title: Higher-order phase shift reconstruction approach.
abstract: OBJECTIVE
  Biological soft tissues encountered in clinical and preclinical imaging mainly consists of atoms of light elements with low atomic numbers and their elemental composition is nearly uniform with little density variation. Hence, x-ray attenuation contrast is relatively poor and cannot achieve satisfactory sensitivity and specificity. In contrast, x-ray phase-contrast provides a new mechanism for soft tissue imaging. The x-ray phase shift of soft tissues is about a thousand times greater than the x-ray absorption over the diagnostic x-ray energy range, yielding a higher signal-to-noise ratio than the attenuation contrast counterpart. Thus, phase-contrast imaging is a promising technique to reveal detailed structural variation in soft tissues, offering a high contrast resolution between healthy and malignant tissues. Here the authors develop a novel phase retrieval method to reconstruct the phase image on the object plane from the intensity measurements. The reconstructed phase image is a projection of the phase shift induced by an object and serves as input to reconstruct the 3D refractive index distribution inside the object using a tomographic reconstruction algorithm. Such x-ray refractive index images can reveal structural features in soft tissues, with excellent resolution differentiating healthy and malignant tissues.
  
  
  METHODS
  A novel phase retrieval approach is proposed to reconstruct an x-ray phase image of an object based on the paraxial Fresnel-Kirchhoff diffraction theory. A primary advantage of the authors' approach is higher-order accuracy over that with the conventional linear approximation models, relaxing the current restriction of slow phase variation. The nonlinear terms in the autocorrelation equation of the Fresnel diffraction pattern are eliminated using intensity images measured at different distances in the Fresnel diffraction region, simplifying the phase reconstruction to a linear inverse problem. Numerical experiments are performed to demonstrate the accuracy and stability of the proposed approach.
  
  
  RESULTS
  The proposed reconstruction formula is a generalization of the transport of intensity equation (TIE). It has the second-order accuracy compared to the linear model used in the conventional phase retrieval approach. The numerical experiments demonstrate that the accuracy and stability of the proposed phase reconstruction method outperforms the TIE-based reconstruction method.
  
  
  CONCLUSIONS
  A novel approach has been proposed to retrieve an x-ray phase shift image induced by an object from intensity images measured at different distances in the Fresnel diffraction region. The authors' approach has the second-order accuracy and is able to retrieve the phase shift of an object stably, overcoming the restriction of slow phase variation assumed by the conventional phase retrieval techniques.
authors.affiliation: Biomedical Imaging Division, School of Biomedical Engineering and Sciences, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061, USA. congw@vt.edu
authors.forename: Wenxiang
authors.initials: W
authors.lastname: Cong
authors.forename: Ge
authors.initials: G
authors.lastname: Wang
mesh_terms: D000465:Algorithms; D055592:Biophysical Phenomena; D006801:Humans; D007091:Image Processing, Computer-Assisted / Q000706:statistics & numerical data*; D021621:Imaging, Three-Dimensional / Q000706:statistics & numerical data; D016014:Linear Models; D011857:Radiographic Image Interpretation, Computer-Assisted / Q000379:methods; D014961:X-Ray Diffraction