<p>This survey examines the mathematical methods employed in image captioning, focusing on key models and algorithms that drive advancements in this field. The primary research question is: How do various mathematical frameworks contribute to the accuracy, creativity, and real-time processing capabilities of automated image captioning systems? We systematically reviewed a diverse range of studies, analyzing foundational concepts such as probabilistic reasoning, optimization strategies, and neural network designs. Our comparative assessment highlights strengths, limitations, and applicability to different types of imagery and contexts, with a specific focus on real-time processing efficiencies. Our findings elucidate the mathematical intricacies of existing methods and discuss emerging trends and potential areas for future research. We identify key unresolved challenges, including contextual understanding, handling abstract concepts, and mitigating data bias. This review provides insights into how mathematical frameworks improve image captioning accuracy, creativity, and real-time performance. It serves as an essential resource for researchers and practitioners, proposing interdisciplinary approaches and advanced AI techniques to enhance the performance and capabilities of automated image captioning systems.</p>