6. Conclusions

This chapter has been devoted to give a short but comprehensive introduction to the basics of the statistical technique known as principal component analysis, aimed at its application to image compression. The first part of the chapter was focused on preliminaries, mean vector, covariance matrix, eigenvectors, eigenvalues, and distances. That part finished bringing up the problems that the Euclidean distance presents and highlights the importance of using a statistical distance that takes into account the different variabilities and correlations. To that end, a brief introduction was made to a distance that depends on variances and covariances.

Next, in the second part of the chapter, principal components were introduced and connected with the previously explained concepts. Here, principal components were presented as a particular case of linear combinations of random variables, but with the peculiarity that those linear combinations represent a new coordinate system that is obtained by rotating the original reference system, which has the aforementioned random variables as coordinate axes. The new axes represent the directions with maximum variability and provide a simple description of the structure of the covariance.

Then, the third part of the chapter was devoted to show an application of principal component analysis to image compression. An original image was taken and compressed by using different principal components. The importance of carrying out objective measures of quality reconstructions was highlighted. Also, a novel contribution of this chapter was the introduction to the study of the periodicity of the principal components and to the importance of the reduction of the first principal component by periodicity. In short, a novel construction of principal components by periodicity of principal components has been included, in order to reduce the computational cost for their calculation, although decreasing the accuracy. It can be said that using the almost periodicity of the first principal component, less information to obtain acceptable reconstructions of the image can be used.

Finally, we would not like to finish this chapter without saying that few pages cannot gather the wide range of applications that this statistical technique has found in solving real-life problems. There is a countless number of applications of principal component analysis to solve problems that both scientists and engineers have to face in real-life situations. However, in order to be practical, it was decided to choose and develop step by step an application example that could be of interest for a wide range of readers. Accordingly, we thought that such an example could be one related to data compression, because with the advancements of information and communication technologies both scientists and engineers need to either store or transmit more information at lower costs, faster, and at greater distances with higher quality. In this sense, one example is image compression by using statistical techniques, and this is the reason why, in this chapter, it was decided to take advantage of statistical properties of an image to present a practical application of principal component analysis to image compression.
