University of Isfahan Mathematics and Society 2345-6493 Articles in Press 2026 06 20 From classical filters to convolutional neural networks: the mathematical foundations and evolution of neural style transfer in images From classical filters to convolutional neural networks: the mathematical foundations and evolution of neural style transfer in images 30519 10.22108/msci.2026.147524.1771 FA Mahmood Amintoosi Computer Science Dept. Faculty of Mathematical Sciences, Ferdowsi University of Mashhad Journal Article 2025 11 20 Neural Style Transfer (NST) stands as a remarkable example of the synthesis of mathematics, statistics, deep learning, and art. In this article, we present a historical and analytical exploration of the mathematical evolution of image-processing tools---from classical and derivative-based filters to frequency-domain transforms and multiscale representations---and trace how these concepts ultimately paved the way for convolutional neural networks, which form the foundation for understanding style transfer. We show how these mathematical developments culminated in the formulation of the NST algorithm. We then examine the mathematical structure of style transfer with a focus on the role of Gram matrices, learned feature spaces, and the optimization-based objective function. A central insight of NST is that its loss function becomes meaningful only through the representational power of deep neural networks: the notions of ``style'' and ``content'' are defined in learned feature spaces rather than in pixel space. Grounded in statistical concepts and solved through classical optimization methods, this formulation illustrates the deep and elegant interplay between applied mathematics, statistics, and deep learning in creating an artistic computational technique. Neural Style Transfer (NST) stands as a remarkable example of the synthesis of mathematics, statistics, deep learning, and art. In this article, we present a historical and analytical exploration of the mathematical evolution of image-processing tools---from classical and derivative-based filters to frequency-domain transforms and multiscale representations---and trace how these concepts ultimately paved the way for convolutional neural networks, which form the foundation for understanding style transfer. We show how these mathematical developments culminated in the formulation of the NST algorithm. We then examine the mathematical structure of style transfer with a focus on the role of Gram matrices, learned feature spaces, and the optimization-based objective function. A central insight of NST is that its loss function becomes meaningful only through the representational power of deep neural networks: the notions of ``style'' and ``content'' are defined in learned feature spaces rather than in pixel space. Grounded in statistical concepts and solved through classical optimization methods, this formulation illustrates the deep and elegant interplay between applied mathematics, statistics, and deep learning in creating an artistic computational technique. https://math-sci.ui.ac.ir/article_30519_267ccfda148bb02a6748ef1c13376f12.pdf