Convexity indicators based on fuzzy morphology

Popov, Antony T.

doi:10.1016/s0167-8655(97)00007-x

Cited by 17 publications

(3 citation statements)

References 7 publications

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“…The integral can be considered as a special case of the defuzzi cation process of the fuzzy indicator. The main di erence of the two indicators is that the subsets in (15) are the -cuts of the fuzzy set A while the subsets in (13) are de ned as the subsets of the support of A where A (x) = , a process that is similar to the extension principle one. In general, the indicator (15) satis es axioms A6 and A7 like inequalities.…”

Section: Comparison With Other Inclusion Indicesmentioning

confidence: 99%

A generalized fuzzy mathematical morphology and its application in robust 2-D and 3-D object representation

Chatzis

Pitas

2000

IEEE Trans. on Image Process.

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In this paper, the generalized fuzzy mathematical morphology (GFMM) is proposed, based on a novel definition of the fuzzy inclusion indicator (FII). FII is a fuzzy set used as a measure of the inclusion of a fuzzy set into another, that is proposed to be a fuzzy set. It is proven that the FII obeys a set of axioms, which are proposed to be extensions of the known axioms that any inclusion indicator should obey, and which correspond to the desirable properties of any mathematical morphology operation. The GFMM provides a very powerful and flexible tool for morphological operations. The binary and grayscale mathematical morphologies can be considered as special cases of the proposed GFMM. An application for robust skeletonization and shape decomposition of two-dimensional (2-D) and three-dimensional (3-D) objects is presented. Simulation examples show that the object reconstruction from their skeletal subsets that can be achieved by using the GFMM is better than by using the binary mathematical morphology in most cases. Furthermore, the use of the GFMM for skeletonization and shape decomposition preserves the shape and the location of the skeletal subsets and spines.

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Section: Comparison With Other Inclusion Indicesmentioning

confidence: 99%

A generalized fuzzy mathematical morphology and its application in robust 2-D and 3-D object representation

Chatzis

Pitas

2000

IEEE Trans. on Image Process.

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“…There exist various papers dealing with extensions of mathematical morphology to fuzzy sets; see for example [l, 21,29,30]. In this subsection, we only wanted to show that the theory of adjunctions on complete lattices is a natural framework for such an extension.…”

Section: Godel-brouwer Implicatormentioning

confidence: 99%

Fundamenta Morphologicae Mathematicae

Goutsias

Heijmans

2000

Fundamenta Informaticae

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Mathematical morphology is a geometric approach in image processing and analysis with a strong mathematical f!avor. Originally, it was developed as a powerful tool for shape analysis in binary and, later, grey-scale images. But it was soon recognized that the underlying ideas could be extended naturally to a much wider class of mathematical objects, namely complete lattices. This paper presents, in a bird's eye view, the foundations of mathematical morphology, or more precisely, the theory of morphological operators on complete lattices.shape processing and analysis known as mathematical morphology. Mathematical morp~ology was enriched and subsequently popularized by the highly inspiring book Image Analysis and Mathematical Morphology by Jean Serra [24]. Today, mathematical morphology is considered to be a powerful tool for image analysis, in particular for those applications where geometric aspects are relevant. The main idea is to analyze the shape of objects in an image by "probing" the image with a small geometric template (e.g., line segment, disc, square) known as the struc~uring element. The choice of the appropriate structuring element strongly depends on the particular application at hand. This however should not be viewed as a limitation, since it usually leads to additional flexibility in algorithm design. The reader will find several examples of this flexibility in this paper, as well as in the various other contributions to this special issue. Morphological Image Operators IntroductionThe basic problem in mathematical morphology is to design nonlinear operators that extract relevant topological or geometric information from images. This requires development of a mathematical model for images and a rigorous theory that describes fundamental properties of the desirable image operators. For example, let us consider the case of binary (black and white) images. Binary images can be mathematically modeled as subsets of a given space E, which, depending on the application at hand, is assumed to possess some additional structure (topological space, metric space, graph, etc.). Thus, the family of binary images is given by P(E), the subsets of E. In this paper, we set E = JRd, the d-dimensional Euclidean space, in which case X is a continuous binary image, or E = zd, the d-dimensional discrete space, in which case X is a discrete binary image. Fundamental relationships between binary images can be mathematically specified by means of set inclusions, unions, or intersections. For example, the fact that image X is hidden by another image Y can be modeled by set inclusion: X ~ Y. If we simultaneously consider two images X, Y, then what we see is their union XUY. The part of an image Y which is not covered by some other image X is their set difference Y\X = Y n xc, where xc denotes the set complement of X, also called the background of X. Some major references in the area of mathematical morphology are [3][4][5][6]8,10,[16][17][18][19][20][22][23][24][25][31][32][33][34]. Complete latticesThe previous discussion c...

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“…Fuzzy or soft morphological operations was proposed in [69] and its ability to reduce effects of impulsive noise was demonstrated. The properties of soft mathematical morphology were further investigated in [70,140,107,142,27,46,46,39]. Chatzis and Pitas [18] presented a generalized formulation of fuzzy mathematical morphology using the notion of fuzzy inclusion indicator and demonstrated its robustness in object representation.…”

Section: Mathematical Morphologymentioning

confidence: 99%

Novel multi-scale topo-morphologic approaches to pulmonary medical image processing

Gao¹

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The overall aim of my PhD research work is to design, develop, and evaluate a new practical environment to generate separated representations of arterial and venous trees in non-contrast pulmonary CT imaging of human subjects and to extract quantitative measures at different tree-levels. Artery/vein (A/V) separation is of substantial importance contributing to our understanding of pulmonary structure and function, and immediate clinical applications exist, e.g., for assessment of pulmonary emboli. Separated A/V trees may also significantly boost performance of airway segmentation methods for higher tree generations. Although, non-contrast pulmonary CT imaging successfully captures higher tree generations of vasculature, A/V are indistinguishable by their intensity values, and often, there is no trace of intensity variation at locations of fused arteries and veins. Patient-specific structural abnormalities of vascular trees further complicate the task. We developed a novel multi-scale topo-morphologic opening algorithm to separate A/V trees in non-contrast CT images. The algorithm combines fuzzy distance transform, a morphologic feature, with a topologic connectivity and a new morphological reconstruction step to iteratively open multi-scale fusions starting at large scales and progressing towards smaller scales. The algorithm has been successfully applied on fuzzy vessel segmentation results using interactive seed selection via an efficient graphical user interface developed as a part of my PhD project. Accuracy, reproducibility and efficiency of the system are quantitatively evaluated using computer-I am indebted to my colleagues

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Convexity indicators based on fuzzy morphology

Cited by 17 publications

References 7 publications

A generalized fuzzy mathematical morphology and its application in robust 2-D and 3-D object representation

A generalized fuzzy mathematical morphology and its application in robust 2-D and 3-D object representation

Fundamenta Morphologicae Mathematicae

Novel multi-scale topo-morphologic approaches to pulmonary medical image processing

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