A Comparison of Rotation-Based Methods for Iterative ... - IEEE Xplore

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IEEE TRANSACTIONS ON NUCLEAR SCIENCE, VOL. 43, NO. 6, DECEMBER 1996

A Comparison of Rotation-Based Methods for Iterative Reconstruction Algorithms E. V. R. Di Bella, Member, IEEE, A. B. Barclay, Member, IEEE, R. L. Eisner, and R. W. Schafer, Senior Member, IEEE

Abstract—Rotation-based projectors and backprojectors offer unique trade-offs between accuracy, computational complexity, and smoothing. We analyzed six different rotation methods for generating projections and iterative reconstructions of tomographic data. Nearest neighbor, upsampled nearest neighbor, bilinear interpolation, and bicubic interpolation were all used to determine the values at rotated grid points. A decomposition of the rotation transformation matrix into three components was also investigated. Linear and cubic interpolation were used in the three-pass method. For all of the methods, mean normalized square errors were computed. Our results demonstrate the tradeoffs associated with the different methods and suggest that the three-pass method with cubic interpolation offers a computationally efficient approach with accuracy between that of bilinear and bicubic interpolation.

I. INTRODUCTION

I

TERATIVE reconstructions in emission computed tomography typically are comprised of repeatedly projecting and backprojecting the data. Rotation-based projectors and backprojectors are gaining in popularity for both sequential [1], [2] and parallel [3]–[7] implementations of iterative reconstruction algorithms. Rotating the image estimate at each view angle simplifies and speeds the implementation of depth-dependent single photon emission computed tomography (SPECT) response correction [1] and scatter compensation [6]. In addition, parallelization of iterative algorithms can take advantage of rotation-based formulations to achieve impressive speed-ups. While several researchers have examined the errors and computational trade-offs associated with various pixel-driven and ray-driven methods [8]–[11], rotation-based methods have not yet been evaluated. Here, we compare the errors, smoothing properties, and computational complexity of different rotation-based methods for projection and reconstruction. Most of the rotation-based reconstructions to date have employed bilinear interpolation. In this paper, we investigate nearestneighbor, upsampled nearest-neighbor, bilinear, and bicubic interpolation rotations. Rotation by the three-pass method of shears [12]–[14] is also studied to determine its utility in parallel and sequential tomographic reconstruction. The three-pass method realizes rotations by shifting each row independently, followed by a shift of each column, and completed by another shift of each row. Equation (1) illustrates the decomposition Manuscript received December 1, 1995; revised March 25, 1996. E. V. R. Di Bella and R. W. Schafer are with the Department of Electrical Engineering, Georgia Institute of Technology, Atlanta, GA 30332 USA. A. B. Barclay and R. L. Eisner are with the Carlyle Fraser Heart Center, Emory:Crawford Long Hospital, Atlanta, GA 30365 USA. Publisher Item Identifier S 0018-9499(96)08967-8.

Fig. 1. 45 rotation by the three-pass shear method. Clockwise from upper left: Original image (128 128), result of first pass, result of second pass, and result of final pass. Our implementation is designed for images confined to a circular region. This example shows the shearing effect more clearly than does an image confined to a circular region (but results in truncation of the image).

2

of the rotation transformation matrix into the multiple shears

(1) An example of the three-pass method is given in Fig. 1. In the discrete domain, interpolation is necessary to realize sub-pixel shears. The three-pass method, however, allows multiple one-dimensional (1-D) interpolations to replace twodimensional (2-D) interpolations for rotation. The three-pass method is also attractive for fine-grain parallel implementations, since the shears of each row may be done independently within each pass. Interpolation schemes ranging from linear shear interpolations to high-order cubic splines have been investigated with respect to providing high quality rotations from multiple pass rotation methods [15], [16]. In this work, linear and cubic interpolations were used with the three-pass method for comparison to the use of the more standard single-pass rotations in computed tomography. Three-pass rotation using nearest neighbor methods did not provide sufficient accuracy at the sampling rate used to warrant further study.

0018–9499/96$05.00  1996 IEEE

Di BELLA et al.: COMPARISON OF ROTATION-BASED METHODS FOR ITERATIVE RECONSTRUCTION

The accumulation of errors with the three-pass method is distinctly different from more standard rotation methods. In general, rotation accuracy degrades with angle. This is due in large part to the effect of the shearing operation on a discrete pixel grid. For example, not only should every row be sheared differently, but the bottom of the row should be shifted differently than the top of the row. If we assume square pixels, this shearing operation within a single row creates parallelograms, which then must be represented by the square pixels. For large angles, square pixels approximate the parallelograms poorly [12]. Also consider that with larger angles, the first shift results in more columns. Each column must be shifted differently, possibly introducing more errors. Errors are minimized in our implementations by rotating angles with a magnitude greater than 45 as a combination of perfect 90 rotations and rotations between 45 and 45 . To give an idea of the differences resulting from the various rotation methods, Fig. 2 shows an example of a 136.5 rotation ( 180 followed by a 43.5 rotation) implemented by the six different rotation methods on a 64 64 noise-free simulation. The figure reveals that the nearest neighbor method produces a jagged, blocky result and happens to completely miss the single pixel hot spot. The other methods all introduce smoothing. The three-pass method with linear interpolation (shears linear) results in the smoothest image, while bicubic appears to result in the least-smooth image (after nearest neighbor). Ringing around high frequency structures is noticeable with the bicubic rotation method and with cubic interpolation in the multiple-pass method. Note that the true value of the isolated hot pixel is 134. All of the rotation methods (except nearest neighbor) perform a smoothing operation, significantly reducing the count level of small structures. In this paper, we seek to evaluate the quality of the various rotation methods for projection and iterative reconstruction of tomographic emission data. Backprojection is not studied explicitly, as it typically has less of an impact on the final reconstruction [1]. II. METHODS A. Rotation Implementations The upsampled nearest-neighbor rotation method employed a fast upsampling technique in which each pixel was implicitly replicated into four sub-pixels. The resulting rotated image was then downsampled by averaging the values of the sets of four sub-pixels. The bicubic interpolation method used , as recommended for use in an image-independent sense [17]. Negative values resulting from the bicubic interpolation were set to zero. The negatives from cubic interpolation were used in the intermediate passes in the three-pass method; any negatives were set to zero in the final, rotated image. To speed implementations of cubic interpolation, the cubic kernel was quantized into 4096 sample values. Rather than computing the function at each point, the program looks up the nearest precalculated value. The quantization of the cubic interpolation kernel is not advantageous for the three-pass rotations, since only two unique coefficients for each row or

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column are needed in each pass. However, the same cubic kernel was used to facilitate comparison to bicubic rotations. Note that for all of the methods under discussion, the interpolation weights may be calculated only once for each desired angle between 45 and 45 . The weights can be stored for subsequent rotations and accessed with a lookup table. Depending on the implementation and the platform, this approach may provide an advantageous memory and speed trade-off. As seen in Fig. 1, the aspect ratio changes at each pass with the three-pass rotation method. More rows or columns become necessary, depending on the angle.1 In our implementation, we assume a circular region of support after the final pass; the final rotated image is cropped such that the center of rotation is at the same location as in the original image. We found that integer crops produced significant errors. More accurate rotations were obtained after modifying the second and third passes to incorporate the sub-pixel shifts required for accurate cropping. Without our modification, the center wanders depending on the rotation angle. In theory, the code could be optimized for circular regions, just as more conventional rotation methods rotate only a circular region. Note also that the final shift of the three-pass method can be omitted when summing to give projections if the locations on the ray of the emitted counts are not needed, for example when SPECT point response, scatter, and attenuation are not accurately modeled. Although, for attenuation compensation, the final pass can still be omitted if the attenuation coefficients are being rotated in the same fashion, since the pixels will correspond. The result will be slightly different due to the lack of interpolation. Also, the first pass in the rotation stage of backprojection will not require interpolation if the first stage of the backprojection results in uniformity within rows. B. Computer Simulation—Projections The differences between the six rotation-based projectors were quantitated by comparing to analytic projections. An analytic 2-D model of circles of varying sizes and ellipses oriented to resemble a body outline and the left ventricle was constructed and then sampled at 256 256. A downsampled version of the model, at 64 64 pixels, is shown in the upperleft corner of Fig. 4. The 64 64 version of the model was used as input to the rotation-based projectors. Projections (32 angles over 180 ) of the continuous model were generated analytically and then sampled (256 bins/angle). The sampled projections were downsampled to 64 bins/angle in the same manner to serve as a gold standard. Normalized square error ( ) was computed for the projections at each angle according to

(2)

1 The integer offset of each row/column can be stored instead. An in-place rotation is possible with a more complex indexing scheme and circular support.

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0

2

Fig. 2. 136.5 rotation of simulation. Nearest 4 the column direction.

refers to nearest neighbor rotation of an image upsampled 2

where is the analytic projection data and is the projections generated with rotation operators. Averaging over . the 32 projections ( ) was done to give the mean

2 in

the row direction and 2

2 in

For comparison, errors were also computed for projections generated from two nonrotation-based projectors. The line length and ray factors projectors from [18] were used.


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Fig. 3. Mean nse for different projectors.

C. Patient Data A test image was taken from 30 iterations of a line length maximum likelihood expectation maximization (ML-EM) reconstruction of clinically acquired cardiac Tl SPECT projections. A measure of accuracy was obtained by rotating the image forward and then backward and comparing to the original. D. Computer Simulation—Reconstructions The same 64 32 analytic projection data set used as the gold standard in Section II-B was input to a rotation-based ML-EM reconstruction routine. The results from 30 iterations, starting from a flat initial estimate, were compared to the 64 64 simulated image in a least-squares sense. A ML-EM reconstruction using a line length projector/backprojector was compared as well. III. RESULTS Fig. 3 plots the as a function of projection angle for 32 angles ranging from 135 to 45 . All of the methods give for the trivial cases of 0 and 90 . The is the same not zero for the trivial cases due to the discretization of the analytic model and projections. Table I reports the theoretical computational complexity and the accuracy associated with the different rotation operators in the context of projecting data from the analytic model. Note that for cubic interpolation with the three-pass method, the number of multiplies is 40% fewer than that required for the bicubic method. Table II gives the results of comparing the rotation-based methods by rotating the ML-EM reconstructions of clinical

TABLE I COMPARISON OF ROTATION-BASED METHODS USING THE ANALYTIC MODEL. THE COMPUTATIONAL COST (PER PIXEL) IS FOR THE ROTATION OPERATION ONLY, AND THE nse WAS COMPUTED BY COMPARING THE GENERATED PROJECTIONS TO THE ANALYTIC PROJECTIONS Method Ray factors Line length Nearest Nearest (upsampled) Shears - linear Shears - cubic Bilinear Bicubic

Projection Data Computational Cost Multiplies/Adds — — 0/0 0/0 3/6 12/9 3/6 20/15

Projection Mean nse 0.0022 0.0020 0.0037 0.0022 0.0024 0.0014 0.0022 0.0013

TABLE II COMPARISON OF ROTATION-BASED METHODS BY ROTATING THE ML-EM RECONSTRUCTIONS OF CLINICAL DATA BACK AND FORTH Method Nearest Nearest (upsampled) Shears - linear Shears - cubic Bilinear Bicubic

Rotation and Unrotation Data Time (s) Rotation Mean nse 0.7 0.029 2.4 0.038 0.4 0.057 0.5 0.037 0.9 0.044 2.5 0.020

data back and forth. The times given in Table II are for rotating 64 image matrix and then unrotating the image for the 64 32 angles on a Sun workstation. The nearest neighbor and the bilinear interpolation take nearly the same time, indicating that data access is more of a bottleneck than the number of multiplies and adds. This point is reinforced by the fact that

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Fig. 4. Thirty iterations of ML-EM on noise-free simulated data. The type of rotation operator is indicated for each case. The contrast has been manipulated in order to bring out small differences (see gray scale on right). The numbers in parentheses are the sum of the squared difference between the reconstructed image and the original image, normalized by the sum of the pixel values squared in the original image.

Fig. 5. Profiles corresponding to Fig. 4.

the shear rotations with linear interpolation are approximately 1.7 times faster than bilinear interpolation. The two methods

require the same number of multiplies, but the data is used in a scan line order for the shear case. These times are a sample comparison obtained from nonoptimized code; the number of multiplies and adds in Table I are the theoretical limits. The results in Tables I and II correspond with a qualitative assessment of the methods. For the noise-free analytic data, the projections generated by all of the methods except the nearest neighbor case looked similar and comparable to the analytic projections. The nearest neighbor case induced noticeable noise and jagged edges. Inspection of the rotations of the noisy ML-EM data, however, revealed little deterioration in the nearest neighbor case. The three-pass method with linear interpolation gave the smoothest rotated versions of the MLEM data. Differences between bicubic and shears with cubic interpolation were difficult to detect. The nearest neighborrotated images showed the most noise. The various reconstructions obtained from 30 iterations of the ML-EM algorithm implemented with the different rotation-based projector/backprojectors and with a line length projector/backprojector are shown in Fig. 4, along with their mean . Fig. 5 displays corresponding profiles.


IV. DISCUSSION A. Projections The projectors that smooth the most—shears linear, bilinear, and ray factors—result in very similar errors, as seen in the lower plot of Fig. 3. In general, smoother interpolators result in higher . Nearest neighbor methods, however, do not correspond to this trend. Also apparent from the plots in Fig. 3 is the similarity between the bicubic method and the shears cubic method. What is not evident here is the decreasing performance of the three-pass method with increasing angle. Two factors contribute. First, the implementation only rotates between 45 . Second, the final interpolation is canceled out by the ray sum process. Errors are only accumulated in the first two passes. In terms of accuracy without excessive smoothing, a rotation-based projector using bicubic interpolation provided the best results, followed by shears with cubic interpolation, line length, upsampled ( ) nearest neighbor, ray factors, bilinear interpolation, shears with linear interpolation, and nearest neighbor. B. Rotations The errors computed from rotating noisy images back and forth revealed trends similar to those seen in the projection stage. The accuracy of the nearest neighbor is much improved relative to the projection accuracy. This is understandable since the pixel values are merely being moved. Although some pixels may be missed, nearest neighbor returns nearly the original image. The relatively higher errors for the three-pass rotation methods may be attributed to the fact that the third shear is necessary for rotations. Noise-free comparisons at a higher resolution have been reported in [16], where they rank the methods similarly with regard to accuracy.

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scatter models may alter the differences between the rotationbased methods. For example, we mentioned that projection errors are less due to the implicit avoidance of the last shearing step in the three-pass method. With inclusion of accurate point response or scatter models, the last shearing step is done explicitly. Results from reconstructing clinical cardiac SPECT projections showed that although rotation using nearest neighbor appears of poor quality (Fig. 2) and gives the worst in the projections and noise-free reconstructions, reconstructions from noisy data using nearest neighbor (not shown) appear little worse than the other methods. This implies, at least qualitatively, that the differences seen in Fig. 4 are small compared to the deviations due to noise common in SPECT imaging. We also considered the effects of aliasing, since a circular field of view does not imply the image is circularly bandlimited. Rotation in the spatial domain corresponds to rotation in the frequency domain. Thus, any sharp diagonal lines resulting in a mix of high horizontal and vertical frequencies may be rotated out of the frequency domain image, only to wrap around and alias. This effect is present even when an ideal interpolator is used. The aliasing effect was noticed in noisefree simulations, but no effects were discernible for noisy fields. Rotation by the method of shears has been shown to exhibit aliasing patterns similar to that of bilinear interpolation [19]. It is also worthwhile to consider that the rotation operation blurs the image in-plane. The 3-D estimate will not be blurred along the -axis (long axis of the body). The 2-D reprojections therefore may not have isotropic resolution. This may slow or alter convergence, depending on the frequency content of the image. Calculation of 3-D point response or regularization blurring kernels should take into consideration the in-plane blurring due to the rotation operator, since uniform resolution is desirable.

C. Reconstructions Since errors and smoothing by the rotations are introduced at every iteration, slight differences in rotation operators may significantly alter the path taken by an iterative algorithm toward convergence. In this sense, we may view smoothing by the rotation operators as a regularization technique. Extensive research remains to characterize appropriate regularization techniques and algorithmic formulations for given tasks. The noise-free iterative reconstructions in Fig. 4 provide an idea of the effect of the different rotation methods on the reconstructions. Except for nearest neighbor, which introduces a mottling of the background ellipse, all of the rotation-based methods and the line-length method appear comparable. The ray factors reconstruction is not shown but closely resembles the bilinear interpolation rotation-based reconstruction. The methods reconstruct all but the smallest structure well. The profiles shown in Fig. 5 echo this interpretation. Further work is needed to determine if the differences in iterative reconstructions, such as seen in Figs. 4 and 5, are significant when applied to tasks of interest. Also, different sampling rates and incorporation of accurate attenuation, point response, and

V. CONCLUSION Almost all of the rotation-based projector/backprojectors performed at least as well as the methods not based on rotations—line length and ray factors. Nearest neighbor and shears with linear interpolation did not provide quality equivalent to line length for noise free data, but may be adequate for tasks employing noisy data. Only 64 64 images, 32 views over 180 were studied, since this represents the fewest samples commonly measured in clinical emission tomography of the heart and brain. More angles and/or a finer grid would be expected to give more accurate results. The computational load is expected to scale linearly with the number of pixels for all of the methods. Our results can be used as a guideline for selecting the appropriate interpolation scheme, depending on the desired trade-offs between accuracy, smoothness, and computation. The three-pass method of shears with cubic interpolation appears to be a particularly efficient method for both sequential and parallel formulations of iterative reconstructions.

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REFERENCES [1] 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, no. 5, pp. 1444–1453, 1992. [2] J. Ye, Z. Liang, and D. P. Harrington, “Quantitative reconstruction for myocardial perfusion SPECT: An efficient approach by depth-dependent deconvolution and matrix rotation,” Phys. Med. Biol., vol. 39, pp. 1263–1279, 1994. [3] M. I. Miller and B. Roysam, “Bayesian image reconstruction for emission tomography incorporating Good’s roughness prior on massively parallel processors,” in Proc. Natl. Acad. Sci., 1991, vol. 88, pp. 3223–3227. [4] C. S. Butler, M. I. Miller, T. R. Miller, and J. W. Wallis, “Massively parallel computers for 3D single-photon-emission computed tomography,” Phys. Med. Biol., vol. 39, pp. 575–582, 1994. [5] G. L. Zeng, Y.-L. Hsieh, and G. T. Gullberg, “A rotating and warping projector/backprojector for fan-beam and cone-beam iterative algorithm,” IEEE Trans. Nucl. Sci., vol. 41, no. 6, pp. 2807–2811, 1994. [6] Z.-W. Ju, E. C. Frey, and B. M. W. Tsui, “Distributed 3-D iterative reconstruction for quantitative SPECT,” IEEE Trans. Nucl. Sci., vol. 42, no. 4, pp. 1301–1309, Aug. 1995. [7] J. L. Cruz-Rivera, E. V. R. Di Bella, D. S. Wills, T. K. Gaylord, and E. N. Glytsis, “Parallelized formulation of the maximum likelihoodexpectation maximization algorithm for fine-grain message-passing architectures,” IEEE Trans. Med. Imag., vol. 14, no. 4, pp. 758–762, Dec. 1995. [8] H. J. Trussell, H. Orun-Ozturk, and M. R. Civanlar, “Errors in reprojection methods in computerized tomography,” IEEE Trans. Med. Imag., vol. 6, no. 3, pp. 220–227, 1987. [9] D.-C. Yu and S.-C. Huang, “Study of reprojection methods in terms of their resolution loss and sampling errors,” IEEE Trans. Nucl. Sci., vol. 40, no. 4, pp. 1174–1178, 1993.

[10] W. Zhuang, S. S. Gopal, Y. Ting, and T. J. Hebert, “Numerical evaluation of methods for computing tomographic projections,” in Conf. Rec. 1993 Nucl. Sci. Symp. Med. Imag. Conf., San Francisco, CA, 1994, pp. 1404–1406. [11] T. M. Guerrero, Y.-C. Tai, A. Chatziioannou, D.-C. Yu, S.C. Huang, M. Dahlbom, and E. J. Hoffman, “Evaluation and application of reprojection methods for 3D PET,” in Conf. Rec. 1993 Nucl. Sci. Symp. Med. Imag. Conf., San Francisco, CA, 1994, pp. 1101–1105. [12] A. W. Paeth, “A fast algorithm for general raster rotation,” Graphics Interface, pp. 77–81, 1986. [13] A. Tanaka, M. Kameyama, S. Kazama, and O. Watanabe, “A rotation method for raster image using skew transformation,” in Proc. IEEE Conf. Comp. Vision Patt. Recog., 1986, pp. 272–277. [14] G. Wolberg, Digital Image Warping. Los Alamitos, CA: IEEE Computer Society Press, 1990. [15] P. E. Danielsson and M. Hammerin, “High-accuracy rotation of images,” in CVGIP: Graphic. Models Image Proc., 1992, vol. 54, no. 4, pp. 340–344. [16] M. Unser, P. Thevenaz, and L. Yaroslavsky, “Convolution-based interpolation for fast, high-quality rotation of images,” IEEE Trans. Image Proc., 1995, vol. 4, no. 10, pp. 1371–1381. [17] S. K. Park and R. A. Schowengerdt, “Image reconstruction by parametric cubic convolution,” in Comp. Vis. Graphics Imag. Proc., 1983, vol. 23, pp. 258–272. [18] R. H. Huesman, G. T. Gullberg, W. L. Greenberg, and T. F. Budinger, RECLBL Library Users Manual—Donner Algorithms for Reconstruction Tomography, Lawrence Berkeley Laboratory, 1977. [19] D. Fraser and R. A. Schowengerdt, “Avoidance of additional aliasing in multipass image rotations,” IEEE Trans. Image Proc., vol. 3, no. 6, pp. 721–735, 1994.