Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2015, Director: F. Javier Soria de DiegoNowadays digital images and videos are used in many fields of our live. From the simpler digital camera to the nets of servers in Internet, all of them work with this kind of objects. Since images are stored in these devices, we need to reduce the size of the pictures in order to accumulate as many images as possible in the minimum storage space without losing quality. This necessity has derived in the appearance of several algorithms dedicated to compress images with or without loss of data.
A simple image is a matrix of points (pixels) each one indicating a level of tone or color at its spatial position. Therefore, an image can be represented mathematically as a matrix with numbers indicating the value of the pixels at each position. There exist several types of representations for images, but we are going to consider gray scale images where each pixel can take a value between 0 and 255, where the 0 value represents the black color and the 255 value represents the white color. More complex images may have several components, for instance in colored images often each pixel value is given by three components R, G, and B corresponding to its tone of red color, green color, and blue color, respectively. Before processing them, the components are decorrelated transforming them into other three components
more suitable to be treated. They are also usually normalized into a symmetric range of values, normally between -1 and 1, to exploit better the capabilities of the ulterior operations. These tasks are performed by the block named Pre-Processing in Figure 1.1, where we can observe the general scheme of an image compression system
