Bayesian Epipolar Geometry Estimation from Tomographic Projections

In this paper, we first show that the affine epipolar geometry can be estimated by identifying the common 1D projection from a pair of tomographic parallel projection images and the 1D affine transform between the common 1D projections. To our knowledge, the link between the common 1D projections and the affine epipolar geometry has been unknown previously; and in contrast to the traditional methods of estimating the epipolar geometry, no point correspondences are required. Using these properties, we then propose a Bayesian method for estimating the affine epipolar geometry, where we apply a Gaussian model for the noise and non-informative priors for the nuisance parameters. We derive an analytic form for the marginal posterior distribution, where the nuisance parameters are integrated out. The marginal posterior is sampled by a hybrid Gibbs–Metropolis–Hastings sampler and the conditional mean and the covariance over the posterior are evaluated on the homogeneous manifold of affine fundamental matrices. We obtained promising results with synthetic 3D Shepp–Logan phantom as well as with real cryo-electron microscope projections.