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When is a Discrete Diffusion a Scale-Space?

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.