1192 E E E TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 40, NO. 4, AUGUST 1993

A Fast Projector-Backprojector Pair Modeling the Asymmetric, Sp~tially Vnryinn Scatter Response Function for Scatter Compensation in SPECT Imaging*

E.C. Frey172,Z.-W. Jul, and B.M.W. Tsui172 Department of Biomedical Engineering 1 and Department of Radiology2

The University of North Carolina at Chapel Hill

Abstract We have previously developed a method, slab derived

scatter estimation (SDSE), for quickly and accurately modeling the asymmetric, spatially varying scatter response function in uniformly attenuating objects with convex surfaces. This model has been implemented in a projector- backprojector that, when combined with iterative reconstruction techniques, provides accurate scatter compensation in single-photon emission computed tomography (SPECT). These iterative reconstruction based scatter compensation techniques have the advantage that they use all detected photons, avoiding the noise amplification occurring with subtraction based schemes. In addition, scatter compensation with iterative reconstruction does not involve the use of arbitrary adjustable parameters. In this paper we present a new fast algorithm for implementing SDSE that reduces computation time by a factor of 16 compared to a direct implementation. Projection data computed with the new algorithm compare well with that from Monte Carlo simulations. When combined with faster computers and iterative reconstruction algorithms that converge rapidly, this fast projector-backprojector pair makes iterative reconstruction based scatter compensation feasible.

I. INTRODUCTION

As a result of the relatively poor energy resolution of current gamma cameras, the measured projection data in single-photon emission computed tomography (SPECT) is contaminated by a significant number of scattered photons. These scattered photons result in a loss of image contrast and quantitative accuracy. The scatter response function (SRF), which describes the distribution of the scattered photons in the projection data, depends in a complex way on the source position, imaging geometry, and composition and shape of the object. Consequently, the SRF is spatially variant and asymmetric. Analytic formulations for the SRF function have proved difficult. Previous investigators have proposed the use of Monte Carlo (MC) simulations to calculate the SRF for each object [ 11.

However, the MC simulations are computationally expensive. In addition, storing the resulting SRFs for general object shapes requires large amounts of memory: 128 MB is required to store the mamx for reconstruction of a 64x64 matrix from 64 projection bins at 128 angles. For this reason, it is desirable to develop a technique to calculate the SRF without resorting to MC calculations and to implement this as a projector-backprojector pair that can be used during the iterative process without requiring large amounts of memory.

*This work was supported by a grant from the Whitaker Foundation and Public Health Service Grant R01 CA39463.

We have previously developed a technique, slab derived scatter estimation (SDSE), that can be used to estimate rapidly the asymmetric, spatially varying SRF in uniformly attenuating objects [2]. In this technique, the scatter response functions of line sources embedded in a slab phantom and knowledge of the shape of the phantom surface (derived from an attenuation map) are used to estimate the asymmetric, spatially variant SRF for more complex object shapes. The SRFs calculated using this technique compare well with direct MC simulations, even for extreme positions near the edge of an object.

In SDSE it is necessary to calculate an array of distances from the surface of the phantom to the line through each source pixel and parallel to the collimator. Since this computation must be performed for each source pixel, it consumes a major part of the computation time. One iteration of the maximum likelihood-expectation maximization (ML- EM) algorithm implemented using this projector- backprojector pair took 12 minutes per slice for the reconstruction of a 64x64 image from projection data with 64 bins at 128 angles on a Kubota Pacific Titan 3000 (32 MIPS) computer.

In this paper we present a fast and efficient algorithm for an SDSE-based projector-backprojector pair. The algorithm is based on rotating the grid used to sample both the source and attenuation distributions at each projection angle such that the rows in the grid are parallel to the collimator face. In this orientation the may of distances from each pixel in the rotated grid to the surface in a uniformly attenuating object can be computed simply. This is done by summing attenuation coefficient values in the columns of the attenuation map and dividing by the value for the object. In addition, the array of distances needs to be calculated only once for all the pixels in a row. This reduces the number of times the array must be calculated by a factor equal to the number of pixels in a row. The new algorithm reduces the computation time to 44 seconds per iteration per slice on the same computer. This is a decrease in computation time of a factor of 16 compared to the previous direct implementation. The computational accuracy of the new algorithm was verified by comparing projection data computed using it to that f" direct MC simulations.

11. METHODS

A . Description of Slab Derived Scatter Estimation The algorithm for SDSE has been described previously [21.

This technique allows one to calculate the asymmetric, spatially varying SRF for a uniformly attenuating object with a more complex shape. The SRF is calculated based on the object's shape and a parameterization of the SRF of a point source in a uniform slab as a function of depth. The method is illustrated in Figure 1. In this figure, each source position and

0018-9499/93$03.00 0 1993 EEE

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(Slab Phantom 2)

Slab phantom SRF for a

lab Phantom 2

I 7 2%:: Figure 1. Schematic of SDSE illustrating the use of slab

phantom SRFs to construct the SRF for a uniformly attenuating object with a convex surface.

viewing angle determines a line, 9, which passes through the source and is parallel to the collimator face. Suppose we wish to calculate the object’s SRF for a given source position at a position on the detector, 9. Let ?B be the point on 3 which projects to P. For a slab phantom, the SRF is a function of three parameters: the source depth; the source-to-collimator distance; and the offset, or distance in the detection plane from the projected source position to the point of interest. The SRF for the object is determined from the slab phantom SRF for a source at a depth equal to the distance from !B to the surface of the phantom along the line connecting !B and P, a source-to-collimator distance the same as for the source; and an offset equal to distance between 9 and the projected position of the source.

For example, in Figure 1 for point Pl we use a slab phantom corresponding to a source at depth dl. We ev_aluate this slab phantom SRF at offset equal to the length of I , and, after normalizing as described below, use this as the value of the SRF for the object SRF for that source position at 8. Similarly, for point V2 we use slab phantom 2 and the value of the slakphantom SRF evaluated at an offset equal to the length of l2 and depth d2.

It should be emphasized that SDSE is not equivalent to using a single slab phantom SRF as the SRF for each source position in the complex object, as has been proposed by previous investigators [3]. This would result in an approximate, symmetric SRF. Instead, for each source pixel SDSE uses a different slab phantom SRF to compute the value of the SRF at each projection bin. In addition, multiple scatter is included parameterization of the slab phantom SRF and the SDSE SRFs do not involve a single-scatter approximation. As a result, the SDSE estimate accurately models true SRF.

Although Figure 1 illustrates the case of two-dimensional (2-D) imaging, we can extend to three-dimensions (3-D). To do so, we think of the drawing in Figure 1 as a cross section through the 3-D object and imaging system in the plane containing: the source position, its projected position on the collimator, and the point at which we wish to evaluate the arbitrarily shaped object SRF.

Mathematically SDSE can be described as follows. For generality we will present the description in 3-D. We will evaluate the SRF for a uniformly attenuating object with a convex surface for a source located at a position defined by

the vector from the origin to the source position, 3. Let T be the vector in the detection plane from the projected position of the source to the point, C? at which we evaluate the SRF. Let 1 be the length of this vector. Let Srf,lab(t D, d ) be the slab phantom SRF normalized to unit volume for a point source at a depth d, distance to the collimator D, and an offset 1 from the projected source position. The line, 3, is the line through

3 parallel to the face of the collimator. Further, let d( I ; I ) be the distance from the point on 3 that projects to Pand the surface of the object. Finally, let p be the attenuation coefficient of the medium composing the object and S(d) be the scatter-primary ratio (SPR) for a source at a depth d in a slab phantom. We wish to compute the scatter contribution,

s( 7; r‘, D), from a source position defined by r‘ to a position

on the detector defined by 7. The scatter contribution is the number of scattered photons measured at a given detector position resulting from photons emitted by a source at a given source position divided by the intensity of the source. For simplicity, we will consider that the source intensity is normalized by the detector system efficiency and the acquisition time; we would detect one photon for a source of unit intensity in air. With these definitions the scatter contribution is:

- - -

47; 7,o) ~ x ~ - ~ a ( ~ , I ) ) . ~ a ( i ; ~ ) ) . s ~ ~ ~ l ; D,a(i;J))7 0) Equation (1) can be applied to 2-D reconstruction (that is,

for sources having infinite axial extent) by using the scatter line eurce response function. In this case r‘ is a 2-D vector and 1 becomes the scalar 1. For 3-D reconstruction we use the scatter point source response function for the slab scatter response function and 1 and 3 are 2-D and 3-D vectors, respectively.

To use Equation (l), we must know the slab phantom SRF as a function of source depth and distance from the collimator. The dependence on source depth was measured by computing the slab SRF using MC simulation techniques for a gamma camera with the collimator resting on the surface of the slab phantom for source depths ranging from 1 to 40 cm. The resulting SRFs were fit with a fitting function consisting of a Gaussian plus an exponential. The fitting parameters required for each slab SRF include: the widths of the Gaussian and exponential components; the ratio of the heights of the components; and the SPR. Each of these fitting parameters was then parameterized as a function of source depth.

The value of the slab SRF for other source-to-collimator distances can be computed using the series equivalent formulation. Let G(v; D) and sRF,lab(V, D, d) be the Fourier transforms of the detector response function, g(I; D), and slab

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phantom SRF, Srfslab(k D, d), respectively, measured at a source-collimator distance D . The detector response function (DRF) includes both the collimator geometric response and the inmnsic resolution of the detector. The Fourier transform of slab phantom SRF for a new sourcecollimator distance, D ’, is given by:

A projector-backprojector pair was developed based on a straight forward 2-D implementation. The scatter response functions computed using this implementation compared well with those from direct MC simulations. However, this implementation required relatively large amounts of computation time, about 12 minutes per iteration of the ML- EM reconstruction algorithm [2]. A large portion of this time

is spent computing the distances d ( l ; r ) , which must be evaluated for each projection bin with respect to the line passing through each source pixel.

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B . Overview of Fast Projector Algorithm The new implementation of the SDSE algorithm is based

on the observation that the distances d( I;r) only need to be calculated once for each row when the rows in the source and attenuation images are parallel to the detector. In this case, the

same values of d( I T ) can be used for a~ the pixels in a row. We can force this to be the case by rotating the gnd used to sample the source and attenuation distributions, as illustrated in Figure 2. Rotation of sampling grid has been proposed previously to compute the projection data including detector response blurring. In this orientation, the blurring can be computed using a convolution [4].

- --

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C . Fast Projector and Backprojector Algorithms The new algorithm for the projector is described below: Compute and store in memory the following tables to reduce processing time. a. The DRF in the spatial domain as a function of offset

and distance from the collimator face. b. The normalized slab phantom SRF (the product of

the SRF normahzed to unit area, SPR and attenuation factor) as a function of offset, source depth, and distance from the collimator.

Set the array of accumulated attenuation coefficients to zero. This array will be used to accumulate the total attenuation coefficient for each column in the rotated sampling grid. It is updated as we move deeper into the object, away from the detector. The resulting values are used both to calculate the distance from the surface and the attenuation factor for each column. Rotate the source and attenuation distribution sampling grids so that the rows are parallel to the detector face. The rotation is performed using bilinear interpolation and gives matrices representing the attenuation and source distributions using the rotated sampling grid. The rotation includes any shift necessary so that pixels in the jth

sourc Attenw tion

Original Sampling Grid

an Attenuation Mau

4.

5.

6.

7.

8.

After Sampling Grid Rotation

Figure 2. Illustration of rotation used in fast projector- backprojector algorithm.

column of the image matrix project into the jth projection bin. Perform Steps 5-9 for each row, starting with the row closest to the collimator. Compute the index into the DRF table for the current row based on the distance from the row to the collimator face. Update the accumulated attenuation array described above. This is done to reflect the attenuation distribution in current row of the rotated attenuation image matrix. It is accomplished by adding the attenuation coefficient values for each column in the current row to the corresponding array element. Compute the unscattered contribution from the current row to the projection data by: a. Convolving the pixel values in the current row of the

source image with the spatial domain DRF for the row.

b. Multiplying these blurred intensities by the attenuation factor for each column in the row. The attenuation factor for a column is computed using the corresponding element of the accumulated attenuation array.

c. Add the resulting values to the corresponding projection bins for the current angle.

Compute the array of distances from the current row to the surface for each column by: a. Dividing each of the elements in the accumulated

attenuation coefficient array by the attenuation coefficient for the object.

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b. Convolving this array with the DRF for this row. 9. For each pixel in the current row, compute the scatter

contribution to jth projection bin from activity in the ith pixel as follows: a. Look up the value in the normalized slab SRF table

corresponding to: a source depth equal to the distance to the surface for the jth column; the distance to the collimator for this row; and the distance from the projected pixel position to the bin. Multiply this value by the activity in the ith pixel. This value is added to the projection data value in the jth bin.

The backprojection algorithm is very similar to the projection algorithm. For each angle we perform the backprojection into a matrix corresponding to a rotated sampling grid, rotate it back to the standard orientation, and add it to the source image. Thus, in Step 3 only the sampling grid for the attenuation image is rotated; the source image is rotated after the loop in Steps 4-9. If the backprojection is performed immediately after the projection at the same angle, as for ML-EM, the same rotated attenuation image mamx can be used and Step 3 can be eliminated. This saves one additional image rotation per view per iteration.

D. Verification The algorithm was verified by comparing projections of

the cold and hot rod phantoms shown in Figure 3. The projection data were computed using the new projector and direct MC simulations. For both phantoms, 2, 4, and 6 cm diameter cylindrical rods were embedded in a uniform cylindrical attenuator 22 cm in diameter. The rods were placed with their centers at equal angular intervals on a 12 cm diameter circle concenmc with the attenuator. For the cold rod

b.

Figure 3. Images of (a) cold and (b) hot rod phantoms.

phantom (Figure 3a), there was no activity inside each rod and the remainder of the attenuator had a constant background activity. For the hot rod phantom (Figure 3b), there was no background activity and the activity concentration was the same for each of the rods.

The MC simulated projection data were simulated using an MC simulation code described previously [5]. The data were simulated for a LEGP collimator, an energy resolution and intrinsic resolution at 140 keV of 11% and 4 mm, respectively, and an energy window of 126-154 keV. The projection data were simulated into 0.62 cm wide discrete projection bins. Components of the projection data arising from scattered and unscattered photon images were stored separately to allow comparison with corresponding components computed using the projector.

It is desirable to compare the absolute quantitative accuracy of the projector and MC simulation. However, the projector does not model the imaging system sensitivity. As a result, the phantom was normalized to an activity value based on the number of emitted used in the MC simulation and the observed sensitivity of the simulated detection system. No further normalization of the projection data was performed.

111 RESULTS AND DISCUSSION

A. Comparison with MC Projection Data Figure 4 shows projection data generated using the new

projector that models the spatially variant scatter response compared with that from MC simulations. Data are shown from both the cold (Figures 4a-b) and hot (Figures 4c-d) rod phantoms. Projection data for each phantom are shown at two different projection angles. Both the scatter only and total (scatter plus primary) components of the projection data are shown. Agreement between the projector generated and MC simulated projection data is excellent.

B . Time Requirements As previously mentioned, one iteration using the ML-EM

algorithm using this projector-backprojector pair takes 44 seconds for reconstruction of a 64x64 image from 128 views and 64 projection bins on a single processor of a Kubota Pacific Titan 3000.

A breakdown of the time spent for one ML-EM iteration is as follows: 9.4% rotating the attenuation and source image sampling grids; 9.0% performing the convolutions in steps 7a and 8b; 3 1.3 % performing the other computations required for projection; and 49.1 % performing other computations required for backprojection.

C . Memory Requirements The memory requirements for this projector-backprojector

pair are relatively modest. In the following, let the number of projection bins and the number of pixels along one side of the reconstructed image both be n, and the number of angles be m. Storing the table of slab phantom SRFs requires

(n/lL)x(JZn)x(Jzn) = n3 floating point elements. The DRF

table requires (n /2)x(nn) elements and the projection data and the rotated and unrotated attenuation and source images require n m and 4n2 elements, respectively. For a 64x64

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: (c) -- 1600- MC Total -

Projector Total - - _ - - -MC Scatter _ _

U) 120&- .......-. Projector Scatter

0

c

3

800-

400-

6 1 ; ' "

:: (a MC Total

-MC Scatter

_- _ _ _ - - Projector Total

. . . . . . . . . Projector Scatter - _

_ _ --

--

_ _

~ 1 ~ ; ~ '

Figure 4. Comparison of projection data obtained using MC simulation and the fast projector that models the spatially variant SRF. The data shown were computed for the cold (a and b) and hot (c and d) rod phantoms for projection angles of 1.4" (a and c) and 37.8" (b and d).

reconstruction from 64 projection bins and 128 views the total memory requirement is 290,000 elements or approximately 1.1 MB. This compares to n3m or 128 MB required to store the entire transition matrix.

The memory savings become even more significant for 3- D reconstruction. Since the slab phantom scatter point source response function and the detector point source response function are radially symmetric, the tables used by this algorithm are the same size as for 2-D reconstruction. In contrast, the memory required to store the full transition matrix increases by a factor equal to the square of the number of slices reconstructed. For example, storing the 3-D transition mamx for a 64x64~64 reconstruction from 64x64 projection images at 128 views requires 512 GB.

D. Applications There are several applications for this fast projector-

backprojector that models the asymmetric, spatially variant scatter response function in SPECT. As mentioned previously, when combined with iterative reconstruction, it can serve as a method for scatter compensation [l], [6]. Iterative reconstruction based techniques have the advantage that, in

contrast to subtraction based techniques, they attempt to return scattered photons to their place of emission. This reduces the noise amplification resulting from the use of subtraction based techniques [61.

The projector-backprojector can be used in several applications where MC calculations have previously been used. These include calculation of the transition matrix required for reconstruction methods based on matrix inversion [7] and generation of simulated projection data for use in observer studies and other purposes.

The new projector-backprojector is also useful in conjunction with subtraction based scatter compensation techniques. For example, it can be used to compute an average SRF for convolution subtraction [8]. It can also serve as the basis for schemes that subtract an estimate of the scatter component of the projection data computed based on the image reconstructed from the original data [9], [lo]. In these methods, an image is first reconstructed from the measured projection data including scatter. Next the scatter component in the original projection data is estimated, for example using the new fast projector, assuming that this reconstructed image represents the true distribution of source intensity. This scatter

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estimate is subtracted from the original data to estimate the scatter-free projection data.

IV. CONCLUSIONS We have presented a new fast projector-baclcprojector that

accurately models the spatially variant aspects of the SPECT imaging system response. The model includes the asymmetric scatter response, detector response, and attenuation in uniform objects with convex surfaces. The projection data computed using this projector-backprojector compare well with that from direct MC simulations. Among other applications, this projector-backprojector can be combined with iterative reconstruction techniques for accurate scatter compensation in SPECT. This algorithm eliminates the need for MC simulations to determine the transition matrix and large memory required to store it. When combined with new faster computers and iterative algorithms, such as the weighted least squares-conjugate gradient algorithm which has a convergence rate 10 times greater than the maximum- likelihood-expectation maximization algorithm [ 113, this projector-backprojector will allow accurate scatter compensation in times that are clinically acceptable. These iterative reconstruction based scatter compensation algorithms have the advantage that, compared to conventional techniques, they more accurately model the spatially variant aspects of the imaging system without the noise amplification found with subtraction based techniques [6].

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E. C. Frey and B. M. W. Tsui, “A practical method for incorporating scatter in a projector-backprojector for accurate scatter compensation in SPECT,” 1991 Nuclear Science Symposium and Medical Imaging Conference, Santa Fe, NM, 1991, pp. 1777-1781.

B. C. Penney and M. A. King, “A projector, back-projector pair which accounts for the two-dimensional depth and distance dependent blurring in SPECT,” IEEE. Tram. Nucl. Sci., vol. 27, no. 2, pp. 681-686, 1990. G. L. Zeng and G. T. Gullberg, “Frequency domain implementation of the three-dimensional geometric point response function correction in SPECT imaging,” IEEE Nuclear Science Symposium and Medical Imaging Conference, Santa Fe, NM, 1991, pp. 1943-1947.

M. Ljungberg and S.-E. Strand, “A Monte Carlo program for the simulation of scintillation camera characteristics,” Comp. Meth. Prog. Biomed., vol. 29, pp. 257-272, 1989.

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M. F. Smith, J. C.E. Floyd, R. J. Jaszczak and R. E. Coleman, “Reconshxction of SPECI‘ images using generalized matrix inverses,” IEEE Trans. Med. Imag., in press.

B. Axelsson, P. Msaki and A. Israelsson, “Subtraction of Compton-scattered photons in single-photon emission

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[lo] M. Ljungberg and S.-E. Strand, “Scatter and attenuation correction in SPECT using density maps and Monte Carlo simulated scatter functions,” J. Nucl. Med., vol. 3 1, no. 9, pp. 1560-1567,1990.

[ l l ] B. M. W. Tsui, X. D. Zhao, E. C. Frey and G. T. Gullberg, “Comparison between ML-EM and WLS-CG algorithms for SPECT image reconstruction,” IEEE. Trans. Nucl. Sci., vol. 38, no. 6, pp. 1766-1772, 1991.\

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