Morphological representation of nonlinear filters

Abstract

Abstract. Morphological operators provide very efficient algorithms for signal (image) processing. The efficiency of morphological operators has been captured by using them as approximations of nonlinear operators in numerous applications (e.g., image restoration). Our approach to the approximation of nonlinear operators is the construction of morphological bounds on them. We present a general theory on the morphological bounds on nonlinear operators, propose conditions for the existence of these bounds, and derive several fundamental morphological bounds. We also derive morphological bounds on iterations of nonlinear operators, which are superior to the original nonlinear operator in some applications. Because obtaining the results of the convergence of iterations of a nonlinear operator is often particularly desirable, we provide morphological bounds on the convergence of such iterations, and propose conditions for their convergence based on morphological properties. Finally, we propose several criteria for the morphological characterization of roots of nonlinear operators.

Composing morphological filters

“…2.1]. It is generally accepted that complete lattices provide a natural framework for the study of morphological operators [2], [6], [8], [9], [11] [12]. A comprehensive treatment of the complete lattice framework for morphology is given in [4].…”