## 1 Introduction
The output intensity, \(I\), in monochromatic case is given by integrating the attenuation through the sample in the direction of X-ray propagation, \(z\), as follows [21]:  
\[I=I_{0}exp^{-\int\ r(z)dz} \tag{1}\]  
where \(I_{0}\) is the incident intensity. For low-density materials, such as soft tissue, the X-ray attenuation generates only relatively low contrast [22]. By comparison, in the hard X-ray regime, the phase shift is typically three orders of magnitude larger than the corresponding absorption. This has led to hard X-ray phase-contrast imaging, which utilizes refraction (i.e.,X-ray phase shifts), becoming an increasingly widespread technique for the nondestructive imaging of low-density samples [23; 24]. X-rays that pass through materials undergo a phase shift, \(\varphi=-2\pi\delta(\lambda)T/\lambda\), where \(T\) is the local sample thickness. Spatial variations of the phase in the object plane affect the intensity in the image plane at a distance z downstream from the object, and can be accurately described by the Transport of Intensity Equation (TIE) [25]:  
\[\frac{\partial I(x,y,z)}{\partial z}+\frac{1}{k}\
abla_{\perp}.(I(x,y,z) \
abla_{\perp}\varphi(x,y,z))=0 \tag{2}\]  
where \(k=2\pi/\lambda\) is the wavenumber, and \(\
abla_{\perp}\) is the 2D differential operator in the plane perpendicular to the optical axis, \(z\). The free-space propagation-based phase-contrast imaging method described by the TIE equation is often easier to implement compared to other phase-sensitive X-ray imaging techniques, due to its straightforward experimental setup, which does not require any optical elements [26; 27]. This method simply allows a spatially coherent wave field to propagate a sufficient distance away from the sample so that the diffraction fringes can be observed. Due to its simplicity, this propagation-based contrast mechanism may be readily applied to 3D CT imaging [5; 28; 29]. In propagation-based phase-contrast computed tomography (PCT), an additional step (compared to conventional absorption-based tomography) is required, during which a suitable phase-retrieval algorithm is applied to the measured intensity to retrieve the phase shifts induced by the sample. There are several approaches that are currently used to solve the phase-retrieval problem for propagation-based contrast in the measured intensity. The method chosen often depends on the imaging regime within which the data has been acquired. Phase retrieval based on the TIE [27; 30] is primarily used for quantitative phase imaging in the near-field Fresnel region. For homogeneous samples, the \"homogeneous\" TIE-based phase retrieval (\"TIE-Hom\") [31] algorithm (also known as Paganin's method) can be applied to each measured 2D propagation-based phase-contrast intensity image, as follows (note that \(z\) is fixed) [31]:  
\[T(x,y)=-\ln\left[\mathbb{F}^{-1}\bigg{(}\frac{\lambda}{4\pi\beta+z\delta \lambda u^{2}}\mathbb{F}\bigg{[}\frac{I(x,y,z)}{I_{0}}\bigg{]}\bigg{)}\right] \tag{3}\]  
where \(u=(u_{x},u_{y})\) is the Fourier conjugate of (x,y), \(\mathbb{F}\) is the Fourier transform operation, and \(\mathbb{F}^{-1}\) is the inverse Fourier transform. The TIE-Hom algorithm is particularly useful for imaging approximately homogeneous low-density objects. However, it can also be applied to objects consisting of several different components with similar X-ray attenuation properties such as breast tissue (which mainly consists of adipose and glandular tissues) [5; 32]. This is due to the fact that TIE-Hom acts as a low-pass filter, and the ratio \(\delta/\beta\) effectively determines the resultant image sharpness. Moreover, for samples containing two homogeneous materials (denoted by 1 and 2), one can justifiably use the ratio of the differences \((\delta_{1}-\delta_{2})/(\beta_{1}-\beta_{2})\), as described by [PERSON] et al. [33] and [PERSON] et al. [34]. The TIE-Hom PCT technique is practicable for biomedical CT imaging as it requires only one projection at each viewing angle to be collected at a single propagation distance.