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MAAPO:an innovative membrane algorithm based on artificial protozoa optimizer for multilevel threshold image segmentation
Xiaopeng Wang, Vaclav Snasel, Seyedali Mirjalili, Jeng-Shyang Pan
TL;DR
Global optimization and multilevel image segmentation require effective search methods for complex threshold-selection problems. The paper proposes MAAPO, combining membrane computing with roulette fitness-distance balance within APO. MAAPO surpasses 12 compared algorithms on CEC2017 and produces competitive segmentation results measured by PSNR, SSIM, and FSIM, but incurs slightly higher runtime and has limited segmentation objectives and datasets.
Problem
The paper targets global optimization and multilevel thresholding image segmentation, where meta-heuristics are used to optimize segmentation objectives such as Otsu and Kapur entropy.
Method
MAAPO combines APO with a parallel-distributed membrane system and roulette-based fitness-distance balance for reference-point selection.
Results
MAAPO surpasses 12 state-of-the-art algorithms on CEC2017 and achieves competitive segmentation results measured by PSNR, SSIM, and FSIM.
Takeaways & Limitations
Membrane separation-merging and reference-point selection are presented as transferable strategies for improving meta-heuristic optimization.
Takeaways & Limitations
The improvements slightly increase runtime, and segmentation experiments use only Otsu and Kapur entropy on popular animal images.
Abstract
from arXiv · showhide
This paper proposes a novel membrane algorithm based on artificial protozoa optimizer (MAAPO) for global optimization problems. The artificial protozoa optimizer (APO) is adopted as the base meta-heuristic algorithm due to its novelty and competitive performance. MAAPO integrates two key innovations:(1) a membrane computing (MC) framework that introduces a parallel distributed paradigm to improve population diversity and search dynamics, and (2) an enhanced autotrophic model within APO that uses a roulette-based fitness-distance balance (RFDB) mechanism for adaptive reference point selection. These strategies collectively enhance the algorithm's exploration-exploitation balance and global search capabilities. To validate its performance, MAAPO is tested against 12 advanced algorithms on the CEC2017 test suite, and further applied to the multilevel thresholding image segmentation problem using Otsu and Kapur entropy as objective functions. The quality of segmented images is assessed using peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), and feature similarity index (FSIM) metrics. Experimental results demonstrate that MAAPO outperforms its counterparts, delivering superior segmentation quality. This research on MAAPO contributes an effective enhancement strategy to meta-heuristic algorithms and introduces a novel, highly applicable approach for complex image segmentation tasks.
1 Introduction
The paper motivates multilevel thresholding as an important image-segmentation problem and proposes MAAPO to improve meta-heuristic optimization for this task. MAAPO combines membrane computing and roulette fitness-distance balance, with evaluations on CEC2017 and image segmentation.
- 1 Introduction: Multilevel thresholding divides images into multiple intensity-based classes and is motivated by the need for precise visual understanding.Thresholding is presented as computationally simple, robust, and effective.
- 1 Introduction: Earlier studies optimized Otsu, Kapur entropy, or cross-entropy objectives with particle swarm, electromagnetism, differential evolution, and other meta-heuristics.The cited progression includes methods published from 2007 through 2025.
- 1 Introduction: MAAPO combines a membrane system with roulette fitness-distance balance in APO to enhance exploration and exploitation.The membrane system provides parallel-distributed search, while RFDB guides reference-point selection.
- 1 Introduction: MAAPO surpasses 12 state-of-the-art algorithms on the CEC2017 test suite.The paper evaluates MAAPO against APO, GWO, IGWO, PSO, PPSO, DE, ARO, FPA, GOA, MAQUATRE, SGA, and AOO.
- 1 Introduction: MAAPO achieves more competitive results than counterpart methods in multilevel thresholding image segmentation.Segmentation quality is assessed using PSNR, SSIM, and FSIM.
2 Preliminaries
The preliminaries introduce APO’s biologically inspired search behaviors and membrane computing’s cell-like parallel structure. These concepts provide the basis for MAAPO’s exploration–exploitation mechanisms and population organization.
- 2.1 Artificial protozoa optimizer: APO models autotrophic and heterotrophic foraging, dormancy, and reproduction to search through complementary exploration and exploitation behaviors.Autotrophic foraging and dormancy support exploration, whereas heterotrophic foraging and reproduction support exploitation.
- 2.1 Artificial protozoa optimizer: APO’s search operators use population positions, neighboring protozoa, bounds, mapping vectors, and random perturbations to update candidate solutions.The notation defines X_i, X_j, neighboring positions, Mf, dim, ps, and the search bounds used by the update rules.
- 2.1 Artificial protozoa optimizer: The probability pah decreases over iterations, shifting protozoa from autotrophic toward heterotrophic foraging.The pdr parameter also depends on fitness ranking, favoring reproduction for superior protozoa and dormancy for inferior ones.
- 2.2 Membrane computing: Membrane computing represents molecular evolution and intercellular communication through parallel-distributed cell-like systems.The system contains membranes, objects, rules, priorities, and an output membrane.
- 2.2 Membrane computing: Cell-like membrane systems use evolution, communication, merging, and separation rules to transform objects and reorganize membranes.Merging combines membranes, while separation divides them into multiple membranes.
3 Proposed MAAPO algorithm
MAAPO combines membrane computing with an APO variant to distribute population search and improve reference-point selection. Its framework monitors population diversity, separates and merges membranes when needed, and uses RFDB-guided APO updates until termination.
- 3 Proposed MAAPO algorithm: MAAPO combines membrane computing’s distributed parallelism with meta-heuristic search efficiency.The algorithm treats solutions as objects in membranes and uses membrane operations to support population evolution.
- 3.1 Framework of MAAPO: MAAPO uses mVOL to assess population diversity and activates separation-merging when mVOL exceeds ε; otherwise, solutions remain in one membrane.Separated membranes are updated independently and then merged, while ε is set to 0.0001 and m is bounded between 1 and 4.
- 3.2 APO variant with FDB: RFDB probabilistically favors individuals with high FDB scores while retaining randomness, and it replaces random Xj selection in APO autotrophic foraging.FDB balances normalized fitness and distance through a weight parameter, while RFDB combines this score with roulette selection.
- 3.1 Framework of MAAPO: The framework initializes and evaluates a population, computes mVOL, updates solutions with the RFDB-based APO variant, updates the global optimum, and repeats until termination.Figure 3 summarizes this sequence, including the conditional membrane structure.
- 3 Proposed MAAPO algorithm: MAAPO has total computational complexity O(itermax · ps · (ps + dim + f(·))).This combines APO search and evaluation, mVOL computation, and RFDB selection costs.
4 Empirical studies
MAAPO combines membrane computing with RFDB-based reference selection to improve APO, and is evaluated through parameter studies, ablations, and CEC2017 comparisons. It ranks first overall and generally matches or exceeds competing algorithms without degradation in the reported ablation comparisons.
- Comparative study: MAAPO ranks first on CEC2017 with a Friedman score of 3.35, ahead of MAQUATRE at 3.37 and APO at 3.55.The benchmark covers 30 unimodal, multimodal, hybrid, and composition functions, using Wilcoxon and Friedman tests.
- Analysis of APO variants: RFDB-APO is the best APO variant, with APO recording 0 wins, 27 draws, and 3 losses against it.RFDB uses roulette selection to choose diverse reference points, unlike the deterministic FDB and DFDB selections.
- Parameter analysis: The recommended membrane settings are ε = 0.0001 and mmax = 4, with mmax = 4 achieving the best average ranking of 3.08.The study reports that dividing the population into membranes improves performance and that dynamic membrane numbers outperform static settings.
- Ablation experiment: Ablation comparisons show MAAPO improves over APO in 8 functions, matches it in 22, and degrades in none.Against MS-APO and RFDB-APO, MAAPO wins 5 of 30 functions, has no significant difference in 25, and is inferior in none.
- Comparative study: MAAPO records 30 wins and no losses against both GWO and SGA, plus 8 wins, 22 draws, and no losses against APO.It also achieves 19/7/4 against DE, 21/7/2 against ARO, 22/6/2 against FPA, and 10/10/10 against MAQUATRE.
- Comparative study: MAAPO shows superior convergence speed and accuracy on most test functions, while competing algorithms tend to converge prematurely.The improved search performance comes with an acceptable running time relative to APO.
5 Multilevel thresholding image segmentation
MAAPO is applied to multilevel color-image thresholding by optimizing RGB-band thresholds with Otsu and Kapur entropy objectives. Segmentation quality is evaluated using PSNR, SSIM, and FSIM, with performance generally improving relative to comparison algorithms as dimensionality increases.
- 5 Multilevel thresholding image segmentation: MAAPO optimizes multiple thresholds for each RGB band using Otsu and Kapur entropy as separate fitness functions, then evaluates the segmented images with PSNR, SSIM, and FSIM.The three metrics are higher-is-better measures of segmentation quality.
- 5 Multilevel thresholding image segmentation: As threshold dimensionality increases, MAAPO outperforms comparison algorithms in most tested instances, whereas results are generally similar at four thresholds.The comparisons include multiple threshold settings under both Otsu and Kapur entropy.
- 5 Multilevel thresholding image segmentation: Under Otsu, MAAPO ranks first in SSIM with an average rank of 3.83, while PSO leads PSNR at 3.17 and DE leads FSIM at 3.58.MAAPO improves on APO under Otsu but does not lead PSNR or FSIM.
- 5 Multilevel thresholding image segmentation: Under Kapur entropy, MAAPO achieves the best overall performance, with average ranks of 3.17 for PSNR, 3.75 for SSIM, and 3.42 for FSIM.This ranking is reported across the three tested images.
6 Conclusion
The conclusion presents MAAPO as an APO variant combining membrane computing with RFDB-guided reference-point selection. Across CEC2017 optimization and multilevel image segmentation, the reported results are competitive, while runtime, objective-function, and dataset scope remain limitations.
- 6 Conclusion: MAAPO combines a membrane computing framework, multidimensional-volume diversity guidance, dynamic separation, and roulette-based fitness-distance balance for reference-point selection.The membrane system introduces parallel population management, while RFDB refines the APO autotrophic model.
- 6 Conclusion: MAAPO is evaluated against 12 state-of-the-art optimizers on CEC2017 and then applied to multilevel thresholding segmentation using Otsu and Kapur entropy.Segmentation quality is assessed with PSNR, SSIM, and FSIM.
- 6 Conclusion: The reported experiments show competitive optimization performance and high-quality segmented images measured by PSNR, SSIM, and FSIM.The conclusion describes MAAPO as identifying superior solutions in global optimization and validating its segmentation performance and general applicability.
- 6 Conclusion: The added improvements slightly increase runtime compared with APO, while segmentation experiments use only Otsu and Kapur entropy on popular animal images.The authors identify broader objective functions and more diverse datasets as future evaluation directions.