## 3 Materials and Methods
### Multimodal Systems and Fusion Strategies  
To enhance recognition accuracy and system robustness, three deep learning-based fusion strategies were implemented: feature-level fusion, which combines feature vectors from FP and FV before classification; score-level fusion, which merges prediction scores from the unimodal models; and hybrid fusion, which integrates both features and scores to exploit the strengths of each modality and optimize overall performance. The feature-level fusion model enhances biometric recognition by combining feature vectors from fingerprint (FP) and finger vein (FV) images before classification, preserving rich discriminative information from both modalities. As illustrated in Figure 4, FP and FV images undergo preprocessing and are processed through the most effective CNN models to extract high-level features, which are then concatenated and fed into a softmax classifier for user identification. The score-level fusion model improves biometric authentication by combining softmax scores from two CNNs: one for fingerprint (FP) and another for finger vein (FV) recognition. Unlike feature-level fusion, which merges features before classification, this method integrates final prediction scores, offering a modular and noise-resilient design. As shown in Figure 5, FP and FV images from the NUPT-FPV dataset are preprocessed separately and passed through dedicated CNNs to produce modality-specific scores. These scores are then fused--typically via summation or averaging--to make the final recognition decision. For score-level fusion, we used simple averaging to combine the softmax output scores from the fingerprint and finger vein models. This means that, for each class, we  
Figure 4: Feature-level fusion model. took the scores from both models and calculated their average. We did not use weighted averaging or trainable weights for this process. Preliminary tests indicated that simple averaging provided comparable or better performance than more complex methods [17; 35]. The hybrid-level fusion model integrates both feature- and score-level fusion to leverage complementary strengths of each. Feature-level fusion captures fine-grained biometric characteristics, while score-level fusion reflects the model's confidence in its predictions. By combining both, the system enhances recognition accuracy and becomes more resilient to noise, spoofing, and modality-specific weaknesses. As shown in Figure 6, after preprocessing, each modality is processed by a shared CNN to produce feature vectors and softmax scores. These are fused--features via concatenation and scores via aggregation--and passed to a joint decision layer for final prediction. Algorithm 1 outlines the process: models \(M_{FP}\) and \(M_{FV}\) output feature vectors (\(F_{FP},F_{FV}\)) and scores (\(S_{FP},S_{FV}\)), which are fused into \(F_{fused}\) and \(S_{fused}\). These are processed by the joint layer to generate predictions \(\hat{y}\), and loss \(L\) is used to update the networks. The final model \(M_{2 LF}\) integrates both levels of information for enhanced recognition. In the hybrid fusion approach, we first concatenate the feature vectors from both modalities. Separately, we also use the averaged softmax scores from each model (as in score-level fusion). These fused features and averaged scores are then combined and passed to a fully connected decision layer that is trained to automatically balance their influence. No explicit or manual weights are set between features and scores; the neural network determines the best balance during training [37]. To verify the performance of the trained models, predicted labels \(\hat{y}\) are compared with true labels \(y\) to compute standard evaluation metrics, including accuracy, precision, recall, and F1-score. In addition, we used biometric-specific metrics--false acceptance rate (FAR), false rejection rate (FRR), false error rate (FER), and equal error rate (EER)--to provide a comprehensive assessment of biometric system performance [46]. Figure 5: Score-level fusion model.