Cone beam geometry for small objects in phase contrast tomography

Abstract

Phase contrast tomography has developed rapidly within the last ten years. The new method enables the reconstruction of the refraction index in addition to the attenuation coefficient and can therefore be very well applied to samples which are only weakly absorbing. First studies in phase contract tomography were done using synchrotron devices which are modeled by the so-called parallel geometry. Samples studied so far are special foams and fiber materials, see Cloetens et al (1999 App. Phys. Lett. 75 2912-4), which give almost no contrast due to absorption but provide excellent images in phase contrast. Recently tubes were successfully applied to a variety of applications. These laboratory devices no longer fulfil the requirement of a parallel geometry but need to be treated as a fan/cone beam geometry. In this paper we derive a mathematical model for cone beam geometry in phase contrast tomography in two and three dimensions for objects small compared to the two distances of object to detector and x-ray source to object. All approximations needed are analyzed and an efficient reconstruction method providing both phase and absorption in a single step is derived, based on the method by Louis and Maaß (1990 Inverse Problems 6 427-39). The reconstruction method is successfully tested using numerical examples with simulated phantom data.

A uniqueness result for propagation-based phase contrast imaging from a single measurement

“…of at most exponential order by theorem 4 and  α ∉ holds by assumption, equation(25) exactly matches the setting of lemma 6 with…”

“…non-divergent incident radiation, whereas state-of-the-art phase contrast imaging setups often use cone beams emanating from a point-like micro-focus x-ray source. However, we emphasize that measurements obtained with cone beam illumination can be associated with an approximate parallel beam setup via the Fresnel scaling theorem [48,49] or incorporated explicitly into the model [25].…”