Sampling the Sinogram in Computed Tomography

Nothing selected for Back to Basics is a proper "cup of tea" for all our regular readers. But I promised a wider variety in coverage of the newer techniques. The Back to Basics article deals with some interesting basics of computed tomography (CT). These basics are much more mathematical than the first Back to Basics on CT (by Frank A. Iddings) in the November 1982 issue. One of these two articles may be your cup of tea for CT. Frank A. Iddings Tutorial Projects Editor

Introduction
Computed tomography (CT) is a nondestructive testing technique, using X-rays or g-rays, that allows an image of the mass density distribution within a cross section, or slice, of an object to be reconstructed (Kak and Slaney, 1988). In the simplest form of CT scanning, the first generation arrangement, a finely collimated beam of radiation is passed through the object in the desired slice plane and the attenuation is measured (Wells et al., 1994).

Figure 1 - Schematic of the CT scanning process.

The attenuation is the natural logarithm of the ratio of the incident beam intensity to the transmitted beam intensity, and is also referred to as a ray-sum. A set of ray-sums, a projection, is obtained by stepping the beam across the object at an interval of Dx. The process is then repeated and a set of projections is acquired as the X-ray beam is passed through the object at different angles. The angular increment between projections is denoted by Df . The total angular range over which these projections are obtained may be restricted to 180 degrees as the same result for a ray-sum is obtained along a given path through the object regardless of which one of the two directions is chosen, although this is only strictly true when using mono-energetic radiation. A schematic of a first generation CT scanning arrangement is shown in Figure 1.

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Some CT practitioners today are still unaware of Rattey and Lindgen's results.

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A reconstructed image of the 2D distribution of the linear attenuation coefficient, µ(x,y), is obtained from these projections, and for the purposes of this article it is sufficient to note that the linear attenuation coefficient is approximately proportional to the mass density. What is not commonly understood is the requirement regarding the relation between the translational and angular increments, Dx and Df, in order to provide an image with a given spatial resolution using the smallest number of ray-sum measurements. In the authors' experience, more measurements are taken than are necessary for the spatial resolution required. As many industrial CT instruments operate in first generation mode, and the data acquisition phase can be very time consuming, it is important to understand correctly the sampling process involved. This paper explains the manner in which first generation CT data should be acquired by going back to the basics of the 2D sampling procedure as originally presented by Rattey and Lindgren (1981). In their paper they state "It is significant to note, however, that to the authors' knowledge, this standard signal processing approach has not been applied to the tomographic imaging problem." It appears that some CT practitioners today are still unaware of Rattey and Lindgren's results.

Data Acquisition
In a CT scan, if the ray-sums could be acquired with an infinitely narrow X-ray beam and the angular increment was vanishingly small the result would be a continuous set of projections, each comprising a continuous set of ray-sums. Displayed as a 2D function this is referred to as the sinogram, or the Radon transform of µ(x,y). One axis of the sinogram is the translational position, x_r, that varies from -R to R, where R is the radius of the object circle that fully encompasses the object and is formed by the rotating X-ray source/detector arrangement. The object circle is included in Figure 1. The other axis is often labeled f, but here it is identified as the angular distance around the object circle, y_f= fR, varying from 0 to 2pR. However, noting the comment above about the identical value of the ray-sum in passing through the object in either direction, the range of y_f may be restricted to the interval 0 to pR. The full 360 degree sinogram of the object in Figure 1 is shown in Figure 2, illustrating these points.

Figure 1 - Schematic of the CT scanning process.

Figure 2 - The sinogram of the object shown in Figure 1.

CT scanning is all about sampling the continuous sinogram. One measure of the spatial resolution, d, in the final reconstructed image is to use d = Dx, while the width of the collimated X-ray beam (or X-ray detector aperture) is denoted by w. Furthermore, it is assumed that, in looking at the sinogram of Figure 2, the spatial resolution in the x_r and y_f directions will be the same; the sampling will be carried out on a square grid with Dx= Dy. A first look at the problem would then suggest that N_x = 2R/d ray-sums per projection should be measured for each of N_f = pR/d projections. This leads to the commonly accepted criterion that Nf » pN_x /2, yielding a total number of data points from a square sampling array of N_square = 2pR²/d².

The only parameter left to determine is the width, w, of the X-ray beam. The 1D sampling theorem (Jackson, 1990) states that in order to correctly sample continuous 1D information the sampling interval, Dx, must be such that 1/Dx ³ 2X_max, where X_max is the largest spatial frequency associated with the physical parameter being measured in the object cross-section - in this instance, µ. In practice X_max is not known; one does not really have any idea about the variations of the linear attenuation coefficient within an object. The way around this problem is to realize that the sampling condition only pertains to a situation where the samples are strictly point samples, while the use of an X-ray beam and/or a detector aperture of width w brings in another factor that helps us over this difficulty and allows the sampling theorem criterion to be met.

Figure 3 - The 1D aperture function of width w (left) and its Fourier transform (right).

A beam of width w can be represented by a 1D rectangular function as shown in Figure 3. Its Fourier transform, also shown in Figure 3, provides information about the manner in which the detector accepts spatial frequency detail from the object function µ(x,y) carried by the attenuated X-ray beam. It can be seen that in the spatial frequency domain W(X), a sinc function, has its first zeros at X = ±1/w. Outside this frequency interval, the spatial frequencies accepted by the detector are markedly attenuated and may be ignored. This use of the finite width X-ray beam (or detector aperture) effectively band-limits the experimental data to the spatial frequency range X = -1/w to +1/w and overcomes the problem about not knowing the spatial frequencies of the original attenuation function. All spatial frequency detail in µ(x,y) greater than | X | = 1/w is effectively lost during the CT scan; the information acquired is band-limited and the sampling theorem is satisfied if, in direct space, Dx £ w/2. This means that by setting d = Dx = w/2, the total number of sample points can be expressed as N_square = 8p R²/w².

The only problem now is that one ends up acquiring twice as many samples as is really needed, and this can be very costly in data sampling time and image reconstruction time. This is explained in the following sections.

Thinking in Two Dimensions
The sinogram as noted above is a 2D function, and the sampling of the sinogram in a CT scan should be considered from this point of view, rather than thinking of it as a set of 1D projections which is commonly done. What has the sampling theorem to say about 2D sampling? If the limits, the boundary, of the spatial frequencies in all directions in a 2D space can be defined, a bounding shape referred to here as S(X,Y) with an origin at (X,Y) = (0,0) is specified. The sampling theorem in 2D simply requires that a periodic array of this shape can be formed without the adjacent shapes overlapping. The centers of these non-overlapping shapes then define a lattice of points in spatial frequency space, and the inverse Fourier transform of this 2D infinite lattice provides the required sampling pattern in direct space. If, within the periodic array, these shapes are made to overlap, then aliasing occurs and high spatial frequency detail can get "folded" into the reconstructed image as low spatial frequency artifacts. A simple example will demonstrate these ideas.

In viewing a 2D scene with an optical instrument that has a circular aperture of diameter w, the shape S(X,Y) that bounds the allowed spatial frequencies, at least out to the first zeros in 2D spatial frequency space, is a circle of radius X_max = 1/w. These circles can be most efficiently packed together, without overlapping, on a hexagonal grid as shown in Figure 4. The corresponding lattice made from the center points of these circles can then be inverse Fourier transformed to arrive at the optimum sampling pattern in direct space (see for example Kittel, 1971). The result, not surprisingly, is another hexagonal lattice with Dx = w/Ö3, Dy = w/2 and this is also shown in Figure 4. This hexagonal sampling arrangement is about 13 percent more efficient than using a sampling scheme based upon a square grid with a sampling step size of Dx = Dy = w/2. This may explain why nature has, in many instances, evolved animal eyes where the visual receptors on the retina are arranged in an approximately hexagonal fashion, rather than on a square array.

Figure 4 - Close-packed circles of radius 1/w in spatial frequency space (left) and the corresponding sampling grid in direct space (right).

The Fourier Transform of a Sinogram
To decide on the optimum sampling pattern of the sinogram during a CT scan seems, at first sight, a daunting problem as knowledge of the Fourier transform of the sinogram of the object is required. However, it turns out to be relatively straightforward as it is not the full Fourier transform that is needed (this will obviously be different for different objects), but rather the bounding shape S(X,Y) in spatial frequency space that encloses all the spatial frequencies of the sinogram for any object.

First, in the X-direction, the width of the X-ray beam or detector aperture defines the band-limiting value of the shape S(X,Y) as | X_max | = 1/w. In the Y-direction the task is more complicated, but progress is made by noting that everything done in a CT experiment and image reconstruction relies on the fact that the steps involved are mathematically linear processes. This means that if the Fourier transform of one possible sinogram (that derived from a very simple object) can be determined, then almost certainly the Fourier transform of the sinogram of any object is likely to be defined, at least as far as the extent of the spatial frequencies are concerned, and that is all that is actually required. The simplest object is a small cylindrical one of uniform density, smaller than any likely sampling interval that might be chosen. This object and its sinogram are depicted in Figure 5, and in this instance, the object is on the object circle for a reason that will become evident.

Figure 5 - A small attenuating object close to the object circle (left) and its sinogram over the range y_f from 0 to pR (right).

Figure 6 - Fourier transform of the sinogram of Figure 5

Figure 6 shows the Fourier transform of the sinogram of Figure 5. The bounding shape S(X,Y) is probably best described as a bow-tie, and this term was used by Rattey and Lindgren (1981) who were the first to look at sampling the sinogram in a CT scan in this manner. As long as N_x and N_f are significantly larger than unity, R » Dx, the bow-tie outline makes an angle of ±45 degrees to the X and Y axes as indicated.

There is a straightforward property of the Fourier transform that states that if the direct space function is compressed then its Fourier transform expands correspondingly, and vice versa. Now consider another small, uniformly dense object, but this time placed closer to the center of rotation and away from the object circle. The sinogram is compressed in the x-direction compared to that shown in Figure 5, so the spatial frequencies expand in the x-direction and are bounded by a bow-tie shape that makes a smaller angle than ±45 degrees to the x-axis. However, these spatial frequencies are still band-limited by the aperture function of the beam/detector. The conclusion is that the 45 degree, 1/w band-limited bow-tie is the required shape inside of which almost all the spatial frequencies of any object must lie. This follows from the linear property of the attenuation function as any object scanned in a CT system can be considered as a weighted, distributed, collection of these fine "point" objects lying within the object circle.

The Sampling Scheme For CT
The remaining task is to see how to pack these bow-ties most efficiently in spatial frequency space. To achieve this, take the resulting lattice of center points of these shapes and Fourier transform to find the best sampling scheme for a CT scan. Bow-tie shapes clearly pack very well together as shown in Figure 7, and the infinite 2D lattice of points generated using the centers of these bow-ties may be described by two primitive translation vectors V₁ = (2/w)i and V₂ = (1/w)(i + j), where i and j are unit vectors in the X and Y directions, respectively. The inverse Fourier transform of this lattice provides another lattice in direct sinogram space described by primitive vectors v₁ = (w/2)(i - j) and v₂ = wj. The resulting set of sample points, also shown in Figure 7, is a non-regular hexagonal lattice with sample points Dx = w apart within a projection, and a spacing of Dy = w/2 between projections.

Figure 7 - Close-packed bow-ties in spatial frequency space (left) and the corresponding sampling grid in direct sinogram space (right). The generating primitive vectors, V₁, V₂, and _V1, _V2 are shown.

 

This result indicates that it is adequate to sample the ray-sums in each projection a distance w (and not w/2) apart, without any loss of spatial resolution, as long as adjacent projections have their ray-sums sampled in an "interleaved" manner. Thus the total number of samples required is N_hex = 4pR²/w², which is half the value quoted previously for N_square. This affords a tremendous saving on data acquisition time in a first generation CT scan, and also a saving on image reconstruction time. But in practice does it actually yield the same spatial resolution?

Experiments and simulations have been performed using both square (w/2 x w/2) and hexagonal (w x w/2) sampling as described above. In each experiment two images were obtained from 8pR²/w² and 4pR²/w2 data points using the square and hexagonal sampling respectively. For these two images the root mean square difference in pixel grey level is typically less than 2 percent per pixel indicating that the saving from hexagonal sampling is well worth it and comes with virtually no loss in spatial resolution. As there are exactly half the number of ray-sums per projection in the hexagonal sampling case the image reconstruction, performed using summation convolution backprojection, takes less time as well. There is, however, a small cost in programming complexity as the interleaved sampling between adjacent projections makes the backprojection procedure slightly less straightforward.

An additional point to note is that in the hexagonal sampling scheme the sampling interval is the same as the X-ray beam width, or detector aperture width. In first generation CT scanning this has no further implications, but in higher generation scanners, detector arrays are used with multiple detection elements where the sampling interval coincides with the element aperture width (assuming w is much greater than the separation between the detection elements).

The above analysis then suggests that maybe some conventional higher generation CT scanners are not making the best use of the available data if they are not using some scheme that approximates hexagonal sampling, but are using a scheme equivalent to square sampling. It seems that one ought to get the maximum spatial resolution from the number of samples acquired as the CT scanning and image reconstruction are expensive enough in time and dollars without unnecessarily throwing resolution away.

References
Jackson, L.B., Signal, Systems and Transforms, 1990, p 171. Addison-Wesley, Reading, MA.

Kak, A.C., and M. Slaney, Principles of Computerized Tomographic Imaging, 1988. IEEE Press, Piscataway, NJ.

Kittel, C., Introduction to Solid State Physics, 4th ed., 1971, p 66. John Wiley and Sons, New York, NY.

Rattey, P.A., and A.G. Lindgren, "Sampling the 2-D Radon Transform," IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. ASSP-29, No. 5, 1981, pp 994-1002.

Wells, P., J. Davis, and M. Morgan, "Computed Tomography," Materials Forum, Vol. 18, 1994, pp 111-133.

 

* Physics Dept., Monash University, Clayton, Victoria 3168, Australia; [61] (3) 9905-3642; e-mail peter.wells@sci.monash.edu.au.

 

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