Automatic deconvolution in 4Pi-microscopy with variable phase

4Pi-microscopy doubles the aperture of the imaging system by coherent addition of the wavefronts for illumination and/or detection through opposing objective lenses. This improves the axial resolution 3-7 fold, but the raw data usually features ghost images which have to be removed by image reconstruction. This straightforward procedure is sometimes precluded by imperfect alignment of the instrument or a specimen with strong variations of its refractive index, because the image formation process now depends on the space-variant phase difference between the counter-propagating wavefronts. Here we present a computationally fast method of parametric blind deconvolution that allows for automatic and robust simultaneous estimation of both the object and the phase function in such cases. We verify the performance of our approach on both synthetic and real data. Because the method does not require a-priori knowledge of the phase function it is a major step towards reliable 4Pi-imaging and automatic image restoration by non-expert users.