In this paper, we study the mathematical imaging problem of optical diffraction tomography (ODT) for the scenario of a microscopic rigid particle rotating in a trap created, for instance, by acoustic or optical forces. Under the influence of the inhomogeneous forces the particle carries out a time-dependent smooth, but complicated motion described by a set of affine transformations. The rotation of the particle enables one to record optical images from a wide range of angles, which largely eliminates the "missing cone problem" in optics. This advantage, however, comes at the price that the rotation axis in this scenario is not fixed, but continuously undergoes some variations, and that the rotation angles are not equally spaced, which is in contrast to standard tomographic reconstruction assumptions. In the present work, we assume that the time-dependent motion parameters are known, and that the particle's scattering potential is compatible with making the first order Born or Rytov approximation. We prove a Fourier diffraction theorem and derive novel backprojection 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.

翻译：在本文中,我们研究了光学分解成像学(ODT)的数学成像问题,这是在微小的硬质粒子在声学或光学力量等创造的陷阱中旋转的情况。在不相容的力量的影响下,粒子进行着一种时间依赖的平稳但复杂的运动,由一系列的形形变所描述。粒子的旋转使得人们能够从一系列广泛的角度记录光学图像,这在很大程度上消除了光学中的“漏线问题 ” 。然而,这一优势在于,这一情景的旋转轴不是固定的,而是不断经历一些变异,而旋转角度也不是同样有空间的,这与标准的成形变形重建假设形成对比。在目前的工作中,我们假设根据时间的运动参数已知度,粒子的散落潜力与第一个顺序相容或Rytov近。我们证明了一个Fourier diffem 的分解,并得出了用于再散射的公式,而后向的公式是一些变形的,而这种变形角度与标准的成形变形法则取决于其内部的变形变形法。