Nuclear Instruments and Methods in Physics Research A242 (1986) 603-609

North-Holland. Amsterdam

Section VIII. Tomography

RESOLUTION, ARTIFACTS AND THE DESIGN OF COMPUTED TOMOGRAPHY

Terence TAYLOR and Lawrence R. LUPTON

Atomic Enrr,g of Canuda Lml~ed, Chalk River Nuclear Laboratories, Chulk Ricer. Ontario. C’amda KOJ IJO

The factors that limit the fidelity of a computed tomography image have been demonstrated by a series of imaging experiments.

The observed spatial resolution. contrast resolution and artifacts are. for the most part, consistent with existing theories.

1. Introduction

The fidelity of a computed tomography (CT) image is very sensitive to the data acquisition and image reconstruction procedures [l]. A thorough appreciation of the many factors that limit the quality of the raw data and the accuracy of the reconstruction is essential to the development of an optimized CT system. In the present work. the influence of the most significant parameters is reviewed from a theoretical perspective and demonstrated by a series of imaging experiments.

2. Procedure

Images were generated with a first generation CT system [2] comprised of a ‘s’Cs radioisotopic source, two collimators of continuously variable width, a NaI(T1) scintillation detector and a computer-controlled stage that rotated and translated a test object through the gamma-ray beam. The test object was a 50 mm diameter brass disk having seven sets of three holes of various sizes with centre-to-centre spacings equal to two diame- ters. The diameters of the holes were 0.50, 0.75, 1.0. 1.5. 2.0, 3.0 and 4.0 mm.

Except for the single parameter that was varied in a given imaging experiment, the data acquisition and image reconstruction parameters were as follows: The full-width at half-maximum (fwhm) of the gamma-ray beam was 1.0 mm at the centre-of-rotation of the sys- tem; The projections were sampled at 128 discrete points separated by 0.50 mm; One hundred and eighty projec- tions were collected at 1.0’ intervals spanning a 180” arc; A total of 6.0 X 10’ photons contributed to each image; The images were reconstructed by filtered back- projection [3,4] of linearly interpolated projections with a Ramachandran-Lakshminarayanan (Ram-Lak) filter [5]; The width of the pixels was 0.5 mm.

To emphasize the noise and the artifacts, the images were displayed using a linear grey scale with upper and

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SYSTEMS

lower thresholds set at 110% and 90% of the attenuation of brass respectively.

As a measure of the spatial resolution, the contrast transfer function (CTF) was plotted versus the spatial frequency for each image. (The CTF for a spatial frequency f is given by the relative amplitude of the oscillations observed in the attenuation profile that passes through the set of holes with centre-to-centre spacing f- ’ as shown at the lower left of fig. 1.) In

addition, to obtain an indication of the contrast resolu- tion, the standard deviation from the mean attenuation was calculated for 349 pixels occupying a featureless region of the image.

Fig. 1. A typical image of the test object is shown at the upper left. The horizontal and vertical lines locate the attenuation

profiles shown at the upper right and the lower left respec-

tively.

0168-9002/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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604 T. Trr)lor, L. R. Lupton / Computed tomogruph~~ systems

3. Results

3. I. Limitations imposed by the data acquisition procedure

3. I. 1. Beam profile The profile of the interrogating photon beam places

a fundamental constraint on the spatial resolution. Two small features separated by less than the fwhm of the beam cannot be distinguished from a single large fea- ture no matter how frequently the projections are sam- pled [6].

Fig. 2 shows images obtained with beams having fwhm at the centre-of-rotation of 0.50. 1.0 and 2.0 mm. (It should be noted that the fwhm of the beam does not correspond, in general, to the collimator width [7]. In fact, for a system with a uniformly distributed source, a detector of uniform sensitivity and identical rectangular collimators at both the source and the detector, the beam profile at the midpoint of the system is triangular with a fwhm equal to half the width of the collimators.) The quantitative evaluation of these three images is summarized in fig. 3. Increasing the beam width from 1.0 to 2.0 mm almost halved the frequency correspond- ing to a given CTF. (i.e. The size of the smallest set of resolved holes increased by a factor of two.) On the other hand, reducing the beam width from 1.0 to 0.50 mm improved the spatial resolution by less than one half. In this case the CTF was limited by the sample spacing and the pixel size as discussed in subsects. 3.1.2 and 3.2.1.

The standard deviation from the mean attenuation did not change significantly with the beam width. (The count rate obviously varied with the width of the col- limators, however, the counting time was adjusted so that equal numbers of photons contributed to each image. Had the counting time been held constant, the noise would have improved by approximately a factor of two for each factor of two increase in the beam width.)

5 - L

I I I I I I

2 4 6 8 10 FRECLUENCY (cm-1)

Fig. 3. Contrast transfer functions and relative standard devia-

tions, 0. for photon beams having fwhm of 0.50, 1.0 and 2.0

mm.

3.1.2. Sample spacing The Nyquist theorem implies that the sampling

frequency should be twice the highest frequency that is to be faithfully reproduced [6,8]. The highest spatial frequency in a projection is limited to the inverse of the fwhm of the beam. Therefore, the projections should be sampled at least twice per fwhm [6,7] (i.e. for the geometry used here, approximately four times per col- limator width).

Images produced with sample spacings of 0.25, 0.50 and 1.0 mm are shown in fig. 4. A sample spacing equal to the 1.0 mm beam width generated inferior spatial resolution and the Moirt: pattern characteristic of inade- quate sampling. However, oversampling at a sample spacing of 0.25 mm did not enhance the image noticea-

a

Fig. 2. Images obtained with photon beams having fwhm of (a) 0.50 mm, (b) 1.0 mm, and (c) 2.0 mm.

a

T. Tuylor, L. R. Lupton / Computed tomogruphy systems

b C

605

Fig. 4. Images measured at sample spacings of (a) 0.25 mm. (b) 0.50 mm and (c) 1.0 mm corresponding to sampling frequencies of 4.

2 and 1 mm-’ respectively.

bly. These observations are confirmed by the CTF shown

in fig. 5. Theoretical arguments suggest that the standard de-

viation from the mean attenuation should vary as the three-halves power of the sampling frequency, assuming that the total number of photons contributing to the image is held constant [3,4,9]. Increasing the sampling frequency from 2 to 4 mm-’ did indeed increase the standard deviation by almost the factor of 2.8 as ex- pected. The noise for a sampling frequency of 1 mm-‘, however, was slightly higher than anticipated, probably because the standard deviation included fluctuations associated with the Moire artifact.

+. l/4 mm [(r= 9.2%)

I I I I I I J 2 L 6 0 10

FREQUENCY (cm-’ 1

Fig. 5. Contrast transfer functions and relative standard devia- tions, 6, for sample spacings of 0.25, 0.50 and 1.0 mm corre-

sponding to sampling frequencies of 4. 2 and 1 mm-’ respec-

tively

3.1.3. Nunlher of projections The requisite number of projections can be deduced

from either of two separate arguments. The first, that the number of independent measurements must at least equal the number of free parameters in the image (i.e. the number of pixels), leads one to conclude that the number of projections must be at least n/4 times the number of samples per projection [3]. The second, that streak artifacts emanating from sharp-edged high-con- trast objects located at the edge of the image must be totally suppressed. implies that the minimum number of projections is 7r/2 times the number of samples per projection [6].

Fig. 6 shows images obtained with 180, 90 and 45 projections (i.e. approximately r/2, a/4 and VT/~ times the number of samples per projection). The anticipated artifactual streaks are clearly visible in the 45 projection image. Ninety projections appear to be sufficient to suppress the streak artifacts in this case. However, the test object used here probably does not provide a very stringent test of the system’s sensitivity to streaking. Although one can argue that, for most cases of practical interest, 71/4 is an adequate ratio, the only advantage of fewer projections is a reduction in the reconstruction time. Provided that the total number of photons con- tributing to the image was held constant, neither the contrast resolution nor the spatial resolution depended on the number of projections. (The analysis is not presented here.)

3.1.4. Angular distribution of projections Under idea1 conditions the projections need only

span 180”; measurements made between 180” and 360” are redundant. However, if the beam is asymmetric with respect to the centre-of-rotation, or if the subject is in motion, or if the centre-of-rotation of the system has been misidentified, then measuring projections over a

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a b

Fig. 6. Images generated by (a) 180 projections, (b) 90 projections, and (c) 45 projections.

a

Fig. 7. Images obtained (a) with the CT system properly aligned. (b) with the centre-of-rotation misaligned by 0.50 mm and the

projections spanning 180”, and (c) with the CT system misaligned and the projections spanning 360”.

full 360” can suppress the artifacts that would otherwise arise [6].

Three illustrative images are shown in fig. 7. In this demonstration, misidentifying the centre-of-rotation by 0.50 mm generated the so called tuning fork artifact (i.e. positive streaks on one side of a feature and negative streaks on the other). The steaks were eliminated by including projections measured between 180” and 360” However, the quantitative analysis of these images (see fig. 8) shows that the price paid for artifact suppression was a significant degradation of the spatial resolution (even though, for an aligned system. the CTF corre- sponding to the 180” and 360” data sets were virtually identical). The contrast resolution was, of course, unaf- fected.

+ ALIGNEO 180° In z 3 8%)

+ ALIGNED 360' lo = 3 6%1

06 t

OL

t 02

0 L Rather than risk an undetected misalignment, pro-

jections should probably be measured over 180’ only. The system can then be adjusted as required.

I I I I I I

2 i b 8 10

FREQUENCY (cm-’ ) 3. I. 5. Number of photons

On theoretical grounds, the noise should vary with Fig. 8. Contrast transfer functions and relative standard devia- the square root of the number of gamma-rays contrib- tions, (r, for projections spanning 180” and 360” with the CT

uting to the image, assuming that the data are limited system aligned and misaligned.

T Taylor, L. R. Lupton / Computed tomogruphy system.r

b C

607

Fig. 9. Images generated with pixel width of (a) 0.25 mm, (b) 0.50, and (c) 1.0 mm

by quantum fluctuations [3,4,9]. The spatial resolution should, of course, be independent of the photon count. These suppositions have been confirmed experimentally. (Neither the images nor the analysis are presented here).

3.1.6. Discrete measurement versus continuous measure-

ment The images shown thus far were obtained by in-

tegrating the signal from the detector for a given time at a fixed position then moving to the next position. However, high-speed CT systems must be continuously in motion, inevitably blurring the image somewhat. The differences between images obtained by these two tech- niques were imperceptible, provided that the sample spacing did not exceed half the beam width. (Neither the images nor the analysis are presented here.)

3.2. Limitations imposed by image reconstruction proce- dure

3.2. I. Pixel size

To ensure that two small features can be resolved, they must be separated by at least two pixel widths. Thus, the pixel width should be no greater than half the fwhm of the beam (i.e. less than or equal to the sample spacing) [7].

Fig. 9 shows images reconstructed on pixels with widths of 0.25, 0.50 and 1.00 mm. A substantial loss of spatial resolution at a pixel width of 1.0 mm was confirmed by the CTF shown in fig. 10. On the other hand, the spatial resolution was not improved signifi- cantly be decreasing the pixel width to 0.25 mm. Fur- thermore, increased correlations between adjacent pixels give the 0.25 mm image a “wormy” texture.

The contrast resolution was not affected by changes in pixel size.

0.8 -

0.6 -

0.4 -

0.2 -

0 -

t I I I I I

2 & 6 8 10

FREQUENCY (cm-‘)

Fig. 10. Contrast transfer functions and relative standard devia- tions, 0, for pixel widths of 0.25, 0.50 and 1.0 mm.

05

- RAM-LAK

‘.‘.” SHEPP-LOGAN

---HANN

0.1 02 03 OL OS

FRECAUENCY Fig. 11. Magnitude of Ram-Lak, Shepp-Logan and Hann filters as a function of relative spatial frequency.

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608 T. Tuylor, L.R. Lupton / Computed tomogruphy systems

3.2.2. Reconstruction filter

Three reconstruction filters are in common use. The Ram-Lak filter [S] is essentially the ramp function that is required to eliminate the blurring inherent to back- projection [3]. The Shepp-Logan filter [4] slightly sup- presses the high spatial frequencies, whereas the Hann filter [8] emphasizes the intermediate spatial frequen- cies. Fig. 11 depicts these three filters in the frequency domain.

Images generated with the Ram-Lak, ShepppLogan and Hann filters are shown in fig. 12 while the corre- sponding analysis is summarized in fig. 13. In accor- dance with theoretical predictions [9], the Hann filter attenuated the noise by about a factor of three at the expense of a significant degradation of the CTF. The Shepp-Logan filter provided a compromise between the extremes of the high spatial resolution Ram-Lak filter and the high contrast resolution Hann filter. One can- not categorically state that a given filter is superior to all others. If large low-contrast features are of interest, the Hann filter is preferable. However, the Ram-Lak filter enhances detectability for small high-contrast fea- tures.

4. Discussion

Although all of the fundamental data acquisition and image reconstruction parameters have now been dealt with, the present study was incomplete in several respects. Firstly, the experiments were conducted with a first generation CT system equipped with a monoen- ergetic source. Systems having multiple detectors and/or X-ray tubes with continuous spectra may be susceptible to geometric and/or beam hardening artifacts not encountered in this work [6]. However, all of the factors considered here apply equally to higher generation sys- tems, provided that the raw data are rebinned. Sec- ondly, artifacts that are inherent in the physics of CT

+- RAM-LAK (u = 38%)

SHEPP-LOGAN b= 3.0%

I I I 1 I 2 1 6 8 10

FREQUENCY (cm-‘)

Fig. 13. Contrast transfer function and relative standard devia-

tions, O. for the Ram-Lak. Shepp-Logan and Hann filters.

have not been touched on. For example, streaks emanat- ing from the straight edges of high contrast features are an inevitable consequence of the finite width of the photon beam [6]. Thirdly, a CT system is more than an imaging device, it is also a precision instrument for measuring photon attenuation coefficients. The factors that influence the accuracy of these measurements have been discussed extensively elsewhere [l].

Nevertheless, a number of firm guidelines for the design of an optimal CT system have been deduced. The fwhm of the photon beam should be equal to the desired resolution. The projections should be sampled twice per fwhm. The projections should number n/2 times the number of samples per projection and should span 180” (unless either the photon beam is asymmetric

a b

Fig. 12. Images generated by (a) the Ram-Lak filter, (b) the Shepp-Logan filter, and (c) the Hann filter.

T Taylor, L. R. Lupton / Computed tomography systems 609

with respect to the centre-of-rotation or the subject is in [2] L.M. Zatz, ref. [l], ch. 116.

motion in which case projections should be measured [3] R.A. Brooks and G. Di Chiro, Phys. Med. Biol. 21 (1976)

over a full 360”). The pixel width should be equal to the sample spacing. The Ram-Lak filter is preferred for small high-contrast features and the Hann filter for large low-contrast features while the Shepp-Logan filter provides a useful compromise.

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151

[61 [71

References

[I] See e.g. Radiology of the Skull and Brain, vol. 5, Technical Aspects of Computed Tomography, eds., T.H. Newton and

D.G. Potts (C.V. Mosby, St. Louis, Toronto and London,

1981).

ISI

[91

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(1974) 21.

G.N. Ramachandran and A.V. Lakshminarayanan, Proc.

Nat. Acad. Sci. US 68 (1971) 2236.

P.M. Joseph, ref. [l], ch. 114.

M.V. Yester and G.T. Barnes, Proc. SPIE, Appl. Opt. Instr.

in Med. VI 127 (1977) 296.

R.K. Otnes and L. Enochson, Digital Time Series Analysis

(Wiley, New York, London, Sydney and Toronto, 1972). D.A. Chesler, S.J. Riederer and N.J. Pelt, J. Comput.

Assist. Tomogr. 1 (1977) 64.

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