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Iterative image reconstruction using geometrically ordered subsets with list-mode data Lucretiu M. Popescu, Samuel Matej and Robert M. Lewitt
Abstract— In positron emission tomography (PET), the format in which the data is stored has a major influence on the image reconstruction procedure. The use of the list-mode format preserves all of the measured attributes of the detected photon pairs but the events are stored in the order that they were measured, which allows only sequential access to the data. This fact limits the number of applicable algorithms and often computing speed or memory capacity constraints require the use of algorithms that do not make full use of the original precise information in the data. In this paper we show how through a change of the format in which the data is stored one can keep all the initial information about the individual events while providing random access to subsets of events belonging to given geometrical regions, thus making possible the use of maximum likelihood ordered subsets (OSEM) type algorithms with data provided as a collection of individual events (list-mode), and facilitating the adaptation of other types of algorithms. The structured data format also allows for more compact (compressed) storage of the information compared to the simple list-mode format.
new format, most often variants and relatives of the maximumlikelihood expectation maximization (ML-EM) algorithm [5]. However, of high practical interest are the ordered subset expectation maximization (OSEM) type (or block iterative) algorithms [6]–[8] due to their faster convergence, and this correspondence seems lost in this case because the list format does not provide random (indexed) access to the events in a given geometrical region. An approach that partially overcomes this drawback of list-mode data storage has been developed by Reader et al. [9] and Levkovitz et al. [10], in which subsets formed by taking a given number of consecutive events in the list are used. This approach is equivalent to using time ordered subsets instead of geometrically ordered subsets. In this paper we will present a method that through a change of the way the list-mode data is organized, enables the use of geometrically defined subsets of events in a similar manner as OSEM type algorithms are performing for binned data.
I. I NTRODUCTION Recording the relevant properties of each detected event in a list has become a common practice in emission tomography applications where it is known as “list-mode” data acquisition and storage. The reconstruction algorithms able to reconstruct images by directly using such a set of individual events, without binning them into a histogram of any type, are generically called list-mode reconstruction algorithms. An important class of such “list-mode” algorithms is based on maximum likelihood estimation of the image given the observed set of events. This is a consequence of the fact that the likelihood function can be defined for any given set of statistically independent events in a similar manner as it is defined for binned data, unlike other statistical estimators, such as the least squares method, which can be defined only for data given in a histogram form. This property (generally acknowledged in the statistics treatises [1]) was formalized to the particular case of emission tomography in [2]–[4]. Given this similarity between the likelihood function formulations for sets of individual events and for binned events the resulting algorithms for finding the maximum likelihood estimation for list-mode data can be seen, in general, as adaptations of the “old” versions developed for binned data to the The authors are with University of Pennsylvania, Department of Radiology, 423 Guardian Drive, 4-th floor Blockley Hall, Philadelphia, PA 19104-6021, e-mail: popescu,matej,robert @mipg.upenn.edu Paper presented at Nuclear Science Symp. Medical Imaging Conf., 2004, Rome, M9-211.
II. T HE
DETECTION STATISTICAL MODEL
Denote by the image space domain, is a point in image space, and by the data space domain, with the time dimension included, is a point in data space. The activity is discretely distribution, the object to be imaged, represented using basis functions , where are real parameters. The probability density of detecting an emission from volume element (voxel) with the attributes is . The total density of detected events is (1)
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01-.' 26 2:9 4 3 * 5 7 3 * 8 .' 2;6 " # 2:9 .73 * 5 -.' ? ( is the “true” events density, and where ' is the background counts density (randoms and scatters) which are events not included in the true coincidences model. The voxel sensitivity — the probability of an emission from / with any attribute @ADFE 2 is BC voxel DFE 0>-.' HtoGI .beThe9detected total number of 2 counts is JK HGIL " #%$'& BMN (O8 J , and P= 73 * 5 Q & 73 * is the detected.'counts distribution. For a set of R events SUTWV+YX[Z X$ &]\ ^ , the likelihood function (the probability density) of S given is _ .S`3 * abcNR3 * d ^ P= X 3 * fe (2) X$'& Qih Q is the probability of detecting R where bc.R3 * g ^kjlm events following a Poisson distribution. The log-likelihood 2
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The ordered subsets method consists in partitioning the data and performing in subsets , domain each iteration only with the events belonging to one subset . For " each subset the quantities have ! their counterparts . Three variants of the update equation for # -th iteration are
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where $ is a relaxation parameter. The subsets are selected according to a given rule. The equation (5) is equivalent with a relaxed version of the update equation used in the OSEM algorithm [6], the equation (6) is the equivalent of the update equation in RAMLA [7], while the equation (7) represents a power relaxed version of the original ML-EM update equation ( $98 ). Other versions of the update equation such that used in RBI algorithm [8] can also be derived. B. Subsets selection strategies
The domain can be expressed as the Cartesian product of at least two sub domains: a geometrical domain : and a temporal domain ; , =
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D. Computation of the attenuation array and sensitivity image The computation of the image element sensitivities represents practically the main difference between maximum likelihood image reconstruction from list-mode data and histogramed (binned) data. Because in the latter case in each iteration the algorithm steps through all concerned geometric positions the values can be automatically computed, while in the listmode case some geometrical positions can be skipped and others visited several times depending on how many events were recorded. Therefore in the list-mode case the voxels sensitivities must be estimated separately prior to the reconstruction. Because of the attenuation, the sensitivity image is object dependent. Analytically estimated or precomputed sensitivity images determined based on the detector geometry can be considered only if the attenuation is neglected in the projector model and corrected by using weights. This practice is generally undesirable because it may considerably alter the true events probabilities. In addition, the attenuation factors should be also precomputed and stored as an array spanning the data space, especially in the case of TOF where for one event one has to visit only the image elements intersected by a limited segment of the LOR. The above procedures, the computation of the attenuation factor array and of the sensitivity image, involve projection and back-projection through attenuation map and, respectively, the image. One way to speed-up this preliminary step, that otherwise may take as long as the reconstruction itself, is to combine the two projection and back-projection steps at the level of the attenuation map as follows. For each position in the attenuation array first project the ray through attenuation map and compute the attenuation factor T T ; then return and update the sensitivity values for all visited voxels in the attenuation map T ? . After the given dataT domain (the whole domain or one subdomain in the case of geometrically defined subsets) has been covered then convert the sensitivity image from the attenuation map format to the reconstruction image format by using a kernel interpolation scheme . The kernel interpolation step can be also used to smooth the initial preliminary sensitivity image in order to eliminate aliasing artifacts due to coarse sampling of the data and image spaces.
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IV. S TRUCTURED LIST DATA AND COMPACT A. Compact storage of event attributes Let us consider a PET event O
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STORAGE
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where is the position of detected photon , ( ) on the cylindrical surface of the detector, its energy, the number of scatters suffered (for simulated data); is the time of flight difference between the two photons, and time of detection. These parameters have values in the intervals min max , min max , min max , 2 D 2 4 , min max , min max . Denote by , , , , the precision required for storing the above data fields, which can be achieved by splitting each , , , , bins. Together of the above intervals in O O O with the scatter information we have & &
& O bins each event being described by the indices & & . U U U U Instead of storing each of the above indices in one byte or two bytes integral type fields (or with a fixed number of bits within a larger bitfield type) one can store a single integral type field large enough to be able to represent any number up to using a codification such as
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B. More compact storage In a previous paragraph we noted that, in general, parameters common to a large group of events do not need to be stored as fields attached to each individual event. This is true for parameters concerning the events within a given temporal range, and inserted as control codes. It should be also true for parameters common to events within a given geometrical region. Such common parameters can be easily found among the more significant bits (the more significant parts) of the position indices. In other words: instead of using a unique list where one needs long bitfields (long numbers) to represent the positions, split the domain in subintervals (corresponding to the more significant part of the position data) and save the events in separate sublists where one needs only short bitfields (short numbers) in order to accurately describe the positions. This procedure is analogous to that described in section III for grouping of the events in subsets. The sublists can be kept as distinct files or merged in a single file using interleaved blocks as shown in Fig. 2, or as an array header
list pointer table
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Fig. 2. Storage scheme for a list of events split into geometrically defined sublists and stored using interleaved blocks.
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For recording parameters that affect whole sets of consecutive events, control codes can be used. That is usually the case with frame number (patient bed position), or in another case can count minor time divisions while major time divisions are inserted in the list as control codes. In general, parameters that affect in bulk a large group of events do not need to be stored as fields attached to each individual event. As an example, let us consider a practical case with , , . , that , , gives . In this case, an address space of 64 bits seems enough for storage of the relevant properties of a PET-TOF event. It is important to note that storing the position parameters of a detected PET event (a LOR) as a quadruple of the O O form is not very compact in the case when many of such combinations are not occurring. A transformation to the usual three-dimensional sinogram format can prove to be more compact, in the sense that, one can achieve similar precision (in average) with , , , O O satisfying 8 . Other transformations, like O O with , , O O O O and , , or something O O intermediate like , might prove to be both compact and convenient to manipulate.
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Fig. 3. Data set size as function of & for a ')(+*,-,/.0*,21 events, data space size 34(65-187:9@?2A and 3B(C5-DE19@D2A respectively, with FG(6H and IJ(6K bytes. On the plotted LM curves are indicated the numbers of bits, in multiples of F , necessary for an event representation as function of & range. The continuous smooth curves correspond to the LN functions.
of lists. In general if the total data domain of size is split in O sub-domains the storage capacity required is PRQ O (12) VO 2W OTS 2U where the term O XW accounts for the overhead storage needs due to the access tables and additional control codes used. and W are expressed in multiples of bits. The optimal O is obtained when the minimum of
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Note that the theoretical O opt does not depend on . Fig. 3 shows the dependence of the storage size of O for two combinations that may occur in practice. The larger corresponds to time of flight, multi-frame and dynamic PET studies, while smaller can represent a more traditional case with no time of flight and without detailed photon pair energies recorded. For the larger the minimum size is obtained using 6 or even 5 bytes per event, giving a compression around 66% from the original list-mode size.
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C. Even more compact storage It has been shown [13] (see also [14]) that a moderate compression efficiency (2.9 bytes/event) can be achieved by applying the Lempel-Ziv algorithm — simply archiving with gzip — to an ordered list of events stored as unique index numbers obtained in a manner similar as in (10). A high compression (1.9 bytes/event) is realized if the index numbers are replaced with the differences between consecutive events,
while the original not ordered list shows poor compression rates. In all cases four bytes integer numbers were used, and the list contained & events. The high compression of the differential format storage is explained by the fact that the differences between consecutive events are usually small, in most cases being enough to represent them with only one byte, the remaining bytes in the storage word being redundant. In fact one can avoid employing a general compression method by using a flexible format of storage of these differences, for example by using the first seven bits from a byte and if those are not enough reserve the 8-th bit to signal overflow and use in a similar manner the consecutive bytes in the list until enough bits are obtained to cover the gap between events. This offers both high compression and fast coding and decoding. However, this format does not offer random access to subsets of events unless it is applied distinctly for events already partitioned in a convenient arangement of subsets. More importantly it requires ordering the events which is a time consuming operation of the order .
*
R GKAL R
V. C ONCLUSIONS The concept of data has a dual nature: on the one hand it refers to the information contained, which in our case consists of the attributes of the detected events and their precision. On the other hand it refers to the method of accessing this information, in the software engineering terminology, the data container, without which the information cannot be stored and manipulated. An algorithm sees a data set from this dual perspective, requiring specific information to be available through certain access methods. In the case of list-mode data, since it contains virtually all the relevant information provided by the scanner, it is the access mode — sequential access only — that dramatically limits the range of applicable algorithms, to the point that if other constraints are imposed, such as computing speed or memory capacity, the information cannot be efficiently used. We have shown how through a change of the format in which the data is stored one can keep all the initial information about the individual events while providing random access to subsets of events belonging to given geometrical and temporal regions. This makes possible the use of ordered subset type algorithms with data provided as a collection of individual events (list-mode), and facilitates the adaptation of other types of algorithms.
Furthermore the structured data format allows for more compact (compressed) storage of the information compared to the simple list-mode format. ACKNOWLEDGEMENTS The authors thank J. S. Karp, S. Surti and M. E. Werner for discussions on reconstruction from list-mode and time of flight data. Lucretiu M. Popescu is grateful to Joel Karp for his support. This work is supported by grants EB002131 and EB002135 from the National Institutes of Health, USA. R EFERENCES [1] W. T. Eadie, D. Dryard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics. Amsterdam, London: North-Holland, 1971. [2] D. L. Snyder and D. G. Politte, “Image-reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci., vol. 30, no. 3, pp. 1843–1849, 1983. [3] H. H. Barrett, T. White, and L. C. Parra, “List-mode likelihood,” J. Opt. Soc. Am. A, vol. 14, no. 11, pp. 2914 – 2923, 1997. [4] L. Parra and H. H. Barrett, “List-mode likelihood: EM algorithm and image quality estimation demonstrated on 2-D PET,” IEEE Trans. Med. Imag., vol. 17, no. 2, pp. 228 – 235, 1998. [5] R. M. Lewitt and S. Matej, “Overview of methods for image reconstruction from projections in emission computed tomography,” Proc. IEEE, vol. 91, pp. 1588–1611, October 2003. [6] H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imag., vol. 13, no. 4, pp. 601 – 609, 1994. [7] J. Browne and A. R. De Pierro, “A row-action alternative to the EM algorithm for maximizing likelihoods in emission tomography,” IEEE Trans. Med. Imag., vol. 15, pp. 687 – 699, 1996. [8] C. Byrne, “Block-iterative methods for image reconstruction from projections,” IEEE Trans. Imag. Proces., vol. 5, no. 5, pp. 792–794, 1996. [9] A. J. Reader, K. Erlandsson, M. A. Flower, and R. J. Ott, “Fast accurate iterative reconstruction for low-statistics positron volume imaging,” Phys. Med. Biol., vol. 43, pp. 835 – 846, 1998. [10] R. Levkovitz, D. Falikman, M. Zibulevsky, A. Ben-Tal, and A. Nemirovski, “The design and implementation of COSEM, an iterative algorithm for fully 3-D listmode data,” IEEE Trans. Med. Imag., vol. 20, no. 7, pp. 633 – 642, 2001. [11] A. J. Reader, R. Manavaki, S. Zhao, P. J. Julyan, D. L. Hastings, and J. Zweit, “Accelerated list-mode EM algorithm,” IEEE Trans. Nucl. Sci., vol. 49, no. 1, pp. 42 – 49, 2002. [12] L. M. Popescu and R. M. Lewitt, “Ray tracing through a grid of blobs,” in IEEE NSS-MIC Conf. Rec., 2004. M10-220. [13] S. Vandenberghe, I. Lemahieu, and R. Van de Walle, “Compression of sorted PET listmode data,” J. Nucl. Med., vol. 44, no. 5, p. 255P, 2003. [14] E. Asma, D. W. Shattuck, and R. M. Leahy, “Lossless compression of dynamic PET data,” IEEE Trans. Nucl. Sci., vol. 50, no. 1, pp. 9–16, 2003.
