A new Bayesian tomographic method for soft-X and bolometer diagnostic systems has been developed. The method is non-parametric in the sense of using Gaussian processes to model ,the underlying emissivity distribution, and the regularization of such a model becomes defined by a multivariate normal distribution at the points where the emissivity distributions should be evaluated. As opposed to currently used methods, e.g. Maximum entropy (MaxEnt) and Equilibrium-Based Iterative Tomography Algorithm (EBITA), to which this method is compared, this method is fully analytical, involving no nonlinear iterations, and so can be feasible for real-time applications. Additionally, uncertainties of the solution, accounting for both measurement uncertainties and ambiguities due to insufficient coverage of sight lines, can be provided by direct sampling from the posterior probability distribution. Describing the emissivity distribution by Gaussian processes has the further advantage that regularization can be expressed in a natural way as correlation length scales of a diffusion process. In particular, the method can locally adapt the length scales to the varying smoothness of the emissivity distribution. This method has been applied to three different experiments: soft-X reconstructions for W7-AS stellarator, and bolometer reconstructions for the WEGA stellarator and the JET tokamak, comparing favourably to currently used methods.