  We study Cauchy-distributed difference priors for edge-preserving Bayesian
statistical inverse problems. On the contrary to the well-known total variation
priors, one-dimensional Cauchy priors are non-Gaussian priors also in the
discretization limit. Cauchy priors have independent and identically
distributed increments. One-dimensional Cauchy and Gaussian random walks are
special cases of L\'evy $\alpha$-stable random walks with $\alpha=1$ and
$\alpha=2$, respectively. Both random walks can be written in closed-form, and
as priors, they provide smoothing and edge-preserving properties. We briefly
discuss also continuous and discrete L\'evy $\alpha$-stable random walks, and
generalize the methodology to two-dimensional priors.
  We apply the developed algorithm to one-dimensional deconvolution and
two-dimensional X-ray tomography problems. We compute conditional mean
estimates with single-component Metropolis-Hastings and maximum a posteriori
estimates with Gauss-Newton-type optimization method. We compare the proposed
tomography reconstruction method to filtered back-projection estimate and
conditional mean estimates with Gaussian and total variation priors.
