3.1 Objective Function

We used binary cross-entropy loss to train our proposed architecture, mathematically defined as follows:

where $y$ represents the ground truth label, $y^{'}$ represents the predicted value, and $c$ denotes the number of categories, which in this case is $c=2$. This loss function measures the discrepancy between the predicted probabilities and the actual labels, penalizing predictions that deviate significantly from the ground truth.

3.2 Training Process of XCNet

Algorithm 1 outlines the training process of XCNet. Initially, the minority-class sensor images are augmented using horizontal and vertical flip methods to create a balanced dataset. This augmentation step ensures that the dataset is suitable for effective model training. Next, training batches are constructed by creating sequences of input sensor images{} and corresponding labels{} for all samples, as described in lines 3 to 8. These batches are stored in a set~$S$.

During the training phase, the model parameters $\theta $ are initialized, as mentioned in line 9. The training process then involves randomly selecting a batch of instances $S_b$ from $S$ and updating $\theta $ by minimizing the objective function $\mathcal{L}\left(\theta \right)$, using a gradient descent-based optimization algorithm like Adam, as in lines 10 to 13. This process is repeated iteratively until the predefined stopping criteria, such as a maximum number of epochs or convergence of the loss function, are met. At the end of the training process, the learned model $f$, represented by the optimized parameter set $\theta $, is produced as the output, as described in line 14.

4.1 Ablation Study

The ablation study presented in Table 3 investigates the impact of varying the number of convolutional blocks and filter configurations on classification performance under different levels of class imbalance. Additionally, it provides a comprehensive analysis of computational efficiency, including model complexity (parameter count and GFLOPs) and inference speed, critical factors for real-time defect detection in industrial settings. The dataset consists of two training ratios, 1:10 and 1:50, representing different levels of minority class underrepresentation.

Four architectures were explored with convolutional blocks configured as [32, 64, 128], [32, 64, 128, 256], [16, 32, 64, 128, 256], and [32, 64, 128, 256, 512]. These configurations progressively increase network depth and filter sizes, enhancing feature extraction and enabling finer defect detection. Larger filters improve receptive fields, capturing structural variations in sensor board defects while maintaining computational efficiency.

For the 1:10 ratio, deeper architectures, such as AN3 ([16, 32, 64, 128, 256]) and A.N. 4 ([32, 64, 128, 256, 512]), significantly outperformed shallower configuration. These models achieved high accuracy (99.09% and 99.55%, respectively) and precision (100%), while also improving recall for the minority class (90.90% and 95.45%). The F1-scores of these configurations (95.23% and 95.45%) highlight their ability to balance sensitivity and precision effectively. Conversely, shallower architectures, like A.N. 1 ([32, 64, 128]), show significant drop in recall, achieving only 40.90% and F1-score of 58.05%, despite maintaining high overall accuracy (94.09%).

For the 1:50 ratio scenario, the performance gap between shallow and deep architectures became even more visible due to the severe class imbalance. A.N. 5 ([32, 64, 128]) struggled, with a minority class recall of 21.05% and an F1-score of 33.33%, demonstrating its limitations in handling extreme imbalance. In contrast, the A.N. 8 ([32, 64, 128, 256, 512]) architecture achieved the better performance, with an accuracy of 98.61%, precision (100%), and an F1-score of 91.42%. This configuration showed substantial improvements in recall (84.21%), indicating its effectiveness in capturing minority class instances even in highly imbalanced scenarios.

Computational efficiency and scalability. Shallower models (A.N.~1, A.N.~5) have fewer parameters (0.2~M) and lower FLOPs (1.18~G) but struggle to capture minority-class instances effectively. Deeper architectures generally require more resources but substantially improve recall. For example, A.N.~3 retains the same 1.0~M parameters like A.N.~2 but reduces FLOPs (0.90~G vs.~1.75~G) and inference time (3.75~ms vs.~4.65~ms), while achieving better performance due to enhanced feature extraction. Building on this, A.N.~4, which employs larger convolutional filters, has 4.0~M parameters and 2.31~G FLOPs, but maintains an inference time of 5.63~ms, making it a viable option for real-time applications while delivering the highest recall and F1-score.

These findings confirm that increasing the network depth and filter size improves performance in heavily imbalanced scenarios but also offer competitive inference speeds, making deeper architectures like A.N. 4 and A.N. 8 suitable for real-world manufacturing environments where real-time defect detection is critical.

4.2 Analysis of Data Distribution on Model Performance

This section provides an in-depth analysis of how variations in the composition of training data affect the performance metrics of the model in two cases, Case I and Case II. In Case I, the model is trained with the original data distribution, while in Case II, the minority class sample is augmented. Details of the dataset are discussed in Subsection 4.1.

Fig. 3 shows four graphs, each illustrates the performance metrics (accuracy, precision, recall and F1-Score) for both cases, plotted against the fraction of the total good-class training images. The x-axis represents the fraction of 800 good-class images used for training, while the number of minority-class images is fixed at 28 for Case I and 80 for Case II. The testing dataset remains constant in both cases. This comparative evaluation demonstrates the significant impact of data distribution, particularly the size of the minority class, on model performance.

Fig. 3(a) illustrates the accuracy metrics used to analyze the impact of data distribution on model performance. In both cases, the model's accuracy progressively improves as the fraction of good-class training data increases. However, Case II starts with a higher initial accuracy (approximately 80%) and reaches near-perfect accuracy (100%) much faster than Case I. This rapid improvement can be attributed to the larger minority-class size in Case II, which mitigates class imbalance and allows the model to effectively learn the dominant class, even with limited good-class training data. In contrast, the improvement in Case I is slower, likely due to its smaller minority class, which creates a greater imbalance and requires additional good-class training data to achieve comparable accuracy. Overall, Case II demonstrates a clear advantage in accuracy across all training levels, highlighting the critical role of managing class distributions to optimize model performance.

As shown in Fig. 3(b), precision follows a similar trend, with Case II achieving near 100% precision early, whereas Case I exhibits a more gradual increase. The superior performance of Case II can be attributed to its larger minority-class size, which helps reduce overall class imbalance. This improved balance enables the model to effectively minimize false positives, even with limited good-class training data. In contrast, Case I, with its smaller minority class, struggles to achieve comparable precision under similar conditions.

Fig. 3(c) highlights model performance in terms of recall stability and sensitivity under different data distributions. Unlike accuracy and precision, which show stable and gradual improvement as training data increases, recall behaves differently. In Case I, recall fluctuates at lower training fractions, with noticeable dips indicating inconsistent sensitivity in detecting the minority class. This instability likely arises from the smaller minority-class size in the training dataset. In contrast, recall in Case II remains consistently high across all training fractions, demonstrating the model's robustness in detecting the minority class. The larger minority class in Case II mitigates class imbalance, ensuring that the good class receives adequate representation during training.

Fig. 3(d) illustrates the F1-score, which combines precision and recall into a single metric to provide a holistic measure of the model's classification performance. Case II consistently outperforms Case I, with a rapid increase in F1-score that stabilizes near 100%, even at lower fractions of good-class training data. This superior performance can be attributed to Case II's larger minority-class size, which enhances the model's ability to distinguish between classes and reduces the trade-off between precision and recall. In contrast, Case I shows a slower, more gradual improvement in F1-score, remaining consistently lower across all training levels. This slower progression reflects challenges in balancing precision and recall caused by the smaller minority class, which intensifies class imbalance and limits the model's ability to optimize performance with less training data.

The key observations and implications are as follows: One significant finding is the impact of minority-class size on model performance. Case II, with a larger minority-class size, consistently outperforms Case I across all metrics, demonstrating that a more balanced class distribution improves the model's learning and generalization. Another important insight is the sufficiency of training data. Case II achieves near-optimal performance with fewer training images, highlighting the efficiency of balanced datasets in achieving high accuracy and other key metrics with reduced data. Both cases show performance improvements as training data increases. However, Case II excels in accuracy, precision, recall, and F1-score, even with smaller training fractions. This stability underscores the importance of balanced class distributions for reliable and consistent model performance.

Fig. 3. Data quality study. The plots show the effect of varying the proportion of good quality training data on (a) Accuracy, (b) Precision, (c) Recall, and (d) F1-Score for two cases: Case I (original data) and Case II (augmentation data).

4.3 Result Analysis

Table 4 compares XCNet with other state-of-the-art (SOTA) models, including VGG16 ^[27], ResNet34 ^[28], ResNet50 ^[28], ViT-Tiny ^[29], and ViT-Base ^[29], all pretrained on ImageNet1k. While these models perform well on large-scale datasets, they struggle to adapt to the small, domain-specific defect dataset, particularly in cases of severe class imbalance. ViT-Base, for instance, with 86.90 million parameters and 17.58 GFLOPs, fails to surpass XCNet's performance, likely due to the limited training data ($<1000$ images) and the domain shift from natural images to industrial defect images.

In contrast, XCNet delivers higher recall and F1-scores under both 1:10 and 1:50 imbalances, attaining 99.45% and 98.62% accuracy, respectively, while also maintaining perfect precision (100%). From a complexity standpoint, XCNet requires substantially fewer parameters (4.0~M) and GFLOPs (2.31~G) than the larger SOTA models---VGG16 with 138.36~M parameters and 15.47~GFLOPs or ViT-Base with 86.90~M parameters and 17.58~GFLOPs. This efficient combination of strong performance and moderate computational requirements underscores XCNet's suitability for real-world manufacturing scenarios, where data are scarce and real-time defect detection is critical.

4.4 Analysis on Overfitting to Synthetic Patterns

Figs. 4 and 5 analyze the potential risk of overfitting, specifically whether the model learns artificial patterns from data augmentation instead of genuine defect features. To investigate this, we applied augmentation only to the minority class in both the training and test datasets. Each augmented set includes both original and modified images, ensuring that synthetic images do not dominate the evaluation.

In Fig. 4, the training loss (showing results from the original training dataset; other graphs with similar trends are omitted for clarity) remains steady and stable, while test losses across various augmentations (e.g., flipped images) show no abrupt spikes---indicating that the model learns robust features rather than memorizing synthetic patterns. Fig.~5 further underscores this robustness by showing steady improvements in recall and F1-scores for each augmented scenario. The fact that performance consistently increases as more augmentations are introduced suggests that XCNet acquires generalizable defect features rather than relying on artificially introduced cues. This combined findings strongly support the effectiveness of our augmentation strategy in helping XCNet learn meaningful defect characteristics without overfitting to synthetic patterns.

Fig. 4. Training and test loss under various augmentation scenarios.

Fig. 5. Recall and F1-Scores comparisons across different augmentation scenarios.

4.5 Adaptability to Other Semiconductor Products

Although our work focuses on sensor board defect detection, XCNet can easily be adapted to other semiconductor components and industrial products. Numerous studies confirm the versatility of CNN-based architectures for various defect detection tasks, from wafer map analysis ^[30] to surface flaw identification ^[31,32]. Transfer learning has further demonstrated CNN adaptability to different data distributions ^[33,34]. Building on these findings, XCNet's emphasis on efficient feature extraction and robust classification requires only minimal adjustments---such as domain-specific data augmentation or slight architectural tweaks---to detect defects across diverse industrial contexts. This adaptability highlights XCNet's potential to make a broader impact on semiconductor inspection and defect detection in a wide range of manufacturing scenarios.