# Semi-Supervised Deep Subspace Embedding for Binary Classification of Sella Turcica
## 2 Materials and Methods
### Proposed Semi-Supervised Deep Subspace Embedding (SSLDSE) Architecture  
Unlike conventional medical classification methods that predominantly depend on fully labeled datasets for accurate predictions, this approach introduces an innovative combination of semi-supervised deep subspace learning targeted at the binary classification of ST anomalies. It employs labeled and unlabeled data, utilizing deep subspace embeddings, KL divergence to refine probability distributions, and manifold learning to capture non-linear relationships within cephalometric data. This method focuses explicitly on binary classification, aiming to differentiate between normal and abnormal ST structures, particularly associated with dentofacial anomalies. Furthermore, using a zero-shot classifier enhances the model's ability to identify new or previously unobserved abnormalities, even with limited labeled data. This approach significantly improves classification accuracy and robustness, especially in scenarios marked by noise and variability in cephalometric images. The flow diagram in Figure 3 illustrates the overall process of the proposed framework. 2.3.1. Exploring Non-linear Embeddings with Kullback-Leibler Divergence  
The Kullback-Leibler (KL) [56; 57] divergence is a quantitative measure that assesses the disparity between two probability density functions (PDF), \(f_{1}(x)\) and, \(f_{2}(x)\). In this context, \(f_{1}(x)\) represents the PDF for bridging sella shapes, and \(f_{2}(x)\) represents the PDF for non-bridging sella shapes. The discrimination information function \(D(f_{1},f_{2})\) is defined as the difference between the two PDFs using \(f_{2}\) as the reference distribution. It is calculated as  
\[D(f_{1},f_{2})=\int_{-\infty}^{\infty}f_{1}(x)\ln(f_{1}(x))\,dx-\int_{-\infty}^{ \infty}f_{1}(x)\ln(f_{2}(x))\,dx, \tag{1}\]  
Figure 3. Process flow diagram of the proposed SSLDSE framework. as described in [6; 29]. It is evident that \(D(f_{1},f_{2})>0\), and this value is only zero when the two PDFs are nearly identical. Therefore, the KL divergence is always non-negative, and higher values signify a more significant distinction between the two distributions [30]. In the present study, KL divergence is employed to map image sets to a non-linear subspace, where each point corresponds to a KL divergence value [58]. The process commences by extracting features from each image and representing them as probability distributions. Initially, various types of features are experimented with, including raw pixel values, histograms, and intermediate outputs from a CNN. Although raw pixel values and histograms are considered, they are ultimately not used in the final KL divergence calculation due to their limitations in dimensionality and lack of spatial information. Instead, intermediate CNN outputs are exclusively used due to their superior ability to capture both low-level and high-level features. Given the computational intensity of calculating KL divergence for all possible pairs of images, particularly in large datasets, our methodology incorporates an efficient strategy to manage the computational load. Instead of calculating the KL divergence for every possible pair of images, a sampling technique is employed to select a representative subset of image pairs. This approach reduces the number of pairwise comparisons without compromising the integrity of the analysis. Additionally, parallel processing is utilized to distribute the computational workload across multiple cores, further accelerating the KL divergence calculations. **[**CNN-Based Feature Extraction and Probability Distribution Estimation**]** The rich representational power of intermediate CNN outputs, specifically from Inception-ResNet-v2 [59; 60; 61], allows for the utilization of these features as the primary inputs for the KL divergence calculations [62]. These CNN-extracted features encapsulate both low-level and high-level morphological traits in a reduced dimensional space, making them particularly suitable for this form of probabilistic analysis.