Fourier reconstruction for diffraction tomography of an object rotated into arbitrary orientations

Abstract

Inverse Problems 37 (2021) 115002 C Kirisits et al a Fourier diffraction theorem and derive novel backpropagation formulae for the reconstruction of the scattering potential, which depends on the refractive index distribution inside the object, taking its complicated motion into account. This provides the basis for solving the ODT problem with an efficient non-uniform discrete Fourier transform.

Total variation-based reconstruction and phase retrieval for diffraction tomography

“…Equation ( 17) is a linear equation system that may be solved in the least-square sense by applying the iterative conjugate gradient (CG) method. This approach has been successfully applied in ODT [25,45] as well as in X-ray imaging [58], magnetic resonance imaging [48], or spherical tomography [38]. We consider the overdetermined case M N d´1 ě K d .…”

“…In this section, we describe the setup in optical diffraction tomography (ODT), cf. [45]. We want to reconstruct an object's scattering potential f : R d Ñ R, where we restrict our considerations to the practically meaningful dimensions d P t2, 3u.…”

“…In general, the identity (7) only holds in a distributional sense [45]. Geometrically, the set thpy 1 q : y 1 2 ă k 0 u is a semisphere or semicircle, whose center is on the negative x d axis, and which passes through the origin.…”

“…For this purpose, we first restrict the total field u tot t or, more precisely, the scattered field u sca t to the cube r´L M , L M s d´1 and then discretize the fields using N P 2N samples for each spatial direction and M P N time steps, cf. [45]. On the uniforms grids…”

“…The respective Fourier space coverage Y in (8) has the form of a horn torus and is depicted in Figure 6c, cf. [45]. The so-called missing cone around the x 2 axis in the Fourier space coverage makes the reconstruction more difficult and promotes certain artifacts, like horizontal stripes at the top and bottom in Figure 7.…”