GLRTA: Extensión del problema generalizado de aproximación matricial de rango reducido a tensores tridimensionales con aplicaciones al procesamiento de imágenes y videos

Image processing is the set of techniques applied to digital images to improve quality or facilitate the search for information. A grayscale image can be represented as a matrix, where each entry can be expressed as a number that takes values ​​in the interval [0, 1]. A color image using the RGB color model can also be represented as an array of numbers, where each entry represents one of three color components: red, green, and blue. Therefore, a color image can be expressed as a three-dimensional array.

One option to study this three-dimensional array of numbers is using tensor algebra. A tensor of order 3 (or three-dimensional tensor) is a multidimensional array of ordered numbers in three dimensions: rows, columns, and tubes. One way to modify a color image is to use tensor algebra and perform tensor operations on said image, improving it and obtaining information about it In the case of grayscale images, the generalized reduced rank matrix approximation problem, also known as the GLRMA (generalized low-rank matrix approximation) problem, allows denoising images using training image bases to obtain a matrix that filters out noise. However, this technique has not been studied in color images in the current state of the art. For this reason, in this project, we present a proposal to carry out an extension to the set of tensors of order three of the GLRMA problem to compress and eliminate noise from color images and videos. This new problem is called the GRLTA (generalized low-rank tensor approximation) problem. We will use the matrix transformation technique to solve the GLRTA problem. Such a matrix representation uses the front faces of the three-dimensional tensors to create a block matrix.

After obtaining the theoretical solution to the GLRTA problem, fast computational implementation will be carried out in Octave, Python, and C++, using algorithms that speed up the calculation of the t-pseudoinverse and the t-SVD, which are generalizations to tensors of the pseudoinverse and the SVD matrix. Finally, we will apply the GRLTA problem to compressing and denoising video and color images, using the GLRTA problem

Desarrollar la teoría del problema GLRTA y su posterior implementación computacional eficiente en el área del procesamiento de imágenes.

Matemática Aplicada: modelación, simulación, inteligencia artificial, análisis de datos, visualización de información, optimización y aplicaciones a la ingeniería y a las ciencias