Partial differential equations and finite difference methods in image processing--Part II: Image restoration

Abstract

Application of Partial Differential Equation (PDE) models for restoration of noisy images is considered. The hyperbolic, parabolic, and elliptic classes of PDE's yield recursive, semirecursive, and nonrecursive filtering algorithms. The two-dimensional recursive filter is equivalent to solving two sets of filtering equations, one along the horizontal direction and other along the vertical direction. The semirecursive filter can be implemented by first transforming the image data along one of its dimensions, say Column, and then recursive filtering along each row independently. The nonrecursive filter leads to Fourier domain Wiener filtering type transform domain algorithm. Comparisons of the different PDE model filters are made by implementing them on actual image data. Performances of these filters are also compared with Fourier Wiener filtering and spatial averaging methods. Performance bounds based on PDE model theory are calculated and implementation tradeoffs of different algorithms are discussed.