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Glaucoma is defined as progressive optic neuropathy that damages the structural appearance of the optic nerve head and is characterized by permanent blindness. For mass fundus image-based glaucoma classification, an improved automated computer-aided diagnosis (CAD) model performing binary classification (glaucoma or healthy), allowing ophthalmologists to detect glaucoma disease correctly in less computational time. We proposed learning technique called fast discrete curvelet transform with wrapping (FDCT-WRP) to create feature set. This method is entitled extracting curve-like features and creating a feature set. The combined feature reduction techniques named as principal component analysis and linear discriminant analysis, have been applied to generate prominent features and decrease the feature vector dimension. Lastly, a newly improved learning algorithm encompasses a modified pelican optimization algorithm (MOD-POA) and an extreme learning machine (ELM) for classification tasks. In this MOD-POA+ELM algorithm, the modified pelican optimization algorithm (MOD-POA) has been utilized to optimize the parameters of ELM’s hidden neurons. The effectiveness has been evaluated using two standard datasets called G1020 and ORIGA with the -fold stratified cross-validation technique to ensure reliable evaluation. Our employed scheme achieved the best results for both datasets obtaining accuracy of 93.25% (G1020 dataset) and 96.75% (ORIGA dataset), respectively. Furthermore, we have utilized seven Explainable AI methodologies: Vanilla Gradients (VG), Guided Backpropagation (GBP), Integrated Gradients (IG), Guided Integrated Gradients (GIG), SmoothGrad, Gradient-weighted Class Activation Mapping (GCAM), and Guided Grad-CAM (GGCAM) for interpretability examination, aiding in the advancement of dependable and credible automation of healthcare detection of glaucoma.