A Fully Discrete Theory for Linear Osmosis Filtering

Abstract

Osmosis filters are based on drift–diffusion processes. They offer nontrivial steady states with a number of interesting applications. In this paper we present a fully discrete theory for linear osmosis filtering that follows the structure of Weickert’s discrete framework for diffusion filters. It regards the positive initial image as a vector and expresses its evolution in terms of iterative matrix–vector multiplications. The matrix differs from its diffusion counterpart by the fact that it is unsymmetric. We assume that it satisfies four properties: vanishing column sums, nonnegativity, irreducibility, and positive diagonal elements. Then the resulting filter class preserves the average grey value and the positivity of the solution. Using the Perron–Frobenius theory we prove that the process converges to the unique eigenvector of the iteration matrix that is positive and has the same average grey value as the initial image. We show that our theory is directly applicable to explicit and implicit finite difference discretisations. We establish a stability condition for the explicit scheme, and we prove that the implicit scheme is absolutely stable. Both schemes converge to a steady state that solves the discrete elliptic equation. This steady state can be reached efficiently when the implicit scheme is equipped with a BiCGStab solver.