Image Compression and Processing Optimization: Integration of Mathematical Methods and Artificial Intelligence.

This work proposes a hybrid approach to image compression, combining classical methods of linear algebra with modern techniques of artificial intelligence. Efficient image compression is essential in applications that involve large volumes of visual data, such as embedded systems, cloud storage, and medical image analysis. The central objective of this research is to reduce the dimensionality of the data while maintaining the visual quality of the image, thus optimizing both storage and transmission.

Initially, Decomposition into Singular Values (SVD) is used, a method established in linear algebra for the analysis of matrix structures. SVD allows you to represent an image by means of eigenvectors and singular eigenvalues, of which only the most significant ones are kept for an approximate representation. The implementation was carried out manually in Julia, respecting the theoretical foundations, which favors the control over the process steps and the fine-tuning of the compression parameters, such as the optimization of the code for greater speed.

In the second step, neural networks, especially convolutional autoencoders, capable of refining the image compressed by the SVD, are explored. Autoencoders learn nonlinear latent representations, aiding in the reconstruction of details lost in compression via SVD. As an additional method, Hilbert curves are implemented, being the tools that rearrange the pixels of the image according to a fractal structure, improving the preservation of spatial locality and optimizing network learning.