A decomposição em valores singulares aplicada à reconstrução e compressão de imagens

Abstract

The objective of this paper is to present the Singular Value Decomposition and develop a computational application for compression and reconstruction of digital images, which can be stored and transmitted using less information than the original image. For this, a theoretical study of the preliminary concepts of Linear Algebra is carried out, and then the Main Theorem of the decomposition is stated and demonstrated, followed by numerical examples. In order to develop the computational applications the RGB color system is considered (red, green and blue). The image is described using three matrices, one for each channel of these colors, and decomposition is applied for each matrix. In case the image is in grayscale, only one matrix is required. Then, the Eckart-Young Theorem (1937) is applied to image reconstruction. Computational programs using Matlab are presented, with numerical examples and applications in images, which are compact and displayed in color and grayscale, requiring a much smaller amount of values to be stored.