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An immediate after-backprojection filtering method with blob-shaped window functions for voxel-based iterative reconstruction

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2006 Phys. Med. Biol. 51 5825 (http://iopscience.iop.org/0031-9155/51/22/007) View the table of contents for this issue, or go to the journal homepage for more

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INSTITUTE OF PHYSICS PUBLISHING Phys. Med. Biol. 51 (2006) 5825–5842

PHYSICS IN MEDICINE AND BIOLOGY

doi:10.1088/0031-9155/51/22/007

An immediate after-backprojection filtering method with blob-shaped window functions for voxel-based iterative reconstruction Bin Zhang1 and Gengsheng L Zeng2 1 Department of Electrical and Computer Engineering, University of Utah, Salt Lake City, UT 84112, USA 2 Department of Radiology, University of Utah, Salt Lake City, UT 84108, USA

Received 9 May 2006, in final form 22 September 2006 Published 24 October 2006 Online at stacks.iop.org/PMB/51/5825 Abstract Spherically symmetric volume elements (blobs) have better resolution–noise performance than voxels because of the overlapping of their rotational symmetric basis functions; however, using blobs is more computationally expensive than using voxels due to blob overlap. In this paper, we propose an immediate after-backprojection filtering method (ABF) with blob-shaped window functions for a voxel-based reconstruction. We compared this method with the general voxel-based method (without filtering), the blob-based method, the voxel-based method with between-iteration filtering (BIF) and with post-filtering (POF), using computer simulations. Both the quality of the reconstruction and the computational cost were evaluated. The reconstruction quality was measured by the contrast recovery coefficient (CRC) versus the background noise. It is shown that images reconstructed using this method are characterized by less image noise and preserved image contrast in comparison with both the general voxel-based method and the voxel-based method with BIF. The improvement in image quality achieved by this method varies with the parameters chosen for the Kaiser–Bessel (KB) windows. As with blobs, wider KB windows achieve better contrast–noise trade-offs in the reconstructed images, but are more computationally expensive. When using a KB window of a = 2.0, α = 10.4 and m = 2, known as the basis function of a ‘standard’ blob, this new method achieves identical CRC–noise features to the blob-based method with ‘standard’ blobs. In addition, the ABF method can be combined with the post-filtering method to achieve better noise–resolution performance than the general voxel-based post-filtering method. The computational cost of the ABF method is slightly greater than that of the general voxel-based method, but much less than that of the blob-based method. (Some figures in this article are in colour only in the electronic version)

0031-9155/06/225825+18$30.00 © 2006 IOP Publishing Ltd Printed in the UK

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1. Introduction For 3D iterative image reconstruction, images are traditionally represented by voxels. Voxels are localized in space and do not overlap each other when representing 3D images. The shortcoming of voxels is the band-unlimited nature of their frequency transforms. This is inconsistent with the finite nature of the real data (Lewitt 1990), and may lead to modelling errors or excessive noise amplification. As iterative reconstruction algorithms are not restricted to any particular discretization of the image (Herman and Lent 1976, Lewitt 1983), many alternatives to voxels, smoothly decaying to zero with increasing distance from their centre, were proposed (Hanson and Wecksung 1985, Lewitt 1990, 1992). When using these volume elements to represent a 3D image, the image value at any point is the sum of contributions from more than one element. In other words, there is overlap between a volume element and its neighbouring elements. A three-parameter family of spherically symmetric volume elements, also known as ‘blobs’, was found to be a good candidate for 3D image reconstruction (Wang et al 2004, Matej and Lewitt 1995). The basis functions of blobs are constructed using generalized Kaiser–Bessel (KB) window functions whose parameters enable the user to control their shape and smoothness, and consequently to control the reconstruction characteristics (Lewitt 1990, 1992). Unlike voxels, the basis functions of blobs are not only localized in space, but also nearly band-limited in the frequency domain. It was shown that blobs are superior to voxels for iterative reconstruction in terms of the noise– resolution trade-off (Wang et al 2004, Matej and Lewitt 1996). The blob-based reconstruction converged to a resolution that was comparable to that of voxels but with a substantially lower noise level. It was also verified that blob-based reconstruction outperformed voxelbased reconstruction post-filtered with a blob-shaped window function (Matej and Lewitt 1996). The performance of blobs varies with the chosen parameters of the blobs. The shortcoming of the blob-based method is that the computational demands for blobs tend to be much larger than that for voxels due to the overlapping of blobs (Lewitt 1992, Matej and Lewitt 1996). The continuous image volume can also be represented by a set of discrete samples on a grid, rather than being composed of superimposed volume elements. To reduce the reconstruction noise, the final result is smoothed using a convolution kernel, called ‘sieves’ (Snyder and Miller 1985, Veklerov and Llacer 1990). These approaches achieve a specified resolution with reduced noise in the reconstructed image, in a similar way to that achieved using those smoothly-varying image basis functions described in the previous paragraph. Another approach to suppress reconstruction noise directly applies smoothing to the reconstructed images either after each iteration (Silverman et al 1990, Slijpen and Beekman 1999) or after the reconstruction is finished (Matej and Lewitt 1996, Nuyts and Fessler 2003). In Silverman’s work (Silverman et al 1990), the conventional maximum likelihood expectation maximization (ML-EM) approach was modified by introducing a simple smoothing step to reduce reconstruction noise at each EM iteration. This so-called expectation maximization smoothing (EMS) method was related to a maximum penalized-likelihood approach. It was evaluated with several simple linear smoothers and local nonlinear filters. The best resolutionto-noise performance was achieved by employing weighted local mean smoothers. Nuyts and Fessler (2003) proposed another penalized-likelihood method, which was compared to the post-smoothed maximum likelihood approach based on resolution–noise performance. It was shown that the noise properties of their method were not superior to those of post-smoothed maximum-likelihood reconstruction with matched spatial resolution. Recently, Yendiki and Fessler (2004) have compared the performance of a post-smoothed blob-based approach with that of a post-smoothed voxel-based method. It was observed that the blob-based approach

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outperformed the voxel-based approach in terms of the bias–noise trade-off in the reconstructed images, but with a higher computation cost. In this work, we propose an immediate after-backprojection filtering method (ABF) with blob-shaped window functions for voxel-based iterative reconstructions. Unlike the post-filtering method (POF: filtering the final image) discussed in Matej and Lewitt (1996), Nuyts and Fessler (2003) and the between-iteration filtering method (BIF: filtering the image generated in each iteration) introduced in Silverman et al (1990), Slijpen and Beekman (1999), this method suggests a filtering operation applied to the image-like backprojection results for each backprojection operation. These images resulting from backprojection are the update factors of the image between iterations. Since filtering only applies to the backprojection results in the reconstruction, it can be regarded as a part of the backprojector. Section 2 introduces some basic concepts and notations for image representation and iterative image reconstruction with unmatched projector/backprojector pairs (Zeng and Gullberg 2000). Iterative reconstruction using the blob-based method is compared with that using the ABF method in section 3. Section 4 shows the simulation studies to compare iterative reconstruction using blobs, general voxels (voxels only), voxels with BIF, voxels with POF and voxels with ABF using a lesion phantom. The trade-offs between image resolution and image noise were studied experimentally for each method with different filters and blob windows. The simulation results are presented in the form of visual images and graphs of the contrast recovery coefficient (CRC) versus background noise at different iterations. 2. Basic concepts 2.1. Image representation A 3D image f (x, y, z) can be constructed as the sum of scaled and shifted copies of a volume element, also called ‘basis function’ , as follows, I −1
ci (x − xi , y − yi , z − zi ) (1) f (x, y, z) = i=1

where (x, y, z) represents the point having coordinates x, y and z along the usual three orthogonal axes. The set {(xi , yi , zi ) | i = 0, 1, 2, . . . , I − 1} denotes the set of I sample points that are the nodes of a uniform grid over a region in the 3D space, and coefficient ci is a scaling factor applied to sample point i. The set of coefficients {ci | i = 0, 1, 2, . . . , I − 1} is stored in the computer to represent the image. A voxel basis function has the form of  1 |x|, |y|, |z|
1/2 v (x, y, z) = (2) 0 otherwise. A spherically-symmetric basis function (e.g. for a blob) can be expressed as    b (x, y, z) = b r = x 2 + y 2 + z2

(3)

where r is the radial distance between point (x, y, z) and the origin. The basis function of the blob proposed in Lewitt (1990, 1992) is a Kaiser–Bessel (KB) window function given as     1 ( 1 − (r/a)2 )m Im (α 1 − (r/a)2 ) for 0
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a bm,α,a (r) = Im (α) (4)  0 otherwise where Im is the modified Bessel function of the first kind of order m, α is a non-negative parameter that controls the shape of the blob, a is the radius of the blob, and m controls the smoothness of the blob at its boundary r = a.

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For conventional voxels, in order to evaluate each projection, the intersection length (for line-integral data) or area (for planar-integral data) of each voxel and each projection path needs to be calculated individually. Unlike voxels, blobs are rotationally symmetric. They provide simple analytical expressions for both the 3D x-ray transform (line-integral Bline (s)) data and the 3D Radon transform (planar-integral Bplanar (s)) data, which are functions of the distance s from the projection path to the centre of the blob: 

  2π  1  a ( 1 − (s/a)2 )m+ 2 Im+ 12 (α 1 − (s/a)2 ) for 0
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a (5) Bline (s) = Im (α) α   0 otherwise  2   2π    a ( 1 − (s/a)2 )m+1 Im+1 (α 1 − (s/a)2 ) for 0
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a Bplanar (s) = Im (α) α (6)  0 otherwise. Equations (5) and (6) are also in the form of Kaiser–Bessel functions but with different parameters and scaling factors (Wang et al 2004). With equations (5) and (6), the 3D projections can be accurately calculated and stored in the computer to save computations of the projection data. Parameters m, a and α control the shape of the blob and the properties of the Kaiser–Bessel window function. A good discussion about how to select these three parameters of the blob is given by Matej and Lewitt (1996). They stated that choosing m = 2 or higher is desirable so that the blob has a continuous first derivative at the radial boundary. If choosing m = 2, the optimal selection of a and α is given by an equation √ 1 α 2 + u2 = , (7) 2π a  where u can be chosen from a series of numbers which corresponds to the first, second, third and fourth zero-crossing of the side lobes of the blob in the frequency space (e.g. u = 6.998, 10.417, 13.698, 16.924, etc). Due to the overlapping nature of blobs (when a > 0.5), different blob parameters can result in different modelling errors when representing a uniform distribution. It was found in Matej and Lewitt (1996) that a blob of m = 2, a = 2.0 and α = 10.4 is a good candidate, and is the so-called standard blob. 2.2. Iterative reconstruction Iterative image reconstruction from projections is intended to solve the system equation Af = p

(8)

where A represents the system matrix which models the projection operator, p is the measured projection data vector and f is the unknown emission distribution in the vector form. In an iterative reconstruction, the kth iteration generates an image fˆ(k) by updating the image fˆ(k−1) produced in the (k − 1)th iteration. The reconstruction is attempting to find an image vector fˆ such that the projections of fˆ are approximately equal (in some sense) to the measured data p. The maximum likelihood-expectation maximization (ML-EM) algorithm (Shepp and Vardi 1982, Lange and Carson 1984, Miller et al 1985) is a well-known iterative method in medical imaging, which is given as follows, fˆjold
gi hij N (9) fˆjnew = hil fˆold i  hi  j i

l=1

l

where gi is the ith measured projection, fˆjold and fˆjnew indicate the estimations of the j th image element in the previous iteration and in the current iteration, respectively. The ML-EM

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algorithm provides a maximum likelihood estimation of the solution of equation (8) (Liew et al 1993). The ML-EM algorithm has many desirable features, such as the non-negative nature of the solution, but it suffers from slow convergence which makes it difficult for practical use. Its variant, the ordered-subset EM (OS-EM) (Hudson and Larkin 1994) algorithm, accelerates the convergence successfully. The basic idea of OS-EM is to break up the full set of projection data into a series of mutually exclusive subsets and apply the reconstruction algorithm to each subset sequentially. Compared with ML-EM, OS-EM achieves a significant acceleration usually on the order of the number of subsets. Note that both the ML-EM and the OS-EM methods carry out backprojection operations in each iteration. The backprojection results at each iteration are the updating factors for the image produced in the previous iteration. Our after-backprojection filtering method with blobshaped window functions is applied to these backprojection results in order to provide similar blob overlapping effects to a voxel-based iterative reconstruction. An ML-EM algorithm incorporating our after-backprojection filtering method can be expressed as  

gi

ζ h w i,j −w N w∈Wb i ˆold l=1 hil f l   (10) fˆjnew = fˆjold

w∈Wb ζw i  hi  ,j −w where Wb denotes the coordinate vector of the centred 3D blob-window filter, and ζw represents the filter coefficient at coordinate w. For each voxel, the filter coefficients of itself and its neighbouring voxels are calculated with post-normalization using the blob function given in equation (4) with r equal to their 3D displacements from the centre voxel. 3. Blob versus after-backprojection filtering In this paper, we will only consider reconstructions using parallel-beam projection data (the 3D x-ray transform). For 3D iterative algorithms incorporating blobs, the 3D parallel-beam projections of the blob are called the footprints of the blob (Matej and Lewitt 1996), which can be calculated using equation (5). For a ray-driven projector, the contribution of any blob to the given projection at any orientation can be computed simply by placing (projecting) the corresponding footprint at the proper position and adding scaled (scale factor fj ) blob footprint coefficients (Bline (s) or Bplanar (s)) to the projection values. For a ray-driven backprojector, the same footprint specifies which backprojection paths (lines) contribute to the given blob value and by what weight. The footprints of the blob can be precalculated on a much refined grid and stored in tables. For each calculation of the projection and the backprojection, the contribution (weight) of each blob to each path can be looked up from the footprint table, and only depends on the distance from the projection or backprojection path to the centre of the blob. Compared with voxels, the precomputed footprints of blobs enable computational savings in the calculation of projections and backprojections. However, the computational demand increase of a blob-based algorithm due to the overlapping of blobs can be much greater than the computational savings achieved by the precalculated footprints, relative to a voxel-based algorithm. For a voxel-based algorithm, the contribution of each voxel to each projection is determined by the intersection length of the voxel and the projection line. Calculating the intersection length for each voxel at each iteration is computationally more expensive than looking up the footprint table for blobs. However, as computer storage capabilities have improved, it is no longer difficult to store the whole projection matrix of a voxel-based reconstruction. To reconstruct a SPECT image of typical size, for example 128 × 128 × 128, the projection matrix will contain approximately about 128 × 128 × 128 × 128 non-zero

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components for the parallel-beam projection data, assuming 128 detector positions are employed. The memory required to store this projection matrix in a double float format is about 1 GB. Therefore, substantial computational savings can also be achieved by storing the entire projection matrix in advance for a voxel-based reconstruction. Because of the nonoverlapping nature of voxels, voxel-based algorithms are more computationally attractive for iterative reconstructions than blob-based algorithms. As mentioned above, a blob-based algorithm achieves better image noise reduction than a general voxel-based algorithm with preserved resolutions (Matej and Lewitt 1996). Here, we propose a new immediate after-backprojection filtering method (ABF) for voxel-based algorithms. Unlike the post-filtering method or the between-iteration filtering method, our method is to filter the backprojection results at each iteration instead of filtering the final image or the output images between iterations. A voxel-based algorithm with this new method tends to have slightly higher computational demands than the general voxel-based algorithm due to the extra filtering operations. However, this ABF method still requires much less computation than a blob-based algorithm. Since filtering only operates on the backprojection results, the filtering of ABF can be regarded as a part of the backprojector. Assuming the projection matrix is H and the blob filter matrix is B, the ABF backprojection matrix can be expressed as BH T . As discussed in Zeng’s work (Zeng and Gullberg 2000), an ML-EM reconstruction may employ unmatched projector/backprojector pairs and still be able to converge. A projector/backprojector pair is valid if the eigenvalues of the product of the backprojection matrix and the projection matrix are positive. Here, the blob filter is a local low-pass filter such that the blob filter matrix B will be a diagonally dominant square matrix with positive diagonal elements. In this case, the eigenvalues of matrix B will be positive (Atkinson 1987, Zeng and Gullberg 2000). Since the eigenvalues of matrix H T H are always positive, matrix BH T H will only have positive eigenvalues, which suggests that the ABF backprojector BH T and the projector H form a valid projector/backprojector pair that guarantees the convergence of the reconstruction (Zeng and Gullberg 2000). 4. Simulation studies In an iterative reconstruction, as the algorithm iterates, image resolution is improved at the cost of increased image noise and degraded detectability of image features. The quality of the reconstructed image is consequently dependent on the resolution–noise trade-off of the reconstruction. Generally, the speed of convergence varies with the iterative method and the size of the object to be imaged. Matej and Lewitt have compared the noise and resolution characteristics for ML-EM reconstructions over a wide range of iteration numbers using three methods: (1) the general voxel-based method (voxels only, no filtering), (2) the blob-based method with various blobs and (3) the voxel-based method with the use of blob functions in post-filtering of the reconstructed image (Matej and Lewitt 1996). The blob-based method was shown to be the best. In the remainder of this section, we will compare the blob-based ML-EM reconstruction using the three blobs employed in Matej and Lewitt (1996) with the voxel-based ML-EM reconstruction using our ABF method with the same three blob window functions. The radial profiles and the relative power spectra of the three blob functions given in Matej and Lewitt (1996) are redrawn in figure 1. We will also compare the ABF method with other voxel-based methods using different smoothing strategies. A 3D lesion phantom was generated for the simulation studies. The geometry of this phantom is illustrated in figure 2. The phantom is of size 64 × 64 × 64 (relative to the voxel size d or the blob grid increment d) and placed on a regular Cartesian x–y–z grid. It consists

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of a centred background sphere (value 100) of radius 30, three ‘hot’ spherical lesions (value 400) of diameters 3, 5 and 7 (in the units of grid sampling increment d), respectively, and three ‘cold’ spherical lesions (value 0) of diameters 3, 5 and 7, respectively. All six lesions are placed on the central x–y plane (z = 0). The noise-free 3D parallel-beam projection data were calculated analytically without considering attenuation and scatter. For a better convergence of the ML-EM algorithm, the detector bin size was set to half of the grid sampling increment (d/2) (Hwang and Zeng 2005). In other words, the 2D detector plane was of size 64 × 64d 2 ,

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but consisted of 128 × 128 detector bins. The projector of the ML-EM algorithm was designed to incorporate the distance-dependent detector response model (Zeng and Gullberg 1991) in the projection simulation. We simulated ML-EM reconstructions using the three blobs and also using our voxel-based ABF method with the three blob functions. For each reconstruction, we employed 120 detector positions that were uniformly distributed from 0 to 178.5◦ . Poisson noise was generated and added to the noise-free projection data before the reconstruction. Noise was generated at two different levels to simulate real SPECT data from a low-count scan (high noise level) and from a high-count scan (low noise level), respectively. The characteristics of contrast recovery and noise propagation of an iterative reconstruction can be shown by the contrast recovery coefficient (Liow and Strother   (CRC) 1991) of each lesion versus the normalized standard deviation Mσback of the background. The CRC is defined as (Mles /Mback )rec − 1 CRC = (11) (Mles /Mback )phan − 1 where Mles and Mback represent the mean of the lesion and the mean of the background, and subscripts ‘rec’ and ‘phan’ denote the reconstruction and the phantom, respectively. In this simulation study, we calculated the CRC for each of the six lesions at each iteration. All the image elements inside each lesion except those at the edge of the lesion were employed to calculate Mles of that lesion. The background mean Mback was computed within the region inside the background sphere but outside all the lesions, excluding the image elements at the boundary of the background sphere and the surface of the lesions. The hot lesionto-background contrast was 4:1, and there was no activity in the cold lesions. Thus, the hot and the cold lesions in our phantom correspond to (Mles /Mback )phan − 1 = 3 and −1, respectively. The noise properties of the images reconstructed from the noisy data were evaluated using the normalized standard deviation of the noise in the background sphere, instead of the general standard deviation of the noise, since for different image intensity levels, the same standard deviation of the background noise can result in different image qualities. The normalized standard deviation Mσback is calculated within the same region in which Mback is calculated using the formula   N
1  σ  = (1/N) (fi − f¯ i )2 (12) Mback Mback i=1 where σ is the standard deviation, N is the number of image elements that are used in the calculation, fi is the value of the ith element of the image reconstructed from noisy data, and f¯ i is the expected mean value of the ith element. We used an isolated noise-free reconstruction to calculate f¯ i . This was to eliminate the influence on the noise measure of nonuniform values of the image within the regions that are supposed to be uniform. Figures 3 and 4 show the CRC versus noise performance of an ML-EM reconstruction, using voxels only, using standard blobs and using voxels with our ABF method, at two different levels of data noise (low noise level for figure 3 and high noise level for figure 4). The six sub-figures in figures 3 and 4 illustrate the CRC versus noise performance of the six lesions, respectively, in the order (a)–(f) indicated in figure 2(b). Each symbol position in each sub-figure represents results for a particular iteration up to 41 iterations, at which the CRCs for all the lesions are close to their convergence. It is shown in these figures that for any fixed value of the normalized standard deviation, both the blob method and the ABF method achieve higher CRC values than the corresponding voxel-only method for all six lesions. Also, for any given value of the CRC, both the blob and the ABF methods are characterized

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by lower image noise in comparison with the voxel-only method. In addition, the CRC–noise performance of the ABF method with the standard blob window is almost identical to that of the blob method with standard blobs. This is also evident from figure 5, which shows the centre slices of the reconstructed 3D images using the three methods. In figure 5, the contrasts of all the three images are about the same, while the images reconstructed by the blob method and the ABF method are obviously less noisy than the image reconstructed by the voxel-only method, and the images reconstructed by the blob and the ABF method are nearly identical. The ABF method was also compared with the between-iteration filtering method (BIF) using the standard blob window function. The simulation results are illustrated in figure 6. Figure 6(a) shows the centre slices of the 3D reconstruction images using the voxel-only method and the two filtering methods after 22 iterations in the high noise level situation. Figure 6(b) illustrates the CRC–noise curves of the three methods. The BIF method with standard blobs provides a less noisy image; however, the contrast of the image is significantly degraded, compared to the ABF method and the voxel-only method. It suggests that using the standard-blob filter for the BIF method will result in oversmoothed images. We also simulated the BIF method with the narrow-blob filter. The images are still oversmoothed (not shown here). To compare the three cases based on the same recoverable CRC, we implemented a 3D linear local smoother (L1) used in Silverman’s work (Silverman et al 1990). Here we set the filtering weights of W −1 for the neighbouring values equal to 1/50 (the optimal

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Figure 5. The centre slices of the 3D reconstructed images using voxels without filtering, standard blobs and voxels with ABF after 22 iterations (enough to recover image contrast according to figure 4) in the two noise level situations ((1) the low noise level situation (figure 3) and (2) the high noise level situation (figure 4)).

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choice given in Silverman et al (1990)). The resulting images and their CRC–noise curves are shown in figures 7(a) and (b), respectively. It shows that in this simulation, the three methods achieve similar CRCs after the reconstruction converges, while the ABF method has better noise performance than the other two methods. In Nuyts and Fessler’s work (Nuyts and Fessler 2003), it was shown that excellent noise performance for a specified resolution can be obtained by running ML-EM for a large number of iterations, then post-smoothing the result to achieve the specified resolution. Figure 8 shows the comparison between the voxel-only method and the post-filtering method (POF) with four different filters (the linear filter (L1) and three Gaussian filters shown in figure 8(a)). The image shown in figure 8(a)(i) was generated after 40 iterations using the voxel-only method. No filtering was applied. The other four images shown in figures 8(a) (ii)–(v) were generated after 100 iterations (to make sure the

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reconstruction thoroughly converged). Each of the four images was post-filtered with one of the four filters before being displayed. Figure 8(b) illustrates the CRC–noise performance of the six lesions ((i)–(vi)). Note that for the voxel-only method, the CRC–noise values are plotted from 1–40 iterations, while for the four post-filtering cases, only the final CRC–noise values are plotted. It can be seen that by using an optimal filter (G3 in this case), the POF method can achieve good resolution–noise performance. But the performance of the POF is dependent on the choice of the filter and the number of iterations being carried out before the filtering. Note that the ABF method can also be combined with the POF method. Figure 9 shows a comparison study of the POF method with filter G3, the ABF method with standard blobs and the combined ABF–POF method with both standard blobs and G3. It is shown that the

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ABF–POF method achieves better performance than the ABF method and outperforms the POF method based on the simulation setups indicated in figure 9(a). The voxel-only method, the blob-based method and the ABF method were evaluated and compared using the narrow and wide blob windows similarly. The CRC–noise performance of the three methods with the narrow and wide blobs is illustrated in figures 10 and 11 (both with high level noise), respectively. It is shown that for both the narrow and wide blobs, the blob and the ABF methods outperform the voxel-only method. The ABF method works slightly better in the case of using the wide blob window function and the blob method is preferable in the case of using the narrow blob; however, the difference is not significant.

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R All the reconstructions were carried out on a Dell workstation installed with a Pentium 4 CPU (3.4 GHz) and 2.0 GB RAM. The projection matrices were precalculated and stored on the hard drive for all reconstructions. Table 1 shows the hard drive space required to store the projection matrices and their corresponding index and header files as well as the computation time needed for running one iteration of the reconstructions for each of the three methods: the voxel-only method, the blob-based method with standard blobs, and the ABF methods. The codes were implemented in C ++ and the computation time was calculated by employing a standard C ++ function ‘clock’.

5. Conclusions Spherically symmetric volume elements (blobs) are alternatives to conventional voxels. It has been shown that blobs possess better image noise properties and preserve image resolution (Lewitt 1990, 1992, Matej and Lewitt 1995, 1996); however, because of the overlapping

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nature of the blobs, the computational demands tend to be too large for blobs to be used in practice. We proposed a new filtering method with blob-shaped windows for voxel-based iterative reconstruction, which can achieve comparable contrast–noise performance to that obtained by blobs, with the computational cost similar to that of a voxel-based method. Unlike the post-filtering method (filtering the final image) (Matej and Lewitt 1996, Slijpen and Beekman 1999, Nuyts and Fessler 2003) and the between-iteration filtering method (filtering between-iteration images) (Silverman et al 1990, Slijpen and Beekman 1999), we filter the backprojection results at each iteration of the reconstruction. The backprojection results are updating factors for the output image of each iteration. Filtering the images resulting from backprojection with blob-shaped windows gives a similar overlapping effect with voxels to that seen with blobs. Simulation studies show that the ABF method achieves faster convergence than the voxelonly method. With a standard blob window function, the ABF method can achieve identical CRC–noise performance to that obtained with a blob-based reconstruction using the standard blob. The performance of our method, relative to that of the corresponding blob-based method, varies with the blob window parameters: for the narrow blob, the blob-based method is preferable, while for the wide blob, our method is slightly better. Overall, the performance of these two methods is comparable. The running time measurements show that the filtering operation in our method causes only a slight increase in computational demand, relative to the voxel-only method. In comparison

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with the blob-based method, our method is computationally inexpensive. Of the three methods, ours is the most efficient method in terms of the performance–cost trade-off. Comparison study between the BIF method, the POF method and the ABF method shows that with the same resolution the ABF method with standard blobs achieves better noise– resolution performance than the BIF method with the optimal local linear filter suggested in the EMS approach. The POF method with a properly designed filter may be able to outperform the ABF method with standard blobs. However, the ABF method incorporating the same postfiltering can outperform the voxel-based POF method. The performance of different methods varies with the filters chosen. Further comparison of these methods based on a wider range of filters should be considered in future work. Acknowledgments The authors gratefully acknowledge Dr Roy Rowley for his help with English correction, as well as the anonymous reviewers for their helpful suggestions. This work is partially supported by NIH Grants NIH-EB001489 and NIH EB003298. References Atkinson K E 1987 An Introduction to Numerical Analysis 2nd edn (New York: Wiley) Hanson K M and Wecksung G W 1985 Local basis-function approach to computed tomography Appl. Opt. 24 4028–39 Herman G T and Lent A 1976 Iterative reconstruction algorithms Comput. Biol. Med. 6 273–94 Hudson H M and Larkin R S 1994 Accelerated image reconstruction using ordered subsets of projection data IEEE Trans. Med. Imaging 13 601–9 Hwang D S and Zeng G L 2005 Reduction of noise amplification in SPECT using smaller detector bin size IEEE Trans. Nucl. Sci. 52 1417–27 Lange K and Carson R 1984 EM reconstruction algorithms for emission and transmission tomography J. Comput. Assist. Tomogr. 8 306–16 Lewitt R M 1983 Reconstruction algorithms: transform methods Proc. IEEE 71 390–408 Lewitt R M 1990 Multidimensional digital image representations using generalized Kaiser-Bessel window functions J. Opt. Soc. Am. 7 1834–46 Lewitt R M 1992 Alternatives to voxels for image representation in iterative reconstruction algorithms Phys. Med. Biol. 37 702–16 Liew S C, Hasegawa B H, Brown J K and Lang T F 1993 Noise propagation in SPECT images reconstructed using an iterative maximum-likelihood algorithm Phys. Med. Biol. 38 1713–26 Liow J S and Strother S C 1991 Practical tradeoffs between noise, quantitation, and number of iterations for maximum likelihood-based reconstructions IEEE Trans. Med. Imaging 10 563–71 Matej S and Lewitt R M 1995 Efficient 3D grids for image reconstruction using spherically-symmetric volume elements IEEE Trans. Nucl. Sci. 42 1361–70 Matej S and Lewitt R M 1996 Practical considerations for 3-D image reconstruction using spherically-symmetric volume elements IEEE Trans. Med. Imging 15 68–78 Miller M I, Snyder D L and Miller T L 1985 Maximum likelihood reconstruction for single photon emission computed tomography IEEE Trans. Nucl. Sci. 32 769–78 Nuyts J and Fessler J A 2003 A penalized-likelihood image reconstruction method for emission tomography, compared to postsmoothed maximum-likelihood with matched spatial resolution IEEE Trans. Med. Imaging 22 1042–52 Shepp L A and Vardi Y 1982 Maximum likelihood reconstruction for emission tomography IEEE Trans. Med. Imaging 1 113–22 Silverman B W, Jones M C, Wilson J D and Nychka D W 1990 A smoothed EM approach to indirect estimation problems, with particular reference to stereology and emission tomography J. R. Stat. Soc. Ser. B 52 271–324 Slijpen E T P and Beekman F J 1999 Comparison of post-filtering and filtering between iterations for SPECT reconstruction IEEE Trans. Nucl. Sci. 46 2233–8 Snyder D L and Miller M I 1985 The use of sieves to stabilize images produced with the EM algorithm for emission tomography IEEE Trans. Nucl. Sci. 32 3864–72

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