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Exact and Approximate Rebinning Algorithms for 3-D PET Data Michel Defrise,* P. E. Kinahan, D. W. Townsend, Member, IEEE, C. Michel, Member, IEEE, M. Sibomana, and D. F. Newport
Abstract— This paper presents two new rebinning algorithms for the reconstruction of three-dimensional (3-D) positron emission tomography (PET) data. A rebinning algorithm is one that first sorts the 3-D data into an ordinary two-dimensional (2-D) data set containing one sinogram for each transaxial slice to be reconstructed; the 3-D image is then recovered by applying to each slice a 2-D reconstruction method such as filtered-backprojection. This approach allows a significant speedup of 3-D reconstruction, which is particularly useful for applications involving dynamic acquisitions or whole-body imaging. The first new algorithm is obtained by discretizing an exact analytical inversion formula. The second algorithm, called the Fourier rebinning algorithm (FORE), is approximate but allows an efficient implementation based on taking 2-D Fourier transforms of the data. This second algorithm was implemented and applied to data acquired with the new generation of PET systems and also to simulated data for a scanner with an 18 axial aperture. The reconstructed images were compared to those obtained with the 3-D reprojection algorithm (3DRP) which is the standard “exact” 3-D filteredbackprojection method. Results demonstrate that FORE provides a reliable alternative to 3DRP, while at the same time achieving an order of magnitude reduction in processing time. Index Terms—Image reconstruction, medical imaging, positron emission tomography, X-ray transform.
I. INTRODUCTION
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OSITRON emission tomography (PET) is one of the medical imaging modalities for which the transition from two-dimensional (2-D) to three-dimensional (3-D) acquisition has been most successful. Following pioneering work in extending PET to 3-D imaging [1]–[3], the development after 1989 of multiring scanners equipped with retractable septa [4]–[6] has led to the present widespread utilization of volume PET scanners [7]–[10]. These scanners have an open, collimator-less cylindrical geometry which allows the
Manuscript received May 8, 1996; revised November 14, 1996. The Associate Editor responsible for coordinating the review of this paper and recommending its publication was C. J. Thompson. Asterisk indicates corresponding author. *M. Defrise is with the Division of Nuclear Medicine, Free University of Brussels AZ-VUB, Laarbeeklaan 101, Brussels B-1090 Belgium (e-mail:
[email protected]). He is also with the National Fund for Scientific Research (Belgium). P. E. Kinahan and D. W. Townsend are with PET Facility, University of Pittsburgh Medical Center, Pittsburgh, PA 15260 USA. C. Michel is with PET Laboratory, Catholic University of Louvain, Louvain-La Neuve 1348 Belgium. He is also with the National Fund for Scientific Research (Belgium). M. Sibomana is with PET Laboratory, Catholic University of Louvain, Louvain-La Neuve 1348 Belgium. D. F. Newport is with CTI, Knoxville, TN 37921 USA. Publisher Item Identifier S 0278-0062(97)02488-9.
measurement of coincidences between all pairs of detectors on the cylindrical surface. These 3-D data approximate line integrals of the radioactive tracer distribution along lines of response (LOR’s) which are not restricted to lie within transaxial planes, in contrast with the 2-D data acquired when the scanner is operated in 2-D mode, with interslice septa. The transition from 2-D acquisition to 3-D acquisition leads to a significant improvement of the scanner sensitivity, due to the increased number of measured LOR’s and to the elimination of the detector shadowing by the septa. Usually, 3-D PET data are reconstructed using the 3-D reprojection algorithm (3DRP), a 3-D filtered-backprojection (FBP) method obtained by discretizing an analytical reconstruction formula and by estimating data not measured by the scanner [11]. Owing to the considerable number of LOR’s measured in 3-D mode, it is not surprising that the 3DRP algorithm is much more time consuming than the 2-D sliceby-slice FBP used to reconstruct data acquired in 2-D mode. A further reason for this increased complexity is that the reconstruction of the 3-D image is not decomposed into the reconstruction of a set of independent slices. Other algorithms relying on exact analytical formulas [12]–[15] have so far been unable to reduce reconstruction time by factors larger than two compared to the 3DRP algorithm. In contrast, significant improvements in the reconstruction speed have been achieved using various combinations of the three following approaches. The first one is the introduction of faster, but often expensive, hardware and the optimization of the implementation of the 3DRP algorithm [16]–[18]. The second approach uses a reduced sampling of the 3-D data to decrease the number of LOR’s which must be backprojected. Reduced sampling is achieved by adding groups of adjacent LOR’s in such a way that the resulting loss of spatial resolution remains acceptable for a given type of study. Finally, the third successful approach to faster 3-D reconstruction has been the introduction of approximate algorithms based on rebinning the 3-D data in plane integrals [19] or in a 2-D data set [20]–[22]. This paper presents new algorithms belonging to this latter family. A rebinning algorithm is defined as an algorithm which sorts (rebins) the 3-D data into a stack of ordinary 2-D data sets, where for each transaxial slice the 2-D data are organized as a sinogram. These rebinned data are equivalent geometrically to data collected in the conventional 2-D mode and can therefore be reconstructed by applying the 2-D FBP algorithm to each slice separately. Thus, rebinning decomposes
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the 3-D reconstruction problem into a set of independent 2D Radon transforms. Provided the rebinning procedure itself is efficient, reconstruction becomes almost as fast as in the 2-D mode, while retaining the increased sensitivity of 3-D acquisition because the complete set of 3-D LOR’s are used in the reconstruction. Rebinning requires a way to estimate the value of the direct LOR’s, which lie within a transaxial slice, from the measured oblique LOR’s which traverse several slices. The simplest way to do this is to neglect altogether the angle between an oblique LOR and the transaxial plane and to reassign that oblique LOR to the slice lying midway axially between the two detectors in coincidence. This is how 2-D data are acquired with multiring scanners operated with interslice septa, although in this case the presence of the septa restricts the angle between the LOR’s and the transaxial plane to small values. The extension of the same principle to 3-D acquisition leads to the single-slice rebinning algorithm (SSRB), which is an order of magnitude faster than the 3DRP algorithm [20]. As can easily be appreciated geometrically, the SSRB approximation is accurate only when the tracer distribution is concentrated close to the axis of the scanner [23]. Thus, accurate SSRB reconstructions have been obtained with the 16-ring ECAT 953B for receptor studies involving the human basal ganglia [24]. For Ffluorodeoxyglucose (FDG) brain imaging, the tracer distribution extends further transaxially, and larger distortions could be expected with the SSRB algorithm, especially when the brain is not well centered in the field-of-view (FOV). Even in this case, the differences with images obtained with the 3DRP algorithm are often minor because the axial aperture of many current PET scanners does not exceed 10 . However, other applications such as whole-body imaging, or brain imaging with wide aperture scanners [10] require either exact 3-D reconstruction or more accurate rebinning methods. An example of such a method is the multislice rebinning algorithm (MSRB), in which an oblique LOR contributes to the sinogram of all slices which it traverses [21], [25]. This algorithm is more accurate than SSRB, but is less stable in the presence of noise [25]–[27]. This paper introduces a class of rebinning algorithms based on a new exact inversion formula and describes in detail the implementation and performance of one particular algorithm [28], the Fourier rebinning algorithm (FORE). The substantial gain in reconstruction time achieved by this algorithm is particularly useful for applications where multiple 3-D data sets must be reconstructed as in dynamic PET studies or in whole-body imaging. The new inversion formula is derived in Section II and leads to an exact rebinning algorithm related to the direct 3-D Fourier reconstruction method [13]. This new algorithm is expected to be faster than the 3DRP algorithm, but its numerical complexity remains high compared to the SSRB algorithm, particularly because the unmeasured projection data which are truncated axially must be estimated as in the 3DRP algorithm. Therefore, this paper focuses on approximate but faster rebinning algorithms which are obtained by expanding the exact rebinning formula in powers of the angle between the LOR’s and the transaxial plane. The lowest order approximation, setting , corresponds to the
IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 16, NO. 2, APRIL 1997
Fig. 1. Geometry of a cylindrical PET scanner. Transaxial view (left) with the standard sinogram variables s and parameterizing the straight line AB. Longitudinal section (right) with the axial variables z and . Note that the x and y axis have been rotated for illustrative purposes.
1
SSRB method. The next approximation, which includes the terms linear in , yields the FORE method. This algorithm is derived in Section III, and an enlightening geometrical interpretation is given in terms of the frequency-distance relation [29]–[31]. The implementation of FORE for multiring scanners is described in Section IV. The performance of the FORE algorithm is illustrated in Sections V and VI using data acquired with the Siemens/CTI ART and 961 (ECAT EXACT HR) scanners, as well as simulated data for a small diameter 32-ring scanner with an axial aperture of 18 . II. AN EXACT REBINNING METHOD FOR 3-D RECONSTRUCTION A. Sinogram Data for a Cylindrical PET Scanner We consider a cylindrical scanner of radius and length , with its axis along the axis which defines the axial direction. Any plane orthogonal to the axis is called a transaxial plane. The distribution of the radioactive tracer is represented by a defined in a cylindrical FOV with a function radius and with the same axis and length as the scanner. We assume for simplicity that , although this condition is more restrictive than necessary. is measured along The integral of the function all straight lines (called lines of response, or LOR’s) joining two detectors on the lateral surface of the scanner, and the reconstruction problem consists in estimating from these data. The line integrals can be parameterized in different ways, which are mathematically equivalent as long as continuous data are assumed, but which result in different data sampling patterns. Although the natural sampling for a multiring PET scanner is a fan-beam or a cone-beam scheme [32], [33], commercial scanners reorganize the data either as a set of 2-D parallel projections, or as a set of oblique sinograms. We consider in this paper this latter sampling scheme, in which the line integral between two detectors and is parameterized
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as (Fig. 1)
(1) where integration
is a unit vector along the line of
(2)
are defined as follows. and the four variables , , , and The variables and are related to the axial coordinates and of the two detectors in coincidence by
and
Thus, is the axial coordinate of the point midway between the two detectors and is the axial spacing between the two detectors. For a fixed value of the pair ( ), the data are parameterized by the same variables and as in an ordinary sinogram: is the distance between the axis and the projection of the line onto a transaxial plane; is the angle between this projection and the axis (Fig. 1). Therefore we define the set of data corresponding to a fixed pair ( ) as an oblique sinogram. When , the sinogram is called a direct sinogram. For a PET scanner with rings, each pair of rings corresponds to a fixed pair ( ), and hence the data acquired in 3-D mode consist of sinograms, in which are included direct sinograms and oblique sinograms. The range of the four variables are , and . Due to the finite length of the cylindrical scanner, the range depends on and does not cover the whole FOV, except for the subset. This reduction in the range with increasing is referred to as the truncation of the 3-D data. The new rebinning formula derived below is based on a slightly different parametrization defined by
(3) where is the tangent of the angle between an LOR and the transaxial plane, and where the variable of integration is now along the projection of the LOR in the transaxial plane at axial position [instead of along the LOR, as in (1)]. In a multiring scanner the variable is proportional to the difference between the indexes of the two rings in coincidence and, therefore,
Fig. 2. The principle of a rebinning algorithm, illustrated for the slice “direct” slices lying sampling typical for existing multiring scanners, with in the plane of the detector rings and 1 “cross” slices lying midway, axially, between adjacent detector rings.
N
N0
N
with a small abuse of terminology, will be called the ring difference. The range of the axial variables are and .1 Building a set of data parameterized according to (3), from the measured data corresponding to (1) requires a one-dimensional (1-D) interpolation across different oblique sinograms. However, when , the angle is approximately constant in a given oblique sinogram and the resampling reduces in this case to a weighting by . With the ECAT EXACT HR, for instance, the variation of in a sinogram does not exceed 1.4 for a 40-cm FOV, and this variation is neglected in most implementations of the 3DRP algorithm (even though it may in some cases exceed the angular data sampling interval). We assume below that the resampling has been achieved, or that it can be neglected. This paper addresses the problem of solving the integral (3). B. What is a Rebinning Algorithm? When a scanner is operated in 2-D mode, the measured LOR’s are restricted to lie in a transaxial plane, so that . Therefore, a 2-D data set is also described by (3), but with (4) These 2-D data contain, for each slice , one ordinary (or direct) sinogram which can be reconstructed by 2-D FBP, independently of the other slices. Not surprisingly, this slice by slice reconstruction of a three-parameter data set is considerably faster than the reconstruction of the four-parameter 3-D data set with the 3DRP algorithm. This observation suggests an alternative approach to 3D reconstruction, in which the 3-D data are not directly reconstructed, but serve to estimate 2-D data from which the image can then be recovered using any 2-D reconstruction algorithm such as FBP (Fig. 2). Therefore, we define a rebinning algorithm as a method to estimate
j j
0j jp 0
can be seen from (3), the actual range is z L=2 R 2 s2 and depends on the variable s. For simplicity we use the maximum range that is independent of s. 1 As
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from . A useful rebinning algorithm should be as follows: 1) fast (otherwise there would be no advantage with respect to other techniques); 2) accurate and ideally based on an exact analytical inversion formula; 3) as stable with respect to noise as the 3DRP method. Note the importance of this last requirement. Noise-free 3-D PET data are redundant in the sense that the direct sinograms, i.e., the sinograms with , are sufficient for an exact reconstruction. Hence, without requirement three, rebinning could be achieved trivially by extracting from the 3-D data the direct sinograms, according to (4). While valid for noise-free data, this approach would not take advantage of the increased sensitivity achieved by a 3-D acquisition. An optimal signal-tonoise ratio (SNR) in the reconstructed image can be obtained only if the rebinning method incorporates the whole 3-D data, as does the 3DRP algorithm. C. Derivation of an Exact Rebinning Formula The inverse problem defined by the integral (3) is invariant for rotations around the axis, and also for translations along the radial variable . These invariances can be exploited by calculating, for each pair ( ), the continuous Fourier transform of the oblique sinogram with respect to the variable and the Fourier series with respect to the azimuthal angle
(5)
to . The 3DRP algorithm overcomes this axial truncation by initially estimating the data for a larger range covering the whole FOV, that is [recall that here represents the axial coordinate of the midpoint of the LOR’s in the sinogram ( )]. This is done by forward projecting an initial image estimate reconstructed from the direct sinograms only. Similarly, we assume here that the data have been extrapolated to all . Invariance for axial translations is thereby restored, and we take the Fourier transform of (8) with respect to
(9) where is the axial frequency and is the 1-D axial Fourier transform of written with cylindrical coordinates
(10) In (9) the integration variable is no longer present in the argument of , and hence can be factored out of the integral. Using the standard integral representation of Bessel functions [34] one has
is the radial frequency and the where Fourier index. As the oblique sinograms are sampled only for , (5) is calculated in practice using the symmetry (11)
(6)
where the arctan is defined in the interval Putting (11) into (9) yields
.
Substituting (3) into (5) one has
(7)
(12)
which can also be written using cylindrical coordinates as The new exact rebinning formula is a direct consequence of (12). Indeed, noting that the second exponential factor as well as the square root are independent of the integration variables and , and rewriting (12) for the particular case of the direct sinograms with (8) Owing to data truncation, (3) is not invariant for axial translations because the measured range of the variable is limited
(13)
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yields the following relation between the 3-D Fourier transforms of oblique and direct sinograms:
(14)
An equivalent formula is obtained by swapping the left- and right-hand side and by replacing by
(15) where the arcsine is defined in the interval [ ].2 Equations (14) and (15) are the main theoretical results of this paper. D. The Truncation Problem: An Implementation of (14) For each value of the ring difference satisfying the condition , (15) provides an independent estimate of . In the absence of noise all these estimates are equal, but when the data are noisy the SNR is increased by averaging all estimates. Thus, the rebinned 2-D data are calculated in (16) shown at the bottom of the page, where (17) and
is the 3-D Fourier transform of the 2-D data, (4)
(18) Only positive values of appear in (16) because the sinograms with negative have been merged when calculating the 2-D Fourier transform (6). An attractive property of (16) is that it only requires a 1-D interpolation along the radial frequency , and a complex multiplication with the exponential factor. On the other hand, a major problem is the assumption in Section II-C that is available for all . As the 2 A remarkable, though purely formal, identity exists between (15) and the solution to the 2-D attenuated radon transform given by Bellini [35] and Metz and Pan [36, eq. 24]. The link is seen by considering (15) for a fixed !z , and by taking i=at , where at is the constant attenuation coefficient.
=
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cylindrical scanner only covers the range , implementation of the exact rebinning formula is possible only if one first estimates over the unmeasured range . Following the same approach as in the 3DRP algorithm, this can be done by using the 2-D data subset corresponding to . The following algorithm results. Algorithm 1—Exact Rebinning: 1) Initialize a stack of rebinned Fourier transformed sinograms . 2) Calculate the 3-D fast Fourier transform (FFT) of the sinograms . This will be measured used to estimate the truncated data. 3) Consider sequentially each value of : a) calculate the 2-D FFT of the data with respect to and to get on the measured range ; from b) estimate using the result of Step 2 and (14); c) take the inverse 1-D FFT of to get an estimate of range ;
with respect to over the missing
d) merge the result of 3a) and 3c) to get on the full range ; e) calculate the 1-D FFT with respect to ;
to get
), use (14) to increment the f) for each sample ( rebinned data with the appropriate values of . This step involves a 1-D interpolation in ; 4) Normalize the rebinned data to take into account the variable number of contributions to each frequency component . The normalization factor is equal to in (17). 5) Take the inverse 3-D FFT of to obtain the stack of rebinned sinograms 6) Reconstruct each slice using any 2-D reconstruction algorithm. This algorithm [cf. Step 3f)] can be viewed as a “data driven” implementation of the integral in (16), and the Steps 3b)–3d) correspond to the reprojection step in the 3DRP algorithm. The numerical complexity is comparable to that of a direct 3-D Fourier reconstruction [13] with the added advantage that it only involves 1-D interpolations. Nevertheless, the numerical complexity remains considerable compared to SSRB, and the actual implementation is beyond the scope of this paper. Rather, considering the need for fast algorithms, the rest of the paper concentrates on approximate but more efficient rebinning methods.
(16)
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III. APPROXIMATE REBINNING ALGORITHMS The aim of this section is to derive from the exact rebinning (14) approximations which lead to fast implementations. The main result will be the Fourier rebinning equation (28). First define the variable
(19) and are the transaxial and axial frequencies, where respectively, and is the angle between the LOR’s and the transaxial plane. The exact rebinning (14) can be written as (20) where , as before, is the 3-D Fourier transform of the data, and we have defined the phase shift (21) and the frequency scaling (22) A class of rebinning algorithms of increasing accuracy can be obtained by considering truncated Taylor expansions in of the phase shift and of the frequency scaling factor. The algorithms incorporating terms up to , , and are discussed below. A. Zeroth-Order: The SSRB Approximation
As the only dependence on the axial frequency is a linear phase shift, it is possible to calculate the inverse 1-D Fourier transform of (27) with respect to (28)
Equation (28) is the Fourier rebinning approximation [28], [37], [38]. It relates the 2-D Fourier transform of an oblique sinogram ( ) to the 2-D Fourier transform of the direct sinogram of a slice shifted axially by a frequency-dependent offset . The implementation of (28) will be discussed in Section IV, but two important properties should be stressed. a) Implementation only involves a 1-D interpolation in . No interpolation is needed in the frequency variables and . b) In contrast with the exact rebinning based on (14), Fourier rebinning does not require taking an axial Fourier transform with respect to . This considerably simplifies the implementation, not only because of the smaller number of FFT, but more importantly because the data need no longer be known for all values of , and hence the truncated data need not be estimated. C. Second-Order: Estimating the Accuracy of Fourier Rebinning The accuracy of Fourier rebinning can be estimated by calculating the second-order approximation. Using (25) and
The zeroth-order approximation is found by setting in (20)
(29) (23)
one obtains
In this limit the inverse 3-D Fourier transform with respect to , , and can be calculated explicitly (24) Equation (24) is the SSRB approximation [20]. Each oblique sinogram ( ) is an estimate of the direct sinogram ( ), and therefore the rebinned sinogram for a given slice is calculated as the average of the oblique sinograms ( ), for all available values of . B. First-Order: Fourier Rebinning Approximation The next approximation is obtained by keeping terms which are linear in . Noting that (25) and (26) one obtains
(27)
(30) Taking the inverse axial Fourier transform and noting that multiplication of the Fourier transform by is equivalent to a second derivative with respect to , one obtains
(31) as before. The second term in the where right-hand side of (31) is a correction to (28), and its magnitude is an estimate of the accuracy of the Fourier rebinning approximation. The behavior of the correction term at low frequencies can be understood by considering the consistency condition
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for the 2-D Radon transform, which is expressed in terms of the 2-D Fourier transform of a sinogram ( ) as (32) is the radius of the FOV [33, Sec. III-3]. Equation where (32) shows that for large values of the consistency condition forces the data to be zero precisely in the low region where the Fourier rebinning approximation would be invalid due to the dependence in the correction term in (31). Therefore, the correction term is significant only in the region of the ( ) plane where both and are small. Three other properties are apparent from (31). 1) The Fourier rebinning is exact if depends linearly on . In this case, is also linear in and hence its second derivative in (31) vanishes. The same property holds for the SSRB algorithm. 2) The second term in the right-hand side of (31) is proportional to the square of the ring difference [recall ]. The Fourier rebinning approximation is therefore expected to break down when the axial aperture of the scanner becomes too large. In this case, the exact rebinning algorithm in Section II-D should be used instead of Fourier rebinning. 3) Actually calculating the correction to improve the accuracy of the reconstruction is probably impractical because the second derivative with respect to would strongly amplify noise. D. Geometrical Interpretation of the Fourier Rebinning Equation The Fourier rebinning equation has been derived in Section III-B by approximating an exact inversion formula, (14). The original derivation of Fourier rebinning [28] was based on the frequency-distance relation, a property of the 2-D Radon transform discovered by Edholm et al. [29]. This alternative derivation provides an instructive geometrical interpretation of Fourier rebinning and is summarized in this section. We first introduce the frequency-distance relation for an oblique sinogram ( )
(33) of this Consider the 2-D Fourier transform sinogram, calculated according to (5). The frequency-distance relation states that the value of at the frequency ( ) receives contributions mainly from sources located at a fixed distance along the lines of integration. Note that is the integration variable in (33) and represents the signed distance measured on a transaxial projection of the LOR, from the midpoint of the LOR. This remarkable approximate property can be viewed as a “virtual time of flight principle” since it provides information about the location of a source along the LOR’s, something that is not possible in non timeof-flight tomography. The price to pay is that this information
Fig. 3. Geometrical interpretation of Fourier rebinning: a virtual source S corresponding to a frequency component (!; k ) is at a transaxial distance t from the midpoint of the LOR AB. The axial coordinate of the source can then be calculated as z 0 z + t tan ().
=
is obtained only after mixing all measured LOR’s when taking the 2-D Fourier transform. Applications of the frequencydistance relation can be found in [29]–[31]. Let us also mention that the consistency condition (32) is a corollary of the frequency-distance relation. For, if the activity is contained within a FOV of radius , then the distance of any source along any LOR cannot exceed , and hence the frequencydistance relation implies that the Fourier transformed data must vanish when . Applications of this property can be found in [39] and [40]. We now apply the frequency-distance relation to the rebinning of 3-D PET data. Consider the contribution of a point source to an oblique sinogram ( ), and assume that the transaxial position of along an LOR is known. Then, using the known angle between that LOR and the transaxial plane, the axial position of can be obtained as
(34) where is the axial coordinate of the point midway along the LOR (Fig. 3). The contribution the source would make for can then be rebinned into a direct sinogram ( ). This method appears impractical because no unique value of can be associated with an LOR. However, after taking the 2-D Fourier transform of the sinogram, the frequency-distance relation gives an estimate of for each “virtual” source corresponding to a frequency ( ). Application of (34) then immediately leads to the Fourier rebinning (28). IV. THE FOURIER REBINNING ALGORITHM (FORE) The main result of the previous section is the relation (28) between the 2-D Fourier transform of an oblique sinogram and the 2-D Fourier transform of a direct sinogram. This section describes the Fourier rebinning algorithm (FORE) based on this relation. The principle of the method is first presented in Section IV-A, then implementation details are given in Sections IV-B and IV-C, and the section concludes with a step by step description of the algorithm in Section IV-D.
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where is determined by the requirement that the oblique sinogram ( ) be measured
(38)
is the radius of the detector ring and is the where largest ring difference included in the reconstruction (the maximum possible value of is ). Only positive values of appear in (37) because the sinograms with negative have been merged when calculating the 2-D Fourier transform (6). In the second high-frequency region 2 the consistency condition (32) is not satisfied and hence Fig. 4. One quadrant of the (!; k ) space, and its partitioning into three subregions. The Fourier rebinning, the consistency condition, and the SSRB are used in regions 1, 2, and 3, respectively.
A. Principle of the Algorithm A rebinning algorithm reconstructs 3-D data by first estimating a 2-D data set containing one direct sinogram for each transaxial slice. As explained in Section II-B, the rebinned data are represented by
or (39) Finally, in the low-frequency region 3, Fourier rebinning is not applicable. Therefore, in this region, the rebinned data are estimated using only the oblique sinograms with a small value of . Owing to the small value of the axial shift can be neglected as in the SSRB approximation, and hence the following estimate is used:
(35) and where and are the usual sinogram variables (Fig. 1) and is the axial coordinate of the slice. The FORE algorithm estimates the 2-D Fourier transform of the rebinned sinogram
(36) As we have seen in Section III-B, Fourier rebinning is based on a high-frequency approximation. Therefore, it is necessary to handle separately the low and high frequencies. This is done by subdividing the ( ) plane into three regions (Fig. 4) defined by two parameters and , and by applying in each region a different method to estimate . In the high-frequency region 1 the rebinned data are estimated using Fourier rebinning. For each value of , (28) provides an independent estimate of . In the absence of noise all these estimates are equal, up to the accuracy of the Fourier rebinning approximation. However, with noisy data the SNR is increased by taking the average of all estimates. Thus, the rebinned 2-D data are calculated as
or (37)
(40)
where . The parameter must be small enough to minimize systematic errors due to the use of the SSRB approximation. Typically, one selects the same as the one which defines the data subset used to estimate the truncated data in the 3DRP algorithm. The algorithm defined by (37)–(40) depends on three parameters , and , which must be chosen so as to ensure a good compromise between the systematic error (accuracy) and the statistical error (noise) in the reconstructions. The possibility to find a set of parameters allowing an image quality comparable to that obtained with the 3DRP algorithm depends on the two following observations, based on a large number of simulations. 1) Although the Fourier rebinning approximation is in principle an asymptotic relation, valid only at high frequencies, it turns out to be surprisingly accurate even at very low frequencies for the PET scanner geometries considered in Section V. 2) The fact that the low-frequency components in region 3 are calculated using a small subset of the available data ( ) has little influence on the SNR because noise in a tomographic reconstruction arises mainly from high-frequency amplification by the ramp filter. Small values of and may therefore be chosen. Thus, with scanners such as the ECAT EXACT HR, good and reconstructions have been obtained by taking where is the frequency sampling. However, there has been no systematic optimization of the parameters
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, and , and more generally, of the partitioning of the ( ) plane. The rest of this section gives details on the implementation of FORE. B. Organization of the 3-D Data The FORE algorithm requires the 3-D data, (3), to be organized as a set of oblique sinograms, each of which is characterized by a pair ( ). The algorithm processes each sinogram independently of the other, and its implementation is therefore independent of the way in which the variables and are sampled. We describe in the rest of this section the particular sampling schemes used by the ECAT scanners described in Section V. Consider a -ring scanner3 with a spacing between adjacent rings. This scanner collects one oblique sinogram for each pair of rings ( ), where the ring indexes and run between zero and . This generates the following sampling of and : where
(41) is the axial angular sampling and where is the axial coordinate of the center of the first ring. The parameter determines the maximum value of in the acquired data. Equation (41) represents the standard sampling scheme on the first generation of volume scanners. To gain both memory and reconstruction speed, the scanners used in this study acquire 3-D data with a reduced axial sampling. The rationale is that, for many PET studies where is not large, the axial angular sampling is finer than required to satisfy Shannon’s sampling condition. Therefore, the following sinogram sampling is used:
(42) is an integer parameter called the span, and . Each discrete sample ( ) is obtained by summing the range of LOR’s about defined by where is defined in (41). The number of LOR’s summed in this way—hence the data compression factor—is approximately equal to . An example is shown in Fig. 5. Note finally that data sampled according to (42) can be organized on disk or in memory either as oblique sinograms characterized by a pair ( ), or as 2-D parallel projections
where
N
Fig. 5. Schematic representation of the 3-D data acquired with a 16-ring scanner. The vertical and horizontal axis correspond, respectively, to the indexes j and i of two rings in coincidence. Each square corresponds to one oblique sinogram (i; j ) in the sampling scheme of (41). In the sampling scheme of (42), sets of oblique sinograms linked by the diagonally oriented line segments are added together. The example shown is for a span S = 5 and dmax = 12.
N =
where
3 Here refers to the number of rings of detector elements, which is 24 for the ECAT scanners in Section V, and not to the number of rings of detector blocks, which is three, with each block divided axially into eight detectors.
characterized by the angular variables ( ). The latter method is used by recent scanners because the 3DRP algorithm requires data in this format. In this case, the application of FORE requires a preliminary resorting of the 3-D data into oblique sinograms. C. The 2-D Fourier Transform The discrete 2-D Fourier transform ) is calculated by grouping pairs of oblique sinograms with opposite values of (6). Each oblique sinogram matrix has radial samples and angular samples, with
where is the number of detectors/ring and the mash factor is the integer factor by which the sampling rate has been reduced in the azimuthal direction . The rationale for this azimuthal sampling rate reduction is the same as for the axial sampling rate reduction in (42). The FFT algorithm is efficient only when the dimensions and are equal to powers of two, or at least to the product of a power of two by some small factor ( 10, say). does not satisfy this constraint, the sinogram matrix can If be padded with zeros to form a larger array with a dimension equal to the power of two nearest to . Unfortunately, zero padding is not possible in the azimuthal direction because the sinogram is periodic in and the transform is a Fourier series. To solve this problem, our implementation of FORE linearly interpolates the samples to obtain a sinogram with
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a dimension equal to the nearest power of two larger than . The rebinned sinograms are then built with the same number of samples . D. The Algorithm Discretizing the integral in (37) using a standard quadrature would not allow each oblique sinogram to be processed independently. Therefore, the implementation of FORE follows the data driven approach described below. For simplicity we have kept the same names for the discretized variables, and the sampling of , , , and is implicit in all expressions. Algorithm 2—FORE: 1) Initialize a stack of rebinned 2-D Fourier transformed sinograms . 2) Consider sequentially each pair of oblique sinograms ( ) and ( ): a) if needed, pad in and interpolate in to get appropriate dimensions and ; b) using (6), calculate the 2-D FFT with respect to and to get ); ) in region 1 ( c) for each sample ( or ) calculate (one easily checks that ) use linear interpolation to increment the two closest sampled slices : add to add to ; , consider each sample ( ) in region 3 d) if ( and ) add to . 3) Normalize the rebinned data to take into account the variable number of contributions to each sample . The normalization factors are determined by applying procedure 2c)–2d) to unit data . 4) Take the inverse 2-D FFT of to obtain the stack of rebinned sinograms 5) Reconstruct each slice with any 2-D reconstruction algorithm. This algorithm can be viewed as factoring 3-D backprojection into a 1-D axial backprojection [Step 2c)] and a 2-D transaxial backprojection [Step 5)], with intermediate data consisting of the rebinned sinograms . Contrary to a real 2-D data set, these intermediate data are not distributed as independent Poisson variables, although their statistical properties (covariance matrix) could in principle be calculated using the normalization factors in Step 3). In Section V the 2-D rebinned data are reconstructed using 2-D FBP with an optional axial smoothing. For low count rate studies, however, Step 5) could also be implemented using iterative methods such as, for example, the ordered subsets expectationmaximization (OSEM) algorithm [41]. V. ILLUSTRATIVE EXAMPLES In this section, we report on two areas where the FORE algorithm is expected to provide the most significant speed advantage compared to the 3DRP algorithm, even when using
the array processors now standard on most commercial PET scanners. The two application areas are high resolution dynamic brain imaging, where multiple time frames (typically 10 to 25) are used to follow the time course of neuroligands, and whole-body oncology imaging, where multiple bed positions (typically 4–10) are used to image extensive regions of the body. We also investigate the applicability of FORE to future generations of volume-imaging PET scanners with large axial apertures. For all studies with measured data, we used the sampling scheme defined by (42). After normalization and correction for attenuation and scatter (using standard ECAT software) the data were reconstructed using either the 3DRP method, or first rebinned with FORE and then reconstructed with the standard 2-D FBP algorithm. It should be noted that the goal of these studies was to investigate the equivalence of images reconstructed by 3DRP and 2-D FBP after FORE rebinning. To reconstruct images with the 3DRP algorithm, a parallel implementation for an array of i860 processors was used, while for the images reconstructed with FORE, a nonoptimized implementation for SUN Sparc stations was used. A. High-Resolution Brain Imaging To illustrate the FORE algorithm we performed two studies on an ECAT EXACT HR scanner [8], which is well-suited for high-resolution dynamic brain imaging. The EXACT HR is a 24-ring scanner with a diameter of 82 cm and an axial FOV of 15 cm. Each ring has 784 detectors providing a radial sinogram sampling 1.65 mm, and the ring spacing is 6.25 mm. For the first study a line source (length 25 cm, internal diameter 1 mm) containing a small amount of Ffluorodeoxyglucose ( FDG) was placed, in air, in a central transaxial section, and positioned radially through the axis of the scanner with one extremity at 20 cm from this axis. Data were acquired with a span and , corresponding to a maximum value of the angle . The raw sinograms were acquired with and . For the FORE reconstruction, the sinogram was interpolated in the azimuthal direction as explained in Section IV-C, to obtain a sinogram with and . The reconstructions were done without apodizing in the axial or transverse directions and with a pixel size of 1 mm. The full-width half-maximum (FWHM) was determined at 1-mm intervals along the line by fitting a Gaussian function to the tangential and axial profiles through the reconstructed line source. For the second study, we extracted the last 10-min frame of a 60-min dynamic FDG brain scan (where 10.1 mCi were injected), acquired with a span and . This frame contained about 127 million coincidences. The sinograms were acquired with and , and were corrected for scatter and for attenuation (using measured transmission data). For the FORE reconstruction, the sinograms were interpolated in the azimuthal direction to . Data were reconstructed with 2-mm pixels, using a Hamming window with a cutoff at 0.24 mm , both in the transaxial and axial directions.
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B. Whole-Body Imaging To evaluate the FORE algorithm for use with whole-body imaging, we performed a study on an ECAT ART scanner, which is designed for cost-effective clinical imaging. The ART scanner consists of two rotating banks of detectors, and is geometrically equivalent to the ECAT EXACT, a 24-ring scanner with 384 detectors/ring . The ring spacing is 6.75 mm and the radial sinogram sampling 3.375 mm. Data are acquired with the standard sampling scheme defined by (42), with and . This corresponds to . Using the ART scanner we measured a torso phantom containing lung (air) and spine (teflon) inserts, and three pairs of hot and cold spheres having volumes of 5 cm , 2.5 cm , and 1.25 cm . The main chamber of the phantom contained 0.8 mCi of FDG, and the activity in the hot sphere inserts was a factor of 3.5 larger than the activity in the main chamber. There were 74 million coincidences (including scatter) collected during a 30 min acquisition. The data were corrected for attenuation and scatter using a 60-min transmission scan. C. Large Aperture Scanners The FORE method is potentially less accurate at larger maximum ring differences, as indicated by (31). Since several volume PET cameras with axial FOV’s of 24 cm or greater have been recently developed [10], we used simulation studies to test the applicability of FORE to scanners with large axial apertures. We simulated a 32-ring scanner with a diameter of 49 cm and an axial FOV of 20 cm, with the sampling scheme defined by (41), with 3.0 mm, 6.25 mm, , and . Reconstructions were done with a maximum ring difference , which corresponds to an aperture of 18 . Data were simulated for a line source defined as a cylinder of length 380 mm and diameter 3.6 mm, positioned with its axis oriented radially in the central transaxial section, and reconstructed without apodizing in the axial or transverse directions and with a pixel size of 1 mm. The effects of attenuation and scatter were not included in the simulation as the goal of these studies was to determine the equivalence of images reconstructed by 3DRP and 2-D FBP after FORE rebinning. To study noise properties, data were also simulated for a uniform cylinder (diameter 20 cm) extending throughout the axial FOV. Noise free data were first calculated as line integrals of the distribution, and Poisson noise was then added, corresponding to a total of 100 million coincidences. VI. RESULTS A. High-Resolution Brain Imaging Fig. 6 presents the transaxial tangential resolution and the axial resolution for the line source measured with the EXACT HR scanner. The structures in the graphs are likely due to
Fig. 6. Tangential and axial resolution in a central slice of the ECAT EXACT HR, for the 3DRP and FORE algorithms. The FWHM of a reconstructed line 0 is plotted versus the radial distance from source placed in the slice z the axis of the scanner.
=
sampling effects, and to small nonuniformities of the width of the line source, which was not deconvolved from the measured profile because we are only interested in comparing the two algorithms. The data demonstrate a small progressive degradation of the axial and transaxial resolution with the FORE algorithm at increasing distances from the axis of the scanner. This loss of resolution does not exceed 0.50 mm for a FOV of 38 cm, and this conclusion was also supported by inspecting the tails of the reconstructed profiles. Note that the line source was located in the plane where the axial aperture, and hence the potential error introduced by FORE, is greatest. Fig. 7 illustrates one central transaxial slice (27 out of 47) of the FDG brain scan acquired with the EXACT HR scanner, and demonstrates the good agreement between the two algorithms. In the absence of a “gold standard” however, it is impossible to analyze in detail the origin of the residual differences between the two images, which are caused partly by the approximate character of the FORE algorithm, and partly by the fact that the two algorithms have different noise behavior, in particular near the edge of the scanner. The histogram of the relative difference for the individual voxels indicates the level of quantitative agreement which can be expected in practice.
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(a)
(c)
(b)
(d)
(a)
(b)
(c)
(d)
Fig. 7. A central transaxial slice of an FDG brain scan reconstructed with (a) 3DRP and (b) FORE. (c) The absolute value of the difference between the two slices is displayed using a linear gray scale scaled to the range [0%, 10%] of the slice maximum, with the 30% contour level from the 3DRP reconstruction. (d) The histogram of the difference between the two slices in percent of the slice maximum.
Fig. 8. A central transaxial slice of the torso phantom reconstructed with (a) 3DRP and (b) FORE. The contour levels at 10% of the slice maximum are superimposed. The absolute value of the difference between the two slices is displayed using a linear gray scale scaled to the range [0%, 10%] of the slice maximum, with the 10% contour level from (c) the 3DRP reconstruction. (d) The histogram of the difference between the two slices in percent of the slice maximum.
B. Whole-Body Imaging
Fig. 10 represents the standard deviation in the reconstruction of the uniform cylinder simulated with the large aperture 32-ring scanner. Following a standard procedure, this was calculated in each slice as the sample standard deviation of the set of voxels contained in a circular region of interest. While the standard deviation reconstructed with the two algorithms is almost identical in the central slices of the scanner, the standard deviation of the FORE reconstruction increases in the external slices, in agreement with the non uniform axial sensitivity profile of the volume scanner. This property of FORE is also observed with the SSRB algorithm and with an exact 3-D algorithm such as Favor [14], which does not involve a reprojection step. In contrast, the 3DRP algorithm results in a more uniform standard deviation over the whole axial FOV. Comparing the similar resolution of FORE and 3DRP reconstructions (Fig. 9) with the standard deviation analysis (Fig. 10) confirms the observation that FORE does not degrade the reconstructed SNR in the central slices of the scanner [37], [38].
Fig. 8 shows one central transaxial section of the torso phantom reconstructed with the 3DRP and FORE algorithms. Also shown is an image of the relative differences between the two images and a histogram of those differences. In addition, a contour level at 10% of the image maximum was superimposed to show the similarity in the low-level structures and noise of the two images. The cold spheres are located at the phantom border, at symmetric positions with respect to the hot spheres, and are partly masked by the overlayed contours. The images and the histogram of the relative differences indicate the good agreement between the two reconstructions.
C. Large Aperture Scanners Fig. 9 shows the transaxial tangential resolution and the axial resolution for the line source simulated with the large aperture scanner. Note that the results should be interpreted with care because this simple geometric simulation does not take into account the finite resolution of the detectors, each LOR being calculated as a single line integral. However, the almost uniform resolution observed with FORE demonstrates the validity of this approximation for scanners with axial apertures as large as 18 . As in the case of the measured line source in Fig. 6, this result was confirmed by inspecting the tails of the reconstructed profiles.
VII. CONCLUSIONS The motivation for this work is the fact that the computational complexity of fully 3-D reconstruction can still be an obstacle to the routine application of 3-D acquisition in PET for specific applications such as multiframe dynamic studies and whole-body imaging. Rebinning algorithms, by
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Fig. 9. Tangential and axial resolution in a central slice of the simulated large aperture scanner, for the 3DRP and FORE algorithms. The FWHM of 0 is plotted versus the radial a simulated line source placed in the slice z distance from the axis of the scanner.
=
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and multislice rebinning. These algorithms, however, make tradeoffs to some extent in the accuracy or in the stability in the presence of noise. We have presented a theoretical analysis of two new rebinning algorithms that avoid these tradeoffs. The first one involves a reprojection step similar to that in the 3DRP algorithm, but is mathematically exact and is expected to be faster than the 3DRP algorithm, with similar noise reduction properties. The second method, FORE,4 is an approximate rebinning technique based on the frequencydistance relation. This algorithm is much more accurate than the SSRB algorithm, without significant degradation of the SNR. For the data presented in this work, no direct comparison of the reconstruction time was possible because the two methods were implemented using different hardware. In a previous study with the 16-ring ECAT 951 scanner, both algorithms had been implemented on the same SPARC II workstation. When applied to data already corrected for scatter and attenuation, the 3DRP and FORE algorithms took 380 min and 28 min, respectively [37]. These figures demonstrate that the amount of computation required to reconstruct 3-D PET data is reduced by more than one order of magnitude by the use of FORE. Further work will be needed to assess the actual reduction of the reconstruction time when the algorithm is implemented on faster platforms and when the time required by model-based scatter correction [42], [43] is taken into account. The performance of FORE has been investigated for the reconstruction of phantom and patient data from the ECAT EXACT HR and ART scanners for scans representative of high resolution dynamic brain imaging and multibed position whole-body imaging. FORE was shown to be a reliable alternative to the 3DRP algorithm, the worst-case ( ) loss of spatial resolution due to the approximate character of the algorithm being smaller than 0.50 mm for a 40 cm FOV. Similar results were demonstrated with the simulated scanner, which has an axial aperture of 18 . This suggests that the FORE algorithm should also be applicable to modern volume scanners with very large axial FOV. ACKNOWLEDGMENT The authors wish to thank L. Byars for his help and for the use of his 3-D reconstruction software. REFERENCES
Fig. 10. Noise properties of the 3DRP and FORE algorithms, for a uniform 20-cm cylinder simulated with the large aperture scanner. The relative standard deviation in a circular region of interest is plotted versus the plane index.
reducing the 3-D data to an ordinary 2-D data set, speed up reconstruction by an order of magnitude compared to 3-D FBP algorithms such as the 3DRP method. This was demonstrated by previously developed algorithms such as the single-slice
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