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Application of Deep Learning-Based Medical Image Segmentation via Orbital Computed Tomography

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A fuzzy soft planar graph with application in image segmentation.

Waheed Ahmad Khan¹, Arsh E Mah Niaz², Trung Tuan Nguyen³

¹Division of Science and Technology, Department of Mathematics, University of Education Lahore, Attock Campus, Attock, Punjab, 43600, Pakistan. sirwak2003@yahoo.com.

Scientific Reports

|July 2, 2025

Summary

This study introduces fuzzy soft planar graphs (FSPGs) to model uncertainty in planar graphs, demonstrating superior image segmentation performance over traditional fuzzy planar graph models.

Keywords:

Fuzzy soft dual graphs Fuzzy soft graphs Fuzzy soft multi-graphs Fuzzy soft planar graphs Image contraction

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Area of Science:

Graph Theory
Image Processing
Fuzzy Set Theory

Background:

Fuzzy sets and soft sets are mathematical tools for modeling uncertainty.
Planar graphs are used in various applications but struggle with vagueness.
Combining fuzzy and soft set theories can enhance graph modeling for uncertain data.

Purpose of the Study:

To introduce and define fuzzy soft planar graphs (FSPGs).
To explore the characterizations and properties of FSPGs, including dual graphs and edge/face types.
To demonstrate the application of FSPGs in image processing, specifically image segmentation and representation.

Main Methods:

Definition of fuzzy soft multi-graphs (FSMGs) and their intersecting values.
Introduction and characterization of fuzzy soft planar graphs (FSPGs).
Analysis of dual FSPGs, various edge types (effective, considerable, non-considerable), and face types (fuzzy soft, strong, weak).
Comparative analysis of Kuratowski's theorem and FSPGs.
Development of an algorithm for converting crisp images to fuzzy soft image pyramids.
Comparative performance evaluation of FSPG-based model versus traditional fuzzy planar graph (FPG) models for image segmentation.

Main Results:

Established key terms and concepts for fuzzy soft planar graphs (FSPGs).
Characterized FSPGs, including their duals, edges, and faces.
Developed and applied an FSPG-based model for image segmentation, outperforming traditional FPG models.
Successfully converted a crisp image to a fuzzy soft image pyramid.

Conclusions:

Fuzzy soft planar graphs (FSPGs) offer a robust framework for handling vagueness and uncertainty in planar graph theory.
The proposed FSPG model demonstrates significant advantages in image segmentation and representation compared to existing methods.
This research opens new avenues for applying advanced graph theory concepts to complex image processing tasks.