  In optical diffraction tomography (ODT), the three-dimensional scattering
potential of a microscopic object rotating around its center is recovered by a
series of illuminations with coherent light. Reconstruction algorithms such as
the filtered backpropagation require knowledge of the complex-valued wave at
the measurement plane, whereas often only intensities, i.e., phaseless
measurements, are available in practice.
  We propose a new reconstruction approach for ODT with unknown phase
information based on three key ingredients. First, the light propagation is
modeled using Born's approximation enabling us to use the Fourier diffraction
theorem. Second, we stabilize the inversion of the non-uniform discrete Fourier
transform via total variation regularization utilizing a primal-dual iteration,
which also yields a novel numerical inversion formula for ODT with known phase.
The third ingredient is a hybrid input-output scheme. We achieved convincing
numerical results, which indicate that ODT with phaseless data is possible. The
so-obtained 2D and 3D reconstructions are even comparable to the ones with
known phase.
