The CT basic physical principle of X-ray attenuation is based on Lambert- Beer’s law of absorption, which is related to the attenuation coefficient. Therefore, it is possible to describe how the materials attenuate the X-ray beam. The interactions responsible for this attenuation are mainly Compton scattering (proportional 1/E) and photoelectric absorption. The contribution of the last effect depends on the effective atomic number Z and it is particularly important when low energies are employed (Z³/E³).

Beer’s law assumes a narrow X-ray beam and a monochromatic radiation. In practice, CT images are obtained with the assumption that some particular effective energy characterizes the X-ray beam as a whole and the attenuation coefficients are discretized into several element volumes. Measurements of this coefficient of numerous ray projections provide sufficient data to solve multiple equations for attenuation parameter. However, the assumption that the X-ray beam is monochromatic, according to the common models, is not realistic because the relationship between the intensity and photon flux is described by a spectral function. In Eq. (1) it is possible to observe the general relationship between the incident energy I_o(E) and the object attenuation parameter µ (x,y,z;E).

In Eq. (1) the range of integration over z covers the entire scanned object. This is the key equation for X-ray imaging via projection radiography in which I_d(x,y) is the projection image of µ (x,y,z;E) and η(E) represents the quantum efficiency of the detector at energy E.

In this sense, the more absorbent the object is, the fewer the X-ray photons detected. Because low energy X-rays are more efficiently attenuated than high energy X-rays, the distribution of energies is slanted toward higher energies, which lead to a beam hardening artifact. In other words, beam-hardening results from the preferential absorption of low-energy photons from the beam.

A major effect of beam hardening is the enhancement of the image edges. This is one of the most difficult image artifacts of CT because quantitative measurements are highly influenced by this problem due to its relation with the attenuation coefficient. In addition the same material can result in different gray levels depending on the surrounding material, which is known as the environmental density artifact, such as bone implant. Figure 2 shows examples of µCT images of several kinds of materials illustrating the mentioned beam hardening effect.

Cormack and Hounsfield received the Noble prize in 1979 for their work with CT and since then their theory is widely used although enormous amount of computation is generally required to generate CT images. Since then CT became one of the most important of all X-ray procedures worldwide.

Over the years, CT scan has been applied in the medical area with success, however, due to the relative low resolution (in the order of mm) of these images, new efforts were made to achieve better image quality. Therefore, X-ray micro-computed tomography (µCT) systems were developed with a resolution down to 1 µm using an assembly of X-ray sources and detectors to achieve that goal. These systems are commonly called industrial scanners, and are intended for the analysis of inanimate objects. They are a little different from medical equipment in some aspects. In medical CT, the X-ray source is linked to the detector, which is located at the other side of the patient. Together, source and detector are rotated around, and translate along, the patient. Unfortunately this design can promote the appearance of an artifact caused by the patient motion, with a negative effect on the image quality. Figure 3 illustrates an example of µCT radius images without (a) and with (b) patient movement. Theoretically, this motion artifact can be reduced by a faster scanning time, tube alignment or post processing of the scan. This issue does not exist in industrial systems because in this case the X-ray source and the detector remain fixed while the object is translated along its z axis to perform the scan.

Requirements for image quality improvement (adequate spatial and contrast resolutions) in the clinical sphere have led to higher patient doses, which eventually becomes a limiting factor in examinations and quality control management. Because industrial µCT does not examine living subjects, the energy of the X-ray source is high, enabling the inspection of dense materials. Furthermore, the X-ray focal spot is reduced in order to increase resolution. In fact, the improvement in the system resolution due to X-ray tube filament is assigned to the brightness factor, which is equal to the electron flux density emitted by the filament in a certain solid angle. However, obtaining small diameters requires special features, since the X-ray spot size is limited by the amount of heat generated at the anode. Therefore, good target high atomic number materials, such as titanium and tungsten are required. The smaller the X-ray spot size the greater the specific heat power for the focus area, leading to establish limits to the capacity of the target. So, the accuracy of high resolution in CT/µCT is strongly influenced by the spatial stability of the X-ray focal spot. The disadvantage of µCT scanner materials is generally the limited size of the samples, which cannot exceed some centimeters.

In this sense; Eq. (1) has to be modified because the construction of a linear model for measurements is needed. By using this model idealized measurements can be expressed as certain averages of attenuation coefficient. While the choice of coordinates is arbitrary, having a fixed reference frame is crucial for any tomography method. By convention the slices are defined by fixing the last coordinate. In this context, Eq. (1) can be rewritten more simply (Eq. 2), where L represents the trajectory of the radiation through the object, dl is its distance increment along L. Also in (2), we have the term , which is called ray sums and represents the contribution of all µ along the radiation path.

A set of ray sums over a given angle, parallel to the beam radiation, forms the projection term P. Each projection is acquired with the object (X-ray tube-detector system) rotated by an angle φ relative to the original position. So, it is possible to obtain a projection for each angle φ.

The information from transmitted X-rays is processed by a computer in order to obtain the CT/µCT images. In order to achieve this goal the theory of image reconstruction from projections is applied. In general words, the attenuation coefficient in each point (x,y) of the scanned object can be found from the projections using the inverse of the Radon transform. There are a number of alternatives to perform the reconstruction, such as the direct Fourier method or iterative approach. Currently, the most used reconstruction algorithms are based on the direct reconstruction method called filtered backprojection algorithm, which is mainly a combination of filtering and a good numerical stability. Basically, the function of the filter used describes a low pass filter that can be used to globally balance the noise and spatial resolution in the reconstruction results. The filtered back projection issue was first described in the 60s but the key theory on CT filtering reconstruction was presented on the 70s and implemented by Hounsfield, who is acknowledged as the inventor of the CT technique. A great evolution in µCT reconstruction theory was achieved when using a series of X-ray cone beam projections directly into 3D density distribution. Cone beam CT is a 3D extension of 2D fan beam CT and has the advantage of the reduction of data collection time, which is particularly important when moving structures are scanned.

The 3D data set of the scanned object is obtained by stacking contiguous 2D images. Here, the source trajectory is a circle and each horizontal row of the detector is ramp-filtered as if it was a projection of a 2D object. Then, the filtered projection data are back projected along the original rays and the middle slice is reconstructed exactly. The 2D algorithms reconstruct a slice of the scanned object. But, if volumetric data knowledge is required, the complete procedure must be performed slice by slice. Description of µCT reconstruction algorithms can be largely found in the literature.

In order to record the transmitted X-ray beam, a detection system must be used. The use of image intensifier (II) with a charge coupled device (CCD) can be found in many old CT systems. The II are closed vacuum tubes amplifying image signals. They are made of glass, aluminum or non-ferromagnetic metal, which allows the flow of electrons from the photocathode to the anode. Input and output phosphorus and electromagnetic lenses are also its constituents. Therefore, they are responsible for the conversion of X-ray photons into light signals and their diameter are generally about 23–57 cm. The function of the input phosphor is to absorb the X-rays and emit light radiation. It is typically made of cesium iodide activated with sodium screen, but can also be made of zinc-cadmium activated with copper. However the first option is better because the crystals are vertically oriented, which helps to channel the light. The electronic signal from the II is captured by the CCD and then sent to a TV monitor resulting in a representation of the radiographic image in real time. In fact, the digitalization can be performed through CCD or by direct capture of the X-ray detector with a flat panel detector.

The CCD cameras are in general composed of amorphous Si with a scintillation layer, which is basically cesium iodide. Silicon has a low X-ray absorption coefficient, which leads to a small number of photons detected by the CCD. This results in a significant quantum noise. In order to decrease this noise, it is possible to increase the dose of radiation or the quantum detection efficiency. As increasing the dose is undesirable, priority is given to increasing the quantum efficiency of radiation detectors. The quantum efficiency of the detector system can be increased by adding a scintillation layer above the CCD. X-rays are absorbed by this layer, which has a high absorption coefficient, and then converted into visible light (wavelength or near-visible).