Higher-order Reconstruction Method of Differential Phase Shift

Biological soft tissues encountered in clinical and pre-clinical imaging are mainly composed of atoms of light elements with low atomic numbers, and its elemental composition is nearly uniform with little density variations. The x-ray attenuation contrast is relatively poor, and cannot achieve satisfactory sensitivity and specificity [1,2]. In contrast, x-ray phase-contrast provides a new mechanism for soft tissues imaging. The x-ray phase shift of soft tissues is about a thousand times greater than that of absorption within the diagnostic x-ray energy range, achieving a greater signal-to-noise ratio than attenuation contrast images. Thus, phase-contrast imaging can reveal detailed structural variation in soft tissues, offering a high contrast resolution between healthy and malignant tissues [3][4][5]. Moreover, the phase-contrast imaging does not intrinsically rely on x-ray absorption in tissues. With increasing x-ray energy, photoelectric absorption of tissues decreases as 3 1 E , while tissue phase shift decreases much slower only as 1 E . Phase imaging would significant reduce the deposited dose in tissues comparing to absorption contrast imaging within hard x-ray spectral range [6].

As it is well known, the x-ray intensity variation is acquired by a detector, while the phase shift of x-ray passing through an object cannot be measured directly. X-ray Talbot interferometry has recently been proposed as a novel x-ray phase imaging method to efficiently extract quantitative differential phase information from the Moiré pattern using a fringe scanning technique [4,7]. However, the data acquisition procedure is quite time-consuming, resulting in an increasing radiation dose. The gratings with large sizes and high slit aspects are difficult to fabricate and model, especially, the analyzer absorption grating consists of Au pillars encased in epoxy and bounded using a frame. This fabrication process is prone to errors and hard to control in the case of gratings of large sizes and high aspects. Xray phase contrast is formed from the propagation of wave field in free space after interaction with the object, creating a quantitative correspondence between the object and the recorded images, which can be used to retrieve the phase shift induced by the object. Several phase retrieval methods in the Fresnel diffraction regime are proposed, such as transport of intensity equation (TIE) method [8,9], the contrast transfer function (CTF) [10], a mixed approach between the CTF and TIE [11], and a general theoretical formalism for the in-line phase-contrast imaging [12]. In this paper, we propose a novel method to reconstruct x-ray differential phase shift of an object. The reconstruction method keeps a higher-order accuracy comparing to the linearized approximation models, helping overcome the limitation in the weak absorption scenario and the restriction of slow phase variation assumed in the conventional phase reconstruction techniques. Using several intensity measurements in Fresnel diffraction region eliminate the nonlinear terms and simplify the phase retrieval to a linear model with respect to the differential phase shift.

The coherent x-ray-tissue interaction causes the x-ray wave phase change because of both x-ray diffraction and refraction effects. The amount of the phase change is determined by the dielectric susceptibility, or equivalently, by the refractive index of the tissue. The refractive index of x-ray is a complex form:

, where the parameters  and  are the refractive index decrement and the absorption index, respectively.  characterizes the x-ray phase shift, while  is related to the attenuation properties. It has been shown that  values (10 -6 -10 -8 ) is about 1000 times greater than  (10 -9 -10 -11 ) in the biological soft tissue over the 10 keV-100 keV range [13]. This implies that tremendous improvement can be achieved in terms of the sensitivity of x-ray imaging to biological soft tissues if x-ray phase information is utilized. When an object is illuminated by a partially coherent xray beam, the wave-object interaction can be described as a transmittance function,

where r denotes the transverse coordinates  

where λ is the x-ray wavelength.

Due to phase shifts caused by an object, the wavefront of x-ray beam propagation is deformed.

According to the Fresnel diffraction theory, the relationship between wave amplitudes in the transverse plane is described by the Fresnel transformation formula,

where

and  denotes convolution over the transverse coordinates r . From Eq. (3), the Fourier transform of a Fresnel diffraction pattern can be written as [10],

where  

A r is the intensity distribution on the object plane. Because the hard x-ray wavelengths are very short in biomedical imaging and the near field setting, we have:

Taking the second-order approximation of the phase term, Eq. ( 4) at the image plane can be approximated by Fresnel diffraction region, which corresponds to fou

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