Fuzzy Geometry and Image Processing Techniques
Summary
Fuzzy geometry integrates the mathematical frameworks of fuzzy set theory with classical geometric constructs, enabling the representation and manipulation of uncertainty in spatial reasoning. By assigning membership functions to points, distances and shapes, fuzzy geometry generalises Euclidean notions to accommodate imprecision inherent in real-world data. This approach provides flexible metrics and topological relations that can model ambiguous spatial boundaries or noisy measurements. In parallel, image processing techniques have increasingly leveraged fuzzy concepts—such as fuzzy masks and fuzzy level sets—to enhance segmentation, registration and feature extraction under conditions of low contrast or inhomogeneous intensity. The synthesis of these fields has yielded robust algorithms for contour detection, sensor network localisation and mesh refinement in robotic vision, demonstrating practical utility in remote sensing, medical imaging and autonomous navigation.
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Recent studies have extended analytic formulations of classical theorems to fuzzy triangles, deriving fuzzy analogues of the Pythagorean theorem and geometric mean theorem. Under prescribed membership functions for triangle vertices, fuzzy distances satisfy subset relations mirroring Euclidean results, thus laying a rigorous foundation for uncertain spatial metrics.
Investigations into fuzzy conic sections have led to the formal construction of fuzzy hyperbolas. By selecting fuzzy-valued points and solving nonlinear systems to determine curve coefficients, these studies elucidate how membership degrees influence the multiplicity and morphology of hyperbolic loci. Applications range from radar system modelling to the thermal and gas distribution in plant canopies.
In the realm of image processing, a fuzzy level set methodology has been proposed that transforms greyscale images into a fuzzy domain via the maximum fuzzy entropy principle. An associated energy functional governs contour evolution through a partial differential equation, achieving robust boundary delineation in images with weak edges or intensity inhomogeneities and outperforming traditional segmentation schemes.
Fuzzy Geometry and Image Processing Techniques publication trend
The graph below shows the total number of articles in fuzzy geometry and image processing techniques across all publications each year (not limited to Nature Index journals).
Technical terms
Fuzzy set: A collection in which elements have graded membership between zero and one, modelling uncertainty.
Membership function: A mapping that assigns to each element a value in [0,1] representing its degree of belonging to a fuzzy set.
Fuzzy geometry: An extension of geometric theory incorporating fuzzy sets to represent imprecision in points, distances and shapes.
Fuzzy level set: A contour representation technique in which image intensities are mapped to fuzzy membership values and evolved via differential equations.
Fuzzy mask: A weighting function for image regions that assigns membership values to pixels, allowing for soft boundaries in segmentation and registration.
References
- Two‐Dimensional Fuzzy Spatial Relations: A New Way of Computing and Representation. Advances in Fuzzy Systems (2012).
- Metric Relations in the Fuzzy Right Triangle. Mathematics (2023).
- Constructing the Fuzzy Hyperbola and Its Applications in Analytical Fuzzy Plane Geometry. Journal of Mathematics (2022).
- Location Discovery Based on Fuzzy Geometry in Passive Sensor Networks. International Journal of Digital Multimedia Broadcasting (2011).
- Satellite Image Georeferencing on the Base of Coastlines’ Fuzzy Masks. MATEC Web of Conferences (2017).
- A Novel Fuzzy Level Set Approach for Image Contour Detection. Mathematical Problems in Engineering (2016).
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