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.

We develop a general theory of spatially-variant (SV) mathematical morphology for binary images in the Euclidean space. The basic SV morphological operators (i.e., SV erosion, SV dilation, SV opening and SV closing) are defined. We demonstrate the ubiquity of SV morphological operators by providing a SV kernel representation of increasing operators. The latter representation is a generalization of Matheron's representation theorem of increasing and translation-invariant operators. The SV kernel representation is redundant, in the sense that a smaller subset of the SV kernel is sufficient for the representation of increasing operators. We provide sufficient conditions for the existence of the minimal basis representation in terms of upper-semi-continuity in the hit-or-miss topology. The latter minimal basis representation is a generalization of Maragos' minimal basis representation for increasing and translation-invariant operators. Moreover, we investigate the upper-semi-continuity property of the basic SV morphological operators. Several examples are used to demonstrate that the theory of spatially-variant mathematical morphology provides a general framework for the unification of various morphological schemes based on spatiallyvariant geometrical structuring elements (e.g., circular, affine and motion morphology). Simulation results illustrate the theory of the proposed spatially-variant morphological framework and show its potential power in various image processing applications.

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.

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.

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.

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.>

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.