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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 44, NO. 1, FEBRUARY 1997 107 An SVD Study of Truncated Transmission Data in SPECT Gengsheng L. Zeng, Member, IEEE, and Grant T. Gullberg, Senior Member, IEEE Abstract— Even though a noiseless, band-limited function is uniquely determined by its values in a local region, single photon emission computed tomography (SPECT) projections are not band-limited, and unmeasured projections may not be possible to be exactly estimated from the measured data. Projections from all views should be considered simultaneously and are modeled as a set of linear equations. The singular value decomposition (SVD) method is used to analyze and solve the equations. It is shown that truncation does not always result in an underdetermined problem, yet the problem may be ill-conditioned. An inaccurate pixel model may cause reconstruction artifacts via mismatch between the measured data and the modeled projections. Index Terms— Incomplete data, SVD reconstruction, truncation. I. INTRODUCTION I N SINGLE photon emission computed tomography (SPECT), projections may be truncated because of the relatively small detector size. The truncation problem is especially severe during transmission imaging of the chest by a three-detector SPECT system with fan-beam collimators [1]. If the reconstruction of truncated projections is improperly formulated, the problem results in solving an underdetermined system of linear equations that leads to artifacts in the reconstruction [2]–[7]. However, if band-limited truncated projections are sampled sufficiently fine and reconstructed into a sufficiently coarse grid using projection models with minimum model mismatch between image and projection, reconstruction artifacts from truncated projections can be reduced. Reconstruction from limited data sampling, such as the reconstruction of truncated projections, has been studied in many fields of engineering [13]–[16], including MRI [17]. Truncated projections were reconstructed initially by using data extension or data extrapolation methods to “complete” the projections [5]–[9]. In the case of the three-detector SPECT system, some have modified the camera system by using fan beam collimation on the simultaneous transmission/emission detector and parallel collimation on the other two detectors to obtain scatter information that can be used to determine the truncation edge [11], [12]. Using information about the Manuscript received December 11, 1995; revised April 1, 1996 and August 19, 1996. This work was supported in part by the NIH under Grant RO1 HL39792 and by Picker International. The authors are with the Department of Radiology, University of Utah, Salt Lake City, UT 84132 USA (e-mail: [email protected]). Publisher Item Identifier S 0018-9499(97)01524-4. truncation edge to estimate the patient contour, a filtered backprojection [11] or an iterative ML-EM emission algorithm [12] was employed to reconstruct the pixels within the contour. The problem with this approach is the reduction in sensitivity when parallel collimators are used. Another approach is to keep the fan beam collimation but acquire two transmission scans wherein, after the first scan, the patient is translated horizontally in the bore of the imager. Another transmission scan is acquired to fill in the missing projection rays [9]. Other approaches involve using asymmetric fan-beam collimators [10] or camera systems, such as the L-shaped camera [8] design, that can obtain transmission data without truncation. Our approach utilized the three-detector SPECT for cardiac imaging to determine the solution with the information available by solving only the system of linear equations given by the available truncated projections [23]. As the accuracy of the support improved, so did the accuracy of the reconstructed transmission image [1]. Here, a support is a preset region where the pixel value is assumed to be zero if the pixel is outside the region. However, in most cases the accuracy of the support was not critical to calculate the attenuation factors within the cardiac region. The attenuation factors were exponential of negative partial line integrals of the reconstructed transmission data. Thus these factors were calculated with fair accuracy because the total line integrals of the reconstructed transmission data closely approximated the measured projection data. This paper investigates the extent to which truncated projections cause underdetermined image reconstruction. The research will show that when projections are truncated, a band-limited image can be reconstructed under certain circumstances. The singular value decomposition (SVD) is used to obtain the reconstructions from truncated projections; this demonstrates that the reconstruction is not an underdetermined problem if the data and reconstruction are sampled properly. However, a fully determined problem can be ill-conditioned, as determined by its condition number. It is demonstrated how the condition number can be reduced to improve the reconstructed images. In SPECT the projections are not band-limited, thus the truncated projections at one view cannot be exactly estimated from the nontruncated part at the same view. Therefore, using the projections from other views can help, by considering a set of linear equations simultaneously (i.e., tomography). An image can be reconstructed by solving the set of linear equations. The SVD method is used to analyze and solve the 0018–9499/97$10.00 1997 IEEE 108 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 44, NO. 1, FEBRUARY 1997 equations with computer simulated projections and physical phantom measurements. II. METHODS A. Linear Equation Method (a) For a band-limited signal the function values at any two different points are correlated. For example, in MR imaging the k-space signal is band-limited. However, in SPECT or X-ray CT it cannot be assumed that the projections are the samples of a band-limited signal because the imaging point response function has a small finite spread. Therefore, the truncated projections may not be recovered at one view from the measured part in the same view, but the projections from other views can be used to help by doing tomography. On the other hand, the final goal in SPECT imaging is to reconstruct the image from projections. The unmeasured projections do not have to be estimated if an image can be reconstructed with the locally sampled data (i.e., truncated projections). We believe that reconstructing the image with truncated projections is equivalent to reconstructing the image with extrapolated data, because extrapolating does not increase information. This paper uses square pixels with line-length weighting to set up imaging equations, with the pixel values as unknowns. Image reconstruction is achieved by solving the linear equations. The singular value decomposition (SVD) method is used to analyze and solve the equations in these studies. Let be a vector of projections and be a vector representing the image. The imaging equation is given as (1) where matrix is called projection matrix, modeling imaging geometry and physics. The SVD of is given by (2) where is an orthonormal matrix and is a diagonal matrix with nonnegative diagonal elements, known as singular values of which are the square of the singular values of The SVD provides a condition number to diagnose the linear equation system. The condition number of the system is defined as the ratio of the largest singular value and the smallest singular value of The condition number indicates how sensitive the solution is to the noise in projections and the model mismatch in A very large condition number also indicates that the measurement may not be enough, while a small condition number indicates that the image may be properly reconstructed. If the condition number is finite, the image can be reconstructed by (3) can also be reconstructed by the SVD of Equivalently, or of Nevertheless, the formulation of (1) has some drawbacks: 1) A linear system model strongly depends on the pixel characteristics (e.g., a uniform square or a Gaussian blob, etc.); 2) the nonnegativity constraint on the pixel values and some other constraints are very difficult; and 3) the Poisson noise is not modeled. (b) Fig. 1. (a) Ideal image. (b) Filtered backprojection reconstruction with truncated data. B. Computer Simulations for the Linear Equation Method Computer simulations are used to demonstrate how the sampling interval size affects the reconstruction if the projections are severely truncated with a fan-beam imaging geometry. In all the simulations the fan-beam focal length was 65 cm. The fan-beam detector size was 28.5 cm. The detector rotated in a noncircular orbit with a minimal radius of 22.7 cm and maximal radius of 28.8 cm. The reconstruction array was 60 pixels 60 pixels, with the pixel size of 0.712 cm. The ideal image and the filtered backprojection reconstruction are shown in Fig. 1. The pseudo-projection data were generated with a line-length weighted projector. The first set of reconstructions used a circular image support (2642 pixels in a 60 60 image array), and the pixel values were assumed to be zero if the pixels were outside the support. The second set of reconstructions used a tight (but not exactly the same as the object outline) image support (1908 pixels in a 60 60 image array). In these studies the number of views were set to 60 and 360, respectively, and the detector bin size was set to the pixel size and half the pixel size, respectively. Finally, reconstructions were performed from data corrupted with random noise and deterministic noise, respectively. In all but the very last study, a line-length weighted projector was used to form the imaging equations which were solved by the SVD method. The last study used an area-weighted projector to form the linear equations. Because the projections were generated with a line-length weighted projector, the mismatch of these different projectors caused deterministic errors. C. Physical Jaszczak Phantom Study The Picker PRISM 3000 system was used to acquire transmission data of a Jaszczak phantom with a fan-beam detector and a line source (see Fig. 2). The system arrangement was the same as in prior computer simulations. Projections were acquired at 360 views over 360 Total scanning time was 210 min. The projections were stored in 56 56 arrays and were converted into line-integrals prior to reconstruction. Both the whole 360 views and a subset of 60 views were used to reconstruct the images. A line-length weighted projector/backprojector pair were used in the reconstructions. III. RESULTS A. Computer Simulations The SVD reconstructions of the computer-simulated data are shown in Figs. 3–5. These simulations showed the following: 1) reducing the number of unknowns by using a tighter support ZENG AND GULLBERG: SVD STUDY OF TRUNCATED TRANSMISSION DATA (a) 109 (b) Fig. 2. (a) The experimental arrangement with the Picker Prism 3000 system. (b) A scheme of a slice of the Jaszczak phantom. Fig. 3. Reconstructions on a circular region (2642 pixels). Fig. 4. Reconstructions on a tight support (1908 pixels). improves the system condition significantly; 2) increasing the sampling density in the linear and/or angular direction also improves the system condition; 3) when the system is fairly well conditioned, the random noise usually does not deteriorate the object boundary, while the systematic errors in modeling degrade the image. B. Physical Jaszczak Phantom Study Fig. 5 shows the reconstructions of one slice of the Jaszczak phantom. Using more views did reduce the condition number, but this approach was not as effective as using larger pixel size and a tighter support. Reducing the number of pixels is a very effective way to reconstruct the image with truncated projections. The number of unknowns (pixels) is suggested to be reduced so that the system of equations has a unique leastsquares solution, and if an iterative algorithm were used to reconstruct the image, the algorithm would always converge. Fig. 5. Reconstructions on a tight support (1908 pixels) with noise. An area-weighted model was used in B. Reducing the number of pixels by increasing the pixel size reduces the image resolution but may cause model mismatch artifacts, which are better known as aliasing artifacts. The model mismatch artifacts are severe in Fig. 5(B) and Fig. 6. These artifacts do not necessarily indicate that the 110 IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 44, NO. 1, FEBRUARY 1997 Fig. 6. Reconstructions of a slice of the Jaszczak phantom. Model mismatch artifacts are severe. A line-length weighted model was used in the reconstructions. projection data are insufficient, but that the nonoverlapped square pixel model is not accurate enough to form imaging equations. IV. DISCUSSION Truncation does not always result in an underdetermined image reconstruction problem. It is known that an analytical function can be uniquely determined by its values within a small interval, and the function values elsewhere can be evaluated by the Taylor series expansion. An example of an analytical function is a band-limited function such as the k-space signal in MRI measurements [17]. However, in SPECT, the projection data cannot be assumed to be bandlimited or analytical because the object (i.e., the patient) to be imaged has a finite support, and the band-width of SPECT projections is infinite. Discrete samples of SPECT signal in one region are insufficient to exactly estimate the signal in a different region. In other words, exact data extension is not possible. On the other hand, using projections at all the views simultaneously (i.e., doing tomography) seems a better approach than estimating unmeasured data at each view individually. Tomographic image reconstruction can be achieved by solving a set of linear equations. This paper has shown that an accurate image can be obtained with truncated projections, provided the projection model is accurate, that is, the coefficients in the linear equations are accurate. The modeling accuracy can be improved by using smaller sampling intervals and smaller image pixel sizes. Due to practical sampling limitations, when one is unable to increase sampling density, one can use larger pixels, resulting in fewer unknowns in the system of equations, to increase the solvability of the system. However, if square pixels are used, bigger pixels introduce larger model-mismatch artifacts. The condition number indicates how sensitive the model-mismatch errors are. In reconstruction using truncated data, will more samples help? The answer is “yes” if the imaging geometry (pixel projection model, scatter, system geometric point response, etc.) is modeled accurately (see results in Figs. 3 and 4). The answer would be “some, not very effective” if the imaging geometry is not modeled accurately, as in a realistic SPECT study (see Fig. 6). As demonstrated in computer simulations, if the imaging geometry is exactly known, the linear equation method works. However, for the real data, the imaging geometry is not exactly known, thus the linear equation method does not work very well for truncated data because the linear system is illconditioned and sensitive to modeling errors [also illustrated in Fig. 5(B)]. Using a tighter support is equivalent to obtaining more measurements (constraining some pixels to have a value of zero). Therefore, using a tighter support helps in image reconstruction. To ensure the convergence, one should always work with an overdetermined system. The number of independent unknowns can be reduced by using better image basis functions (i.e., alternative forms of pixels) or a priori constraints (e.g., nonnegativity, smoothness etc.). We believe that an image can be accurately modeled by fewer pixels as long as the pixels (i.e., basis functions) are optimally determined. Finding the optimal image basis functions is an active research area [20]–[22]. It may be promising to use overlapped, smooth-edged pixel models to form the projector to eliminate or significantly reduce the model mismatch artifacts. 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