. A static PET scan acquires a static frame over a fixed length of time, whereas a dynamic scan collects a sequence of dynamic time frames in order to follow the temporal evolution of the concentration of the tracer. Figure 1.4 illustrates the differences between static and dynamic PET [38]. 1.2 Types of Coincidence Events The coincidence events collected by detectors can be categorized into four types, namely true, scattered, random and multiple [44]. The first three types are illustrated in Figure 1.5 [41]. 4

9 Figure 1.4: The differences between static and dynamic PET. The static PET scan acquires a static frame over a fixed length of time, whereas the dynamic scan collects a sequence of dynamic time frames. This figure is taken from [38]. A true coincidence is an event in which the two detected annihilation photons originate from the same radioactive decay and have not changed direction before being detected. A scattered coincidence is an event where one or both of the two detected annihilation photons interacts in the subject and changes direction prior to detection, mainly through Compton scattering interactions. This results in both a loss of the energy and mispositioning of the event. A random coincidence is generated by two photons originating from two separate annihilations. It contributes no spatial information to reconstruction and results in a background noise. A multiple coincidence is the event where three or more photons are detected simultaneously. It is normally rejected. The total number of events collected by a PET scanner are called prompt coincidences, which consist of true, scattered, and random coincidences. Scattered and random coincidences have a degrading effect on the measurement and need to be corrected in order to obtain a more accurate distribution of the radiotracer concentration. The degrading effect is significant, especially in 3D PET. Taking scattering as an example, the scattered to true coincidence ratio is about 10% in 2D PET, but is over 30% in 3D PET [47]. 5

10 Figure 1.5: Three types of coincidence events: true, scattered, and random. This figure is taken from [41]. 1.3 Attenuation As the scattered events occur, these events are removed from the original LOR, which results in a loss of counts in that LOR. This is called attenuation. Such events are probably collected as scattered events by detectors in a different LOR. Thus, attenuation and scatter are manifestation of the same physics process [44]. Attenuation leads to the radiotracer concentration inside the object being underestimated, because the photons inside have to pass through more material in order to reach the detectors than the photons outside. This effect needs to be corrected. 1.4 Image Reconstruction The goal of image reconstruction is to provide an image of radiotracer distribution in the subject, using the detected coincidence events and mathematical algorithms. There are two main approaches for image reconstruction, namely analytic and iterative. Analytic reconstruction is based on the mathematics of computed tomography, which assumes that the total number of coincidence along a LOR is approximately the line integral of the radiotracer concentration through the subject, as Figure 1.6 shows. The set of the line integrals at different angles form a Radon transform. The concentration distribution can then 6

11 Figure 1.6: The line integral of the radiotracer concentration through the subject. be calculated using the inverse Radon transform, for which the filtered back-projection (FBP) algorithm is an implementation. Iterative reconstruction aims to model the data collection process in order to iteratively find a solution image that is most consistent with the measured data. This is usually formulated as an optimisation problem, which aims to maximise a likelihood function of the image estimate. Analytic approaches are much faster than iterative approaches. However, they do not take into account the statistical variance inherent in the process of data collection. The resulting noise is suppressed using a filter with a cut-off frequency, at the expense of spatial resolution. Since iterative approaches are able to model the statistical variance as well as many other factors in data collection, such as scattered and random coincidences, they provide a more accurate reconstruction. However, iterative approaches are computationally more expensive. 1.5 The Motion Blur Problem A typical PET scan takes minutes and represents the sum of information over the whole period of acquisition. Movement of the subject is inevitable during this period. It has a degrading effect on reconstructed PET images. The information in the images will be dispersed over an area proportional to the magnitude of motion, resulting in motion blur. 7

12 The effective spatial resolution can be approximated using the following formula [19], d e = (d 2 s + d 2 m) 1/2 (1.3) where d e is the effective spatial resolution measured using full width at half maximum (FWHM), d s is the spatial resolution of the scanner, and d m is the FWHM of the position distribution of the object moving around a mean position. Regarding the magnitude of motion, the diaphragm moves about mm due to respiration [57], the base of the heart was found to move 12.8±3.8 mm towards the apex in a MRI study of 19 normal subjects [54]. While today s high resolution PET scanner can provide a spatial resolution of 1.5 mm FWHM [3], motion becomes a significant factor in PET imaging, which greatly limits the image quality of reconstruction. In clinical diagnosis, blurred images may obliterate small but important functional information. A study showed that motion of lungs in PET imaging might lead to incorrect judgements about lesions [11]. Another study showed that respiration introduced artefacts in the majority of the reconstructed images in study [5]. In radiotracer kinetic modelling, blurred images may yield inaccurate kinetic parameters of the radiotracer. Therefore, motion correction, which studies how to reduce motion blur, is receiving increasing attentions. Motion correction may increase the quality of image reconstruction and provide better images for subsequent analysis. The goal of this project is to develop computational algorithms to correct for such motions and reduce blur. Figure 1.7 illustrates motion blur and the effect of motion correction [6]. The images are reconstructed PET images of the Hoffman brain phantom [23]. Figure 1.7(a) shows a reconstruction of a static phantom. Figure 1.7(b) shows the reconstruction of a moving phantom. It is difficult to discern the details of the brain due to motion blur. Figure 1.7(c) shows the reconstruction of the moving phantom after motion correction. As it shows, after motion correction, the brain structure is clearly discernible. The motion corrected image, Figure 1.7(c), compares favourably with the static image Figure 1.7(a). 8

13 (a) The reconstruction of a static phantom (b) The reconstruction of a moving phantom (c) The reconstruction of a moving phantom after motion correction Figure 1.7: Illustration of motion blur and the effect of motion correction. The images are reconstructions of the Hoffman brain phantom. This figure is taken from [6]. 9

14 Chapter 2 Literature Review Section 2.1 reviews a series of corrections necessary for PET data. Then, Section 2.2 gives several commonly used image reconstruction algorithms. Finally, Section 2.3 reviews a number of motion correction methods proposed in recent years. 2.1 Data Correction In order to produce an image which represents the true radiotracer distribution, a number of corrections need to be applied to the raw PET data [44]. These corrections are typically applied to PET data prior to image reconstruction Attenuation Correction Attenuation refers to the loss of photons for a LOR due to interactions in the object, mainly through Compton interactions. Consider a photon pair along a LOR. The survival probability that the photon pair can pass the object and reach both detectors is, µ(x)dx µ(x)dx P l = e l1 e l 2 = µ(x)dx e l 1 +l 2 = e l (2.1) 10

15 where l is the LOR, l 1 and l 2 are respectively the paths of the two photons, and µ(x) is the attenuation coefficient at x. As Equation 2.1 shows, the survival probability is independent of the location along the LOR where annihilation occurs. This equation forms the basis for attenuation correction methods. There are two major methods, measured attenuation correction and calculated attenuation correction. Measured attenuation correction determines the attenuation map through direct measurement [43]. Because the survival probability is independent of the annihilation location, a radiotracer source outside the object will result in the same amount of attenuation as a source in the object. This method places a source outside the object, then performs a blank scan (without the object) and a transmission scan (with the object). The survival probabilities for the LORs are determined from the ratio of the counts of the transmission scan to the blank scan. The attenuation correction factors (ACFs) are the reciprocals of the survival probabilities. The difficulty with this method is to collect enough counts per LOR to reduce the statistical noise, especially for the whole body scan where the majority of the photons are attenuated through the body. As well, it takes additional time to perform a transmission scan. Calculated attenuation correction assumes that the shape of the object and the attenuation coefficients are already known. The attenuation map is calculated from Equation 2.1 and used to correct the PET data. The shape of the object is either already known, e.g. for the phantom, or segmented from the emission or transmission scan [63, 68]. This method can produce noise-free attenuation maps. A drawback is that the assumption about the actual attenuation coefficients must be made and population averages are usually used. With the introduction of the PET/CT scanner, several questions have been addressed. CT provides higher resolution than PET so more accurate anatomical structures of the subject can be obtained. Also, the attenuation coefficients can be determined from the CT images. As a result, the ACFs can be calculated more accurately [28]. 11

16 2.1.2 Correction for Random Coincidences The two annihilation photons in a random coincidence come from different radioactive decays and contribute no spatial information to reconstruction. The random coincidences can be assumed to be distributed uniformly across the field of view (FOV) and result in a background noise. There are two main approaches to correct for random coincidences in PET. One way is to estimate the random counting rate from the singles counting rate using the following equation, N r = 2τN 1 N 2 (2.2) where N r is the counting rate of random coincidences, 2τ is the coincidence timing window, N 1 and N 2 are the rates of single events in a pair of detectors. N r can be determined if the singles counting rate for each detector is measured and τ is known. Then the count of random coincidences is calculated and subtracted from the PET data. This method has the advantage that the singles counting rates are much higher than the random counting rates for the detectors, therefore the statistical quality of the estimate of N r will be good. The difficulty is that the coincidence timing window needs to be known accurately for each detector, otherwise systematic bias is introduced. An alternative is to directly measure random coincidences using a delayed coincidence window. The signal from a detector is delayed so that the coincidences in that detector pair contain only random coincidences, without true coincidences. The random coincidences in a delayed coincidence window have the same probability distribution as those in the prompt coincidence window. They are then subtracted from the PET data. This method is free of systematic bias but the statistical quality is poorer than the singles estimation method, because the random counting rates are lower than the singles counting rates. This method is implemented and available in most modern PET scanners. The subtraction is performed online and the scanners output the randoms-precorrected data. Note that the randoms-precorrected data may have negative values. The prompt and 12

17 delayed coincidences both follow the Poisson distribution, y prompt Poisson{n + r + s} (2.3) y delay Poisson{r} (2.4) where n, r, and s respectively denote the net true, random and scatter events. Correction for random coincidences involves a subtraction of two Poisson variables, y = y prompt y delay (2.5) which does not lead to a Poisson variable and can result in negative values. The negative values will cause divergence in some iterative reconstruction algorithms, such ML-EM and OS-EM [1]. Zero thresholding has been proposed to apply to the precorrected PET data in order to avoid divergence. However, this will introduce a positive systematic bias in the estimated images. A non-negativity constraint in the image space has been proposed which significantly reduces such bias [50, 51]. Alternative methods to address this problem include the shift Poisson approximation and saddle point approximation methods [1, 69], which approximate the distribution of precorrected data y. The log-likelihood function is derived based on the approximation and then a modified EM update equation is obtained. These methods also solve the divergence problem but require an additional calculation of the expected random counting rates r Scatter Correction Scatter results in mispositioning of a coincidence event. A scattered event is indistinguishable from a true event except for its energy. However, there are two difficulties associated with energy discrimination. One is the limited energy resolution of the detectors, which makes it hard to separate scattered events from net true events. The other is that the net true events will only deposit a proportion of their energy in the detectors, so they fall in the same energy range as scattered events. Therefore we need other methods to correct for scattered events. 13

18 There are three main approaches to do this, respectively analytic, dual energy, and simulation methods. The analytic method assumes that the scatter distribution varies slowly across the field of view. The distribution is approximated using Gaussian fitting of the scattered counts outside the object, i.e. the tails of the projection [8, 64]. This method is fast and gives a smooth scatter distribution. However, it fails in whole body scans, where the scatter tails available are very short because the body occupies a large proportion of the FOV. In addition, it is also not suitable for PET scans where the radioactivity is highly localized so the scatter distribution contains more structure. The dual energy method uses two energy windows for coincidence collection, e.g., kev and kev. The idea is that the high energy window contains both scattered and unscattered events and the low energy window only contains scattered events. A scaled subtraction of the two windows can correct for the scattered events [21]. This method performs correction from direct measurement. The difficulties are how to determine the scale and the fact that actually both windows contain scattered and unscattered events. The simulation method first reconstructs the image and the attenuation map, and then simulates the scatter based on the reconstructions, either using a simple single scatter model, or using a Monte Carlo simulation [35, 42, 67]. This method is very accurate and accounts for the activity distribution and attenuation coefficients. The drawback is that it is very computationally expensive and time consuming, especially when using a Monte Carlo simulation Normalisation The nonuniformities in detector efficiencies, geometrical variations, and detector electronics in different LORs result in different sensitivities. The process to correct for such nonuniformities is referred to normalisation. A straightforward correction method is to measure the nonuniformities of different LORs through a PET scan using a source of uniform radioactivity. The reciprocals are the normalisation correction factors. The difficulty with this method is to collect enough counts per LOR to reduce the statistical variance. An alternative is the component-based method, which factorises the normalisation cor- 14

19 rection factor for each LOR into components of single detector efficiencies and geometrical variations [2, 24]. The number of detectors is much smaller than the number of LORs and the counts per detector is much more than per LOR. Therefore the statistical noise is largely reduced. The geometrical variations are typically determined once for a particular scanner using a very high count acquisition and assumed to be constant Dead time Correction Dead time correction aims to correct for the loss of coincidence events due to detector and system dead time. The main source of dead time in most PET systems is the processing of each event in the detector front-end electronics [44]. Usually the measured and true counting rates are modeled using either a paralysable or nonparalysable dead time model. The parameters of the model are determined by an experiment involving repeated measurements of a decaying source. The model is then used to restore the true counting rates from the measured rates in PET data. 2.2 Image Reconstruction Image reconstruction aims to provide an image of radiotracer distribution in the subject, using the detected coincidence events and mathematical algorithms. As noted earlier, there are two main approaches for image reconstruction, namely analytic and iterative Analytic Reconstruction In the absence of an attenuating medium and assuming perfect detectors, the projection along a LOR is approximately the line integral of the radiotracer concentration through the subject, as Figure 1.6 shows. The set of the line integrals at different angles form a Radon transform. It is formulated as, P θ (t) = f(x, y)δ(x cos θ + y sin θ t)dxdy (2.6) 15

20 where f(x, y) is the object, θ is the angle of the radial line, t is the coordinate along the radial line, and P θ (t) is the projection. The image f(x, y) can be reconstructed from the inverse Radon transform of P θ (t), which is usually performed using the filtered backprojection (FBP) algorithm [27, 29]. The Fourier Slice Theorem is the most important relationship in analytic image reconstruction. Theorem 1 (Fourier Slice Theorem) The 1D Fourier transform of a projection of an image f(x, y) at an angle θ is equivalent to a slice through the center of the 2D Fourier transform F (u, v) at the same angle, formulated as, S θ (w) = F (u, v) u=w cos θ,v=w sin θ (2.7) where S θ (w) is the 1D Fourier transform of the projection P θ (t). The theorem relates the Fourier transform of a projection to the Fourier transform of the object so that we can then calculate the object using the inverse Fourier transform. f(x, y) = F (u, v)e j2π(ux+vy) dudv (2.8) Exchanging the rectangular coordinate system in the frequency domain, (u, v), for a polar coordinate system, (w, θ), and performing a few operations, it is straight forward to show, f(x, y) = π 0 Q θ (x cos θ + y sin θ)dθ (2.9) where Q θ (t) = S θ (w) w e j2πwt dw (2.10) Equation 2.10 represents a filtering operation, where the frequency filter is given by w. Q θ (t) is called a filtered projection. Equation 2.9 represents backprojection, where Q θ (t) is backprojected to the image f(x, y). Given θ and t, Q θ (t) will make the same contribution to all the points along the line x cos θ+y sin θ = t. This algorithm is called filtered backprojection 16

21 (FBP). In practice the ramp filter w is often modified to suppress the noise in the reconstructed images. A cut-off frequency w cut off is used to remove high frequency noise. However, high frequency signal information is also removed so that the resolution is degraded. Besides, the Hann filter is often used instead of the ramp filter. Different cut-off frequencies and filter types will achieve a different trade-off between the signal to noise ratio (SNR) and spatial resolution Iterative Reconstruction Analytic reconstruction is fast, however, the accuracy of the reconstructed images is limited by the approximation inherent in the line integral model on which the reconstruction algorithm is based. In contrast, iterative reconstruction accurately describes data acquisition, modeling the probability that a photon pair emitted at a certain location in the object is detected by a given detector. A second limitation of analytic reconstruction is that it does not take into account the statistical variability inherent in the imaging system. The iterative reconstruction allows explicit modeling of the statistical variability associated with photon detection [34]. Let f = {f j, j = 1,..., N} be the image of radiotracer concentration, where the pixel value is proportional to the number of radionuclides. Let y = {y i, i = 1,..., M} be projection data, which record the number of detected photon pairs in each detector bin. y and f are related through a projection matrix, or system matrix, P = {p ij } M N, where p ij represents the probability that a photon pair emitted from the jth pixel is detected at the ith bin. The mean of y is, ȳ = E{y} = P f (2.11) The emission of positrons from a large number of radionuclides is known to follow a Poisson distribution. The sum of independent Poisson random variables is still a Poisson random variable. The projection data y i in the ith bin collects the emission of annihilation photon pairs from all the pixels, therefore it also follows a Poisson distribution. The probability of y 17

22 conditioned on f is given by, M ȳ y i i e ȳ i P (y f) = y i! i=1 (2.12) This is called the likelihood function. The corresponding log-likelihood function, after dropping the constants, is, M log P (y f) = {y i log ȳ i ȳ i } (2.13) i=1 Thus reconstruction becomes an optimisation problem which aims to maximise log P (y f). The log-likelihood has been proved to be concave so that a global maximum exists [62]. The optimisation problem can be solved effectively under an EM framework [10, 33, 62]. The solution is an elegant closed-form update equation, j = f (k) j f (k+1) i p ij i p ij y i l p ilf (k) l (2.14) Note that Equation 2.14 naturally preserves the non-negativity of the estimate if y 0. This algorithm is called the maximum likelihood EM (ML-EM) algorithm. The ML-EM algorithm models the detection process more accurately and handles the statistical variance in photon detection as well. In practice it shows considerable improvement in image quality over the FBP algorithm. However, there are two disadvantages of the ML-EM algorithm [22], 1. Convergence is very slow. 2. Due to the ill-conditioned nature of the reconstruction problem, the reconstruction tends to have a high variance as the ML solution is approached. Acceleration Convergence can be accelerated using the ordered subset EM algorithm (OS-EM) [25]. The projection data is grouped into an ordered sequence of subsets. The standard EM algorithm is then applied to each of the subsets in turn, using the rows of the system matrix corresponding to these projections. The resulting reconstruction becomes the initial value for the next subset. An iteration of OS-EM is defined as a single pass through all the subsets. 18

23 Let {S i } p i=1 be a disjoint partition of the projection index set {1,..., M}, where p is the number of subsets. Let k denote the index for an OS-EM iteration and i denote the index for a sub-iteration. Define f (k,0) = f (k 1) and f (k,p) = f (k). Then the update equation for OS-EM is given by, j = f (k,i 1) j f (k,i) i S i p ij i S i p ijy i l p ilf (k,i 1) l (2.15) It has been recommended that balanced subsets should be chosen, which means that the pixel activity contributes equally to any subset. With regard to the order of the subsets, it is encouraged that substantial new information is introduced as quickly as possible [25]. For example, the subsets can be chosen to have maximum separation in angle between successive subsets. Because the computational cost of the EM algorithm is proportional to the number of projections, one OS-EM iteration will have similar computation time to one ML-EM iteration. But one OS-EM iteration consists of a number of sub-iterations, each of which converges similar to one ML-EM iteration. Therefore the OS-EM algorithm accelerates convergence by a factor proportional to the number of subsets [25]. Regularisation The high variance problem in the ML-EM algorithm can be addressed either by terminating the algorithm before convergence [66] or post-smoothing the reconstruction [36]. OS-EM suffers similar high variance problem as ML-EM and they can be addressed similarly. Beside early termination and post-smoothing, an alternative is to explicitly introduce a prior distribution into the likelihood function. This is the maximum a posteriori (MAP) algorithm [22]. The probability distribution of the image vector f conditioned on the data vector y is formulated using Bayes rule, P (f y) = P (y f)p (f) P (y) (2.16) where P (f) is the prior distribution. Taking the logarithm and dropping the constants, reconstruction of the image f is an 19

24 optimisation problem as follows, max log P (f y) = log P (y f) + log P (f) (2.17) f The function of the prior P (f) is to regularise the unsmooth image. The Gibbs distribution is often used as the prior. By the Hammersley-Clifford theorem [4], this corresponds to a Markov random field defined on the lattice of the image, which is suitable for describing the local correlation of the pixels. The Gibbs distribution has the form, P (f) = 1 Z e βu(f) (2.18) where U(f) denotes the Gibbs energy function defined as a sum of potentials, each of which is a function of a clique. A clique is a set of pixels such that each pixel is a neighbor of all the other pixels in the clique. The Gibbs energy functions most commonly used are those defined on pair-wise cliques of neighborhoods, formulated as, N U(f) = V (f j, f k ) (2.19) j=1 k N j where N j is the neighborhood of j, V (f j, f k ) defines the potential function. A first-order neighborhood results in 4 pixels in 2D and 6 pixels in 3D. A number of potential functions have been proposed which attempt to produce local smoothing while not blurring the boundaries in the image [16, 17, 20]. With a Gibbs prior, the reconstruction problem in Equation 2.17 can now be written as, max log P (f y) = log P (y f) βu(f) (2.20) f The log-likelihood in Equation 2.13 is concave, so log P (f y) is concave if the Gibbs energy function U(f) is convex. In this case, a global maximum will be found. Otherwise, local maxima may exist. Compared to ML-EM, a regulariser term is added to the likelihood function, which imposes 20

25 the constraint of local smoothness. The parameter β controls the degree of regularisation. While β 0, it gradually reduces to the ML-EM reconstruction. While β is increased, the regulariser becomes more significant and has stronger smoothing effect on the reconstruction. To solve the optimisation problem in Equation 2.20, the generalized EM (GEM) framework is used [22]. Because of the addition of the prior, usually we can not get a closed-form update equation. The update equation has the form, f (k+1) = arg max f j {f (k) j i p ij y i l p ilf (k) l log(p ij f j ) f j p ij } βu(f) (2.21) i Note that this is still an optimisation problem. Since f (k) j is determined, it is easier to solve this one. A gradient search can be applied to the optimisation problem. However, it is difficult to guarantee the non-negativity of f. The iterated coordinate ascent (ICA) method updates the pixels sequentially, which makes imposition of the non-negativity constraint easy [12, 22, 56]. Using ICA the optimisation problem becomes a 1D gradient search. The selection of the hyperparameter β is similar to that of the cut-off frequency in the FBP algorithm. It is a trade-off between the variance and spatial resolution. Apart from subjective selection, some selection methods based on objective measures of the image quality have been proposed [13, 49]. When β is adjusted suitable, the MAP reconstruction can provide an improvement over the ML-EM reconstruction. The disadvantage of the MAP algorithm is that it is computationally more expensive due to calculation of the potential functions in the prior. 2.3 Motion Correction Motions can be grouped into two categories, voluntary and involuntary [6, 65]. Voluntary motions include unpredictable movements of the subject during data acquisition. For example, a subject lying on the couch in the PET scanner repositions the body or moves the head to relieve pain or pressure points. Involuntary motions include periodic movements of the organs, such as cardiac and respiratory motions. Some voluntary motions can be reduced. For example, a number of supports have been 21

26 designed to maintain the position of the head during a neurological PET scan. However, these may not be comfortable for the subject during the acquisition (30-60 minutes). In addition, not all motions can be reduced, especially the involuntary ones. Therefore motion correction is necessary to reduce the degrading effect of motion on reconstructed images. Motion correction consists of two separate stages. First, motion information is either detected using an external motion tracking system or estimated from reconstructed images. Then it is applied to PET data to compensate for motion and to form motion corrected data Motion Detection One method for motion detection is to use an external motion tracking system to record the subject movements. The most common system attaches at least 3 targets to the subject and uses 2 video cameras placed orthogonally to locate the 3D coordinates of the targets [18, 45]. Because three independent points are sufficient to specify the position and movement of an object, the system is able to track the 3D translations and rotations of the subject in 6 degrees of freedom. A commercial optical motion tracking system, Polaris (Northern Digital, Waterloo, Canada), is available and widely used to track head motion [6, 15, 14, 37]. The advantage of this approach is that motion information can be accurately determined by direct measurement. The disadvantage is that the motion tracking system is not suitable for tracking respiratory and cardiac motions, which are inside the body. The other method is to estimate motion information from multiple acquisition frames generated by gating. The coincidence events are gated into different frames with the aid of some external signal, such as monitoring the position of the chest, the pressure around the chest, or the temperature of the exhaled air for respiratory motion [7, 31, 40], and ECG signal for cardiac motion [31]. The frames are reconstructed individually. Then motion is estimated from successive reconstructed frames by image registration. The advantage of this method is that it is able to track respiratory and cardiac motions. However, it assumes that each frame is static and does not account for movement during the individual frames. When the amplitude of movement is large, many frames with short durations will be produced in order to satisfy the assumption. Thus the statistics of the count for each frame is poor and the spatial 22

27 resolution of the reconstruction will be low. Since the accuracy of motion estimation depends on the spatial resolution of successive frames, in this occasion the estimation error will be increased. Another assumption underlying this method is that the radioactivity distribution does not change significantly, so that image registration is valid Motion Compensation There are two major methods to compensate for motion, namely LOR rebinning and the multiple acquisition frame (MAF) methods. The LOR rebinning method is performed in the PET data space, while the MAF method is done in the reconstructed image space. The LOR rebinning method rebins each event from the original LOR to a new one by a geometrical transformation according to measured movement [6, 39]. This method has the potential to be implemented in real time [26]. However, there are several problems when using this method. First is the loss of events. After transformation, some events may fall out of the FOV. These events are discarded and can not contribute to the data. Second is the normalisation problem. When applying normalisation to the PET data, the rebinned event will be weighted with the normalisation factor which belongs to the transformed LOR, instead of with the factor for the original LOR. This problem can be overcome by performing normalisation prior to rebinning [6]. The third problem is the rebinning round-off error. The transformed LOR probably lies between two adjacent LORs. It is solved either by assigning the event to the nearest LOR, or by contributing the event to both LORs by weights according to the distances. Both solutions result in round-off error. The MAF method transforms all the reconstructed frames to a reference frame according to the estimated motion [9, 15, 30, 46]. The transformed frames are summed, resulting in the final reconstruction. This method is usually used when the motion information can only be estimated from successive frames. A major drawback is that it does not account for movement during an individual frame. In addition, interpolation errors will be introduced in transformation. In a comparison study, the LOR rebinning method was found to be more accurate than the MAF method because of its ability to react to motion throughout the scan [14]. 23

28 Apart from the above two major methods, a method using a time varying system matrix has been proposed [48]. This method introduces motion information into the system matrix and modifies the likelihood function in iterative reconstruction. The advantages of this method are that it reacts to motion through the scan by varying the system matrix and it also uses all the events, unlike the LOR rebinning method. A drawback is that this method is computationally expensive. 24

29 Chapter 3 Work to Date In order to study the motion blur problem and develop motion correction techniques, two preparatory steps are PET data acquisition or simulation, and implementation of reconstruction algorithms. Only once we have got PET data with motion, as well as relevant tools to reconstruct images, can we begin to look into the degrading impact of motion on reconstruction and then develop techniques to reduce this impact. During the first nine months of my D.Phil., I used a simulator PET-SORTEO to generate PET simulation data. I implemented attenuation correction and applied the correction to data. In terms of reconstruction, I implemented and tested several of the most common reconstruction algorithms, including FBP, ML-EM, OS-EM, and MAP. I used these algorithms to reconstruct images from the simulation data and compared the results. In order to generate data with motion, I used a moving phantom NCAT as input to the simulator PET-SORTEO. This work is still in progress and not finished yet. 3.1 PET Data Generation Real PET data is expensive to acquire (about 1200 per scan). It is both time and money consuming to generate a real PET database for study. Thanks to the simulator, it overcomes the above problem and lends us the ability to generate realistic PET data on the computer. Using a simulator, we can easily change the phantom or adjust the parameters as we like, and generate loads of PET data for free. In addition, a great benefit of using a simulator is that 25

30 the ground truth is available for validation. PET-SORTEO is a Monte Carlo-based simulation tool that can be used to generate realistic PET projection data, which are especially useful for designing and validating correction and reconstruction methods, for subsequent data analysis, and for performance prediction of PET prototypes [52, 53]. It is installed on the PC cluster in our lab. Three major steps are involved in data generation using PET-SORTEO. First, a text protocol file is created, describing the scanner properties, the acquisition configuration, the emission and attenuation volumes etc. Second, the command CompileProtocol processes the text protocol file and generates a binary protocol file. Finally, the command sorteo takes the protocol file as input and generates the PET data. The Perl script SubmitSORTEO simplifies the last two steps using just one command. Given a voxelized phantom, PET-SORTEO is able to simulate the emission, transmission, and blank scans. It can work in both 2D and 3D modes. The output supports both sinogram and list-mode data. The format of the phantom file and the data file is ECAT7. Currently, I use 2D sinogram data for reconstruction. PET-SORTEO is a multiprocess program, which means that the simulation can be launched on multiple processors in order to increase the speed. There are 40 nodes on the cluster so that the maximum process number can be set to 40 if all the nodes are available. A high process number will increase the speed of simulation. However, it also occupies more computational resources on the cluster. A uniform square phantom and the sinogram generated by PET-SORTEO are illustrated in Figure 3.1. The transverse view is shown. The phantom is a D array. The voxel size is cm 3. The transverse cross section of the array is a square with uniform radiotracer concentration of 200 nci/ml. The length of the PET scan is 600 seconds. The resulting sinogram is a D array. There are 63 planes. For each plane, there are 288 angles and 288 detector bins. The plane separation is cm. The bin size is cm. I adjusted the process number used by PET-SORTEO and recorded the computation time for generating the sinogram. Table 3.1 lists the computation time with regard to the 26

31 (a) The square phantom (b) The sinogram Figure 3.1: Illustration of a uniform square phantom and the sinogram generated by PET- SORTEO. The transverse view is shown. process number. Figure 3.2 illustrates the trend. The data shows that the computation time decreases inverse proportionally to the process number. In my experiments, was a suitable process number, which gave an acceptable computation time and occupied less than half of the resources on the cluster. Table 3.1: Computation time with regard to the process number Process number Time (min) Data Correction In order to produce an image which represents the true radiotracer distribution, the PET data need to be corrected. There are a number of corrections, including attenuation correction, correction for random coincidences, scatter correction, normalisation, and dead time correction. The random coincidences are precorrected by PET-SORTEO. Also, I have implemented an attenuation correction algorithm. Experiments show that it effectively improves the quality of reconstruction. 27

32 Figure 3.2: Computation time with regard to the process number. The data points are taken from Table Precorrection for Random Coincidences Most existing PET scanners correct the random coincidences online using the delayed coincidence window and directly output the randoms-precorrected data. PET-SORTEO is consistent with current PET scanners and also produces the precorrected data Attenuation Correction As we noted earlier, attenuation leads to the radiotracer concentration inside the object being underestimated. A measured attenuation correction method is applied, which determines the attenuation correction factors (ACFs) from a blank scan and a transmission scan. The blank and transmission scans are generated using PET-SORTEO. The ACFs are then determined from the ratio of the counting rates of the blank scan and the transmission scan, acf i = yb i y t i (3.1) where acf i denotes the ACF along the ith bin, y b i and yt i denote the counting rates in this bin in the blank scan and transmission scan respectively. 28

33 (a) The uncorrected sinogram (b) The ACF map (c) The corrected sinogram Figure 3.3: Illustration of the uncorrected sinogram, the ACF map, and the corrected sinogram. The transverse view is shown. The uncorrected sinogram is taken from Figure 3.1(b). The ACFs are used to correct the emission sinogram, y c i = y uc i acf i (3.2) where y c i and yuc i denote the corrected and uncorrected counting rates respectively. Figure 3.3 illustrates the uncorrected sinogram, the ACF map, and the corrected sinogram. The ACF map (Figure 3.3(b)) shows a vertical bright band, which means that the bins in the center of the FOV are compensated. There are two bright peaks along the band, corresponding to 45 and 135, where the photons need to travel along the diagonals of the square thus the greatest correction is needed. Figure 3.4 gives reconstructions using FBP without and with attenuation correction. In 29

34 (a) Uncorrected (b) Attenuation corrected Figure 3.4: Comparison of the FBP reconstructions without and with attenuation correction, using a Hann filter with the cut-off frequency at 0.5/pixel. the FBP algorithm, a Hann filter was used with a cut-off frequency at 0.5/pixel. In the reconstruction without correction (Figure 3.4(a)), the region inside the square looks darker than the edge. The reason is that the photons inside the square need to cross more material than the photons along the edge. In addition, the corners of the square are not very sharp. After correction (Figure 3.4(b)), the square region looks more uniform. The corners of the square look sharper. Besides, the contrast between the foreground and background becomes stronger. Figure 3.5 shows the profiles along the horizontal central lines in the reconstructions in Figure 3.4. From the profiles, we can also see that attenuation correction yields a more uniform square region and increases the contrast between the foreground and background obviously. The experimental results show that attenuation correction leads to an improvement in reconstruction and results in a more accurate distribution of radiotracer concentration Other Corrections I have not yet had time to implement scatter correction, normalisation, and dead time correction. Of these, scatter correction is probably the most difficult that is required in PET [44]. In 2D PET the presence of the septa help reduce the scattered coincidences to a relatively low level (10% fraction to the true coincidences), thus many workers ignore scatter correction. However, in 3D PET the scattered to true coincidences ratio increases to over 30%, so scatter 30

35 Figure 3.5: The profiles along the horizontal central lines in Figure 3.4. correction becomes an important factor in order to yield accurate PET images. As I am working on 2D PET, I currently ignore scatter correction. Normalisation is to correct for the nonuniformities in detector efficiencies, geometrical variations, and detector electronics. Dead time correction is to compensate for the losses of events due to detector and system dead time. I will try to address these issues in the future if I have sufficient time. However, because the focus of the project is on motion correction, I propose to pay more attention to the problem of motion blur and developing motion correction algorithms, as long as the quality of reconstruction without these corrections is still satisfactory and does not affect the study on motion correction. 3.3 Reconstruction Algorithms Reconstruction is crucial for PET imaging in order to visualize the distribution of radiotracer concentration from the raw data. I implemented both analytic and iterative reconstruction algorithms, including FBP, ML-EM, OS-EM, and MAP. They were applied to the simulation PET data and the results of reconstruction were compared. 31

36 3.3.1 Analytic Reconstruction The FBP algorithm was implemented according to the discrete version of Equation 2.9 and The implementation is shown in Algorithm 1. Algorithm 1 FBP for each projection do Take the 1D Fourier transform of the projection. Multiply by the filter function. Take the inverse Fourier transform. Backproject to the 2D image space. end for I used the ramp filter and the Hann filter for reconstruction of the uniform square phantom, with the cut-off frequency at 0.5 and 1/pixel. Reconstructions using different configurations are shown in Figure 3.6. As we can see, there is a trade-off between the signal to noise ratio (SNR) and the spatial resolution as the filter is changed and the cut-off frequency is varied. FBP using the Hann filter with the cut-off frequency at 0.5/pixel (Figure 3.6(d)) gives the most consistent result with the original phantom (Figure 3.1(a)) Iterative Reconstruction The update equations of the ML-EM and OS-EM algorithms were given by Equation 2.14 and The system matrix was calculated using the area-weighted model, where the contribution of a pixel to a bin is the intersection area of that particular pixel with the projection tube [32]. In addition, the update equation in the ML-EM algorithm, Equation 2.14, is reformulated by introducing a non-negativity constraint [50], j = f (k) j f (k+1) i p ij [ i p ij y i l p ilf (k) l ] + (3.3) where [x] + = x if x > 0 and equals 0 otherwise. The OS-EM update equation is reformulated similarly. Finally, the implementation is shown in Algorithm 2. Figure 3.7 demonstrates the effect of the non-negativity constraint. Figure 3.7(a) and 3.7(b) are respectively reconstructions using OS-EM without and with the non-negativity constraint. The number of subsets is 12, with 20 iterations. The figure shows that without 32

37 (a) Ramp, 1/pixel (b) Ramp, 0.5/pixel (c) Hann, 1/pixel (d) Hann, 0.5/pixel Figure 3.6: Reconstructions of the uniform square phantom, using different filters and cut-off frequencies. Algorithm 2 ML-EM and OS-EM Construct the system matrix. for each iteration do Apply the update equation. Apply the non-negativity constraint. end for 33

38 (a) Without the non-negativity constraint (b) With the non-negativity constraint Figure 3.7: The effect of the non-negativity constraint on reconstructions using OS-EM. 12 subsets, 20 iterations. the constraint, OS-EM tends to diverge. Another experiment for ML-EM showed similar tendency. However, it took more iterations for ML-EM without the non-negativity constraint to diverge (500 iterations in the experiment). Figure 3.8 shows reconstructions of the uniform square phantom using ML-EM and OS- EM. OS-EM uses far fewer iterations than ML-EM to obtain a similar quality of reconstruction. Because the computational cost of one OS-EM iteration is similar to that of one ML-EM iteration, it achieves the effect of acceleration. Compared to Figure 3.6, we can see if ML-EM or OS-EM reconstruction stops at an early stage (Figure 3.8(a) or 3.8(b)), the quality of reconstruction shows an obvious improvement over that of FBP. As the number of iterations increases, the reconstructed image shows a higher variance. The update equation of the MAP algorithm was given by Equation It is not a closeform equation. I have solved it using the ICA method given by [22]. With regard to the potential function in the Gibbs energy function, I used the squared difference given by [16], V (f j, f k ) = (f j f k ) 2 (3.4) The implementation of MAP is shown in Algorithm 3. In the MAP algorithm, there is a hyper-parameter β which controls the strength of regularisation. Reconstructions using different values of β are illustrated in Figure 3.9. When 34

39 (a) ML-EM, 12 iterations (b) OS-EM, 12 subsets, 1 iterations (c) ML-EM, 36 iterations (d) OS-EM, 12 subsets, 3 iterations (e) ML-EM, 60 iterations (f) OS-EM, 12 subsets, 5 iterations Figure 3.8: Reconstructions of the uniform square phantom, using ML-EM and OS-EM. 35

40 Algorithm 3 MAP Construct the system matrix. for each iteration do Apply the usual ML-EM algorithm. The result is the starting value for the next step. for each pixel do repeat Calculate the gradient of the log-likelihood function. Apply the gradient ascent search method. Check if the result is non-negative. Check if the value of the log-likelihood function has increased. until the two checks above are passed end for end for β = 0, there is no regularisation and MAP reduces to ML-EM (Figure 3.9(f)). As β increases, the strength of regularisation also increases and reconstruction tends to be uniform (Figure 3.9(a)). Using a suitable value of β, MAP achieves a good balance between the SNR and spatial resolution (Figure 3.9(c) or 3.9(d)), outperforming ML-EM reconstruction. Table 3.2 lists the computation time to reconstruct a image from a slice of sinogram using different reconstruction algorithms. The programs are implemented in the C language. It is performed on a Dell Precision workstation with a dual-core Intel Pentium(R) 3.20GHz CPU and 2GB RAM. As shown in the table, the FBP algorithm is the fastest. The reason is that FBP only takes backprojection once but ML-EM, OS-EM and MAP are iterative and take projection and backprojection for every iteration. Among the iterative algorithms, OS-EM uses much less time than ML-EM to achieve a similar quality of reconstruction due to the use of ordered subsets. MAP is the slowest. It takes far longer for each iteration than ML- EM and OS-EM. The reason is that compared with ML-EM and OS-EM, MAP needs extra computation for calculating the potential function, checking whether the likelihood function has increased, and adjustment of the step size in the gradient search method. Table 3.2: Computation time using different reconstruction algorithms Algorithm Parameters Time (sec) FBP Hann, 0.5/pixel 0.3 ML-EM 36 iterations 53.8 OS-EM 12 subsets, 3 iterations 7.5 MAP β = 0.001, 20 iterations

41 (a) β = 1 (b) β = 0.01 (c) β = (d) β = (e) β = (f) β = 0 Figure 3.9: Reconstructions of the uniform square phantom, using MAP with different values of β. 20 iterations. 37

42 (a) Anterior view of the NCAT phantom (b) Cardiac and respiratory motion models of the phantom Figure 3.10: The NCAT phantom and the motion models. This figure is taken from [58]. 3.4 Moving phantom In order to generate PET data with motion, we require a moving phantom. The 4D NCAT phantom provides a realistic and flexible model of the human anatomy and physiology for use in nuclear medicine research [59, 60, 61, 65]. It also models common motions in the human body, such as cardiac and respiratory motions. As a result, the 4D NCAT phantom provides an excellent tool with which to study the effects of anatomy and patient motions on SPECT and PET images. Figure 3.10 shows the NCAT phantom and the motion models. Given a parameter file, the NCAT program produces a series of dynamic frames. Each frame is a static 3D volume, where the voxel value represents the label of an organ. It means a continuous motion is separated into a number of static phases. If the duration of each phase is small enough, the continuous motion can be approximated accurately. The type of motion (cardiac, respiratory, or both), the magnitude and period of motion, the labels of organs, the size of the volume etc. are all defined in the parameter file. Because each frame is a voxelized phantom, it can be accepted by PET-SORTEO as input. A static PET scan for that frame can be generated. The sum of a series of PET scans form the PET data with motion. Figure 3.11 shows reconstruction of a static phantom. The phantom was a frame generated 38

43 (a) Transverse (b) Sagittal (c) Coronal Figure 3.11: Reconstruction of a static phantom. The phantom was a frame generated using the NCAT program. The reconstruction algorithm was FBP, using a Hann filter with the cut-off frequency at 0.5/pixel. using the NCAT program. The concentration of 18 F radioisotope in the lung region was a uniform distribution of 200 nci/ml. There was no radioisotope in any other region of the phantom. The length of the PET scan was 600 seconds. The FBP algorithm was used for reconstruction, using a Hann filter with the cut-off frequency at 0.5/pixel. Figure 3.12 shows reconstruction of a moving phantom. In total 10 frames were generated using the NCAT program. Respiratory motion was modeled, with a period of 5 seconds and a maximum diaphragm motion of 4.0 cm. The 10 frames form a complete respiratory period and each frame represents a phantom in a specific phase. PET-SORTEO generated sinograms for each phantom. The length of scan per phantom was 60 seconds. All the sinograms were added together to form one sinogram. So the total scan length was also 600 seconds. Finally the FBP reconstruction was performed on the resulting sinogram, using a Hann filter with the cut-off frequency at 0.5/pixel. Comparing Figure 3.12 with Figure 3.11, the reconstruction in Figure 3.12 is blurred due to respiratory motion. From the transverse view, we can see that the resolution becomes 39

44 (a) Transverse (b) Sagittal (c) Coronal Figure 3.12: Reconstruction of a moving phantom. The phantom was the sum of 10 frames generated using the NCAT program. The reconstruction algorithm was FBP, using a Hann filter with the cut-off frequency at 0.5/pixel. worse. In addition, the size of the lung region is overestimated, obviously larger than that in static reconstruction. Some streaks appear in the sagittal and coronal view, which should not belong to the lung but actually comes from the sum of different motion phases. The quality of reconstruction is degraded because of motion. 40