∂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.
3.2 Propagation-Based Phase-Contrast Methods
Of all the developed phase-contrast methods, propagation-based phase-contrast imaging or in-line phase-contrast imaging, sets the least amount of requirements for the imaging setup as it does not require the use of any additional optical devices. The method was introduced by A. Snigirev in 1995 on a synchrotron source and since then, the method has been shown to be applicable also on conventional polychromatic X-ray sources. [14, 45, 54] The method is based on the Fresnel diffraction phenomenon, which results in the formation of an holographic image. As the X-ray beam interacts with the sample, some of the incident photons are refracted. The refracted photons interfere at some distance from the sample, which results in the formation of the hologram. [14, 54]
The setup required for in-line phase-contrast imaging is the most simple of all the available phase-contrast imaging methods. The setup, depicted in Figure 5, consists of the X-ray source, the object to be imaged and the detector. For phase-contrast to be visible, the distance from the object to detector needs to be sufficient, greater than what is required for conventional absorption imaging [57]. In-line phase-imaging only requires spatial coherence from the X-ray beam, with chromatic coherence not being an essential requirement [44]. The spatial coherence lengthlcoh is defined as
lcoh= λR1
s , (14)
whereR1 is the distance from the source to the object andsis the focal spot size of the X-ray source [7, 14, 57]. In order for phase-contrast to be visible, the lateral coherence length needs to be larger than the detail imaged [7]. As it can be seen from Equation (14), the coherence length decreases with increasing energy for a particular setup. Therefore increasing the distanceR1 or decreasing X-ray energy will result in better phase contrast.
Figure 5: Imaging setup for in-line phase-contrast imaging, consisting of the X-ray source, object and the detector. W1 and W2 are the wavefronts between the source and object and between the object and the detector, respectively. R1 and R2 are the distances between the source and object and between object and detector, respectively. The interference of the X-rays results in the intensity profileI at the detector. Modified from [57]
In order to reconstruct a tomographic image, the phase information in the hologram needs to be understood. A basic laboratory µCT typically reconstructs the tomographic image using only the intensity profile of the hologram and a conventional reconstruction algorithm.
As phase changes occur mainly at boundaries, phase images without phase retrieval enhance the boundaries of the sample [7]. For a higher quality reconstruction, the phase information of the hologram can be used to reconstruct a map of the refractive index in a process called phase retrieval. The theoretical background has been formulated by Pogany, Gao and Wilkins and they used Fourier optics in their formulation. [44] They proposed that the attenuation and phase shift of an X-ray wave can be described by a 2D transmission functionT(x, y)
T(x, y) =eiϕ(x,y)−µ(x,y)2 , (15) whereϕ(x, y) and µ(x, y) are the phase and linear attenuation coefficient of the object for a wave propagating in the z-direction [14, 44, 54]. When a spherical plane wave travels through the sample and arrives at the detector from distancezfrom the source, the diffracted X-ray f(x, y;z), under the paraxial Fresnel diffraction theory, is given by
f(x, y;z) = i λzeikz
∫ ∫
T(x0, y0)ei2zk[(x−x0)2+(y−y0)2], (16) where k = 2π/λ is the wave number and (x0, y0) are the coordinates on the object plane [14, 54]. The calculation of the integral is very difficult and in the first formulation of the theory, Pogany et al were only able to solve for samples that are weakly attenuating [µ(x, y) ≪ 1] and weak phase objects [|ϕ(x, y)| ≪1] [44]. Using these approximations, the transmission function can be approximated by [14, 44, 54]
T(x, y)≈1 +iϕ(x, y)−µ(x, y)
2 . (17)
Using this, the intensity at the image plane is obtained as I(x, y, z)≈1 +λz
2π∇2ϕ(x, y), (18)
where∇2 is the Laplacian operator defined as ∇2 =∇ · ∇ and ∇= (∂x∂
1, . . . ,∂x∂
n) [14, 44].
As a result, the contrast in the hologram is formed by the intensity changes due to the distortions in the wave front. Additionally, as the intensity is related to the Laplace of the phase shift, propagation-based phase contrast imaging is typically used for edge enhancement as the second derivative measures, in this case, how the rate of rate of change of the refractive index changes.
Since these approximations are derived from the assumptions that µ(x, y) ≪ 1 and
|ϕ(x, y)| ≪ 1, the equations above are not valid in many cases and cannot be used to ac-curately reconstruct a tomographic image [14, 54]. As a result, a new formalism has been introduced as a basis to deal with attenuation and the phase-shift by Gureyev et al. [23].
The new formalism is based on the transport of intensity equation (TIE), which is derived from the paraxial wave equation approximation. The TIE links the phase to the derivative of the intensity along the direction of the wave propagation. The TIE is given by
∂I(x, y, z)
∂z =− λ 2π∇ ·[
I(x, y, z)∇ϕ(x, y, z)]
(19) where the ∇operator is with respect to x and y. [23, 48, 54] For the TIE to be valid, the paraxial approximation needs to hold but in addition, Gureyev et al. had to make a near field approximation
whereI(x, y,0) is the intensity at the object plane. Using these assumption it was determined that Equation (21) is based on the TIE and it is valid when the near-field criterion is met. As with Equation (18), the assumption made do not hold in all cases and as a result, additional formalisms have been proposed for phase retrieval and reconstruction of the tomographic image. [23, 54] Many of these methods rely on the TIE as a basis but other methods aim to better model the Fresnel diffraction of the X-ray wavefronts.
Paganin et al. used the TIE as a basis for the derivation of a new method to obtain the intensity and phase information generated by a homogeneous object of interest. The intensity and phase at the object exit surface (z= 0) are related to the projected thickness, T(x, y), of the object on the plane in which the image is taken by
I(x, y, z = 0) =Iine−µT(x,y) (22) ϕ(x, y, z= 0) =−2π
λ δT(x, y), (23)
where Iin is the incident intensity. These two equations are then substituted into the TIE, Equation (19), which is then simplified and estimated. The intensities at z=0 and z = R2 are then represented as Fourier integrals and can be used to obtain the projected thickness T(x, y) after a few steps. From Equations (22)-(23), the intensity and phase can be obtained at the exit surface of the object. [39] This formalism was originally developed for X-ray microscopy used to image in the soft X-ray region but has been recently implemented in
various attempts at phase retrieval using the TIE [24, 29, 58]. Gureyev etal. combined the use of Fourier optics and the TIE for phase retrieval. This combination still results in a linear technique, beneficial especially for computation purposes. They were also able to reconstruct images with higher quality using the combined formalism compared to reconstructions using either Fourier optics or TIE. [25]
Wu etal. developed a new formalism for in-line phase contrast imaging using the Fresnel diffraction theory. The formalism was derived for spherical waves, both in the real space and frequency space. A brief description of the formalism follows. First, the spherical X-ray wavefront arriving at a location on the detector plane is written as a Fresnel-Kirchoff integral, which is then related to the intensity at the detector plane. Next, the Fourier transform is taken of the intensity after which a general formula for in-line phase-contrast imaging is obtained. During the derivation of the formalism, it is assumed that the phase varies only moderately in a small distance, which is reasonable especially in clinical X-ray imaging, as the X-ray wavelengths are short. The general formula can also be simplified depending on the application and in addition, the results of Pogany etal. were obtained as a special case of the new formalism. The intensity in real space can be found by using the convolution theorem of the Fourier space and the retrieval of attenuation information is also possible. [54] In another study Wu etal. derived a phase retrieval algorithm based on the phase-attenuation duality. The duality is a result of both the phase and attenuation being determined by the electron density of the sample assuming that the X-ray energy is between 60-150 keV. Wu etal. tested their phase retrieval using computer simulations and obtained enhanced image contrast. [55]
Chen et al. proposed a new formalism to improve contrast in phase-contrast images, in which the projections have been obtained using a source with a finite size and non-uniform distribution. The theory is based on the Huygens-Fresnel principle and the Fresnel-Kirchhoff integral. Chen etal. were able to validate the method experimentally. [13]
A completely different approach to the reconstruction of the phase image was presented by A. Bronnikov. The method does not require a phase retrieval process and the reconstruction of the refractive index distribution is done directly from the intensity distribution measured from a single plane in the near-field region. The theory behind the method is based on the relationship between the 3D Radon transform of the object function and the 2D Radon transform of the intensity distributions. [8] The original theory has been extended and implemented by others [9, 18, 21, 22, 29].
Arhatari et al. obtained similar expressions as A. Bronnikov with more general con-siderations [1, 8, 9]. The phase retrieval and reconstruction algorithm was developed for weakly absorbing, homogeneous objects and it is based on the use of the TIE. The benefit of the algorithm is that only one tomographic data set in the Fresnel region is required for reconstruction. This is because some Fourier space operations, done both in phase retrieval and FBP reconstruction, can be combined. The process has added benefits as it limits the radiation dose and the amount of data acquired is smaller, increasing computation efficiency.
[1] In another study, Arhatari etal. developed a phase retrieval method using a laboratory X-ray source with a large bandwidth. This method is also based on the TIE. [2]
3.3 Interferometric Phase-Contrast Methods