Computed tomography (CT) is a development of x-radiography that dates back to the 1920s. The basic idea behind tomography is to present a 2-D image of a 2-D cross-section, plane, or slice, through a 3-D object. This is what distinguishes CT from radiography, where in radiography the final 2-D image is of a 3-D object. Radiography loses "depth" information; CT attempts to overcome this.
CT has gone by various names since the pioneering medical diagnostic instrument work of Godfrey (later Sir sGodfrey) Hounsfield; computed axial tomography, or computer assisted tomography (CAT) is the most common. However, CT is the usual acronym used nowadays. Hounsfield worked for the UK recording company EMI (strange place to develop medical instruments!) and the first medical CT scanner was built in 1971. This was the so-called 1st generation scanner. Within a short time the 2nd generation scanner was built by EMI, the CT1010 scanner, that during the 1970s was able to provide two adjacent 2-D cross-sectional images of a patient's head in about 5 minutes. Each image was of a section through the patient of a slice about 5-10 mm thick. Modern medical scanners are the so called multi-slice helical/spiral/volume CT instruments that are based on the 3rd or 4th generation designs. The generation design will be mentioned a little later in this article.
In 1979 Hounsfield and Allan Cormack (a South African mathematician) were awarded the Nobel Prize in Medicine/Physiology in 1979.
In November 1895 Wilhelm Conrad Roentgen (also a Nobel Prize recipient in Physics in 1901) discovered x-rays in his laboratory in Wurtzberg. Within a year or two x-ray film radiographs s as we know them today were routinely taken in hospitals in the USA, UK and Europe. The use of film radiography as an industrial tool for non-destructive testing (NDT) soon followed.
Figure 1: Wilhelm Roentgen.
By the 1920s the major limitation of plane film radiography (that of acquiring a 2-D image from a 3-D patient/object) was being seriously addressed. In 1920 Bocage (Figure 2) developed the idea of moving the x-ray source (tube) and a film about the object in such a way as to keep "in focus" only one plane within the object, with the other regions of the object being "blurred". This was partially successful and was referred to as classical tomography and led to a range of techniques including laminography. About the same time an Austrian mathematician, Johann Radon published a paper on a method of reconstructing information by acquiring partial information along multiple paths (at the time this was a purely mathematical exercise and Radon had no knowledge of how this might eventually lead to CT as we know it today).
Figure 2: André Bocage.
In the 1960s Ron Bracewell, an Australian electrical engineer used Radon's idea to produce an image derived from radio-astronomy signals. As we shall see, CT is a computationally intensive process and with the advent of the digital computer the earlier ideas of Radon, Bocage and Bracewell (and many others) could be developed and lead to the first CT prototype of Hounsfield.
Figure 3 shows the essential difference between radiography and CT. Notice that in radiographic procedure the x-rays are basically along a direction perpendicular to the imaging plane, while in CT the x-rays are parallel to this plane.
Figure 3: The essential difference between radiography and CT.
The basic physics that underpins CT is precisely the same as that for radiography. In both instances the simplest way to start is to consider a very fine beam, a pencil beam, of x-rays. If this fine beam passes through an object the incident beam direction defines the ray-path as illustrated in Figure 4. As the x-ray photons (the "bundles" of x-ray energy) pass along this ray-path there is a probability that these x-rays will either be absorbed (by photoelectric absorption) or scattered (by Compton scattering). These are the 2 main interactions with x-rays (or gamma-rays) that have energies in the range 10 keV to 1 MeV. This energy range covers the normal diagnostic film radiography/mammography regime from about 20 to 100 keV, the diagnostic CT x-rays energies of about 80 keV to 150 keV, and the nuclear medicine region of about 100 keV to 600 keV. The industrial use of x-rays and gamma-rays also involves photons in the energy range 100 keV to about 1 MeV, or higher.
Figure 4: X-ray intensity decrease on transmission.
Above 1 MeV, used in radio-therapy as well as some industrial NDT, there is an additional photon-material interaction called pair-production where the incident x-rays interact with the atomic nuclei. All these interactions decrease the intensity of the x-ray beam by scatter and/or absorption. If we take the incident x-ray intensity as I0, then after transmission through a thickness t of material the intensity of the beam continuing in the direction of the incident beam decreases through these absorption and scattering processes, and is denoted by It.
The attenuation of the beam is indicated by the ratio of It/I0, where the word attenuation is used to cover the intensity loss by both absorption and scattering. This x-ray attenuation is a function of three things; the energy of the x-ray photons, E, the atomic number of the material, Z, and the density of the material, r. As a general rule the relative intensity of the transmitted beam decreases (the attenuation increases) for increasing Z and increasing density, but increases (the attenuation decreases) as the x-ray photon energy is increased. The higher the photon energy the more translucent the material becomes. The fraction of attenuation due to absorption compared to scatter increases with increasing atomic number.
A measure of the attenuation in material is the linear attenuation coefficient, m. There is a relatively simple equation that relates the intensity change over a thickness t of uniform (homogeneous) material as
It = I0 e(-μt)
where the "e" denotes the exponential function. The transmitted intensity, It, is always less than I0 and this follows from the negative sign in the above equation. For example, if t = 10 cm and for the material m = 0.2 cm-1 (m has units of inverse length) we can see that
It/I0 = e(0.2×10) = e-2=0.8
This tells us that only 80% of the initial bean intensity survives the passage through 10 cm of this material, in the direction of the incident beam. The product mt, a dimensionless quantity, is sometimes referred to as the attenuation.
Radiography
In radiography you can think of the diverging beam from the x-ray tube or gamma-ray source as being made up of many narrow beams, each being attenuated as they pass through the object under study, so that a reduced intensities of all these beams finally exposes the film and provides the necessary brightness and contrast.
There are, however, a few complications as in some applications (and certainly in all medical diagnostic cases) the x-rays come from an x-ray source where the x-rays have a range of energies; a spectrum. This is not a problem in many industrial situations where the radiation source in mono-energetic (a line spectrum); a Cs-137 isotope of 662 keV or a Co-60 source of about 1.2 MeV. Another common industrial source is the short-lived isotope Ir-192 (with a half-life of 74 days) and the gamma-ray photon energies involved does complicate things with E ranging from about 150 keV to over 600 keV. The complication arises as the linear attenuation coefficient is a function of energy, m = m(E, Z, r), so the simple attenuation equations above do not work quite so straightforwardly. In these cases we can always define an effective x-ray/gamma-ray energy, E (a sort of average across the energy spectrum) and an effective average linear attenuation coefficient, m.
The other obvious complication, and the whole reason for undertaking radiographic NDT in the first place, is that the material is not uniform with different regions of varying thickness and different linear attenuation coefficients (due to different Z values), and different densities. This makes the simple equation It = I0 e(-μt) more complicated as the bit in the square brackets, the attenuation, now has to be a sum of the products mt over all the separate regions and thickness of the object from the source side to the film side. For the mathematically inclined, the expression in the square brackets is an integral of m(t) as a function of t through the full thickness of the material.
Again, notice that in radiography, the resultant transmitted intensity for one ray-path is dependent on the attenuation (absorption and scattering) properties of the inhomogeneous material and this is where the depth information is lost, or at best scrambled. A simple example in Figure 5 will illustrate that the same x-ray intensity change will occur through the 3 different objects, all with the same overall attenuation.
Figure 5: Three "objects" with the same overall attenuation; It is 0.67% of I0.
The NDT radiographic film image tells us about variations in the object by changes in the exposure of the film, the different intensities that reach the film along a multitude of ray-paths. A region of uniform film grey-ness (film optical density) does not indicate that the material is necessarily homogeneous; it just indicates that, on average through the material, there is no difference in overall attenuation. This problem is less likely to be a concern for thin sections, but for thicker objects the problem is compounded. For thick objects you will need higher x-ray/gamma-ray energies to get sufficient transmitted intensity in order to expose the film in a reasonable time. As you go up in energy the relative difference between the linear attenuation coefficients of low Z and high Z material regions becomes less, so the contrast diminishes (see Figure 6). As a general rule, if you can afford the time, use as low a photon energy as possible to maximise the contrast.
Figure 6: A hand radiograph at 50 kV (left) and 120 kV (right).
For photon energies above about 150 keV, so certainly in most industrial NDT applications, the linear attenuation coefficient used above is best written as the product of the mass attenuation coefficient, mm and the density, r, as m = mm r. The value in doing this is that mm is fairly constant for almost all materials of industrial/technological interest at a given photon energy. While mm still decreases as we increase the photon energy, E, at one energy, for example, mm for Al is very close to that for Fe, Cu, etc. This can make life a little easier in interpreting radiographic film, and certainly helps with CT images as we shall see. Roughly speaking, a CT image is a map of the density distribution.
The dependence of mm on energy for air (an average based on 80% N plus 20% O) and lead is illustrated in Figure 7.
Figure 7: Photon energy dependence of the mass attenuation coefficient (Graham, 1999).
Here PE, CS and PP stand for photoelectric, Compton scatter and pair-production respectively.
Notice that these 2 plots also demonstrate that at a particular energy the mass attenuation generally
decreases with increasing photon energy, at least up to about 21 MeV where pair-production kicks in.
Spatial resolution is all about being able to distinguish fine object features (voids, cracks, de-lamination, inclusions etc.) in the resultant image. This is determined essentially by the size of the radiation source (focal spot size of the x-ray tube, for example) and the characteristic size of the detecting system used, and in film this is of the order of around 50 micron or so. As the extent of the radiation source is generally going to be around a mm or so, this is often a guide as to the ultimate spatial resolution. An extreme example is illustrated in Figure 8.
Figure 8: High (left) and low (right) spatial resolution radiographs.
Contrast and spatial resolution are also affected by "noise", and in a radiographic context this will be contributed to by scatter (within the object as one of the results of attenuation) and also by background radiation in the environment used to do the exposure. Detectors and any further electronic or image processing also add to the image noise.
In CT scanning the object has a multitude of x-ray/gamma-ray beams passed through it in the slice plane of interest. In Hounsfield's 1st generation CT scanner these were separate beams that were passed through the specimen by moving a single narrow pencil beam across the object and also rotating the object as well (Figure 9). A 1st generation CT scanner uses a fine pencil-beam of radiation along with a single detector. The source-detector arrangement is translated across the objects from air on one side to air on the other side. This forms one projection. The source-detector arrangement is then rotated by a small angle and the translate process repeated. This continues until at least a total angular interval of 180 degrees gas been covered. This is often a very slow process, but does produce excellent reconstructed images.
Figure 9: First generation, translate-rotate, CT design (Bushberg et al, 2002).
Figure 10 shows the 1st generation prototype CT scanner built by Hounsfield based on a lathe bed.
Figure 10: Godfrey Hounsfield and his prototype CT scanner (Webb, 1990).
Modern 3rd and 4th generation CT scanners use a thin fan-beam of radiation that encompasses the object and is then incident on a detector array. The fan-bream is often about 1-5 mm thick and defines the slice thickness over which the object attenuation is averaged. In 3rd generation scanners the detector array covers an arc (often around 60 degrees) of a circle, usually with the centre of the arc coincident with the photon source, and the source and detector array both rotate about the object as a single unit. In the 4th generation design there exists a complete circle of detector elements, centred about the centre of the object aperture. Here only the source rotates about the object; the circular detector ring is stationary. Both these designs are illustrated in Figure 11.
Figure 11:Third (rotate-rotate) and fourth (rotate only) CT generation designs (Bushberg et al, 2002).
X-ray tubes and linear accelerators can provide a very intense beams of radiation but do suffer from the broad spectrum of output photon energies. A modern medical diagnostic CT scanner will acquire the data for one cross-sectional image in about 0.5 - 2 seconds. Isotope sources can be used in tomography (just as in industrial radiography) but they have far less intense outputs (less brightness) and CT scans using isotope sources can take a long time to collect the data, sometimes many hours.
Whatever scheme is used, the idea is to ensure that within the plane, or cross-section, of the object there are many parallel ray-paths forming a projection at a given angle, and many projections. As we noted above, the ultimate spatial resolution is going to be determined by the source/detector element characteristic size. In CT this means the width of the effective pencil beams that strike each detector element. Typically the effective detector element width in a medical CT scanner is around 0.5 mm (maybe a little less) and this gives rise to the normal spatial resolution of around 0.4 mm. Much higher spatial resolution system do not use discrete detector elements but use CCD systems or imaging plates. The spatial resolution can always be improved using "optical" magnification techniques by having the detector a greater distance from the object than the source.
Although the few industrial CT scanners around differ widely in specifications (and they have to as they are custom made for a particular materials and objects), the number of effective parallel ray-paths, N, is just the number of detector elements, and the larger the better. It is not unusual to have N = 500 or more in a modern medical CT scanner, and in some industrial scanners there can be over 2000 effective ray-paths in a projection. The number of views or projections, M, each comprising N ray-path attenuation values, should also be large, and if you take data every 0.5 degree you will end up with 360 views in a total angular scanning range of 180 degrees.
What is actually measured and stored in the scanner computer for a given ray-path from x-ray source to detector element is the ray-sum, p. This is the natural logarithm of the ratio of the intensities, so a ray-sum is simply the measured attenuation, defined by
p = ln(I0/It)
Modern scanners using thin fan-beams actually have to scan over more than 180 degrees, and with fan angles typically about 60 degrees these days a full scan is normally taken over 360 degrees. As an example, using 720 projections we end up with M = 720 and N = 500 (say). This means that a total of about 720 x 500 = 360,000 ray-path attenuation values, ray-sums values of ln(I0/It), are acquired for a single cross-sectional CT image. In this processing the natural logarithm, the "ln", is used to make the data stored in the computer look just like the sum of [mt] across the object, rather than retain the exponential function.
It is from these 100,000s of separate ray-sum, or raw attenuation data, values that the CT image is reconstructed. Note that the extent of the object can always be determined from the ray-sums as for air there is essentially no attenuation, It = I0, and p = ln[1] = 0. Beam attenuation leads to positive (only) values for the ray-sum. Many scanners effectively take over 1 million attenuation measurements in order to reconstruct a single CT image, so you can see where the C in CT comes from.
While CT image reconstruction is not a particularly complicated process the full details go a bit beyond the scope of this brief introduction. A feel for CT image reconstruction can be gleaned by taking each projection (and you can think of a projection as a very narrow strip radiograph) and back-projecting this across a 2-D image space. This is repeated for all projections with the back-projection being done at the angle across the image space equivalent to the angle at which the projections was acquired in the first place. This sounds a bit complicated but the process can be viewed interactively at the following web-site; Physics 2000.
The end result of this back-projection process is, in fact, a somewhat blurred image (as you will se if you go to the website above), but a little clever digital image processing built into the reconstruction process will present you with a 2-D image where each pixel (picture element) represents the effective linear attenuation coefficient, m = mmr, of the small corresponding volume in the object slice plane. And the pixel truly does represent a volume, often referred to as a voxel, as the detector element determines the in-plane pixel size (and the spatial resolution) while the fan-beam "thickness" provides the other dimension. A CT image provides a 2-D image averaged over a slice thickness "cut" through the object by the x-ray beam.
As most industrial CT scanning is performed using x-rays or gamma-rays in the energy range 200 keV to 1 MeV (and more), the pixels in the CT image can be reasonably reckoned to be proportional to density, and this is why most CT images are readily interpretable to the lay-person as most of us have a feel for the density of things. Medical CT images are calibrated against water that has a density of 1 g/cm3, or 1000 kg/m3, although the pixel values in a medical CT image are given in Hounsfield units (H) or CT numbers (CT#), where very approximately
H, CT# = r (in kg/m3) - 1000
Figure 12 shows the scanning of a hardwood log on a 3rd generation Hitachi medical CT scanner in the author's laboratory. Many adjacent slices allow for a full 3-D digital image to be produced. In this instance this 3-Dc image was used to design algorithms to automatically find and classify defects and features like cracks, knots, growth rings, gum pockets and rot, as well as to track moisture content. An example of the separate CT slices and the composite 3-D image is depicted in Figure 13.
Figure 12: Scanning a hardwood log.
Figure 13: A few contiguous 2-D CT slice images (right) and the composite 3-D image (left).
The images have been pseudo-coloured to make then "look" like timber.
As with all imaging procedures things can go wrong, and lack of image quality can often be worked out from a range of typical artefacts that might arise. Some of the common ones are listed below.
speckled appearance - not enough photons counted in the transmitted beam, so increase counting time or use a more energetic photon beam.
streaks - due to inclusions/regions in the object where the linear attenuation coefficient (read density) changes dramatically from the neighbouring areas. If the inclusions are high density, use a more penetrating (higher energy) photon beam.
poor contrast - use a lower energy photon source, or get better counting statistics by recording the transmitted beam intensity over longer periods of time.
poor spatial resolution - use a finer radiation source extent (smaller spot size on the x-ray tube, for example), or more closely spaced detector elements, or both.
cupping artefact - the m values, or density values, in the image for a uniform test object are a bit lower than they should be towards the centre of the image compared to the outer regions. This may be caused by objects that have excessive scatter contributions to the attenuation. This can sometimes be overcome by special collimation on the detectors, but is more normally overcome by fudging it in the reconstruction software!
partial volume effect - the density value of a fine feature in the object is too low (because you have a test object to check this). This is because the fine feature in the object is smaller than the pixel/voxel and the image result for any pixel is the average over the full voxel volume involving material in the object of (usually) lower density than the high density feature. In the same way fine cracks may not "show" the density of air in the image if the crack is narrower then the pixel dimension.
Medical scanners are expensive, around $1 million upwards. All medical scanners effectively have the same basic specifications as they are designed to scan humans. The relatively few industrial CT scanners are often very expensive as they are all purpose-built with vastly different specifications. There is not a lot that is similar in the CT scanner that NASA uses to scan 3 m diameter rocket motors and re-entry vehicles and the scanners that have been constructed to look at the geometry and architecture of pores in oil bearing rock samples that have a spatial resolution of maybe 5 micron.
One thing that is often not a problem in industrial CT scanning is dose or time constraints. CT scanners have been constructed to image cross-sections of timber (even growing trees) and wood objects with a spatial resolution down to 10 microns, to scan archeological artefacts (wood, brass, iron rock and ceramics), metal and alloy castings with a resolution of maybe 0.1 mm upwards, nuclear waste containers, human tissue like bone at high spatial resolution to look at the way osteoporosis affects bone. There are an amazing list of objects that have been scanned using CT, but in many cases a specially built CT scanner had to be constructed. About the only things from the above list that can routinely be scanned on a medical CT scanner are wooden objects from with typical dimensions ranging from about 1 cm up to about 50 cm (the length being irrelevant as long as you can get it in the scanner room!)
The final image, Figure 14, shows the digital radiograph and typical CT slice through a small acetylene gas bottle. The internal structure of the acetylene bottle and the high and low density variations in the filler are clearly seen.
Figure 14:CT digital radiograph, scoutscan (left) and one CT slice image (right) through a small acetylene gas bottle.
Websites
Some examples of industrial CT scanners, images and applications may be found on the following websites:
The following references may also be of interest.
Other references/bibliography
ASTM, E1441-97, Standard Guide for Computed Tomography (CT), American Society for Testing and Materials, 1997.
Blanck, C., Understanding Helical Scanning, Williams & Wilkins, Baltimore, 1998.
Bryant, L.E. and McIntire, P. (eds.), Non-Destructive Testing Handbook, American Society for Nondestructive Testing: USA, Volume 3, 1985.
Bushberg, J.T., Seibert, J.A., Leidholdt, E.M. and Boone, J.M., The Essential Physics of Medical Imaging, Lippincott Williams & Wilkins: Philadelphia, 2nd edition, 2002.
Dendy, P.P. and Heaton, B., Physics for Diagnostic Radiology, Institute of Physics: Bristol, 2nd edition, 1999.
Graham, D.T., Principles of Radiological Physics, Churchill Livingstone: New York, 3rd edition, 1999.
Kak, A.C. and Slaney, M., Principles of Computerized Tomographic Imaging, IEEE Press, 1988.
Kak, A.C. and Slaney (the electronic version of the above text, 1990)
Webb, S., From The Watching of Shadows; The Origins of Radiological Tomography, Adam Hilger, Bristol, 1990.
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