## 4 Proposed Methodology: Dynamic Context-Aware Transformer (DCAT)
### System Model
#### 4.1.4 Dynamic Attention Mechanism  
In DCAT, the dynamic attention mechanism is a critical innovation that enhances the model's ability to generate contextually and linguistically coherent captions. It consists of spatial attention for processing visual inputs and temporal attention for sequential text generation.  
Spatial Attention in ViT Encoder  
Spatial attention within the ViT encoder allows the model to focus on the most significant parts of the image, enabling it to capture relevant visual details crucial for accurate captioning.  
\[a_{i} =\sigma(f(x_{i};W_{\text{ViT}})) \tag{17}\] \[\dot{a_{i}} =\frac{\exp(a_{i})}{\sum_{j=1}^{N}\exp(a_{j})} \tag{18}\]  
Here, \(a_{i}\) represents the unnormalized attention scores calculated by a neural network function \(f\), parameterized by weights \(W_{\text{ViT}}\). The sigmoid function \(\sigma\) ensures that the attention scores are scaled appropriately. The normalization of these scores across all patches ensures that the model focuses on the most relevant parts of the image.  
Temporal attention is utilized in the GPT-3 decoder to ensure that each word in the generated caption is contextually aligned with the preceding words. This mechanism is vital for maintaining coherence and relevance throughout the caption.  
\[b_{t} =g(y_{t},y_{1:t-1};U_{\text{GPT-3}}) \tag{19}\] \[\hat{b_{t}} =\frac{\exp(b_{t})}{\sum_{k=1}^{t}\exp(b_{k})} \tag{20}\]  
In these equations, \(b_{t}\) represents the attention scores calculated for each word in the caption sequence. The function \(g\), parameterized by \(U_{\text{GPT-3}}\), takes into account both the current word and the preceding words in the sequence. Normalizing these scores ensures that the model appropriately weights the influence of each word when generating the subsequent word in the caption.  
The integration of spatial and temporal attention mechanisms allows DCAT to generate captions that are not only accurate in describing the visual content but also fluent and coherent. The spatial attention ensures that the model focuses on the most informative parts of the image, while the temporal attention maintains the contextual flow of the caption.  
This dynamic attention mechanism is a cornerstone of the DCAT model, enabling it to adaptively adjust its focus and generate high-quality captions that accurately reflect both the visual content and the contextual nuances, particularly in the Tamil language.  
The DCAT model, through the integration of ViT and GPT-3, augmented by the Context Embedding Layer, delivers a novel approach to image captioning. It not only addresses the linguistic and contextual complexities inherent to Tamil but also sets a precedent for similar low-resource language applications. By incorporating the dynamic spatial and temporal attention mechanisms, DCAT enhances its ability to generate captions that are not only accurate but also contextually rich, reflecting a deep understanding of both the visual and textual aspects of the input data. This dual attention approach is pivotal in achieving superior performance in image captioning tasks, particularly for complex languages like Tamil. The seamless amalgamation of advanced image processing and natural language processing techniques within DCAT paves the way for groundbreaking advancements in AI-driven language generation and computer vision.