The statistical reconstruction of lattice models of real porous media is one of the basic engineering problems in the theory of porous structure and has a variety of applications in the study of transport in mineral processing, and in material characterization. A systematic analysis of the reconstruction problem of porous media is presented, as well as a closed-form solution. The solution is presented in the form of a linear filter acting on Gaussian processes by means of a superposition of elementary correlated processes with prescribed correlation properties, and in the form of a memoryless process recalling the theory of Khinchin on the properties of correlation functions. The connection of this approach with models of correlated percolation is also discussed.