Projection Space Image Reconstruction Using Strip Functions To ...

24

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

Projection Space Image Reconstruction using Strip Functions to Calculate Pixels More “Natural” for Modeling the Geometric Response of the SPECT Collimator Yu-Lung Hsieh, Gengsheng L. Zeng, Member, IEEE, and Grant T. Gullberg,* Senior Member, IEEE

Abstract— The spatially varying geometric response of the collimator-detector system in single photon emission computed tomography (SPECT) causes loss in resolution, shape distortions, reconstructed density nonuniformity, and quantitative inaccuracies. A projection space image reconstruction algorithm is used to correct these reconstruction artifacts. The projectors F use strip functions to calculate pixels more “natural” for modeling the two-dimensional (2-D) geometric response of the SPECT collimator transaxially to the axis of rotation. These projectors are defined by summing the intersection of an array of multiple strips rotated at equal angles to approximate the ideal system geometric response of the collimator. Two projection models were evaluated for modeling the system geometric response function. For one projector each strip is of equal weight, for the other projector a Gaussian weighting is used. Parallel beam and fan beam projections of a physical three-dimensional (3-D) Hoffman brain phantom and a Jaszczak cold rod phantom were used to evaluate the geometric response correction. Reconstructions were obtained by using the singular value decomposition (SVD) method and the iterative conjugate gradient algorithm to solve for q in the imaging equation FGq = p, where p is the projection measurement. The projector F included the new models for the geometric response, whereas, the backprojector G did not always model the geometric response in order to increase the computational speed. The final reconstruction was obtained by sampling the backprojection Gq at a discrete array of points. Reconstructions produced by the two proposed projectors showed improved resolution when compared against a unit-strip “natural” pixel model, the conventional image pixelized model with ray tracing to calculate the geometric response, and the filtered backprojection algorithm. When the reconstruction is displayed on fine grid points, the continuity and resolution of the image is preserved without the ring artifacts seen in the unit-strip “natural” pixel model. With present computing power, the geometric response correction using the proposed projection space reconstruction approach is not yet feasible for routine clinical use. Index Terms— Collimator, geometric response, natural pixels, projection space image reconstruction, single value decomposition, SPECT. Manuscript received October 17, 1996; revised September 10, 1997. This work was supported in part by the National Institutes of Health under Grant RO1 HL 39792 and by Picker International. The Associate Editor responsible for coordinating the review of this paper and recommending its publication was M. W. Vannier. Asterisk indicates corresponding author. Y.-L. Hsieh and G. L. Zeng are with the Medical Imaging Research Laboratory, Department of Radiology, University of Utah, Salt Lake City, UT 84108 USA. *G. T. Gullberg is with the Medical Imaging Research Laboratory, Department of Radiology, University of Utah, CAMT, 729 Arapeen Drive, Salt Lake City, UT 84108 USA (e-mail: [email protected]). Publisher Item Identifier S 0278-0062(98)02994-2.

I. INTRODUCTION

I

T has been a goal for some time to implement models of the geometric response of the collimator in order to compensate for the spatially varying blurring in tomographic reconstructions that occur in single photon emission computed tomography (SPECT). Blurring is one of the major factors that degrade the resolution and produce shape distortions and quantitative inaccuracies in SPECT images [1], [2]. In most of these models it is assumed that the reconstructed image is pixelized. A more “natural” approach would be to represent the image as a continuous function and the projections as discrete measurements corresponding to the finite resolution of the collimator-detector system. The reconstruction is formulated as a projection space image reconstruction where the discrete projections are related to the continuous image distribution function by a spatially varying photon distribution function (PDF) that models the geometric response of the SPECT collimator. The aim of this paper is to explore the development of a more “natural” pixel representation of the geometric response of the collimator to enable better correction of reconstruction artifacts that result from the spatially varying response of the collimator, with the eventual goal of modeling a true “natural” pixel representation of data acquired by SPECT systems. The true “natural” representation should model the effects of collimator response, scatter, and attenuation in addition to the intrinsic resolution of the camera system. Various image space reconstruction methods have been proposed to compensate for the spatially varying geometric response of the collimator. The proposed analytical methods use the filtered backprojection algorithm with restoration filters [3]–[6] which approximate the blurring effect by a spatially invariant blurring function or they use the frequencydistance principle [7]–[10] which incorporates the distancedependent collimator blurring into the Fourier transform of the sinogram. Iterative methods assume that the continuous activity distribution is either digitized or uses an interpolated projector–backprojector. For the pixelized image model, raydriven, and pixel-driven projector–backprojectors that model the geometric response, have been implemented. It was assumed in the ray-driven models that the geometric response could be approximated by a fan [11], or a cone [12] of rays emanating from the projection bin with each ray weighted to match the geometric response of the collimator. The pixel-

0278–0062/98$10.00  1998 IEEE

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

driven methods used a geometric response function, which was a function of the distance from the center of the pixel to the detector. In some cases, the geometric response was measured from either a point source or line source fitted to a Gaussian function [13]. In other cases the geometric response was based upon a distribution function calculated from a Monte Carlo simulation [14], or from an actual geometric formulation [11], [12] based upon the collimator hole shape and size [15]. The speed of implementing the pixel-driven projector–backprojector was improved by rotating the image matrix so that the face of the imaging volume remained parallel to the face of the detector. Each plane in the volume parallel to the detector is blurred, plane by plane, by a convolvor that is a function of the distance from the detector. Results are summed along projection and backprojection rays using equal pixelized weighting [16], [17]. For converging geometries (cone and fanbeam), the projection and backprojection operations were sped up by warping the image space to a parallel geometry [18]. More recent implementation of geometric response correction has increased the speed calculating the projector–backprojector by using small convolution kernels from which the blurring function is accomplished by starting with the farthest slice parallel to the detector, convolving its intensities, then adding the result to the intensities of the next slice and continuing convolutions of the intensities for the following slices after the result of the convolution of the previous slice has been added to it. By this incremental blurring, one concatenates the small convolution kernel in a way that represents the actual blurring that occurs as distance from the detector is increased [19]. Concurrent with the development of more efficient models for modeling the geometric response has been the development of faster iterative reconstruction algorithms such as the ordered-subset–expectation-maximization (OS-EM) algorithm [20]. Unfortunately, most of these models do not model the spatial variance of the geometric response across projection bins when the collimator septa is thick, or the collimator hole size is large. For example, this occurs with 511-keV collimators (3.43 mm septa thickness and 5.08-mm collimator hole size). In this paper, projection space (instead of image space) reconstruction methods are used. Pixels more “natural” for modeling the two-dimensional (2-D) geometric response of the collimator are utilized in order to preserve the continuity of the activity distribution and to better model the spatially varying PDF in the projector–backprojector. An initial inclination might be to propose that the geometric response be modeled as strip functions that diverge from the sampling bin; the divergence accounts for the resolution fall-off with increased distance from the face of the collimator. The strip functions are planar cones, with the vertex at the sampling bin and with varying weights established such that the integral over every cross-sectional area of the cone parallel to the detector has equal area no matter what distance the plane of the crosssectional area is from the collimator. This means that the strip function is no longer a “natural” pixel as originally conceived (that is, a characteristic function of one and zero); but instead, it decreases in value with increased distance from the collimator. The problem with this representation

25

is the difficulty in calculating the intersection of the strip functions, which is necessary to formulate the inverse tomography problem. We overcome this difficulty by using smaller parallel strips to approximate the geometric “natural” pixel. The intersection of parallel strips can be calculated by using straightforward analytical expressions. The system geometric response function is defined by summing the intersection of an array of multiple strips rotated at small, equal angles over the acceptance angle of each collimator hole for one projection bin. For one projector, each collimator hole is considered separately within the projection bin. For two other projectors, the holes are not considered separately, but rather, the sum of the holes are considered in one projection bin. In one projector, each strip is of equal weight, and in the other, a Gaussian weighting is used. As many as 2001 strip functions pass through one collimator hole in order to calculate the geometric response of the collimator. The model builds upon the concept of “natural” pixels originally conceived as unit-strip functions by more accurately representing the “natural” pixel for collimator holes and the functional variation of the geometric response with increased distance from the collimator. In previous work [21] it was found that the implementation of the projection space reconstruction method could be made more efficient if the image was represented by the backprojection of a vector along “natural” pixels different from those used in defining the projections. This was a generalization of the “natural” pixel representation, which was implemented in [21] for fan beam and truncated projection geometries and is anticipated to be a more efficient method for correcting attenuation and geometric response. Previous to this, in our application of “natural” pixels to reconstruct exponential [22] and variable attenuated Radon projections [23], the transpose of the “natural” pixel projection operator was used as the backprojector. The implementation was extremely complicated and inefficient for addressing the variable attenuation case. A backprojection operator that does not have to model the attenuation or the geometric response would improve efficiency. In this paper, we present results of geometric response correction where the geometric response is modeled only in the projector. The reconstruction is formulated as a solution , wherein to the imaging equation is the projection measurement, is the unknown vector (analogous to a modified filtered projection vector in a filtered is the projector backprojection reconstruction algorithm), is a generalized that models the geometric response, and backprojector defined so that the reconstructed image is an expansion of “natural” pixels with coefficients equal to the components of the solution vector . It has been common practice to define the backprojection operator as the adjoint of the projection operator. However, can be defined quite arbitrarily; for example, a uniformly weighted “natural” pixel (no geometric response modeling) could be used instead of the adjoint of the projector for the backprojection. Thus the reconstruction problem is formulated as determining a least squares solution from a vector space different from that of the projection measurements . Using the singular value decomposition (SVD) method or an iterative algorithm to

26

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

invert the equation , a different solution for can be determined for different choices made for the backprojector . Assuming the image can be expanded by using a generalized natural pixel basis, the final image is , using these generalized obtained by backprojecting “natural” pixels and sampling at a discrete array of points. In this paper we outline the development of a 2-D system geometric response function that simulates the septa thickness, the collimator hole size, and the distribution of the collimator holes in one projection bin in order to model the 2-D spatially variant photon distribution transaxially to the axis of rotation. First, a general projection model is developed for modeling the geometric response of the collimator based upon the actual physical characteristics of collimator holes by using many small strip functions to calculate a more “natural” representation of the geometric response. From this, two simpler models are developed that simplify the calculation over individual collimator holes into a one for one projection bin: One projection model assumes equal weighting for each strip function, whereas the other projection model assumes a Gaussian weighting. Expressions are given for the elements in the imaging equation . of the matrix This equation is solved using the SVD method or the iterative conjugate gradient algorithm. The reconstruction is obtained by backprojecting the solution . Parallel beam and fan beam data of a physical three-dimensional (3-D) Hoffman brain phantom and a Jaszczak cold rod phantom are used to evaluate the correction for the spatially varying geometric response in the reconstructions. II. THEORY A. More “Natural” Pixel Representation A “natural” pixel basis arises “naturally” from the scanning geometry without digitizing the image [24]–[29]. If photon scatter, geometric response, and attenuation are ignored, the basis forms a decomposition of the image plane into a set of overlapping pixels, which are strips uniquely defined by the paths of the projection rays for a specified geometry. Each path corresponds to a “natural” pixel, which forms the support of a characteristic function. The collection of all of the characteristic functions forms a “natural” pixel-basis set. For example, a “natural” pixel corresponding to the angle index , and the projection bin index , is shown in Fig. 1(a), with the characteristic function , where is a vector in the – plane. The value of the characteristic function is one for in the strip corresponding to the natural pixel, and zero otherwise. The integral of the activity distribution (considered to be continuous) over the “natural” pixel with index is the projection of the activity within the particular strip corresponding to the “natural” pixel, into the corresponding projection bin . This paper considers a more “natural” representation for the basis set so that it includes the 2-D geometric response of the collimator. The more “natural” representation is shown in Fig. 1(b), with photon distribution function PDF . The function PDF is a probability density function which

specifies the probability that a photon emitted at location will be detected in the th projection bin at projection view . The support (projection ray) of the photon distribution function PDF is the more “natural” pixel. The more “natural” pixel diverges from the projection bin and corresponds to the support of an aperture function for a series of collimator is not holes within a projection bin. The function PDF a characteristic function (i.e., zero and one), it is a spatially varying photon probability distribution function that specifies the photon fluency at the point that is expected to project into the projection bin at projection view . The projection operator is defined as a vector whose [30]. Each functional components are functionals: maps the continuous function (a distribution of isotope by concentration) to a measured projection value integrating the function weighted by the function PDF over the more “natural” pixel that corresponds to the angle index and the projection bin index . The functional is defined as PDF

(1)

where the integral over is restricted to a finite support and the photon distribution function PDF not only specifies the path of the photon rays that will project into the projection bin at angle index , but also characterizes the physics of the detection of photons within the support. Each distribution function for the projection bin and the angle index is confined at a certain range and is defined according to how the projection measurement is obtained. The collection of all forms a more the photon distribution functions PDF “natural” pixel-basis set. deIn general, the photon distribution function PDF pends upon several physics factors involved in the image detection process. These include the attenuation and scatter of photons emitted from injected tracers as well as the geometric response of the collimator. In our previous work [22], [23] “natural” pixel models were used to reconstruct exponential and variable attenuated Radon projections. Other additional physics factors of the system can influence the blurring of the system response as well. These include: scattering within the scintillation crystal, light output from the crystal, the photoelectron yield of the photomultiplier tubes, and the accuracy of the electronic - positioning circuitry which are intrinsic factors of the system. When modeling the total blurring effect by using a “natural” pixel representation, these intrinsic physical factors of the system should also be considered. In Section II-B, we only address the modeling of the effect of the collimator. B. The System Geometric Response Function The system geometric response is only one aspect of the total PDF and its determination depends upon the collimator structure in terms of the arrangement of the collimator holes, their size, length, and shape. It is difficult to derive an analytical function of the geometric response for any projection bin because each bin has a different distribution of collimator holes; in some cases the lead septa actually project

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

27

(a)

(b) Fig. 1. Parallel beam geometry. (a) A natural pixel strip is illustrated as the support for the characteristic function jm (r ) corresponding to the angle index m and the projection bin index j . The value of the characteristic function is one for r inside the strip in the x–y plane. (b) A more “natural” pixel representation is illustrated as the support for the characteristic function PDFjm . This is the PDF, which is a probability distribution that considers all of the physics factors of the imaging detection process. The integral of  over the “natural” pixels within the support D maps the continuous function  into a measurement pjm .

an opaque image. However, the system geometric response can be calculated by considering the actual geometry and arrangement of the collimator holes through the use of multiple strip functions. In this section, a formulation which considers each collimator hole is developed, whereas in Section IIC, an approximation is derived that allows a more efficient calculation of the geometric response without the necessity of considering each collimator hole. To obtain some heuristic perspective, let us first consider the fan beam case where a parallel strip focused to the focal point (but with equal width from the detector bin to the focal point) was used to represent the fan beam “natural” pixel [21]. Initially, this relationship may not be obvious because it may

seem more natural to use a unit-strip function with support that diverges from the focal point to a width at the detector equal to the width of the bin. However, these pixels are required to have a nonuniform weighting distribution so that the line integral perpendicular to the central axis of the projection bin would be equal for every position along the ray in order to preserve count density. Instead the parallel unit-strip satisfies the equal weighting requirement and provides a much easier means for calculating the intersection of overlapping pixels. The parallel strip is also a good approximation for fan beam geometry: It better represents the physics of the collimated detector used in SPECT in that a collimator hole does not see rays converging to a focal point at some distance from the detector, but, because

28

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

(a)

(b)

(c)

(d)

Fig. 2. Using multiple strip functions to calculate the system geometric response function for an array of collimator holes arranged within one projection bin. (a) The dimensions of an array of collimator holes within one projection bin: binw is the projection bin width, ColL is the collimator length, Colth is the thickness of the collimator hole, and Colw is the inner flat-to-flat distance of the hexagonal collimator hole. (b) The cross-sectional slice through the array of collimator holes. The location is at the central cross-section of the projection bin in the  -plane. (c) The maximum acceptance angle thmax for one collimator hole and one strip s. (d) An array of strip functions Uh; s used to calculate the system geometric response function for one projection bin. Note that the widths of the strips vary with rotation angle.

of the finite aperture, sees rays that diverge from a focus at the detector, much like an inverted fan beam or cone beam geometry. Therefore using uniform parallel strips as natural pixels for fan beam geometry was physically more realistic and seemed to perform some resolution recovery in phantom studies [21]. Even so, the unit-strip function is still not sufficiently “natural” to model the true spatial photon distribution. By using several unit-strip functions, fanned in such a way that the strip function rotates through the acceptance angle of the aperture of the collimator hole, the weighting for the geometric response of the collimator can easily be calculated from the summation of the intersectional areas of overlapping strips. This results in a more “natural” formulation of the geometric response. The ideal collimator in Fig. 2(a) is mounted to the detector. It is assumed that the gap between the collimator and the detector is ignored. Since in most low-energy collimators the design parameters are chosen to allow less than 5% of total photons to penetrate the lead septa and to reach the crystal, photon penetration is ignored. The collimator has

, the septa thickness , the inner the hole length , flat-to-flat distance for the hexagonal collimator hole . One cross-sectional view and the projection bin width of one projection bin is shown in Fig. 2(b). The maximum acceptance angle of the photons detected for one collimator atan , as shown in Fig. 2(c). hole is Unit-strip functions shown in Fig. 2(d) are rotated through the acceptance angle of the collimator hole in order to estimate the system geometric response function. is defined to be A unit-strip function if is inside the strip otherwise

(2)

where the strip is tipped at the angle from a line perpendicular to the detector at the center of the collimator hole for collimator hole . An example using only five unit-strip for each collimator hole is shown in functions Fig. 2(d). Note that the width of the strip, , varies depending on the angle of the

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

29

(a) Fig. 3. (a) Parallel beam geometry. A strip of constant width is shown for the characteristic function n jm . The projection pm is rotated by m with unit vector  m situated parallel to the detector. The center of the j th projection bin for pm is pj . The bin width for pm is wp . The center line for the strip, n , is rotated n from the line perpendicular to the center of the projection bin at  . which is the domain of the characteristic function jm pj

1

strip. When is equal to zero, the strip perpendicular to the collimator face has a maximum width equal to the width of the hole. The strip rotated at the maximum angle has the smallest width and is determined so that the strip at that angle can pass between the two septa without penetrating the septa and yet still intersecting the crystal. Each strip, which has width that depends on the collimator hole length, the collimator hole size, and the orientation of the strip, is the region in which photons emitted at an angel can pass through the collimator hole. For one particular collimator hole, the geometric response function for a photon emitted at is equal to the normalized integration over all angles of strips that include the point up to the acceptance angle of the collimator hole. Thus, for one projection bin, the spatially varying geometric response function SGRF at the spatial position is calculated by summing the integrals for each collimator hole within the bin and normalizing by the total integral along the direction

SGRF

where in order to make numerical calculations, we discretize

the angles using the index

SGRF

(3)

The index ranges from to . When equals zero, the strip is perpendicular to the collimator hole. The angle of rotation between the individual strip functions is equal to , where is the total number of unit-strip functions. The expression in (3) represents a 2-D system geometric response function. Conceptually, it can be extended to a 3-D system geometric response function where, for example, strip functions are replaced by cylindrical characteristic functions. The system geometric response function derived here is used as a reference in the calculation of the following approximate formulations. C. New Projectors for Modeling the System Geometric Response Since the collimator holes may not align the same way for each projection bin, formulation of the system geometric response function in (3) is complicated and time consuming to calculate. We propose two new projectors that approximate

30

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

(b) . The projection pm with focal length Fig. 3. (Continued.) (b) Fan beam geometry. A strip of constant width is shown for the characteristic function n jm F and radius of rotation R is rotated by m with unit vector  m parallel to the detector. The pseudo focal length for the nth strip is Fpj; n . The center n of the projection bin for pm is pj , and the bin width is wp . The center line for the strip, which is the domain of the characteristic function jm , is rotated n from the line situated perpendicular to the center of the projection bin at pj . The width of the strip is equal to the width of the projection bin wp multiplied by the sine of the angle between the rays of the strip and the detector.

~

1

the system geometric response function in (3) and simplify the calculation so that each collimator hole need not be considered. First, strip functions of constant widths equal to the bin width will be defined. These differ from the strip functions , which have variable widths. These new strip functions rotate through a sufficient angle (empirically determined) so that the new formulation is a good approximation of the system geometric response function (3) developed in the previous section. Fig. 3(a) shows the schematic diagram for the parallel beam geometry with one unit-strip function illustrated. The strip has a width equal to the projection bin width and a line passing through the center of the strip intersects the center of the projection bin at the projection view . Note, refers to the natural pixels in the projection, later we use to refer to the natural pixels in the backprojection. The th strip is rotated by an angle relative to the central line of the projection bin. At the rotation view and the projection bin , the th strip function is defined as

where

(5)

cot The unit vector

is parallel to the face of the detector and

the unit vector

is orthogonal to

in Fig. 3(a).

is the

rotation radius, is the position vector of a point source, and is the projected location of the point source in the detector coordinate. When the projected location is inside the projection bin, the characteristic function is weighted as one, otherwise, it is weighted as zero. Fig. 3(b) shows the schematic diagram for the fan beam geometry. The unit-strip function is defined as if otherwise (6) where

if otherwise

(7) (4)

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

(a)

31

(b)

N=

2 in (8). Fig. 4. The strip functions used to model the geometric response for the two new projectors. (a) Five equally weighted strip functions for = 2 in (9). A portion of each Each strip is bounded by parallel rays and each pair is separately identified. (b) Five Gaussian-weighted strip functions for strip is shaded according to the weight for the strip. As indicated, the overlap of the shaded strips gives a somewhat Gaussian shape in terms of intensity.

N

cot acot is the pseudo focal length for the th strip function and is the focal length for the fan beam geometry as shown in Fig. 3(b). Note that (6) has the same form as in (4), but in (7) differs from that in (5). Using these definitions, two projectors that model the system geometric response are proposed. For one, each strip is of equal weight. For the other, the strips are weighted by a Gaussian distribution so that the weights for the strips represent the fall-off of the geometric response as the acceptance angle increases. 1) Equally Weighted Strip Distribution Function: A distribution function is defined by summing equally weighted strips which rotate at equal angles of atan from the angle atan to atan relative to a line perpendicular to the projection bin. The inner flat-to-flat distance of the hexagonal collimator hole and the collimator hole length are defined in Fig. 2(a). The rotation angle atan , which is smaller than half the acceptance angle of the colli, was found to provide a good mator hole atan approximation for the system geometric response function in (3). The distribution function is defined as

(8)

An illustration of this distribution function with shown in Fig. 4(a).

is

2) Gaussian-Weighted Strip Distribution Function: The second distribution function is defined by summing Gaussian-weighted strips rotated at equal angles of atan from the angle atan to atan relative to a line perpendicular to the projection bin. The rotation angle atan , which is used to define the extent of the boundary of the more “natural” pixel, is equal to half the acceptance angle of the collimator hole. The distribution function is defined as (9) where the Gaussian weighting function is defined as

(10) is An illustration of this distribution function with shown in Fig. 4(b). When is very small, such as 0.3, the distribution function is very close to a single unit-strip function. When is large, such as 300, the distribution function is similar to that of the equally weighted strip function. In order to approximate the system geometric response function, is chosen empirically.

32

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

Using one of these distribution functions, the projection is a vector whose components are functionals: operator , such that (11) The integral over is restricted to a finite support , and the distribution function specifies the path of the photon rays that will project into bin at angle index . The funcis a probability distribution that considers only tion the geometrical aspects of the collimator. Each distribution is confined function for projection bin and angle index to a certain domain. The collection of all of the distribution forms what we term a more “natural” functions pixel basis set.

1) Parallel Beam Geometry: First, we consider a parallel geometry that has strip functions for both the projection and backprojection operations, but which are not necessarily of equal width. The projector has a bin width and the back. The angle difference between projector has a bin width and that of the the rotation angle of the backprojector is . The number of strips for projector and the number for the backprojector the projector is . is For the equally weighted strip backprojector–projector, the integral in (15) can be derived to obtain the following result:

(16) D. Backprojector maps vectors from a vecThe backprojection operator tor space of discrete projection values to continuous image operates on a vector , with functions. The backprojector elements , giving (12)

, used in the backprojection where the function operation, has supports corresponding to “natural” pixels that may be different from those of used in (11) [21]. Thus, maps a projection vector into a continuous function , multiplied by the that is the summation of the elements corresponding function . E. Backprojector–Projector—The

-Matrix

where is the intersection area of a projector and a backprojector strip function [21]. The incremental rotation angle for the strips is defined above. The formulation in (16) is the summation of several strip intersections of the geometric response for the projection operator and those representing the geometric response for the backprojection operator. The overlap of a strip of width with a strip of width , represented by (16), is limas given in (15). Therefore, when ited to the support is equal to zero, (16) becomes the and another collinear intersection of one strip with width restricted to the support . For example, strip of width and . For the Gaussian-weighted strip backprojector–projector, are the elements of the matrix

Combining the expressions for the projection operation in (11) and the backprojection operation in (12), the backprojection–projection operator produces a real number for each functional (17) (13) In matrix form the backprojection–projection operator is (14) where is the measured projection data and each element of is the integral in (13) (15) Expressions for the elements of the backprojecdescribed in (15) can be tion–projection operator formulated for parallel beam and fan beam geometries using the following described technique.

where , and are defined in (10). The adjustable factor is chosen to approximate the system geometric response function. The incremental rotation angle is defined above. 2) Fan Beam Geometry: For fan beam geometry, it is assumed that the focal lengths for the projection and backprojection are and , respectively. The location of the center and of the projection bin and the backprojection bin is , respectively. The projector has a bin width and the . The number of strips for backprojector has a bin width , and the number for the backprojector the projector is . The angle difference between the projection angle is . and the backprojection angle is is defined In the formulation, a pseudo focal length for projection bin and strip , and a pseudo focal length

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

SPECIFICATIONS Field of view at 100 mm 240

2240 mm2

Maximum Useful Energy

Hole Shape

160 KeV

Hexagonal

is defined for projection bin acot

FOR THE

Colw

(Flat to Flat Dis.) 1.40 mm

33

TABLE I HIGH-RESOLUTION PARALLEL BEAM COLLIMATOR

ColL

(Hole Length) 27.0 mm

and strip (18)

Colth

(Septa Thickness) 0.18 mm

System FWHM at 0 cm 3.5 mm

Number Bins/View

(19)

The incremental rotation angle is defined above. Each element of the -matrix for the equally weighted strip backprojector–projector has the expression (20) shown at the bottom of the page. For the Gaussian weighting, the pseudo focal lengths for the strip functions are acot

(21)

acot

(22)

and

Again, the incremental rotation angle for the strips is defined above. -matrix for projection bin at Each element of the projection view and backprojection bin at backprojection has an expression [as shown in (23) at the bottom of angle , and are defined the page], where in (10). The adjustable factor is chosen to approximate the simulated spatial PDF. III. METHODS Correction for the geometric response was evaluated for the equally weighted strip and Gaussian-weighted strip projectors for parallel beam and fan beam geometries. Modeling for attenuation was not included in the evaluation. The physical 3-D Hoffman brain phantom was used to collect data for both

System Sensitivity 245 cpm/Ci

TABLE II GEOMETRIC SPECIFICATIONS FOR THE PARALLEL BEAM PROJECTION DATA OF THE 3-D HOFFMAN BRAIN PHANTOM

and acot

System FWHM at 10 cm 8.0 mm

Number Views/180

Bin Width

Rotation Radius

Original Projections

60

60

5.35 mm

212.7 mm

Processed Projections

40

60

5.35 mm

212.7 mm

geometries and the Jaszczak cold rod phantom was used to collect data for fan beam geometry. A. Parallel Beam Geometry (3-D Hoffman Brain Phantom) The system geometric response function in (3) and the two approximated formulations in (8) and (9) were generated according to the parallel collimator specifications in Table I and then plotted. A total of 2001 strips for each collimator hole were used in (3) to calculate the system geometric response function for one projection bin. The approximated system geometric response function for the equally weighted strip in (8) and the Gaussian-weighted strip model in (9) were in (10) also generated by using 2001 strips. The factor was chosen to be 3.20 in order to approximate the system geometric response function in (3) at the central cross-section of the projection bin, 5–30 cm away from the detector face. A 3-D Hoffman brain phantom (Data Spectrum Corporation, Hillsborough, NC) was used to evaluate the reconstruction of actual data collected using a SPECT system. The phantom was injected with 20 mCi of 99m Tc, and the projection data were collected using the PRISM 3000 (Picker International, Cleveland, OH) with high-resolution parallel collimators consisting of hexagonal holes. Specifications for the parallel collimators and the projections used in the reconstruction are listed in Tables I and II, respectively. A total of 120 64 64 (5.35 mm/bin) projections of 190 s per view were acquired over 360 . The rotation radius was 212.7 mm. To reduce

(20)

(23)

34

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

SPECIFICATIONS Field of view at Maximum 100 mm Useful Energy 240

2 300 mm2

160 KeV

Hole Shape Hexagonal

Colw

FOR THE

(Flat to Flat Dis.) 1.40 mm

TABLE III HIGH-RESOLUTION FAN BEAM COLLIMATOR

ColL

(Hole Length) 27.0 mm

computation time and memory requirements, projection data for one slice were generated by truncating the 64 bins into 40 projection bins (5.35 mm/bin), and by using only 60 views over 180 . The total counts for one slice summed over these projections was equal to 9.7 10 . The reconstruction was confined to a circular support with a radius of 102 mm. Within the circular support, each of the (40 bin 60 angles) elements of the -matrix were calculated by using (16) and (17) for the equally weighted strip projector and the Gaussian-weighted strip projector, respectively. The projector had 2001 strips and the backprojector was a single unit-strip function. The SVD method [21]–[23] was used in the equation . The number of to solve for singular terms was determined from the observation of the reconstructions for different numbers of singular terms. The number that provided the best tradeoff between resolution and noise was selected. The final images were obtained by backprojecting to a discrete array of points that comprised a grid of 40 40 (5.35 mm between grid points). A filtered backprojection algorithm was also performed. A ramp filter was applied to each of the 60 projections sampled over 180 . Each projection had 40 projection bins (5.35 mm/bin). The filtered projections were then backprojected into a 40 40 array (5.35 mm between grid points).

B. Fan Beam Geometry (3-D Hoffman Brain Phantom) Geometric response functions were also generated for a fan beam collimator according to specifications in Table III. As for the parallel beam geometry, a total of 2001 strips for each collimator hole were used to calculate the system in (10) geometric response function. However, the factor for the Gaussian-weighted modeled was chosen to be 2.6 in this case. A 3-D Hoffman brain phantom study was repeated using a high resolution fan beam collimator of 65-cm focal length (Table III). The longer than normal focal length collimator was used to insure that the phantom data were not truncated. A total of 120 64 64 (5.35 mm/bin) projections of 150 s per view were sampled over 360 . The rotation radius was 207.5 mm. Projection data of 32 bins (10.7 mm/bin) for one slice were then generated by combining two projection bins into one (Table IV). Of the 120 views, only 60 views over 360 were used in the reconstruction by selecting the even number views. The total counts summed over these projection data were equal to 6.2 10 . The reconstruction was confined to a circular support with a radius of 161.2 mm. Each of the (32 bin 60 angles) elements of the -matrix was calculated using (20) and (23) for the equally weighted strip projector and the Gaussianweighted strip projector, respectively. The projector had 2001

Colth

(Septa Thickness) 0.18 mm

System FWHM at 0 cm 3.5 mm

System FWHM at 10 cm 8.0 mm

System Sensitivity

Focal Length

300 cpm/Ci

65 cm

TABLE IV GEOMETRIC SPECIFICATIONS FOR THE FAN BEAM PROJECTION DATA OF THE 3-D HOFFMAN BRAIN PHANTOM Number Number Bins/View Views/360

Bin Width

Rotation Radius

Original Projections

64

120

5.35 mm

207.5 mm

Processed Projections

32

60

10.7 mm

207.5 mm

TABLE V GEOMETRIC SPECIFICATIONS FOR THE FAN BEAM PROJECTION DATA OF THE JASZCZAK COLD ROD PHANTOM Number Bins/View

Number Views/360

Bin Width

Rotation Radius

Original Projections

64

120

7.12 mm

207.5 mm

Processed Projections

64

120

7.12 mm

207.5 mm

strips and the backprojector was a single unit-strip function. The SVD method was used to solve for in the equation . The choice of the number of singular terms was determined by the observation. The final images were obtained by backprojecting to a discrete array of points that comprised a grid of 64 64 (5.35 mm between grid points) or 256 256 (1.34 mm between grid points). For comparison, a ramp filter was used in a filtered backprojection algorithm to reconstruct a 64 64 image (5.35 mm between grid points) from the same 60 projections (Table IV). C. Fan Beam Geometry (Jaszczak Cold Rod Phantom) The Jaszczak cold rod phantom (Data Spectrum Corporation, Hillsborough, NC) was also used to acquire fan beam projection data. This was used to eliminate the third dimension in order to better evaluate the implementation of the 2-D geometric response correction. The phantom was injected with Tc, and the projection data were collected as 10 mCi of before with the 65-cm focal length high-resolution fan beam collimator (Table III). The radius of rotation was 207.5 mm. A total of 120 64 64 projections of 180 s per view were sampled over 360 . All projections with 64 projection bins (7.12 mm/bin) were used in the reconstructions (Table V). The total counts summed over the projection data were equal to 3.2 10 for one slice. The geometric response correction using the more “natural” pixel model presented here was compared with the geometric response correction using a conventional pixelized image model. In the “natural” pixel approach, each of the (64 bin 120 angles) elements of the -matrix were calculated using (23) for the Gaussian-weighted strip projector. Eleven strips were used in the projector and 11 strips in the backprojector. Reconstructions were confined to a circular support with a radius of 136.6 mm. The conjugate gradient algorithm

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

35

(a)

(b)

(c)

Fig. 5. Parallel beam collimator (see Table I). For one projection bin (a) the system geometric response function SGRFbin in (3), (b) the distribution function jm in (8), and (c) the distribution function jm in (9).

with 100 iterations was used to solve for in the equation . The final images were obtained by backprojecting to a discrete array of points that comprised a grid of 64 64 (4.27 mm between grid points) and then a Chang attenuation correction was applied [31]. For the image pixelized approach, the geometric response was modeled using a ray-driven projector–backprojector with an 11 11 cone of rays to model the geometric response [12]. The expectation-maximization–maximum-likelihood (EM–ML) algorithm [32] with 100 iterations was used to obtain the reconstruction. The Chang attenuation correction was applied to obtain the final result. A filtered backprojection algorithm with ramp filter was also used to reconstruct a 64 64 array (4.27 mm between grid points) from the same 120 projections (Table V). Also, the Chang method was used to correct for attenuation. IV. RESULTS A. Parallel Beam Geometry (3-D Hoffman Brain Phantom) Fig. 5(a) shows the system geometric response function of one projection bin for parallel beam geometry. Fig. 5(b)

and (c) shows the approximated system geometric response function for the equally weighted strip projector and the Gaussian-weighted strip projector. The system geometric response function in Fig. 5(a) shows multiple peaks at distances close to the collimator. The method used to model the geometric response involves rotating strips over the aperture of the collimator holes. Multiple peaks are apparent in the response function at short distances due to the width of the septa and diminishes with increased distance from the collimator. The width of the septa restricts the range of the rotation of the strips from adjacent collimator holes so that there is a limit in the range of rotation of possible overlapping strip functions. As a result, voids are created. The intrinsic resolution of the detector was not included in the system geometric response function in (3). We suspect that with the inclusion of the intrinsic response these peaks will diminish. The distribution function that used equally weighted strip functions, shows a trapezoidal appearance to the response for all distances from the collimator. Using a Gaussian weighting in Fig. 5(c) results in response functions that appear more like the expected true response in Fig. 5(a), except that they do not model septal effects at short distances. The differences between these two approximate models and the system geometric response model

36

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

(d)

(e) Fig. 5. (Continued.) (d) the difference between the distribution functions in (b) and (a) and (e) the difference between the distribution functions in (c) and (a) is plotted at 5, 10, 15, 20, 25, and 30 cm away from the collimator face and at a vertical location through the central cross-section of the collimator holes shown in Fig. 2(a). Peaks are seen at 5 and 10 cm in the system geometric response function due to collimator septum. The equally weighted strip and the Gaussian-weighted strip distribution functions were generated with 2001 strip functions. A factor  3:20 was chosen for the Gaussian distribution in (10), so that the distribution function jm in (9) would correspond to the SGRFbin in (a) at 5–30 cm away from the detector.

=

based upon the collimator holes is better illustrated in the difference profiles in Fig. 5(d) and (e). Fig. 6(a) and (b) shows the reconstructed images produced by the filtered backprojection algorithm with ramp filter and the SVD method with the unit-strip projector–backprojector. Fig. 6(c) demonstrates reconstructions using both unit-strip backprojector and the equally weighted strip projector and Fig. 6(d) demonstrates reconstructions using unit-strip backprojector and the Gaussian-weighted strip projector. No filtering was performed in order to preserve the maximum resolution. The reconstructions in Fig. 6(b)–(d) are displayed on grid points of 40 40 (5.35 mm between grid points) and were generated using the unit-strip natural pixel backprojector. Results show that the reconstructions made using the new proposed projectors that model the geometric response [Fig. 6(c) and (d)], have higher resolution than the filtered backprojection in Fig. 6(a) and the SVD reconstruction with the unit-strip projector–backprojector in Fig. 6(b). Even though the geometric response functions differ somewhat for the Gaussian-weighted strip projector and the equally weighted strip projector, Fig. 6(c) and (d) shows little difference in the quality of the reconstruction.

Fig. 7 compares a 40 40 (5.35 mm between grid points) parallel geometry reconstruction of the Hoffman brain phantom using the Gaussian-weighted strip projector with 11 strips with that using 2001 strips. From the intensity profiles of Fig. 7(a) and (b), the reconstructed image that uses 2001 strips is almost identical with the one that uses 11 strips. This is better demonstrated in the difference image in Fig. 7(c) using a different gray scale. B. Fan Beam Geometry (3-D Hoffman Brain Phantom) Fig. 8(a) shows the geometric response function for one projection bin for the fan beam collimator. Fig. 8(b) and (c) shows the approximated system geometric response function for the equally weighted strip projector and the Gaussianweighted strip projector. Results similar to those for the parallel beam collimator in Fig. 5 are shown. However, here the profiles are at different distances from the collimator than the profiles in Fig. 5. The reconstructed images obtained using the filtered backprojection algorithm, the unit-strip projector model, the equally weighted strip projector model, and the Gaussian-weighted strip projector model are shown in Fig. 9(a)–(d), respectively.

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

37

(a)

(b)

(c)

(d)

Fig. 6. Parallel beam geometry. Reconstructions of the Hoffman brain phantom by using (a) the filtered backprojection algorithm, and the SVD method with (b) 990 terms (unit-strip projector–backprojector model), (c) 690 singular terms (equally weighted strip projector of 2001 strip functions and unit-strip backprojector model), and (d) 680 singular terms (Gaussian-weighted strip projector of 2001 strip functions and unit-strip backprojector model). All images are displayed on a 40 40 square grid (5.35 mm between grid points). Intensity profiles are shown for three locations.

2

In all cases, the array size was 64 64 with 5.35 mm between grid points. Again, no filtering was performed. The results show that the reconstructions produced by the new projectors with geometric response modeling have improved resolution. Fig. 10 shows the same reconstruction (i.e., the same solution ) backprojected onto 64 64 (5.35 mm between grid points) and 256 256 (1.34 mm between grid points) grids. was determined using In Fig. 10(a) the solution to the Gaussian-weighted strip projector with 11 strip functions and the unit-strip backprojector. In Fig. 10(b) the solution was determined using Gaussian weighting with 11 strip functions in both the projector and backprojector. Even though the 64 64 (5.35 mm between grid points) reconstructions are similar for both unit-strip backprojector and the Gaussianweighted strip backprojector, the latter does not show ring

artifacts in the 256 256 display, and shows the detail of the continuous image better than the unit-strip backprojector. The ring artifacts are caused by the aliasing in backprojecting a small number of large unit-strip natural pixels into a finely sampled 256 256 image array.

C. Fan Beam Geometry (Jaszczak Cold Rod Phantom) Fig. 11 shows 64 64 reconstructions (4.27 mm between grid points) of the filtered backprojection, geometric response correction using a conventional image pixelized model, and geometric response correction using the natural pixel model. Fig. 11(a) shows the result of the filtered backprojection algorithm and Fig. 11(b) shows the result of the image space reconstruction with 3-D geometric response correction using

38

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

(a)

(b)

(c) Fig. 7. Parallel beam geometry. Reconstructions of the Hoffman brain phantom by using the SVD method with the Gaussian-weighted strip projector and the unit-strip backprojector. In (a) 11 strip functions and 670 singular terms were used, and in (b) 2001 strip functions and 680 singular terms were used to model the geometric response and to represent the reconstruction. Image (c) shows the difference between images (a) and (b). To emphasize the structural difference, image (c) is scaled between 175 and 163, which is not the same scaling used in the profiles in images (a) and (b).

0

100 iterations of the EM–ML algorithm with a ray-driven geometric response model. Fig. 11(c) shows the results of geometric response correction using 100 iterations of the conjugate gradient algorithm with the unit-strip projector–backprojector model, and Fig. 11(d) shows it with the Gaussian-weighted strip projector–backprojector model (11 strips in the projector, 11 strips in the backprojector). The projection space reconstructions that were produced using the Gaussian-weighted strip projector–backprojector provided the best resolution. The iterative ML–EM algorithm was used for the image space reconstruction, while a conjugate gradient algorithm was used for the projection space reconstruction. In comparing projection space methods to image space methods, there is not a direct correlation in convergence rate, even if the same algorithm type are used. For the results presented here, the iteration number was chosen such that the reconstructions stabilized before the algorithm started to diverge. More iterations were chosen for the conjugate gradient algorithm applied to the projection space reconstructions than would have been chosen if applied to the image space reconstruction, because it was found that the image space algorithms tended to converge faster.

V. DISCUSSION A projection space image reconstruction method was implemented to correct for the spatially varying system geometric response in SPECT. The spatially varying system geometric response is modeled by summing the intersection of an array of multiple strip functions (as many as 2001) rotated at equal angles over the acceptance angle of each collimator hole in a projection bin. The system geometric response function is related to septal thickness, collimator hole size, and distribution of the collimator holes in the projection bin. Two projection models were developed to approximate the system geometric response function for both parallel and fan beam geometries. For one projector, each strip is of equal weight, for the other, a Gaussian weighting is used. Reconstructions show that use of the SVD method, or an iterative conjugate gradient algorithm with projectors that model the geometric response of the collimator, results in reconstructions with better detail than those produced using the filtered backprojection algorithm, the SVD method with a unit-strip projector–backprojector, or the iterative EM–ML algorithm with conventional geometric response modeling using image space pixelization. The models presented extend the original concept of natural pixels to pixels

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

39

(a)

(b)

(c)

Fig. 8. Fan-beam collimator (see Table III). For one projection bin (a) the system geometric response function SGRFbin in (3), (b) the distribution function jm in (8), and (c) the distribution function jm in (9) is plotted at 0, 10, 20, 30, 40, and 50 cm away from the detector and at a vertical location through the central cross-section of the collimator holes shown in Fig. 2(a). Peaks are seen at 0 and 10 cm in the system geometric response function due to collimator septum. The equally weighted strip and the Gaussian-weighted strip distribution functions were generated with 2001 strip functions. A factor  2:6 was chosen for the Gaussian distribution in (10) so that the distribution function jm in (9) would correspond to the SGRFbin in (a) at 10–40 cm away from the detector.

=

more “natural” for modeling the geometric response of the SPECT collimator. In addition to producing higher resolution reconstructions and revealing more structural detail, the new projector–backprojectors produce a better model of the continuity of the image than the unit-strip projector–backprojector and the images do not show ring artifacts. For example, the unitstrip projector–backprojector is adequate for reconstructing projection data of 60 projections of 32 projection bins and for displaying the image on a 64 64 grid. However, if the image is displayed on a finer grid of 256 256, ring artifacts occur. The ring artifacts are caused by the aliasing in the backprojection of a small number of large unit-strip natural pixels into a finely sampled 256 256 image array. Better detail, without ring artifacts can be produced by using the Gaussian-weighted strip backprojector to backproject the estimate onto the finer grid. This type of image presentation helps solve the pixelized partial volume problem. In practice, this can be useful when the physician needs more detailed information. A finer local image can be reconstructed by applying Gaussian weighting and directly backprojecting the

estimate onto a local region of finer grid points without any further SVD reconstruction. A major concern at present, is the practicality of applying the proposed method due to the large number of computations involved. A great deal of computer time was required just to produce the 2-D implementation presented in this paper and it is difficult to determine the potential for extension of the method to include scatter and to model 3-D system geometric response. The method has the potential to produce benefits when faster computational speed is available. Several investigators [11]–[13], [16]–[19] have been actively pursuing methods to efficiently and effectively implement image pixelized models of the spatially variant collimator–detector response in projector–backprojector pairs. By combining these projector–backprojector pairs with fast iterative reconstruction algorithms, such as the OS-EM algorithm [20], effective collimator detector response compensation can be accomplished in much shorter time and with less memory requirements than projection space reconstruction methods. In addition, the image pixelized models have been shown to allow 3-D modeling of both the geometric response and the scatter response. In this

40

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

(a)

(b)

(c)

(d)

Fig. 9. Fan beam geometry. Reconstructions of the Hoffman brain phantom by using (a) the filtered backprojection algorithm and the SVD method with (b) 690 terms (unit-strip projector–backprojector model), (c) 350 singular terms (equally weighted strip projector of 2001 strip functions and unit-strip backprojector model), and (d) 350 singular terms (Gaussian-weighted strip projector of 2001 strip functions and unit-strip backprojector model). All images are displayed on a 64 64 square grid (5.35 mm between grid points).

2

work, we used three data sets to investigate the potential of recovering the 2-D geometric response transaxial to the axis of rotation without scatter compensation. The 40 bins 60 angles of sampled data for the parallel geometry reconstructions were formed by truncating 24 projection bins that extended beyond the projection of the phantom from the 64 sampled projection bins for each of the 120 views sampled. The projection data were formed from the first 60 views sampled over 180 , so that the spatial and angular sampling approximated that used in conventional clinical practice. The results for one slice were obtained in 4 h on a SUN SPARCstation 2000 (40-MHz superSPARC processor, 1-GB of memory). For the fan beam reconstructions of the Hoffman brain phantom, the 32 60 data were formed from a coarse sampling of 60 views over 360 . For each projection two of the 64 original sampled bins

were formed into one bin. The results were obtained in less time (3 h on the same SUN computer) but did not recover the full resolution potential because of the poorer sampling resolution. The results of the Jaszczak cold rod phantom study were more representative of the clinical potential of the proposed method. The reconstruction was obtained using 64 projection bins of 7.12 mm and 120 projection views sampled over 360 . The results for one slice were obtained in 1.5 h using an iterative conjugate gradient algorithm and they demonstrated a significant improvement in resolution recovery over an image space reconstruction method and they illustrated that with better sampling resolution the reconstructions can better recover the resolution of the collimator–detector system. In an earlier paper we found that a mismatch in the projector–backprojector was useful for speeding up the cal-

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

41

(a)

(b) Fig. 10. Fan beam geometry. Comparison results between use of a unit-strip backprojector and a Gaussian weighted strip backprojector displayed on 256 256 (1.36 mm between grid points) and 64 64 grids (5.35 mm between grid points). (a) Reconstructions using the Gaussian-weighted strip projector with 11 strip functions and the unit-strip backprojector (350 terms). (b) Reconstructions using the Gaussian-weighted strip projector and backprojector with 11 strip functions in both (440 terms).

2

2

culations of the elements of the -matrix [21]. Generalized “natural” pixels were used in the backprojector, which were supports for basis functions that were different than those used to define the projector. In the application presented in this paper, the projection model for the geometric response was calculated from as many as 2001 strip functions. In many of the results presented, the backprojector was a single unit-strip function that did not model the geometric response, which saved considerable computation time. In general, different generalized “natural” pixel bases can be selected and the reconstruction problem formulated as a least squares problem where the solution is determined from a vector space that differs from that of the projection measurements [21]. A least

to the equation (for squares solution example, is a backprojector with a single unit-strip function and is a projector with geometric response calculated using 2001 strip functions) will not necessarily be the same least to . Generally, the squares solution will depend upon the solution to the system choice of . However, we have found qualitatively that if is chosen carefully the reconstruction given by can for the least squares be similar to the reconstruction to . solution An iterative algorithm or an SVD method can be used in the equation . With the SVD to solve for method it was found that aliasing is sensitive to the number

42

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

(a)

(b)

(c)

(d)

Fig. 11. Fan beam geometry. Reconstructions of the Jaszczak cold rod phantom by using (a) the filtered backprojection algorithm, (b) using 100 iterations of the EM–ML algorithm with seven rays for the ray-tracing square pixel model, and 100 iterations of the conjugate gradient algorithm with (c) the unit-strip projector–backprojector model and (d) the Gaussian-weighted strip projector–backprojector model (11 strips in the projector and 11 strips in the backprojector). Four profiles through the rods along the outer circumference for each section are shown below each image. All images are displayed on a 64 64 square grid (4.27 mm between grid points).

2

of singular terms and depends upon the particular model and the number of strips used to define the geometric response. For example, one would anticipate that the Gaussian-weighted strip projector more accurately models the geometric response than the equally weighted strip projector in the reconstructions, however, they appear identical until a high number of singular terms are chosen; then reconstructions produced by the equally weighted strip projector show more aliasing artifacts than those produced by the Gaussian-weighted strip projector. A similar result is observed with the choice of the number of strips to be used to model the geometric response. The use of 11 strips

rather than 2001 strips does not produce large differences in the reconstructions until more singular value terms are chosen. Reconstructions that utilize a small number of strip functions to model the geometric response show more aliasing artifacts with a high number of singular terms. Projection space reconstruction using the SVD method requires much more memory and computation speed than traditional image reconstruction methods, which in recent years, there has been significant progress made in computational efficiency in both calculating the geometric response models and in reducing the number of steps in the iterative

HSIEH et al.: MODELING THE GEOMETRIC RESPONSE OF THE SPECT COLLIMATOR

reconstruction algorithm [20]. Generally, for a 2400 2400 -matrix (2400 40 bins 60 angles) that has 11 strip functions for the projector and the backprojector, the number of zero entries equals to 1.918 504 10 , which gives a sparsity of approximately 33%—less than the 45% sparsity for the unit-strip backprojector–projector. The calculation of requires 156 s of computation time on a SUN the matrix SPARCstation 2000 (40-MHz superSPARC processor, 1-GB of memory). It then takes 3.75 h to perform the SVD of . If 2001 strips are used in the projector and a unit-strip function is used in the backprojector for a 2400 2400 -matrix, the number of zero entries equals to 2.268 120 10 , a sparsity of requires approximately 38%. The calculation of the matrix 7487 s of computation time on the same SUN SPARCstation 2000. If 2001 strips are used in both the backprojector and projector, it would take approximately 170 days to calculate the -matrix. If the modeling of attenuation is included with the geometric response, it is estimated that the calculation of the -matrix alone would require about 121 days using only 11 strip functions in both the projector and the backprojector. An analytical solution for the backprojector–projector operator could tremendously reduce the computation time for was block constructing the matrix . Also, if the matrix circulant, computation time of the SVD could be reduced by solving the SVD of small blocks of submatrices. However, noncircular orbit of the SPECT data acquisition, nonuniform attenuation, spatially varying geometric response, and scatter virtually eliminate the block circulant property. Therefore, the development of new SVD algorithms need to concentrate on matrices that do not have the block circulant property. As the number of projection bins and projection angles increase, it becomes more difficult to calculate the SVD’s of the -matrix. Since present computing capacity cannot efficiently make these calculations we must resort to an iterative approach like the conjugate gradient algorithm, but it could be an EM–ML algorithm as well. In the reconstruction of the Jaszczak cold rod phantom, the conjugate gradient algorithm is used instead of an SVD method. In comparison, with the image pixelized model, it was found that 100 iterations of the EM–ML algorithm with image space geometric response modeling (49 rays to simulate the point spread function) took 45 h to reconstruct 64 slices on a SUN Ultra Enterprise 3000 (167 MHz superSPARC processor, 1 GB of memory), while 100 iterations of the conjugate gradient algorithm with a projection space natural pixel model (11 strips in the projector and 11 strips in the backprojector) took 1.5 h to reconstruct one slice on the same machine. It needs to be emphasized that the ray driven method for geometric response modeling is not the most efficient geometric response correction for image space reconstruction. Diffusion methods presently being developed [19] are much faster and are expected to reduce the above reconstruction time from 45 h to less than 30 min. The image space reconstruction approach does not require the calculation of an -matrix. Instead, the elements of the matrix are calculated efficiently during each projection and backprojection operation thus reducing the memory storage requirements. The projection space approach requires a significant amount of time to calculate the -matrix and requires

43

large amounts of memory to store it. The calculations become more computationally intensive as the strip function sampling becomes finer and the number of samples increase. The image space reconstruction approach of correcting for geometric response is currently not practical for routine application in the reconstruction of clinical data. The results presented in this paper were based upon a 2-D model of the geometric response function. It has been shown that a 3-D model for the geometric response is more accurate and can significantly improve the image quality [2]. The extension of the natural pixel approach to model both scatter and geometric response, and to extend this to three dimensions, is computationally nontrivial. The computation and memory requirements increase tremendously, not only for calculating the elements of the matrix , but also for performing the SVD for the 3-D image reconstruction. Models used to estimate the scatter response for square pixels presumably could be adapted to model scatter for natural pixels. It is recognized that the PDF will not be the same for each projection bin. This lack of any symmetry will increase the computations required to evaluate the elements of the -matrix, especially since no block circulant properties can be utilized. However, as computing power increases and better techniques are developed for evaluating the elements of the -matrix and for calculating its SVD, this technique will become more feasible. The goal is to eventually develop sufficiently “natural” pixels and basis functions that better model the physics of the imaging detection process such as attenuation, 3-D geometric response, and 3-D scatter. The model developed in this paper is a better approximation of the geometric response aspect of the sufficiently “natural” pixel for SPECT. In this paper the more “natural” pixel representation is not a unit-strip function as it has been in the past, but rather, it is a probability measure with its domain defined by the apertures of the collection of collimator holes. This structure still defines functions that map the continuous distribution function of isotope concentration into a set of discrete real number measurements. The reconstruction problem is formulated as a projection space least squares problem. Future work will focus on investigating weighted least square solutions. The ultimate goal is to possess the capability to correct for both the 3-D geometric and scatter response using a projection space least squares reconstruction. The techniques that we have explored here illustrate that if such a goal is obtained, a projection space reconstruction method may offer a more accurate way for compensating for geometric response and scatter. ACKNOWLEDGMENT The authors would like to thank the Biodynamics Research Unit of the Mayo Foundation for the use of the Analyze software package. They would also like to thank D. Tavares and S. Webb for carefully proofreading the manuscript. REFERENCES [1] R. J. Jaszczak, R. E. Coleman, and F. R. Whitehead, “Physical factors affecting quantitative measurements using camera-based single-photon emission computed tomography (SPECT),” IEEE Trans. Nucl. Sci., vol. NS-29, pp. 69–80, 1981.

44

IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 17, NO. 1, FEBRUARY 1998

[2] B. M. W. Tsui, E. C. Frey, X. Zhao, D. S. Lalush, R. E. Johnston, and W. H. McCartney, “The importance and implementation of accurate 3-D compensation methods for quantitative SPECT,” Phys. Med. Biol., vol. 39, pp. 509–530, 1994. [3] M. A. King, P. W. Doherty, R. B. Schwinger, D. A. Jacobs, R. E. Kidder, and T. R. Miller, “Fast count-dependent digital filtering of nuclear medicine images: Concise communication,” J. Nucl. Med., vol. 24, pp. 1039–1045, 1983. [4] M. A. King, R. B. Schwinger, P. W. Doherty, and B. C. Penny, “Twodimensional filtering of SPECT images using the Metz and Wiener filters,” J. Nucl. Med., vol. 25, pp. 1234–1240, 1984. [5] K. Ogawa, S. Paek, M. Nakajima, S. Yuta, A. Kubo, and S. Hashimoto, “Correction of collimator aperture using a shift-variant deconvolution and filter in gamma camera emission CT,” in SPIE Proc. Med. Imag. II: Image Formation Detection, Processing, and Interpretation, 1988, vol. 914, pp. 699–706. [6] D. R. Gilland, B. M. W. Tsui, W. H. McCartney, and J. R. Perry, “Determination of the optimum filter function for SPECT imaging,” J. Nucl. Med., vol. 29, pp. 643–650, 1988. [7] R. M. Lewitt, P. R. Edholm, and W. Xia, “Fourier method for correction of depth-dependent collimator blurring,” in SPIE Proc. Med. Imag. III: Image Processing, 1989, vol. 1092, pp. 232–243. [8] L. Van Elmbt and S. Walrand, “Simultaneous correction of attenuation and distance-dependent resolution in SPECT: An analytical approach,” Phys. Med. Biol., vol. 38, pp. 1207–1217, 1993. [9] S. J. Glick, B. C. Penney, M. A. King, and C. L. Byrne, “Noniterative compensation for the distance-dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imag., vol. 13, pp. 363–374, 1994. [10] W. Xia, R. M. Lewitt, and P. R. Edholm, “Fourier correction for spatially variant collimator blurring in SPECT,” IEEE Trans. Med. Imag., vol. 14, pp. 100–115, 1995. [11] B. M. W. Tsui, H. B. Hu, D. R. Gilland, and G. T. Gullberg, “Implementation of simultaneous attenuation and detector response correction in SPECT,” IEEE Trans. Nucl. Sci., vol. 35, pp. 778–783, 1988. [12] G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, and J. A. Terry, “Threedimensional iterative reconstruction algorithms with attenuation and geometric point response correction,” IEEE Trans. Nucl. Sci., vol. 38, pp. 693–702, 1991. [13] A. R. Formiconi, A. Pupi, and A. Passri, “Compensation of spatial system response in SPECT with conjugate reconstruction technique,” Phys. Med. Biol., vol. 34, pp. 69–84, 1989. [14] C. E. Floyd, R. J. Jaszczak, and R. E. Coleman, “Inverse Monte Carlo: A unified reconstruction algorithm for SPECT,” IEEE Trans. Nucl. Sci., vol. NS-32, pp. 779–785, 1985. [15] B. M. W. Tsui and G. T. Gullberg, “The geometric transfer function for cone and fan beam collimators,” Phys. Med. Biol., vol. 35, pp. 81–93, 1990. [16] G. L. Zeng and G. T. Gullberg, “Frequency domain implementation of the three-dimensional geometric point response correction in SPECT imaging,” IEEE Trans. Nucl. Sci., vol. 39, pp. 1144–1452, 1992.

[17] A. W. McCarthy and M. I. Miller, “Maximum likelihood SPECT in clinical computation times using mesh-connected parallel computers,” IEEE Trans. Med. Imag., vol. 10, pp. 426–436, 1991. [18] G. L. Zeng, Y.-L. Hsieh, and G. T. Gullberg, “A rotating and warping projector–backprojector for fan-beam and cone-beam iterative algorithms,” IEEE Trans. Nucl. Sci., vol. 41, pp. 2807–2811, 1994. [19] G. L. Zeng, G. T. Gullberg, P. E. Christian, F. Trisjono, C. Bai, J. W. Tanner, E. V. R. DiBella, and H. T. Mogan, “Iterative reconstruction of fluorine-18 SPECT using geometric point response correction,” J. Nucl. Med., vol. 39, pp. 124–130, 1998. [20] H. M. Hudson and R. S. Larkin, “Accelerated EM reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imag., vol. 13, pp. 601–609, 1994. [21] Y.-L. Hsieh, G. T. Gullberg, and G. L. Zeng, “Image reconstruction using generalized natural pixel bases,” IEEE Trans. Nucl. Sci., vol. 43, pp. 2306–2319, 1996. [22] G. T. Gullberg and G. L. Zeng, “A reconstruction algorithm using singular value decomposition of a discrete representation of the exponential Radon transform using natural pixels,” IEEE Trans. Nucl. Sci., vol. 41, pp. 2812–2819, 1994. [23] G. T. Gullberg, Y.-L. Hsieh, and G. L. Zeng, “An SVD reconstruction algorithm using a natural pixel representation of the attenuated Radon transform,” IEEE Trans. Nucl. Sci., vol. 43, pp. 295–303, 1996. [24] F. Natterer, “Efficient implementation of optimal algorithms in computerized tomography,” Math. Meth. Appl. Sci., vol. 2, pp. 545–555, 1980. [25] M. H. Buonocore, W. R. Brody, and A. Macovski, “A natural pixel decomposition for two-dimensional image reconstruction,” IEEE Trans. Biomed. Eng., vol. BME-28, pp. 69–78, 1981. [26] F. Natterer, The Mathematics of Computerized Tomography. Chichester, U.K.: Wiley, 1986, pp. 144–150 and 156–157. [27] J. R. Baker, “Spatially variant tomographic imaging: Estimation, identification, and optimization,” Ph.D. dissertation, Univ. California at Berkeley, 1991. [28] J. R. Baker, T. F. Budinger, and R. H. Huesman, “Generalized approached to inverse problems in tomography: Image reconstruction for spatially variant systems using natural pixels,” Crit. Rev. Biomed. Eng., vol. 20, pp. 47–71, 1992. [29] F. Noo, C. Bernard, F. X. Litt, and P. Marchot, “A comparison between filtered backprojection algorithm and direct algebraic method in fan beam C. T.,” Signal Proc.: The official pub. Euro. Assoc. Signal Processing (EURASIP), vol. 51, pp. 191, 1996. [30] M. Bertero, C. De Mol, and E. R. Pike, “Linear inverse problems with discrete data. I: General formulation and singular system analysis,” Inverse Problems, vol. 1, pp. 301–330, 1985. [31] L. T. Chang, “A method for attenuation correction in radionuclide computed tomography,” IEEE Trans. Nucl. Sci., vol. NS-25, pp. 638–643, 1978. [32] K. Lange and R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr., vol. 8, pp. 306–316, 1984