Efficient convolution kernels for computerized tomography

ULTRASONIC IMAGING 1, 232-244 (1979)

EFFICIENT CONVOLUTION KERNELS FOR COMPUTERIZED TOMOGRAPHY

Surender K. Kenue and James F. Greenleaf

Biodynamics Research Unit Mayo Clinic

Rochester, MN 55901

Three concepts are presented: 1) Extended kernels: The Ramachandran- Lakshminarayanan convolution kernel has one zero between each non-zero value in the spatial domain. By extending the kernel in Fourier space, it is shown that each extension leads to two additional zeros in the spatial domain of the convolution kernel, thus decreasing the required number of multiplications necessary for convolution. Any known kernel can be extended and in the limit of extension a simple backprojection reconstruction is obtained. 2) Binary kernels: A technique for generating binary approximations to any convolution kernel is described. Excluding the central element, all other elements of the kernel are approximated by an even power of two; thus, multiplications are replaced by shift operations in the convolution procedure. 3) Recursive convolution: It is shown how additions can be saved by using a recursive formulation which generates new elements in the convolution procedure utilizing only a few summation steps. Results from both extended kernels and their binary approximations are described for simulated phantoms and ultrasound data obtained from breast scans of patients.

Key words: Computerized tomography; convolution kernels; hardware; image reconstruction; ultrasound; x-ray.

Introduction

Computerized tomographic reconstruction of the distribution of "density" within objects requires projection profiles of the object from several angles of view. Given a set of one-dimensional or two-dimensional projections of the object, the problem is to estimate the internal density distribution arising from the given set of profiles or projections. This can be achieved by a wide variety of methods (see [l] for a review and applications). Ramachandran and Lakshminarayanan [2] showed that by applying a convolution theorem, a reconstruction method can be derived which is both efficient and accurate. This key observation has led to the so-called convolution/backprojection (or filtered backprojection) technique which is most widely used in the commercial x-ray scanners available today.

The method of filtered backprojection consists of two steps--first, the profiles are convolved with a kernel and second, the modified profiles are backprojected into the image plane and summed onto a grid of pixels representing the final image. Since the computation time of this technique is approximately equally divided between the convolution and backprojection steps, any success at increasing the speed of the convolution step will increase substantially the overall speed of the algorithm. The purpose of this paper is to describe several methods for increasing the speed of the convolution step by up to 80 to 90 percent.

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Copyright @ 1979 by Academic Press. Inc. All rights of reproduction in any form reserved. 232

EFFICIENT CONVOLUTION KERNELS

The kernel devised by Ramachandran and Lakshminarayanan is ideal for low noise data; however, one encounters noise in practice which is due to electronics, detector thickness, sampling of the data, limited views, etc. The gray levels of the reconstructed pictures using the Ram-Lak kernel are noisy unless some kind of smoothing is done. Shepp and Logan [3] proposed another convolution kernel which reduces noise but at the cost of some image resolution. Tanaka and Iinuma [4] have proposed an optimized convolution kernel based on a given signal-to-noise ratio. Kwoh et al. [5] have proposed a variable filter function, which can be tuned to the user's requirement.

In this paper, techniques are presented for introducing zeros in the spatial domain representation of the convolution kernels. Since zeros need not be multiplied by their related terms in the profile, this increases the speed of convolution and leads to faster computation. The Ram-Lak kernel already has zeros alternating with non-zero elements. The new convolution kernels with many zeros (also called fractional band kernels by Barry K. Gilbert [6]) have many applications depending on the information content of the data being convolved.

The concept of binary kernels was suggested to us by Lewitt [7]; however, the algorithm proposed here for obtaining binary approximations of any given kernel was derived independently and is somewhat different from that of Lewitt [12]. We report here an additional technique which results in a recursive equation for calculating convolutions and is a direct consequence of the fact that the binary approximation to any kernel has many values which are represented by the same power of two. This recursion relation for convolution saves many additions and increases the speed of computation.

Fractional Band Kernels

Although the convolution operation can be done much faster using the Fourier transform technique available on most computers, most commercial companies are now using hardwired mathematical units to obtain very high- speed reconstructions. However, computing Fourier transforms, even with hardwired mathematical computer boxes, is very difficult and costly. Hard- wired convolution is much simpler; thus, techniques for increasing the speed of the convolution method for reconstruction have applicability in designing a hardwired convolver. The Dynamic Spatial Reconstructor (being built at Mayo Clinic) is a very high-speed data-gathering device which can obtain data for more than 3,600 x-ray reconstructions in one second and will require a very high-speed hardwired reconstruction unit (see Gilbert et al. [81).

Two techniques for increasing the speed of convolution are: 1) introducing more zeros in the kernel and 2) truncating the kernel to fewer total values so that the convolution takes less time. The effect of truncating the kernels is described by Gilbert et al. [8] and Lewitt [lo].

The Ram-Lak kernel for either parallel or divergent beam geometries is a half-band kernel, that is, every other element in the spatial domain representation of the kernel is equal to zero. Simple extensions of this kernel can yield additional kernels having 3, 5, 7, 9, 11, . . . zeros between their non-zero values. For example, the fan beam kernel [ll] is given by the following equation.

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KENUE ANTI GREENLEAF

0.50

: c” -1 0.25

I

A ’ I

RAM-LAK, ’ I

/ I

- P” / I

--- i3mry 03 / I /

/ I

_A* 0 0.25 0.50

FREQUENCY

Fig. 1. Modulus of Fourier transform of Q3, Q5, Q7 and binary Q3 kernels. For binary Q5 and Q', the shape is similar to that of binary Q3. The binary versions of Q5 and Q7 are as similar to their full value versions as the binary Q3 is to the full value Q3 shown. Because of symmetry, only one-half of the Fourier domain is shown. (Reproduced with permission [9]).

1/(8a2)

Q(m) -1 = 2[asin(ma>]2

0

The extended kernels are defined as:

m=O

m = 1,3,5,7,...

m = 2,4,6,...

1/(8na2) m=O

Q2”-‘(m) = -1

2n[*sin(ma)12 m # 0 (see below)

Q2n-1(m) kernels yield the Ram-Lak kernel for n=l and Q3(m), Q5(m), Q7(m), Q'(m) for n = 2, 3, 4, 5, respectively with non-zero values at m as defined by:

Q3(m): m=2,6,10,14,... with 3 zeros between values. Q5(m): m=3,9,15,21,... with 5 zeros between values. Q'(m): m=4,12,20,28,... with 7 zeros between values. Q'(m): m=5,15,25,35,... with 9 zeros between values.

The frequency representation of these fractional band kernels are shown in figure 1. It is seen that Q5 and Q7 kernels should be used with caution since they have low responses in the upper portions of the pass band. Profiles having information in the middle to hi her frequency portions of the pass band should not be convolved using Q 5 or Q', since information in the region of high frequencies will be suppressed. However, in many applications, most of the information is concentrated in the very low

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part of the spectrum and only noise occurs in the upper frequencies. Thus Q5 and Q7 kernels should work well in these situations. It can be seen that such kernels will have strong smoothing effects, perhaps obviating the necessity of filtering the projection profiles prior to convolution.

Since each extension adds two more zeros between non-zero values of the kernel, thus increasing the speed, the Q5, Q7, Qq,..., kernels could be used in a "search" mode in which large regions of the reconstruction area are searched quickly at low resolution. Then regions of interest might be reconstructed using zoom reconstructions done with the high resolution kernels. For instance, the entire heart may be reconstructed using a coarse reconstruction but certain regions of the coronary arterial tree could be reconstructed in the zoom mode using the increased resolution of the Q' or Q3 kernels.

Binary Kernels

Convolution time can also be decreased by using binary kernels since elements in the kernel are represented by powers of two and thus multipli- cation of the kernel with the profile consists merely of binary shifts, a very simple operation to accomplish using hardware. This technique has been reported by two separate groups, Kenue [9] and Lewitt [12]. The method presented in this paper is an extension of the method previously reported by Kenue [9].

Let K (I), I = -M,...O then for l&ge H,

,...M, be the original convolution kernel;

; Ko(I) = L = 0. -M

However, since M is usually finite (say 256 or 512); the sum L of the original kernel is not zero because the exact representation of the kernel has been truncated. Thus, as a first step, the central value of the kernel is modified to give

Kc(O) = Ko(0) + L.

Let S be a scale factor and be equal to the binary approximation of e the normalized kernel with respect to K (0). In the

the binary kernel K (I) is obtained from theCnormalised kernel having ;ull digital values ii the following manner:

KB(I) = 2"

for n such that Is(I)-2nl < IKN(I)-2ml for all m # n and for I # 0.

M SincexKB(I) = 0 for the kernel, one sets

-M

~(0) = - &(I) - ;KB(I). -M +1

The binary kernel is then obtained by multiplying KB(I) by the scale parameter S (which is just a power of 2).

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KENDE AND GREENLEAF

The modified central value K' of the binary kernel is now given by KR(O)-L. The profiles then are scaled by a factor K,(O)/K'. In Lewitt's algorithm the non-zero element nearest the central element is normalized but the kernel values are not summed to obtain a normalization factor. The technique reported in this paper is an extension of the previous algorithm [9] in which the central element was not adjusted and normalized.

The values of the modified binary kernel occur in groups, each group having like powers of 2; therefore, one can save shift operations by sum- ming those elements of the profile having like powers of 2 in the kernel. Let P(I), and KR(I), I = O,... M, lution kernel, respectively.

be the projection profile and thenconvo- Then for the convolved partial sum g

1' the

following holds:

N

(P*KB) j = c g'; 2" + P(j)K,(O), n=l

where

; P(T), T=R

and R,S are the lower and upper indices in the kernel of like terms of power 2" and N is the highest power of two.

Note also that gn+l changes by only two elements from gn and that for the tail ends of the 8 inary kernel, S-R+1 can be very large t see table in 191). Therefore, the following relationship can be derived,

n n gj+l = Pj - P(R) + P(S+l).

Therefore, except for the initial value gy, the sum g" can be updated by simply subtracting the previous element and adding the ?lext element of the profile. This procedure can be evaluated in one addition and one sub- traction as compared to S-R+1 additions for each different j. The resul- ting number of operations for both the standard and binary forms of the Shepp-Logan kernel and a list of the terms having like powers of 2 is given elsewhere [9].

Experiments and Discussions

A comparison of the kernels proposed by Ram-Lak [2] and by Shepp and Logan [3] is shown in figure 2 where the modulus of the Fourier transform is plotted versus frequency. Also shown are the Fourier transforms of the binary representations of these two kernels. The.response of the binary version of the Ram-Lak kernel is increased in the upper regions and decreased in the lower regions of frequency, while the binary version of the Shepp and Logan kernel has a somewhat higher response in the lower frequencies. The extended kernels are shown in figure 1 where the modulus of the Fourier transform of the Q3, Q5 and Q7 kernels and the binary representation of the Q3 kernel are illustrated. One can see that the extended kernels are of triangular geometric forms repeating more often as the extension increases. One can also see that the Fourier transform of the binary representation of Q3 is very close to the exact representation.

Reconstructions obtained using the extended kernels and their binary versions were compared to reconstructions obtained with the Ram-Lak kernel and its binary version using data obtained from a thorax phantom. The

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FREQUENCY

Fig. 2. Modulus of Fourier transform of convolution kernels. The kernels shown are Ram-Lak and Shepp-Logan and their binary versions. (Reproduced with permission [9]).

Fig. 3.

Fig. 4.

Top images: Mathematically simulated human thorax and Shepp- Logan head phantom, digitized into 117 x 117 matrix with pixel sizes of 3.2 mm and 2 mm, respectively. Bottom images: Recon- struction of the human thorax by fan beam convolution method. The images are reconstructed using Ram-Lak kernel and its binary approximation.

Top images: The images are reconstructed using the Q3 kernel and its binary version. Bottom images: The images are reconstructed using the Q5 kernel and its binary version.

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KENLIE AND GREENLEAF

Fig. 5. Top images: The images are reconstructed using the Q7 kernel and its binary version. Bottom images: Reconstruction of the human thorax with 5 percent Gaussian noise (variance 5 percent) added to the profiles, using 120 views. were Ram-Lak and Q5.

The convolving kernels

thorax phantom is designed to test the results of reconstruction algorithms to be used in an x-ray reconstruction system currently being built at Mayo Clinic and termed the DSR (Dynamic Spatial Reconstructor [S]).

The source-to-detector spacing and source-to-center spacing were 203 cm and 145 cm, respectively. Divergent beam geometry is used. The total number of views were 360, collected every degree over 360°, and the number of samples in each projection profile was 255. The size of the reconstruction images were 117 x 117 pixels of 3.2 mm square. Figures 3-5 compare the images obtained using data from the thorax phantom. One can see that the Q3 kernel gives results strikingly similar to the Ram-Lak kernel and that the binary version of Q3 and of the Ram-Lak kernel are each virtually indistinguishable from the standard versions. The binary version of the Q3 kernel may give somewhat enhanced images, as can be seen in figure 4. The Q5 and Q kernels are also very useful for reconstruction, since they give a more smoothing effect than the Q3 kernel. Once again, the difference between the non-binary and the binary versions of these two kernels is very difficult to detect. The reconstructions obtained with binary approximations of the kernels exhibited a slight shift in dc value (3 units out of 1,000) and an alteration of gain (17 percent) which were corrected in all images shown.

In order to characterize the stability of these reconstruction kernels in the presence of noise, Gaussian noise was added to each profile with a variance equal to 5 percent of the range of the values in the profiles and reconstructions were done over a range of 120 views, each separated by three degrees. Results for the Q5 kernel are shown in the lower panel of figure 5 and are compared to the results obtained using the Ram-Lak kernel. One can see that the Ram-Lak kernel is more sensitive to noise and has a weaker filtering function than the Q5 kernel and that resolution is main- tained under these conditions even using the Q5 kernel.

Figure 6 illustrates a comparison between the reconstructions of the thorax phantom using the Ram-Lak and the binary Ram-Lak convolution kernels

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RAM-LAK Kernel Binary RAM-L AK Kernel

1 or<q,no,. - Or,qfn~l: - RAN-IAK Kernel: - mnory RAM-LIK Kerns,: -

0.4 -

I 30 60 90 117 I 30 60 90 117

PlXEL NUMBER PIXEL NUMBER

Fig. 6. Comparison of thorax reconstruction with Ram-Lak and binary Ram- Lak convolution kernels. The line plot represents the gray levels of the image matrix through column 59 in units of relative x-ray attenuation (air = 0.0).

0.4

k G

5 Q 0.2

0.0

I-

/

Q3 Kernel

OriqinLv: - 03 Kerns,: -

I 1 I

30 60 90

PIXEL NUMBER

4;; O.O I

0.4

0.2

Binary Q3 Kernel

oriqin*,: - emry 0’ Kernel: -

c 30 60 90

PlXEL NUMBER

117

Fig. 7. Same format as figure 6, except that convolving kernels were Q3 and its binary version.

for values in the image along column 59 in the image matrix. Figures 7, 8 and 9 are of the same format as figure 6, except that the convolution kernels were Q3, Q5, and Q7, respectively. One can see that as the extension of the kernel increases, the low pass filter characteristics of the kernel become stronger. The binary version usually has somewhat enhanced edge response and sometimes overshoots as can be seen in figure 6. Gener - ally, however, the binary versions of the kernels very faithfully reproduce the standard full valued versions of the kernel.

239

KENUE AND GREENLEAF

O5 Kernel Binary O5 Kernel

0.0’ -l 1 I I b J I I I I 30 60 90 117 0.0: 30 60 90 117

PIXEL NUMBER PIXEL NUMBER

Fig. 8. Same format as figure 6, except that convolving kernels were Q5

0.4

: G

5 0 0.2

0.0 I

and its binary version.

0’ Kernel

ongino,: - 07 Kernel: -

PIXEL NUMBER PIXEL NUMBER

I

Binary 0’ Kemr/

s 1 1 - 30 60 90 117


Figure 10 illustrates reconstructions of the Shepp and Logan head phantom [3] which has different characteristics from the thorax phantom. The source-to-center and source-to-detector distances were the same as those used for the thorax phantom and the number of views were the same. The reconstructions are 117 x 117, with a pixel size of 2 mm. The reconstructions of the Shepp and Logan head phantom using the Ram-Lak and Q3 kernels and their binary versions are shown in figure 10. A narrow range of densities, about 5 percent of the total range, is shown to enhance

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Fig. 10. Reconstruction of the Shepp-Logan head phantom. Left images are reconstructed using Ram-Lak and Q3 kernels; right images are reconstructed using their binary approximations.

0.10 -

0.08 -

?I I- s 0.06 -

> I- R 0.04 -

z 0

0.02 -

RAM-LAK Kernel

0.00 L ” L I I i I 30 60 90 I,;

oripinor: - RIM-‘AK I(em*,: -

PIXEL NUMBER

0.08

0.06

Bin~y RAM-LAK Kernel

1 I

.50 6” 90

PIXEL NUMBER

f/7

Fig. 11. Comparison of Shepp-Logan phantom reconstruction with Ram-Lak and binary Ram-Lak convolution kernels. The line plot represents the gray levels of the image matrix through column 59. Units are (u~-~~)/IJ~ where uw = x-ray attenuation coefficient of water.

differences in the images. One can see that the reconstruction using the Q3 kernel is somewhat smoother and that the binary versions have very slightly sharper edges in addition to slight ring artifacts. A comparison of the values along column 59 in the images of figure 10 is shown in figure 11 for the image reconstructed using the Ram-Lak kernel and its binary version.

Figure 12 is the same format as figure 11 for the Q3 kernel and its binary version. One can see that the distribution of gray levels of the original phantom are very faithfully reproduced by both the Ram-Lak and the Q3 kernel even though the Q3 kernel has many more zeros and would be

241

KENUE AND GREENLEAF

O3 Kernel Binary O3 Kernel

0.06

30 60 90 2 OoO PIXEL NUMBER

I 30 60 90 117

PIXEL NUMBER


Fig. 13. Reconstruction of the acoustic speed of in vivo human breast via ultrasound time-of-flight profiles. ted using Ram-Lak and Q5 kernels.

Left images are reconstruc-

using Q3 and Q7 kernels. Right images are reconstructed

The white mass was found to be fibroadenoma, a benign fibrous lesion with high acoustic speed.

much faster to implement. In addition, the binary version of the kernels show low frequency errors near the skull, which may be corrected by some optimization of the binary kernels mentioned later.

Reconstructions of acoustic speed were obtained using a scanner described previously by Greenleaf et al. [13]. The source-to-center distance for these reconstructions was 210 mm and 60 ultrasound time-of-flight profiles were collected every 6" around the range of 360°, with 400 samples in each profile. The dimensions of the reconstruction Images are 118 x 118, with a pixel size of 1.2 mm. Figure 13 illustrates reconstruction of the distribution of acoustic speed within the breast of a patient as part of an examination to detect breast lesions. Reconstructions are illustrated for the Ram-Lak kernel and the Q3, Q5 and Q7 extended kernels.

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One can see that because of the low resolution data, even the Q7 kernel does very well in reconstructing the distribution of acoustic speed within this plane. The circular region at nine o'clock was found after surgical breast biopsy to be a fibroadenoma, a benign fibrous lesion having high acoustic speed.

Conclusions

The "fractional band" kernels are useful in image reconstruction under conditions in which the distribution of information as a function of frequency is known and matches that of the convolution kernel. These kernels have inherent smoothing properties often desirable in situations where data are noisy. Because of their efficiency and speed of computation, the lower resolution extended kernels could be used to search large regions of interest as a preliminary step in evaluating the data. High resolution reconstruction could then be accomplished on regions of interest.

The binary versions of all of the kernels exhibited here yield results similar to their full value representations. Low frequency errors caused by the binary versions of the kernels were observed near the skull in the Shepp and Logan phantom. In general, however, edges were enhanced slightly in the binary kernels relative to the full value kernels.

Reconstructions calculated with the full value kernels and those calculated with their binary approximations are very similar, as would be expected from their frequency domain representations. However, some optimization of the binary kernels may still be possible since in certain elements of the kernel the full value of the original kernel is halfway between two different powers of 2 (see also [12]). In this situation a choice can be made as to which binary value can be assigned to the element. In the current implementation, only the nearest power of 2 is used for the approximation. It may be more useful to use another criteria for choosing the power of 2 representation of each element, thus optimizing in some sense the response of the kernel.

The proposed binary and non-binary fractional band kernels are under consideration for implementation in hardware for application with the Mayo Clinic DSR machine [6]. The algorithms have been implemented in software using shift instructions in place of multiplications with the binary kernels and a detailed study of the savings of computer time using these kernels on a general-purpose computer is in progress.

Acknowledgments

The authors gratefully acknowledge the help of Arnold Lent, Barry K. Gilbert, Robert Lewitt, David E. Gustafson, Steven A. Johnson, Al Chu, Balu Rajagopalan, Paul Thomas, and Bill Samayoa. Thanks are also due to Elaine Quarve for typing and to Marge Engesser for art assistance. This research was supported in part by Grants HL-04664, RR-00007, Hv-7-2928, HL-00060, HL-00170, and HL-07111 from the National Institutes of Health, and NCI-CB-64041 from the National Cancer Institute.

References

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[2] Ramachandran, G. N. V. and Lakshminarayanan, A. V., Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms, Proc. Natl. Acad. --- Sci. USA 68 2236-2240 (1971). - --)

[3] Shepp, L. A. and Logan, B. F., The Fourier reconstruction of a head section. IEEE Trans. Nucl. Sci. NS-21, 21-43 (1974). ----

[4] Tanaka, E. and Iinuma, T. A., Correction functions and statistical noises in transverse section picture reconstruction, Comput. Biol. Med. 5, 295-306 (1976).

[5] Kwoh, Y. S., Reed, I. S. and Truong, T. K., A generalized Iwl-filter for 3-D reconstruction, IEEE Trans. Nucl. Sci. NS-24, 1990-1998 (1977). ----

[6] Gilbert, B. K., Schwartaw, W. K., Chu, A., and Beistad, R. D., Implementation of computation-Intensive reconstruction algorithms for x-ray computed tomography, IEEE Trans. Pattern Analysis and Machine Intelligence (submitted 1979).

[7] R. M. Lewitt, State University of New York, Buffalo, New York. Personal communication.

[8] Gilbert, B. K., Krueger, L. M., Chu, A., Ritman, E. L., Swartzlander, E. E., Jr., and Atkins, D. E., Application of optimized parallel processing digital computer and numerical approximation methods to the ultra high-speed three-dimensional reconstruction of the intact thorax, Int. J. Biomed. Camp. 10 --~ - (1979). (Inpress).

[9] Kenue, S. K. and Greenleaf, J. F., High-speed convolving kernels having triangular spectra and/or binary values, IEEE Trans. Nucl. -- - Sci. NS-26, 2693-2696 (1979).

[lo] Lewitt, R. M., Aspects of the Convolution Method for Image Recon- struction from Projections, in Application of Optical Instrumen- tation in Medicine VII, J. Gray, A. Haus, WrR. Hendee, and R. Rossi,&., SPIE 173 @PIE, Redondo, California, 1979). -- (In press).

[ll] Herman, G. T., Lakshminarayanan, A. V., and Naparstek, A., Convolution Reconstruction Techniques for Divergent Beams, s. Biol. @., 5, 259-271 (1976).

[12] Lewitt, R. M., Ultra-fast convolution approximations for computerized tomography. IEEE Trans. Nucl. Sci. NS-26, 2678-2681 (1979). ----

[13] Greenleaf, J. F., Johnson, S. A., Lee, S. L., Herman, G. T., and Wood, E. H., Algebraic Reconstruction of Spatial Distributions of Acoustic Absorption Within Tissue from Their Two-dimensional Acbustic Projec- tions, in Acoustical Holography, P. S. Green, ed., Vol. 5, pp. 591-603 (Plenum Press, New York, 1974).

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Efficient convolution kernels for computerized tomography

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