Mohammed Charif-Chefchaouni

Spatially Variant Morphological Restoration and Skeleton Representation

The theory of spatially variant (SV) mathematical morphology is used to extend and analyze two important image processing applications: morphological image restoration and skeleton representation of binary images. For morphological image restoration, we propose the SV alternating sequential filters and SV median filters. We establish the relation of SV median filters to the basic SV morphological operators (i.e., SV erosions and SV dilations). For skeleton representation, we present a general framework for the SV morphological skeleton representation of binary images. We study the properties of the SV morphological skeleton representation and derive conditions for its invertibility. We also develop an algorithm for the implementation of the SV morphological skeleton representation of binary images. The latter algorithm is based on the optimal construction of the SV structuring element mapping designed to minimize the cardinality of the SV morphological skeleton representation. Experimental results show the dramatic improvement in the performance of the SV morphological restoration and SV morphological skeleton representation algorithms in comparison to their translation-invariant counterparts

Spatially-variant mathematical morphology

In this paper, we present a comprehensive theory of spatially-variant (SV) mathematical morphology. A kernel representation of increasing operators in terms of the union (resp., intersection) of SV erosions (resp., SV dilations) is provided. A representation of algebraic openings (resp., algebraic closings) in terms of the union (resp., intersection) of SV openings (resp., SV closings) is also provided.<<ETX>>

Morphological representation of order-statistics filters

We propose a comprehensive theory for the morphological bounds on order-statistics filters (and their repeated iterations). Conditions are derived for morphological openings and closings to serve as bounds (lower and upper, respectively) on order-statistics filters (and their repeated iterations). Under various assumptions, morphological open-closings and close-openings are also shown to serve as (tighter) bounds (lower and upper, respectively) on iterations of order-statistics filters. Simulations of the application of the results presented to image restoration are finally provided.

Morphological representation of nonlinear filters

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.

On the invertibility of the morphological representation of binary images

We investigate the invertibility of the morphological representation of binary images. A criteria for the invertibility of the morphological representation of binary images is proposed. Necessary and sufficient conditions for the exact reconstruction of a binary image from its morphological representation are provided.

Morphological bounds on order-statistics filters

In this paper, we investigate the morphological bounds on order- statistics (median) filters (and their repeated iterations). Conditions are derived for morphological openings and closing to serve as bounds (lower and upper, respectively) on order- statistics (median) filters (and their repeated iterations). Under various assumptions, morphological open-closings (open- close-openings) and close-openings (close-open-closings) are also shown to serve as (tighter) bounds (lower and upper, respectively) on iterations of order-statistics (median) filters. Conditions for the convergence of iterations of order-statistics (median) filters are proposed. Criteria for the morphological characterization of roots of order-statistics (median) filters are also proposed.

Morphological bounds on nonlinear filters

In this paper, we present a general theory on the morphological bounds of nonlinear filters. Conditions for the existence of various morphological bounds on nonlinear filters are proposed. Several fundamental morphological bounds on nonlinear filters are derived. Extensions of morphological bounds on the iterations of a nonlinear filter are also derived. Criteria for a root of a nonlinear filter are derived. Finally, conditions for the convergence of the iteration of a nonlinear filter are proposed.

M-Idempotent and Self-Dual Morphological Filters

In this paper, we present a comprehensive analysis of self-dual and m-idempotent operators. We refer to an operator as m-idempotent if it converges after m iterations. We focus on an important special case of the general theory of lattice morphology: spatially variant morphology, which captures the geometrical interpretation of spatially variant structuring elements. We demonstrate that every increasing self-dual morphological operator can be viewed as a morphological center. Necessary and sufficient conditions for the idempotence of morphological operators are characterized in terms of their kernel representation. We further extend our results to the representation of the kernel of m-idempotent morphological operators. We then rely on the conditions on the kernel representation derived and establish methods for the construction of m-idempotent and self-dual morphological operators. Finally, we illustrate the importance of the self-duality and m-idempotence properties by an application to speckle noise removal in radar images.

Comments on "On the invertibility of morphological representation of binary images" [with reply]

The authors comments that Charif-Chefchaouni and Schonfeld (see ibid., vol.3, no.6, p.847, 1994) investigated the invertibility of a morphological representation of binary images and determined the necessary and sufficient conditions for its inverse. The authors show that one of the derived necessary conditions is not valid. A counterexample is given to illustrate our observations. Charif-Chefchaouni and Schonfeld reply that the new sufficient condition is proposed for the invertibility of the morphological image representation. A modification of its inverse is subsequently used to derive a new necessary condition for the invertibility of the morphological image representation. A composition of these conditions is finally used to provide a new necessary and sufficient condition under some restrictions for the invertibility of the morphological image representation. These necessary and sufficient conditions form a revision of one of the necessary conditions for the invertibility of the morphological image representation stated in the original paper.

On the convergence and roots of order-statistics filters

We propose a comprehensive theory of the convergence and characterization of roots of order-statistics filters. Conditions for the convergence of iterations of order-statistics filters are proposed. Criteria for the morphological characterization of roots of order-statistics filters are also proposed. >