Evaluation of an Efficient Compensation Method for ... - IEEE Xplore

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IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 24, NO. 2, FEBRUARY 2005

Evaluation of an Efficient Compensation Method for Quantitative Fan-Beam Brain SPECT Reconstruction Tianfang Li*, Junhai Wen, Guoping Han, Hongbing Lu, and Zhengrong Liang

Abstract—Fan-beam collimators are designed to improve the system sensitivity and resolution for imaging small objects such as the human brain and breasts in single photon emission computed tomography (SPECT). Many reconstruction algorithms have been studied and applied to this geometry to deal with every kind of degradation factor. This paper presents a new reconstruction approach for SPECT with circular orbit, which demonstrated good performance in terms of both accuracy and efficiency. The new approach compensates for degradation factors including noise, scatter, attenuation, and spatially variant detector response. Its uniform attenuation approximation strategy avoids the additional transmission scan for the brain attenuation map, hence reducing the patient radiation dose and furthermore simplifying the imaging procedure. We evaluate and compare this new approach with the well-established ordered-subset expectation-maximization iterative method, using Monte Carlo simulations. We perform quantitative analysis with regional bias-variance, receiver operating characteristics, and Hotelling trace studies for both methods. The results demonstrate that our reconstruction strategy has comparable performance with a significant reduction of computing time. Index Terms—Brain imaging, fan-beam collimation, quantitative reconstruction, reconstruction evaluation, ROC, SPECT.

I. INTRODUCTION

S

INGLE PHOTON emission computed tomography (SPECT) is well known for its functional imaging capability with fairly good quality and relatively low cost. It is one of the most important modalities in demonstrating the blood flow, perfusion, and oxygen and glucose metabolic activity of the brain, breasts and other vital organs. However, it currently provides clinicians with qualitative data that mainly

Manuscript received May 25, 2004, revised September 28, 2004. This work was supported in part by the National Institutes of Health (NIH) National Heart, Lung and Blood Institute under Grant HL54166 . The work of T. Li was supported in part by the Society of Nuclear Medicine under a Student Fellowship. The work of H. Lu was supported in part by the National Nature Science Foundation of China under Grant 30170278. The Associate Editor responsible for coordinating the review of this paper and recommending its publication was J. A. Fessler. Asterisk indicates corresponding author. *T. Li is with the Departments of Radiology and Physics and Astronomy, State University of New York, Stony Brook, NY 11794 USA (e-mail: tfli@mil.sunysb.edu). J. Wen is with the Department of Radiology, State University of New York, Stony Brook, NY 11794 USA (e-mail: wenjh@mil.sunysb.edu). G. Han is now with Computer Associates Inc. at Island, NY 11749 USA (e-mail: ghan@suffolk.lib.ny.us). H. Lu is with the Department of Biomedical Engineering, Fourth Military Medical University, Xi’An, Shaanxi 710032, China (e-mail:luhb@ fmmu.edu.cn). Z. Liang is with the Departments of Radiology, Computer Science and Physics and Astronomy, State University of New York, Stony Brook, NY 11794 USA (e-mail: jzl@mil.sunysb.edu). Digital Object Identifier 10.1109/TMI.2004.839365

allows a visual inspection of the reconstructed images. Although in some situations relatively quantitative analysis (semiquantitation) is used, SPECT images are, in their present form, not adequate for accurate absolute evaluation due to the inherent degrading factors in currently available systems, which include the quantum noise, the Compton scattering and photoelectric absorption of -rays, the spatially variant collimator/detector response or point spread function (PSF), etc. Developing efficient reconstruction algorithms that accurately compensate for all these degradation factors has been a very challenging task. Nonparallel collimations such as fan-beam geometry designed to improve the trade-off between sensitivity and resolution make the image reconstruction task even more complicated. To date, the most accurate methods dealing with all the degrading factors for fan-beam SPECT systems are based on those iterative algorithms, which incorporate an accurate model of photon transport from the body to the detector [1]–[5]. These algorithms often require certain regularization for convergence and intensive computing efforts for the ray-tracing through the field-of-view (FOV) in the iterative cycle of forward- and back-projections, which are quite time consuming, even with some sophisticated convergence speedup strategies [6]–[9]. The aim of this paper is to develop and evaluate an accurate as well as efficient approach, which addresses the unique problems of a specific application—brain SPECT. For brain SPECT, the quantitative reconstruction requires compensation for photon attenuation within the head. If the attenuation coefficient map of the head is known, the tracer distribution can be accurately reconstructed using iterative methods or a recently derived analytical algorithm [10]. However, these methods require the nonuniform attenuation coefficient distribution map, which is usually obtained by transmission scans. This requirement complicates the imaging protocols and causes additional radiation exposure for the patient. Since the attenuating medium of the head can be quite accurately modeled as a uniform part (the brain tissues) surrounded by a nonuniform shell (the skull and the skin), and the radioactive tracers are negligible in the nonuniform shell (in the skull/skin), an alternative approach has been developed which utilizes an enlarged head boundary to define a uniform attenuation map equivalent to the actual nonuniform attenuation map of the head [11]. In order to satisfy the assumption that the source is distributed within a convex region of the uniform attenuating medium (brain tissue), the head must be positioned in such a way that most of the brain tissues (surrounded by the skull) lie in the transaxial sections across the FOV with minimal sinus cavities inside the tissue region. This can be done by lowering the patient bed by 4 cm and rising the head accordingly with a pillow [11]. Quantitative evaluation

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LI et al.: EVALUATION OF AN EFFICIENT COMPENSATION METHOD FOR QUANTITATIVE FAN-BEAM BRAIN SPECT RECONSTRUCTION

showed only a 1% error due to this approximation. With this alternative approach the attenuation can be determined with only the head boundary information [12] and, therefore, the transmission scans can be eliminated. Compensation for uniform attenuation has been mathematically established by the inversion of the exponential Radon transform for parallel-hole collimation, see [13], [14]. Extension of the inversion to include distance-dependent PSF has been developed via the frequency-distance relation (FDR) [15]–[21]. Efforts to extend the inversion from parallel-hole collimation to fan-beam geometry have been also reported in the literature [22]–[25]. However, the FDR approximation in fan-beam collimation is not as good as in parallel geometry due to the spatially variant properties of the collimator response in the nonparallel geometry [25]. Nevertheless, when the spatially variant PSF is combined with the uniform attenuation and the scan orbit is circular, we found that the system imaging matrix has a unique property of circular symmetry. Based on this property, a circular harmonic decomposition and conjugate gradient (CHD-CG) algorithm was derived in this paper for image reconstruction with simultaneous compensation for uniform attenuation and spatially varying PSF. This approach models the imaging process accurately, and can be applied to any geometry with a circular scan orbit. It completely eliminates the tracing for the PSF and attenuation factors through the FOV—the most computing intensive part of the previously reported iterative image reconstruction methods. In addition to the effects of photon attenuation and PSF, other degrading factors such as the Poisson noise and the inclusion of scattered photons must be considered. In this paper, we adapted our previously developed methods to combat these factors step-by-step and analytically. For Poisson noise treatment, the Anscombe transform was performed first to change the noise into a Gaussian distribution with a constant variance, and then the Karhunen–Loève (K–L) transform was performed to find the spatially correlative information among all slices. Since the transformed data (components) in the K–L space are ordered by the eigenvalues that are known to be equivalent to the signal-to-noise ratios (SNRs), a Wiener filter can be accurately designed to treat the noise adaptively across all the principal components in the K–L domain [26]. For scatter treatment, a modified triple-energy-window (TEW) method was shown to be very accurate and efficient in estimating the portion of scattered photons [5], [27], [28] and was adapted in this paper. The proposed image reconstruction procedure can be summarized as follows: first the Poisson noise was treated, and then the scatter contribution was estimated in the sinogram space. Finally the image reconstruction with simultaneous compensation for attenuation and PSF was performed by the CHD-CG algorithm also in the sinogram space. By a conventional statistical iterative approach, the CHD-CG may be replaced by a penalized least-square algorithm – which would require a large computational effort to trace the attenuation and PSF factors through the FOV in the forward- and back-projection cycle. In the following sections, we first introduce the basic theories in detail, then present several experimental results and quantitative analyses, and finally discuss the advantages and limitations of the proposed efficient reconstruction method.

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II. THEORY A. Poisson Noise Treatment The fluctuation of photon measurements in SPECT projections is commonly believed to be Poisson distributed [29], and the task of spatial filtering of the signal-dependent Poisson noise can be greatly simplified by applying the Anscombe transform to all the sinogram data, which converts Poisson distributed noise into approximately Gaussian distributed noise with a constant variance. That is, if is Poisson distributed with mean equal to , then can be approximated as and its Gaussian distributed with its mean equal to variance equal to 0.25. Although it is known that the Anscombe transform introduces bias when the Poisson parameter (mean value or counts) is small, its variance stabilization property works quite well, even for very low counts (typically not less than 4) [30], [31]. After the Anscombe transform, the sinogram noise becomes nearly space-invariant Gaussian white noise. The next step is the K–L transform (also called principal component analysis) along the direction of slices. Recently, K–L transform has been extensively studied in the field of tomographic reconstruction [26], [32]–[34]. It simplifies the complex task of three-dimensional (3-D) filtering into two-dimensional (2-D) process by extracting the correlative information (spatial covariance) among slices. By applying this combined transform, we can construct an accurate 2-D Wiener filter for each sinogram principal component in the K–L domain as (1) where is the 2-D discrete Fourier transform (FT) of the at position in the th prinK–L domain data reflect the frequency coordinates of cipal component , and is the constant noise variance equal to 0.25. It has been shown that the K–L domain Wiener filter smoothes each of the components adaptively according to their SNRs, which leads to an improved restoration of SPECT projections [26]. B. Scatter Estimate by a Triple-Energy Window Acquisition Protocol for Tc-99 m Radiotracer Ogawa’s TEW method has been shown to generate good results in a relatively simple and efficient manner [35], [36]. But its use of the window above the photo-peak and its assumption for scatter characteristics within the photo-peak are questionable, see Fig. 1. By neglecting the upper window data, a better performance was demonstrated[37], but its overestimate of scatter fraction in the photo-peak remains because of the inadequate assumption. Bourguignon et al. [38] tried to correct for the over-estimation by subtracting the portion determined by the two satellite windows below and above the photo-peak, instead of adding that portion as Ogawa did, but obtained incorrectly low (under-estimation) results. By a more accurate modeling on the scatter characteristics inside the photo-peak as well as the satellite windows, an excellent agreement between the true scatter portion and the estimated result has been demonstrated by Monte Carlo (MC) simulation [5], [27], [28], [39]. With this

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Fig. 1. MC simulated Tc-99 m source energy spectrum in SPECT.

modeling of the scatter, the formula for estimating the scatter portion inside the photo-peak becomes (2) where and are the total photon counts inside the lower, , and upper-energy satellite windows, respectively, and and are the widths of the photo-peak (main) energy window and the two satellite energy windows, respectively (see Fig. 1). and in the abFactor is the ratio between sence of attenuation and scatter, which removes the inadequate assumption that primary photons inside both the satellite windows are the same and can be measured by a point source in air. It is noted that the SNR in the satellite windows is very low. Accurate treatment of the noise by the strategy of Section II-A is, thus, essential. C. Image Reconstruction With Simultaneous Compensation for Uniform Attenuation and Spatially Variant PSF in the Fan-Beam Geometry After the noise and scatter treatments, the corrected projections are approximated to an attenuated Radon transform with PSF blurring, which can be expressed as shown in (3) at the is the attenbottom of the page, where the integrand denotes the PSF of the imaging system uation term and (see Fig. 2). In the following, we will derive a reconstruction algorithm based on this approximated model. The final reconstructed results will be examined by comparison with the wellestablished ordered-subset expectation-maximization (OSEM) method which is based on an accurate physical model and provides an integrated approach for compensating for all the physical effects and noise. In general, the object (brain) will not be a sphere or a cylinder, but its boundary can be considered nearly convex. To develop a , we first reconstruction method for the source activity map increased the attenuation map to a larger virtual uniform disk for each slice as shown in Fig. 3(a). It is easy to see that a modified

Fig. 2.

Attenuated Radon transform with PSF blurring in fan-beam geometry.

projection using the virtual attenuation map can be obtained from the projection by the following equation: (4) is the distance from the object boundary to the where enlarged circle boundary with radius along the projection ray in slice . can be regarded as the reNow the modified projection sults of the source activities emitted inside a uniform attenuation cylinder. If we sample the disk at each slice into polar grids as shown in Fig. 3(b) and let the system response of “point source” at each voxel be measured as the imaging matrix , then

(3)


Fig. 3. (a) Modification for the uniformly attenuated projection from each slice: p (s; ; z ) = e measuring the imaging kernel matrix k .

the modified projections can be derived in the polar coordinate system by the following discrete formula:

(5) is the volume of a finite element at location , and is the measured collimator/detector response of the point source inside the uniform attenuation media. Using the polar grid strategy for fast image reconstruction was also reported in [40]. Note that the imaging matrix is shift-invariant along the -axis and circularly symmetrical, i.e., the response of the deis the same tector at view angle for a point source at as the response for the point source at ( , 0, ) with the detector rotated to . These properties reduce the size of the system to for an -element imaging matrix from array detector. To solve (5), the CHD-CG strategy, developed by You et al. [23] and Li et al. [24] can be adapted. First, a 2-D FT is performed on (5) with respect to and , respectively, and we get

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p(s; ; z ). (b) Circular sampling grids (; ') for

, (the subscripts and Equation (6) is equivalent to and the superscript are omitted for simplicity), where is , and a complex matrix and can be written as and are the complex image vector and projection vector in , and terms of the harmonic coefficients, . Define

where

(10) where the superscript denotes the transpose of a matrix or a vector. Solving (6) is equivalent to solving the following equation:

. Since matrix

is symmetric positive, a

conjugate gradient algorithm can be applied for the solution in steps for an matrix) [41]. finite iterations (at most be the -step estimate of . We have the Let following iterative procedure in the FT sinogram space

(6) (11)

where

(7) (8) (9)

where

,

, and

. After

is calculated, the source activity distribution is then uniquely determined by inverse FT. To display in Cartesian coordinates, a further step of interpolation from polar coordinates to Cartesian coordinates should be performed, which can be done by bilinear interpolation with over-sampling.

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III. EXPERIMENTS AND RESULTS A. Reconstruction With Simultaneous Compensations for Attenuation and PSF by CHD-CG Algorithm for Noiseless Data First, we show the property of the CHD-CG algorithm for the noiseless case. In order to validate the proposed algorithm of (11), we analytically simulated 128 projections of 128 64 array size evenly distributed on a circular orbit, including uniform attenuation and spatially varying PSF effects. The simulation process is as follows. First, a system imaging kernel matrix of 128 128 64 size was constructed by the SIMIND MC simulation program [42] using a point source and fan-beam geometry, which was modified from the original SIMIND with fan-beam geometry written by Dr. E. Frey and Dr. D. Lewis at University of North Carolina. The detailed interaction inside the collimator was not modeled using the MC photon tracking, but rather by using a geometric transfer function [43]. The lowenergy high-resolution fan-beam collimator has a regular focal length of 80.0 cm, and hexagonal holes of size of 1.20 mm (flat to flat) and thickness of 4.0 mm. The point source placed on each polar grid inside a uniform water phantom was simulated to obtain the corresponding system responses, i.e., the imaging kernel matrix. The TEW method was used to remove the scattering portions in the kernel matrix as described in Section II. Therefore, the attenuation and PSF effect were inherently built in the kernel matrix . The projections were then calculated by convolving the imaging matrix with the Shepp–Logan phantom as described by (5). These projections were regarded as noise-free data and reconstructed by the proposed CHD-CG algorithm. The result is shown in Fig. 4 along with the original phantom. The profiles of the horizontal lines across the three tiny objects in the two images are also plotted in Fig. 4(c).

Fig. 4. Reconstruction with compensation for the uniform attenuation and spatially varying PSF by the CHD-CG algorithm. (a) Shepp–Logan phantom. (b) Reconstructed image. (c) Horizontal profile across the three tiny objects (solid line—phantom, dotted line—reconstructed image).

B. Comparison With OSEM Using Monte Carlo Simulations With All Degrading Factors Included In addition to the validation of CHD-CG algorithm, our proposed image reconstruction procedure was evaluated by MC simulations including all the degradation factors, such as Poisson noise, scatter, nonuniform attenuation, and PSF. The projections were simulated from the 3-D digital Hoffman brain phantom of 128 128 64 array size. The nonuniform attenuation map was constructed from a scaled CT head image (see Fig. 7). High counts projection data were generated with 10 M photon histories per view on an array size of 128 64. Noisy data equivalent to 100 K counts per view were generated later by simulating Poisson noise using the high counts projection data and scaling. A total of 128 projections were simulated as evenly distributed on a circular orbit. The scatters were included up to the third-order and collected by the triple energy window protocol [44]. The collimators were the same as described previously. The reconstruction was compared with OSEM algorithm as described in the flowchart of Fig. 5. The OSEM algorithm was implemented based on the recursive ray-tracing technique of Siddon [45]. The implementation aimed to optimize the technique for the maximum speed [46], [47], in addition to the use of the projection symmetries. Given

Fig. 5. Flowchart for our proposed reconstruction strategy (Method 1) and the OSEM (Method 2). In Method 1, the projections were corrected first by noise reduction using the K–L domain Weiner filter and by scatter estimation using the modified TEW method, and then reconstructed by the CHD-CG algorithm for simultaneous compensation of attenuation and PSF. In Method 2, all physical factors and noise were considered simultaneously by the OSEM reconstruction algorithm.

the 128 projections, an ordered subset of size 4 or 8 is an optimal choice to take the advantage of projection symmetries. Both choices of 4 and 8 did not show significant difference in


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Fig. 6. Comparison of different reconstruction methods. Each row from top to bottom reflects: (a) Five slices of the 3-D digital Hoffman brain phantom. (b)FFBP reconstructed images without any compensation (i.e., the Ramp filer at the Nyquist frequency cutoff). (c) Reconstruction by our proposed Method 1. (d) Reconstruction by the iterative OSEM Method 2. The display is in gray scales from 0 to 5.

reconstruction accuracy [5], [39], so a size of 8 was chosen for this comparison study. In Fig. 6, we show some reconstruction results using different methods. The top row is the original phantom slice images and the other rows are the reconstructed images. The second row shows the results of the conventional FFBP (fan-beam filtered backprojection) method without any compensation, the third row represents the images of our proposed method, and the last row reports the images of the OSEM algorithm. It should be noted that our proposed Method 1 used an enlarged uniform attenuation map approximation [12], while the OSEM used the true nonuniform attenuation map in image reconstruction. From these pictures, it is clearly shown that the improvement with compensation for the degrading factors is dramatic. Both Method 1 (proposed) and Method 2 (OSEM) have very similar image qualities by visual judgment. IV. QUANTITATIVE EVALUATIONS To further evaluate the performance of our proposed approach and the well-established OSEM algorithm, quantitative measures were documented by three figures of merit: 1) Regional bias-variance (RBV). A better reconstruction algorithm should give the least bias with the least variance at the same time. 2) Receiver operating characteristic (ROC) curve. The reconstruction algorithm which generates a larger area under

Fig. 7. Defect in the brain phantom (left) and illustration of the selected regions of interest for the bias/variance analysis (right).

the curve (AUC) usually reflects a better “detectability” on abnormality. 3) Hotelling observer. The figure of merit of Hotelling observer (or ideal observer) performance is the Hotelling trace (HT), which measures the class separability of imaging systems without human interaction. In these studies, a Hoffman brain phantom with defect was first generated by adding a small cold spot to the original normal phantom [see Fig. 7(a)]. For the RBV analysis, the MC simulated high-counts (10 M per view) data from the Hoffman brain phantom with defect were sampled 500 times by Poisson noise to mimic the 500 observations of the same event, and each has a noise level of 100 K counts per view. The RBV plot was drawn

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for selected regions from the 500 reconstructed images of each method. Similarly, for the ROC and HT studies, two sets of projection data were generated by the MC simulation process. One set (of 500 observations) was simulated from the normal Hoffman brain phantom, and the other set (of 500 observations) was simulated from the Hoffman brain phantom with defect. A better reconstruction algorithm should be able to detect the samples with defects from all datasets. A. Relative Regional Bias and Variance Study The regional bias and variance are calculated from realizations of reconstructed images by (12)

(13)

Fig. 8. Examples of reconstructed images. Top row shows images from the proposed approach, and bottom row shows images from OSEM algorithm; left column shows the normal image, right column images have the defect. The display is in gray scales from 0 to 5.

(14) is the number of realizations, is the region of inwhere is the number of voxels in that region, is terest (ROI), the random variable for the value of voxel in the -th reconis the mean of the true image in region structed image, , i.e., (15) and

is the mean of

, i.e., (16)

Then the relative regional bias and variance are expressed in terms of percentages as follows: (17) (18) In this study, 18 small ROIs—each containing 7 voxels— were placed in each of the 500 reconstructed volumetric images of each algorithm, to evaluate the regional bias and variance, as illustrated in Fig. 7(b). These ROIs were evenly distributed in the image which can cover a large range of bias/variance variation and different brain tissues such as the white matter (WM), the gray matter (GM), and the cerebrospinal fluid (CSF). The results from these two different reconstruction schemes show a very similar regional performance. For the case of noise level at 25 K counts per view, their variances varied similarly from 1% to 15%. For the case of noise level at 100 K counts per view, their variances varied similarly from 0.5% to 5%. For the case of noise level at 1 M counts per view, their variances varied similarly from nearly zero to 1%. Their biases for all the three

noise levels also varied similarly from nearly zero to 50%. The phantom emission density is from 0 to 4 unites per voxel. The ROI means are in this range. B. ROC Study In the clinic, the reconstructed SPECT images are inspected by physicians to detect abnormalities. One accepted method for the evaluation of a medical imaging system or procedure is the ROC curve. In such study, the observers are asked to rate their confidence in the presence or absence of any abnormality. After that, a ROC study is performed to analyze the ratings. More specifically, the true positive fraction (TPF) (also called sensitivity) and the false positive fraction (FPF) (also referred to 1-specificity) are calculated at a number of confidence levels. Then the TPF is plotted versus the FPF and the so-called ROC curve is drawn or fitted using the obtained data for TPF and FPF. A common merit for . In the binormal model comparing ROC curves is the AUC fitting for ROC curves (justified by the central limit theorem when a sufficiently large number of ratings and a sufficiently can be calculated by large sample of images are used), (19) where

is the cumulative standard normal distribution of (20)

and and are parameters that best fit the equation relating the statistics of TPF and FPF (21) In this study, a total of 1000 projections, (500 with and 500 without the defect) were generated, where the noise level is equivalent to 100 K counts per view. They were reconstructed


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The mathematical formalism for calculating the HT is outclasses of images, random samples are lined below. For be the -th sample from drawn to calculate the HT. Let class , where and , and be the number of samples from class . Also suppose that the probability density function of occurrence for the is . Define the interclass scatter matrix and sample intraclass scatter matrix as follows: (22) (23) where Fig. 9. Results of ROC evaluation and the fitted ROC curves of the binormal model. The solid line and cross marks represent the CHD-CG reconstruction and the dashed line and circles stand for the OSEM result.

using each of the two different methods, resulting in a total of 2000 images or samples. The images were randomly ordered when presented for evaluation. The evaluators were asked to classify the images into one of five categories: 1) definitely or almost definitely negative; 2) probably negative; 3) uncertain; 4) probably positive; and 5) definitely or almost definitely positive. During the evaluation, one slice of the volume image containing the lesion (cold spot) was presented. The lesion has a size of 0.75 cm and intensity of 2, (intensity range of the phantom is 0 to 4). The location is specified. The observers were allowed to adjust the gray level and size of the display window. Examples are shown in Fig. 8. Three observers read the images independently. The total rating data were analyzed using the CLABROC code [48], resulting in an averaged ROC curve with 4 pairs of FPF and TPF points for each method, as plotted in Fig. 9. The fitted ROC curves of the binormal model were also drawn in that figure. The area under the fitted curves is 0.8934 for the CHD-CG reconstruction algorithm and 0.8940 for the OSEM iterative reconstruction method. The one-tailed -value is 0.175 (a difference in AUCs will be deemed statistically significant if the -value is less than 0.05). The FPF and TPF values and the fitted ROC curves of the binormal model indicate that both methods have a similar performance in terms of defect detectability. C. Hotelling Trace Calculation Computer-based observers mimicking human observers have been proposed to alleviate the need for observers’ interactions. One of these methods is the Hotelling observer for comparison of two imaging systems. In this method, the HT, which is a measure of object class separability based on first and second-order statistics, is used to evaluate the ability of a system in separating between classes. In fact, Fiete et al. [49] found that the HT correlates with human performance in liver defect detection and Wollenweber et al. [50] recently implemented the HT to evaluate the potential increase in defect detection in myocardial SPECT using high-resolution fan-beam collimator versus parallel-hole collimation.

(24) is the a priori probability of occurrence of class samples

in all the

(25) is the mean image of class (26) is the mean of mean class images which is also the grand mean of all images, and (27) is the covariance matrix for the -th class. With the above definitions, the HT is defined as (28) From the above definitions, the dimension of the scatter matrices is equal to the number of pixels (or voxels) in the image, which is huge. The main difficulty in calculating value, thereis singular if the fore, lies in inverting the matrix . Matrix number of sample images is less than the number of voxels in the image. The requirement that the number of sample images be not less than the number of voxels in the image would consume too much disk storage or computer memory. Abbey et al. [51] suggested the use of subsets of the images, instead of the whole images, to reduce the amount of required space or data to calculate the value. The HT value was then calculated using the above formalism with the images reconstructed in the previous section. The two classes of reconstructed images are those with and without the defect, respectively. In one study, the CHD-CG reconstruction method was used, and in the other study, the OSEM iterative reconstruction method was used for comparison. The results are shown in Table I. The HTs were calculated for two

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TABLE I HOTELLING TRACE J FOR SPECT IMAGES RECONSTRUCTED BY CHD-CG AND OSEM ALGORITHMS

high-resolution (over 256 cubic array size) and dynamic (4-D or 5-D) quantitative SPECT imaging. The fan-beam collimated SPECT system and the developed image reconstruction algorithm are readily applicable to 3-D SPECT mammography. ACKNOWLEDGMENT

regions: one is around the defect spot, and the other is in complete normal tissue. Each region contains 125 voxels. As shown in Table I, both the CHD-CG algorithm and the OSEM iterative method roughly give the similar HT values, which indicate that the performance of the CHD-CG method is comparable to the OSEM algorithm in terms of defect detectability. V. DISCUSSION AND CONCLUSION An efficient statistical approach was investigated for quantitative SPECT image reconstruction in a specific case, where fan-beam collimators are proposed for brain studies to improve the sensitivity and resolution, and a uniform attenuation approximation is suggested to accurately model the head characteristics, which avoids the transmission scan and reduces the patient radiation exposure. Other imaging degradation factors including noise, scatter and variable collimator response or PSF were investigated and incorporated in a coherent manner. is simulated by MC code in The system kernel matrix this paper. For a real SPECT system, it should be measured before the proposed method can be applied. The accuracy of the proposed method is dependent on this measurement, as well as others. The proposed CHD-CG method showed satisfactory reconstruction quality of the phantom datasets. The computation time was approximately 10 minutes for a 128 128 64 volumetric image on a PC Pentium 2.4 GHz platform with 2.5 GB RAM (Window XP operating system). The reconstruction time for the OSEM was approximately 38 minutes per iteration on the same platform when the attenuation was turned off. It should be noted that both the uniform attenuation assumption and the trick of the virtual attenuation circle can be incorporated into the system model used by OSEM as well [40], which would benefit from the same advantages and save the computational time. However, the saving is limited, because the most computing intensive part of iteratively tracing the PSF and attenuation during each forward- and back-projection cycle through the image domain remains. Acceleration of the OSEM method is an interesting topic and beyond the scope of this work. Both the OSEM and the presented methods model the Poisson noise accurately, as well as the scatter, attenuation and PSF. We would expect that they provide a similar reconstruction quality. Both the visual inspection and quantitative measures such as RBV plot, ROC curve and HT calculation on the 2000 reconstruction studies concurred with our expectations. However, the computational advantage of the presented method is clearly seen. The gained speed could be very useful for

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