Noise

Noise is present in all radiographic images and there are several sources of noise including the detector system as well as the subject of interest. Image noise degrades the image quality but cannot be completely removed. Noise resulting from the scattering of X-rays as described in Section 2.1.1, is present in all radiographic images but can be significantly reduced with the use of grids placed in front of the detector [11, 12, 49]. Another common source of noise is electronic noise, which is present in all electronic systems and is typically additive [11, 12].

If the real or interesting signal level is low and electronic noise level is high, the resulting signal consists mainly of noisy signal [11]. This degrades image quality and details in the area where signal level is low will most likely not be resolved.

Another type of electronic noise is structured noise, which is due to the electronic systems of the detector. As the detector pixels are read by their own amplifier circuits, and they cannot be tuned to match the other amplifiers, it will cause in the different detectors to have different settings resulting in structured noise. Due to the fact that the noise is constant over time, structured noise can be corrected for. Additionally, the sample to be imaged gives rise to anatomical noise, which is the structure seen on the anatomic image but are not relevant to diagnosis or are outside the area of interest. Anatomical noise does not add any useful information to the final image but can partially be removed using subtraction imaging.

CT imaging significantly reduces the presence of anatomical noise as overlapping anatomic structures can be separated. [11]

3 Phase-Contrast Imaging

A method for providing enhanced contrast to radiographic images of soft tissues and other low density samples has been the realization and use of various phase-contrast imaging techniques.

In contrast to conventional radiography, where the attenuation of X-rays is used to produce contrast, the phase information of the wave is used in phase-contrast imaging to provide contrast. The phase-contrast techniques are generally divided into three categories. The techniques vary in their experimental setup, feasibility, and complexity, but are based on the use of phase information to generate contrast in a radiographic image. [20]

3.1 Physics and Basic Principle

Phase-contrast imaging methods use the phase information of the wave to generate an image of the sample. Phase contrast arises from phase changes, which are the result of refraction of the X-rays due to the properties of the sample. As refraction takes place for all wavelengths of electromagnetic radiation, the relationships that are familiar for visible light also apply for X-ray radiation. [40]

The refraction of X-rays, far from absorption edges, in a sample can be described using a complex index of refraction

n= 1−δ−iβ, (6)

where n is the refractive index, δ is the real part of the refractive index decrement and is related to the phase shift of the wave,iis the imaginary unit, and β is the complex part of the refractive index decrement related to the absorption properties of the sample [40, 50].

The refractive index can also be expressed as

n= 1−δ+iβ (7)

depending on how the electromagnetic wave propagating in the z-direction is expressed as [14, 54]. The complex part of the refractive index β is related to the linear absorption coefficientµby

β = λ

4πµ, (8)

where λ is the X-ray wavelength [40, 50]. On the other hand, δ is related to the X-ray wavelength by

δ = r₀ρN_AZ

2πA λ² (9)

where r0 is the classical electron radius, ρ is the density of the sample material, N_A is Avogadro’s constant, A is the atomic weight of the material, and Z is the atomic number [14, 32, 40]. Additionally,β can be expressed in terms of electron densityρ_e as

β = ρeλ

4πZ(σpe+σc) (10)

whereσ_pe is the photoelectric cross section and σ_c is the Compton scattering cross sections [14]. Equation (9) can also be written in terms of electron densityρe as [14]

δ = ρer0λ²

2π . (11)

As the imaginary part of the refractive index decrement decreases faster thanE⁻⁴due to the photoelectric effect and Compton scattering, and as is seen from Equation (11) the real part of the refractive index decrement decreases only withE⁻² (E∝1/λ), significantly enhanced contrast could be achieved for soft tissues even with higher energies [7, 14]. Figure 4 depicts the ratio between the imaginary and real part of the refractive index for carbon, oxygen and hydrogen, which constitute most of soft tissue [4]. It can be clearly seen that at diagnostic energies, the ratio is still over one thousand.

Figure 4: Energy dependence of the ratioδ/β for carbon (red), oxygen (blue), and hydrogen (green).

The phase shift that a homogeneous sample introduces in an incident ray is dependent on δand the thickness of the sample. An inhomogeneous sample can be treated to be composed of many infinitesimally thick homogeneous mediums and the amount of X-ray shift introduced in the incident wave is given by

ϕ= 2π λ

∫

δ(l)dl=r₀λ

∫

ρe(l)dl, (12)

where the integral is over the path, which the ray has traveled through. [14, 46, 50, 54]

It is clearly seen from Equation (11) that the phase shift introduced in the wave is only dependent on the distribution of electrons in the sample. In contrast, from Equation (10) it is obvious that the imaginary part of the refractive index is dependent on both electron density and atomic number and as a result, the attenuation coefficient is also dependent on both variables. 3D imaging can also be applied for phase-imaging by measuring the phase shifts at various different angles around the sample and therefore, the resulting image is a depiction of the electron density distribution in the sample [14].

Measuring the phase shift is very difficult in practice and therefore other measurable quantities directly related to the phase shift are often used to determine the amount of phase

shift introduced. If the incident wave travels along the z-axis, the refraction can be observed in the xy-plane. The refraction angle Θ is directly related to the phase shift, and it can be calculated by

Θ(x, y) = λ 2π

∂

∂xϕ(x, y) (13)

This has been used in various optical applications to study properties of tissues and materials.

Due to the refraction angle being in the range of microradians, it is extremely difficult to apply the technique for X-ray applications. [14] As the measurement of phase information of X-rays is difficult in practice, various setup have been developed for the purpose, all with their own requirements. The setups differ in the requirements for stability, the components of the system and X-ray source, as well as the feasibility of the system and even the quantity that is measured.