Necessary and sufficient conditions are discussed which state when the Euler-inspired forward diffusion in a discrete space-time is a scale-space in the sense of both the total and sign variation diminishing. We emphasize that the problem is algebraic and reduces to characterization of the elements of the generalized Laplacian so that the diffusion propagators are positive definite. As a key-product, explicit analytical expressions are found for the principal minors of the frequently-applied class of tridiagonal (Jacobi) matrices. Further generalizations are outlined by introducing novel techniques of evaluating matrix determinants.

Nonlinear diffusion filtering can be improved if viewed as Bayesian Gaussian process regression. We relate the covariance functions of the diffusion process outcome to the spatial diffusion operator and show how Bayesian evidence criterion can he utilized to determine the parameters of the nonlinear diffusivity and the optimal diffusion stopping time. Computational example is given where the nonlinear diffusion filtering outperforms typical Gaussian process regression.