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The design and implementation of COSEM, an iterative algorithm for fully 3-D listmode data

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In this paper,we present coincidence-list-ordered sets expectation-maximization (COSEM), an algorithm for iterative image reconstruction directly from list-mode coincidence acquisition data. The COSEM algorithm is based on the ordered sets EM
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  IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 7, JULY 2001 633 The Design and Implementation of COSEM, anIterative Algorithm for Fully 3-D Listmode Data Ron Levkovitz, Dmitry Falikman*, Michael Zibulevsky, Aharon Ben-Tal, and Arkadi Nemirovski  Abstract— In this paper,we present coincidence-list-orderedsets expectation-maximization (COSEM), an algorithm for iter-ative image reconstruction directly from list-mode coincidenceacquisition data. The COSEM algorithm is based on the orderedsets EM algorithm for binned data but has several extensions thatmakes it suitable for rotating two planar detector tomographs. Wedevelop the COSEM algorithm and extend it to include analyticcalculation of detection probability, noise reducing iterativefiltering schemes, and on-the-fly attenuation correction methods.We present an adaptation of COSEM to the Varicam VG cameraand show results from clinical and phantom studies.  Index Terms— 3-D reconstruction, expectation maximization, it-erative algorithms, PET. I. I NTRODUCTION T HE MAXIMUM-likelihood expectation-maximization(ML-EM) iterative reconstruction algorithm [9] is consid-ered as a “golden standard” in medical image reconstruction;its high computational demands, however, prevented it frombeing widely accepted for industrial and clinical use. In 1994,Hudson and Larkin [5] presented an accelerated version of theEM algorithm based on an ordered sets approach. Many inde-pendent tests have shown that the ordered-sets EM (OSEM)produces images which are similar in quality to those producedby the EM algorithm in a fraction of the time [4], [7]. TheOSEM and the EM algorithms use binned data (e.g., projectionsets) to reconstruct the image. This poses a severe limitationon the utilization of OSEM for reconstructions of three-dimen-sional (3-D) acquisition images from scanners with wide axialapertures, high resolution and low statistics. These scanners arecharacterized by a huge number of bins (hundreds of millions)and relatively small data sets. The resulting projection set willbe huge, and mostly empty.Scanners with wide axial apertures, such as revolvingtwo-head gamma camera scanner or positron emission tomog-raphy (PET) with large numbers of rings, typically producedata in list mode format. List-mode data format is an event by Manuscript received January 8, 1999; revised April 14, 2001. This work wassupported in part by the European Commission PARAPET ESPRIT Project,in part by the Israeli Ministry of Science, and in part by ELGEMS LTD. TheAssociate Editor responsible for coordinating the review of this paper and rec-ommending its publication was C. Thompson.  Asterisk indicates correspondingauthor. R. Levkovitz, M. Zibulevsky, and A. Nemirovski are with the Minerva Opti-mization Centre, Technion, Haifa 32000, Israel.*D. Falikman is with the Minerva Optimization Centre, Technion, Haifa32000, Israel (e-mail: dmfalikman@ie.technion.ac.il).A. Ben-Tal is with the ITS Faculty, Department of Control, Risk, Opera-tions Research and Statistics, Technical University of Delft, 2628CD Delft, TheNetherlands.Publisher Item Identifier S 0278-0062(01)05522-7. event account of the acquisition process. The events are, there-fore, ordered according to the time of acquisition. Standardreconstruction methods bin these events to sinograms and usethe conventional reconstruction techniques. In this paper, wedevelop a coincidence-list-ordered sets expectation-maximiza-tion (COSEM) algorithm. The COSEM algorithm is a full 3-Dalgorithm, particularly suited for large axial aperture tomo-graphs. It avoids the binning problem by directly processingthe coincidence list information. The direct processing of thetime-based coincidence list instead of the geometric-basedordered sets used by the standard OSEM provides a naturalframework for evolving images and dynamic reconstruction.This algorithm is similar in approach to the algorithms in-dependently developed by Barret  et al.  [1] and Reader  et al. [10]. A purely theoretical algorithm is presented in [1]. Animplementation and further development of the algorithmpresented in [1] is described in [2], but it is still far from beingimplemented for real clinical studies. In contrast, COSEMincludes many features (correction factors, exact probabilitycalculation, iterative Metz and Gaussian filtering schemes), isclinically approved and is now in routine clinical use.II. T HE  M ATHEMATICAL  M ODEL  A. The EM Algorithm Consider a body with a variable radiation density containedinthediscretizedcube .Let denotetheemissiondensity of each unit box (voxel) in the cube. Theradiation emitted by the body (pairs of photons produced bythe corresponding positron–electron annihilations) is detectedby the encirclingdetectors. If two photons are“simultaneously”detected by two detectors and if the line of response (LOR),defined by the two sets of detection coordinates, intersects thebody, we consider this as a coincidence acquisition (event). Inessence, all coincidence acquisition reconstruction algorithmstry to determine the unknown activity distribution in the cubegiven the list of detected events.The PET scanner is built of rows of detector rings wherebyeach detector is regarded as a discrete unit. All the events de-tected simultaneously by the same two detectors and arecollected in the single bin . Thus, each bin defines a singleLOR (a tube, in fact) and every event detected in this bin is as-sumed to have srcinated along this LOR.Let represent the voxels of the field of view (FOV) cube and let independent Poisson variables withunknown means represent the number of unknownobserved emissions in each of the voxels. Suppose that an 0278–0062/01$10.00 © 2001 IEEE  634 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 7, JULY 2001 emission from the voxel is detected in bin with proba-bility . The transition matrix(likelihood matrix) is assumed known. We observe the totalnumber of events detected by each bin and wish toestimate the unknown , . For each ,the observed data has the conditional probability (likelihood)(1)where is the mean of the Poisson variable , that is(2)The ML estimate of is(3)For numerical calculations it ismore convenient to work withthe  log-likelihood function (4)where is a constant which does not affect maximization. Theoptimality conditions for to achieve the maximum of (4) are, after simple some algebraic manipulations [9](5)where istheprobabilityofdetectinganeventemittedfromthe voxel , i.e.,(6)The EM algorithm can be viewed as an iterative algorithm forsolving the fixed-point equations (5) and (6)(7)  B. The Ordered Sets EM (OSEM) Using Projection Bins The OSEM algorithm [5] is a relatively straightforward mod-ification of the conventional EM in (7). The projection data aregroupedinto subsets,denotedby .Thesub-sets normally consist of projection views separated by somefixed angle about the object.In the OSEM algorithm, the log-likelihood fixed point equa-tions are iterated for each of the subsets; using the final result of the previous subset as a starting point.Therefore, an OSEM subiteration is(8)where is the current outer iteration and the current subset.The last subiteration in gives the starting point for the nextouter iteration : . This subiteration procedureis repeated until all subsets have been exhausted. Such a cycleis considered a single iteration of OSEM. The cycle is repeatediteratively until a satisfactory reconstruction is obtained. Usu-ally, 1–6 outer iterations (an outer iteration consists of a singlepass on all available data) and 16–32 subiterations/iteration (asingle pass on part of the data) are sufficient for a good qualityreconstruction.In this framework, EM is a special case of OSEM where thenumber of subsets is set to 1. For the case of noiseless pro- jections, it has been shown that each OSEM estimate, based ona subset of projections, converges as fast toward an ML solutionasafulliterationofEMusingallprojections[5].Inotherwords,if the projection data are divided into subsets, then once allprojections have been used in a single iteration of OSEM anestimate has been produced which is similar to m iterations of EM. It is this property that gives OSEM considerable accelera-tion compared with EM. It was found, however, that if the datais divided into too many subsets the results deteriorate as noiseincreases [5]. C. COSEM: An Ordered Sets EM Algorithm for List-Mode Data The list-mode data is a list of coincident gamma pairs that areserially stored in the chronological order of their registration. Itis possible to consider the sinograms as a dense representationof the registered data and the list mode as a sparse representa-tion of the same data. When the number of registered events islimited (several millions, say), the list mode representation ismuch more efficient compact than the sinogram representation.This is because the sinogram representation does not depend onthe number of recorded events but is fixed according to the ge-ometry of the system and can include hundreds of millions of bins. The sinogram element represents a single bin tube (LOR).If such a tube is infinitely thin then the probability of detectingtwo events in the same bin is negligible. The values of  in(7) are, in this case, either one or zero, where one denotes thata coincidence was detected in this particular LOR. It is easy tosee that in the EM algorithm, when 0, the whole termvanishes for this bin. Therefore, the EM algorithm becomes(9)In the COSEM algorithm, the above assumption is used. Thebins “thickness” is determined according to the detection reso-  LEVKOVITZ  et al. : COSEM: AN ITERATIVE ALGORITHM FOR FULLY 3-D LISTMODE DATA 635 lution and, therefore, there is no further rebinning of data. Thisallows us to perform the calculation of for each bin“on-the-fly” without compromising computational efficiency.This method has three major advantages:First, the algorithm uses the exact detection points and doesnot compromise the accuracy by binning the data. Second, thenumber of operations is dependent on the number of detectedevents. Third, there is no need to hold the (very large) four-di-mensional sinogram representation required for true 3-D recon-struction using binned data.The extension of (9) to ordered sets is straightforward andnatural whereby a subset is made of groups of events in thelist-mode data. Such a group represents, for example, acquisi-tion along during a certain period of time, randomly selectedevents, events acquired at certain angles, etc. Let be the listof events and let us split into sequential sets .The COSEM algorithm parallel similar to (8) is(10)where is the current outer iteration and the current subset.The last subiteration in gives the starting point for the nextouter iteration : .III. C ORRECTION  F ACTORS The successful implementation of reconstruction algorithmsrequiresthecorrectionofobserveddata,whicharedistorteddueto physical and geometrical phenomena that are the resultingfrom of factors like the machine geometry, physical properties,and the radiation process, etc. Most phenomena, detector effi-ciency for example, are normally corrected directly on the pro- jection set (random subtraction, attenuation weight multiplica-tion,smoothlow-passfiltering,etc.).Indeed,inmanyPETscan-ners, projection data are automatically pre-corrected during thecreation of the projection set. In the 3-D acquisition mode of the dual-head scanner, the projection set is not created. Thus,we must correct the influences of these phenomena either on in-dividual LORs or to take them into account when calculatingvoxel-detection probabilities.The probability for LOR detection depends on the relativesensitivity of the tomograph’s crystals at the actual detectionpoints, on the geometrical probability of detection, on the bodyattenuation, and on body and crystal scatter. The COSEM im-plementationdescribedinthispaperincludesanexactanalyticalcalculation of each voxel’s geometrical detection probability.Other phenomena are taken into account as a global weightassigned to the LOR, . A modified iteration scheme of COSEM is, thus(11)The assigned LOR weight, , is composed of three ele-ments: where includesnecessary corrections such as uniformity, rate-dependent eventloss, isotope decay, and washouts(12)represents a weight inversely proportional to the proba-bility of an event to be attenuated by the body. is the inter-section length of the LOR and the voxel, and is the map of registered attenuation coefficients.is a weight given to events according to the detectedenergy windows. This is a simple dual window method to sepa-rate scattered events from photopeak events. The scatter weightis currently calculated as: 1 when both photons areregistered in the high energy windows; 0.25 whenone of the photons is registered in the high energy window andthe other in the Compton energy window; otherwise,0. These values for where fixed empirically.Applying corrections to the LOR is an alternative approachto account for the same effects via the probability of detectionmatrix . It is easily seen that it is simpler and more com-putationally efficient to correct effects like attenuation via LORweights. It is equally easy to see that those LORs which willreceive the highest weights are of the lowest probability andthese are also those most likely to contain a large noise segmentcomponent. For example, correcting attenuation effects via thevoxel space requires the calculation of the probability of atten-uated emission for each voxel. If a projection set exists, thenthiscan beachievedbyanoperationcomputationallyequivalentto forward and back projection operations. With the absence of projections, this can be done by simulation of emission in allpossible directions; such a task can be computationally moredemanding than the entire reconstruction process.We have, therefore, developed a mixed approach wherebythe system-dependent corrections are calculated as voxel-detec-tion probabilities while other corrections, mainly body-depen-dent corrections, are still applied to LORs. calculation,including the geometric system corrections, is described in theSection iV.Thepresenceofrandomcoincidencesinthedataalsorequiresaspecialtreatment.Normally,therandomcoincidencesaresub-tracted fromtheprojectiondata setduring binning.Thismethodoften creates negative values in bins due to Possion statisticscount fluctuation and is, therefore, theoretically and practicallyproblematic for algorithms like the EM. In the absence of pro- jections, however, it is impossible to subtract randoms. Instead,we have devised a new method to handle randoms. The randomcoincidences, measured in a delayed time window, are used toreconstruct an alternative image. Using the property that thisimage is featureless, it is filtered This image is filtered witha strong low-pass filter and is subtracted from the real image,voxel by voxel.  636 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 20, NO. 7, JULY 2001 We also note that a possible method for body scatter correc-tion is to reconstruct the events in the body scattered dominatedenergywindowseparately, tolaterdeconvolvethereconstructedimage with the appropriate scatter point-spread function (PSF)and to add it to the srcinal image. This approach is not imple-mentedintheversionofCOSEMdescribedhereandwillbedis-cussed in a separate paper devoted to noise correction methods.IV. C ALCULATION OF THE  V OXEL  P ROBABILITIES  M ATRIX  A. Calculation of for the Dual Head Scanner  Adetectorinthedual-headscannerisbuiltoftwosurfaces.Themain one is the crystal, typically an  NaI   crystal with thicknessranging from around 9 mm (3/8 ″  ) to 25 mm (1 ″  ). On top of thiscrystalitiscommontomountagradedabsorber,madeofseverallayers of metal, to prevent low-energy photons resulting fromCompton scatter in the body or lead X-rays from reaching thecrystal.Assume, for simplicity, that the graded absorber is a singlelayer one. We can characterize the detector by the followinggeometrical parameters (see Fig. 1):where, , and distances from the center of rotation to thegraded absorber surface, crystal surface,and crystal end, respectively;half the detector size in the axial dimen-sion;half the detector size in the transaxial di-mension.Define for 0, 1 the two sets(13)The detectors in Fig. 1 are defined as the union of the graded absorber layer and the crystal layer. In other words, if is a rectangulardetector , then is the graded absorber layer andis the detector layer of the two detector plates . Finally,, defines the tomograph as a union of the tworectangular detectors and .Thegradedabsorberandcrystalarecharacterizedbythephys-icalattenuationconstants .Therefore,theseconstantsalsocharacterizethedetector layers , 0,1.During acquisition, the tomograph detectors revolve aroundthe central axis. Denote by the rotation function around theaxis on angle then(14) Fig. 1. Dual-head detector system: dimensions and geometry of the revolvingsystem. (top) Dimensions of the system. (middle) LOR emission angles—axialview. (bottom) LOR emission angles—transaxial view. Equation (14) defines the transformation between a line of therevolving coordinate system attached to the detectors and thestandard coordinate system.Let be the maximal axial detectionangle of the system. Consider the following subset of theunit sphere, parameterized by angles(15)Take a point : and con-sider a random vector , uniformly drawn on ,and a random angle uniformly drawn on the interval .Now, consider a LOR of an emission event(16)
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