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Article

Swarm-Optimized Explainable Attention–Transformer Networks for Bacterial Colony Segmentation and Quantification

by
Najla Sassi
1,* and
Moulay Ibrahim El-Khalil Ghembaza
2
1
Department of Management Information Systems, School of Business, King Faisal University, Al Ahsa 31982, Saudi Arabia
2
Department of Computer Science, College of Engineering and Information Technology, Onaizah Colleges, Qassim 56447, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(12), 2104; https://doi.org/10.3390/math14122104 (registering DOI)
Submission received: 8 May 2026 / Revised: 1 June 2026 / Accepted: 6 June 2026 / Published: 12 June 2026
(This article belongs to the Section E1: Mathematics and Computer Science)

Abstract

For microbiological diagnostics, accurately counting and segmenting microbial colonies is extremely important. However, manual methods are labor-intensive and yield inconsistent results. We develop a hybrid model using swarm intelligence, combining a convolutional transformer with nested skip connections and global context with channel and spatial attention. Parameter tuning is supported by a variety of swarm optimization algorithms (e.g., Particle Swarm Optimization, Quantum-behaved Particle Swarm Optimization, and Differential Evolution Particle Swarm Optimization). Morphological refinement, including a further watershed transform, an attention graph, and post-processing, enhances colony boundaries by separating them. Grad-CAM++, Integrated Gradients, and temperature scaling provide a transparent and trustworthy model through explainability and post hoc calibration. The proposed model was extensively tested on the Microbial Colony Recognition and Circular Bacterial Colony Datasets, achieving a Dice score of 94.2%, an Intersection over the Union of 88.6%, and a mean absolute counting error of 2.7 colonies. These results significantly outperform several baseline models, including U-Net (88.1%), U-Net++ (89.7%), Attention U-Net (90.6%), and Swin-Unet (91.4%). Statistically significant improvements were confirmed (p < 0.01). A cross-dataset analysis demonstrates the framework’s robustness and cross-domain applicability, and positions it as a trustworthy, explainable automated model for assessing microbial colonies in laboratory and clinical settings.
Keywords: swarm intelligence optimization; image segmentation; hybrid attention–transformer networks; post-processing refinement; microbial colony counting swarm intelligence optimization; image segmentation; hybrid attention–transformer networks; post-processing refinement; microbial colony counting

Share and Cite

MDPI and ACS Style

Sassi, N.; Ghembaza, M.I.E.-K. Swarm-Optimized Explainable Attention–Transformer Networks for Bacterial Colony Segmentation and Quantification. Mathematics 2026, 14, 2104. https://doi.org/10.3390/math14122104

AMA Style

Sassi N, Ghembaza MIE-K. Swarm-Optimized Explainable Attention–Transformer Networks for Bacterial Colony Segmentation and Quantification. Mathematics. 2026; 14(12):2104. https://doi.org/10.3390/math14122104

Chicago/Turabian Style

Sassi, Najla, and Moulay Ibrahim El-Khalil Ghembaza. 2026. "Swarm-Optimized Explainable Attention–Transformer Networks for Bacterial Colony Segmentation and Quantification" Mathematics 14, no. 12: 2104. https://doi.org/10.3390/math14122104

APA Style

Sassi, N., & Ghembaza, M. I. E.-K. (2026). Swarm-Optimized Explainable Attention–Transformer Networks for Bacterial Colony Segmentation and Quantification. Mathematics, 14(12), 2104. https://doi.org/10.3390/math14122104

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