  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.
