Theoretical Foundations of Spatially-Variant Mathematical Morphology Part II: Gray-Level Images

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 30; no. 5; pp. 837 - 850
• Main Authors: Bouaynaya, N., Schonfeld, D.
• Format: Journal Article
• Language: English
• Published: Los Alamitos, CA IEEE 01.05.2008; IEEE Computer Society; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Artificial Intelligence; Color; Colorimetry - methods; Computational modeling; Computer science; control theory; systems; Computer Simulation; Euclidean space; Exact sciences and technology; Filtering; Filtering theory; Filters; Image analysis; Image Enhancement - methods; Image Interpretation, Computer-Assisted - methods; Image processing; Kernel; Kernels; Mathematical models; Mathematical morphology; Models, Theoretical; Morphological; Morphology; Nonlinear systems; Operators; Pattern Recognition, Automated - methods; Pattern recognition. Digital image processing. Computational geometry; Representations; Reproducibility of Results; Sensitivity and Specificity; Signal analysis; Signal processing; Signal Processing, Computer-Assisted; Studies; Vanadium; Morphological; Filtering; Invariant; Grey level image; Image processing; upper semicontinuous functions; Linear time; Time varying system; adaptive order- statistic filters; Behavioral analysis; Systems engineering; Operating system kernels; Euclidean space; Pattern analysis; Geometrical theory; gray-level morphology; Computer vision; Statistical analysis; Probabilistic approach; Adaptive filter; Mathematical morphology; Image analysis; Spatially variant mathematical morphology; Automatic translation; Linear filter; Signal processing; System representation; linear-time-varying systems; Artificial intelligence
• ISSN: 0162-8828; 1939-3539
• DOI: 10.1109/TPAMI.2007.70756

In this paper, we develop a spatially-variant (SV) mathematical morphology theory for gray-level signals and images in the Euclidean space. The proposed theory preserves the geometrical concept of the structuring function, which provides the foundation of classical morphology and is essential in signal and image processing applications. We define the basic SV gray-level morphological operators (that is, SV gray-level erosion, dilation, opening, and closing) and investigate their properties. We demonstrate the ubiquity of SV gray-level morphological systems by deriving a kernel representation for a large class of systems, called V-systems, in terms of the basic SV gray-level morphological operators. A V-system is defined to be a gray-level operator, which is invariant under gray-level (vertical) translations. Particular attention is focused on the class of SV flat gray-level operators. The kernel representation for increasing V-systems is a generalization of Maragos' kernel representation for increasing and translation-invariant function-processing systems. A representation of V-systems in terms of their kernel elements is established for increasing and upper semicontinuous V-systems. This representation unifies a large class of spatially-variant-linear and nonlinear systems under the same mathematical framework. The theory is used for analyzing special cases of signal and image processing systems such as SV order rank filters and ' linear-time-varying systems. Finally, simulation results show the potential power of the general theory of gray-level SV mathematical morphology in several image analysis and computer vision applications.