## 4 Proposed Methodology: Dynamic Context-Aware Transformer (DCAT)
### System Model
#### 4.1.3 Context Embedding Layer  
The Context Embedding Layer in DCAT is a pivotal component that synergizes the visual features extracted by the Vision Transformer (ViT) with the language generation capabilities of the Generative Pre-trained Transformer (GPT-3). This layer is essential for contextualizing visual data, enabling the generation of semantically rich captions that are deeply aligned with the image's content.  
Feature Weighting and Aggregation  
The primary function of the Context Embedding Layer is to aggregate the visual features extracted by ViT and weigh them according to their relevance. This is achieved through a dynamic attention mechanism that focuses on significant aspects of the image.  
\[a_{i}=\sigma(f(x_{i};W_{\text{ViT}})) \tag{11}\] \[\hat{a_{i}}=\frac{\exp(a_{i})}{\sum_{j=1}^{N}\exp(a_{j})} \tag{12}\]  
Here, \(a_{i}\) represents the attention weights for each patch \(x_{i}\) of the image, with \(\sigma\) denoting the sigmoid activation function and \(f\) representing a neural network function parameterized by \(W_{\text{ViT}}\).  
Contextual Feature Synthesis  
Once the attention weights are computed, the Context Embedding Layer synthesizes a contextual feature vector \(C\). This vector is a weighted sum of the image patches, emphasizing regions of the image deemed most relevant for caption generation.  
\[C=\sum_{i=1}^{N}\hat{a_{i}}\cdot x_{i} \tag{13}\]  
This step ensures that the subsequent language model focuses on the most pertinent aspects of the image, thereby enhancing the relevance and accuracy of the generated captions.  
Context Embedding Generation  
The synthesized contextual feature \(C\) is then transformed into a context embedding \(E\). This embedding is formulated to encapsulate the high-level contextual information gleaned from the image, serving as an input to the GPT-3 decoder.  
\[E=h(C;\Theta_{\text{context}}) \tag{14}\]  
In this equation, \(h\) denotes a neural network function that transforms the contextual feature \(C\) into the embedding \(E\), parameterized by \(\Theta_{\text{context}}\).  
Integration with GPT-3  
The context embedding \(E\) is subsequently utilized by the GPT-3 decoder. It enriches the language model with visual context, enabling it to generate captions that are not only linguistically coherent but also contextually aligned with the visual content of the image.  
\[Z=\text{Concat}(E,Y_{\text{partial}}) \tag{15}\]  
\[Y_{\text{generated}}=\text{GPT-3}(Z,\Theta_{\text{GPT-3}}) \tag{16}\]  
Here, \(Y_{\text{partial}}\) represents the partially generated caption, and \(Z\) is the final input to the GPT-3 model, encompassing both the contextual embedding and the textual input.  
Through these steps, the Context Embedding Layer plays a crucial role in the DCAT architecture, ensuring that the visual context is accurately and effectively translated into the linguistic domain. This intricate process enables DCAT to produce captions that are not only descriptive but also deeply rooted in the contextual nuances of the images, particularly enhancing its applicability to Tamil language captioning.