3.2. Dual-Energy Projection Pre-Processing

X-ray imaging can be used to create a projection of the studied sample. The value of every pixel in such resulting projections depends on the integral absorption of the object’s matter across the corresponding trajectory. The main object and a foreign object absorb radiation differently, and this leads to differences in pixel intensities. However, the shape of the studied sample is not known in advance. Thus, the pixels in the object region do not have constant pixel intensities, as their values depend on the sample thickness. Two images acquired under different voltages provide additional information since material absorption depends on the X-ray photon energy.

A dual-energy projection of a sample with a defect can be segmented as a two-channel image. The X-ray absorption rate in a pixel depends on both material attenuation properties and the thickness of the object. The absorption rate is given by

where M(x) is the absorption rate computed for the detector pixel x, P and F are projection and flatfield pixel intensities, I0(E) is a spectrum of the X-ray tube, κ(E) is a material absorption curve, and L(x) is a profile of thickness along the ray. The argument x refers to a detector pixel, and every pixel has a corresponding X-ray beam trajectory from the source to this pixel. The absorption curve κ does not depend on x since the material is assumed to be homogeneous. If scattering is not considered, attenuation properties of the material are defined by X-ray absorption, and the attenuation rate can be calculated according to Equation (1).

If the tube spectrum is monochromatic, then I0(E)=I0δ(E−E0), where δ(x) is a Dirac delta function. Equation (1) can be simplified:

In this case, the two channels of the dual-energy image are linearly correlated. If a homogeneous material is scanned with two monochromatic beams of energies E1 and E2, the corresponding absorption rates are M1(x)=κ(E1)L(x) and M2(x)=κ(E2)L(x). The ratio between M1 and M2 is constant, does not depend on the thickness, and is defined by the ratio of attenuation coefficients for two X-ray energies. As a result, two different materials can be easily separated using a dual-energy projection.

In most CT applications, a beam is usually polychromatic since it is produced by conventional X-ray tubes. In this case, the attenuation rate depends on material thickness according to Equation (1). If the thickness L(x) is small, an effective attenuation coefficient κeff can be computed as a first-order approximation [21]:

However, the attenuation rate does not linearly depend on L(x) in general. Thus, a ratio of attenuation rates is no longer a material characteristic and it depends on the thickness L(x).

A nonlinear dependency of attenuation rate on material thickness is visible in the simulated data. For the simulation, the main object is assumed to be a skeletal muscle with an attenuation curve taken from the NIST database (ICRU-44 report [22]). The tungsten tube spectrum for the voltages of 40 and 90 kV is computed according to the TASMIP data [23]. Figure 2a shows how attenuation rates for two different voltages of the tube correspond to each other. On this plot, thickness changes from 0.1 mm to 20 cm, and the attenuation rates are calculated according to Equation (1). The correlation between the two values is almost linear as if both of them depend linearly on thickness. However, the ratio of two attenuation rates changes with thickness, as shown in Figure 2b. This change is not significant; therefore, it is not visible on a correlation plot between two intensities for different voltages. The ratio dependency on thickness can be calculated as follows:

where M1(x) is the attenuation rate computed for the voltage of 40 keV, M2(x) corresponds to 90 kV, and I1(E) and I2(E) are the tube spectra for voltages of 40 and 90 kV

Correlation between the absorption of skeletal muscle for the X-ray tube voltages of 40 and 90 kV (a). The ratio between the attenuation rate is drawn as a function of thickness. The ratio is not constant due to a polychromatic spectrum (b).

In the real scan, the nonlinear dependency is further complicated by the noise presence. A mass attenuation function is usually unknown for many food industry materials. Therefore, the thickness dependency of the ratio values cannot be predicted beforehand and should be extracted from the data. Experimental measurement produces distributions of M1(x) and M2(x) for two different tube voltages. The quotient distribution R(x) can be computed as follows:

The thickness profile L(x) is unknown. As shown in Figure 2a, attenuation rate M(x) is almost proportional to L(x). Thus, in a data-driven approach, the dependency of quotient values on thickness can be studied as a dependency of R(x) on either M1(x) or M2(x). Values of M2(x) are lower than M1(x) for the same x since the voltage of M2 is higher. Therefore, M2(x) has a lower absolute value of the error and is used as an argument in the function R(M2). The function R(M1) can be studied as well. The dependency R(M2) is further replaced by a polynomial approximation since a high noise level makes it impossible to recover true function from the data without any additional information.

The order of the polynomial chosen for a function approximation depends on the data quality. In the experimental data used in this work, a linear approximation of R(M2) is not sufficient and leads to significant discrepancies between the acquired data and the fit. High orders of the polynomial are prone to noise, and the fit does not always converge as a result. The quadratic approximation was chosen as a middle ground since it provides a sufficiently good representation of the data and has a low noise sensitivity. This approximation is given by

where M1(x) and M2(x) are pixel values in the respective channels of the experimentally acquired projection and where a, b, and c are fit coefficients. The polynomial regression is performed for all pixels of the object simultaneously.

When the dependency of R(x) on M2(x) is extracted from the data in the form of polynomial approximation, the effect of thickness dependency can be reduced. After a polynomial fit, the distribution of R′(x) can be computed as follows:

where R′(x) is a corrected quotient distribution. If the sample consists of a homogeneous material, R′(x) is close to zero regardless of the thickness. However, inclusion of a defect with different absorption properties affects both R(x) and R′(x). R′(x) is easier to use for defect detection since the form variation of the object does not significantly influence this distribution.