Acknowledgements

5.3. Reduction of the first principal component by periodicity

134 Statistics - Growing Data Sets and Growing Demand for Statistics

principal components, the results shown in Figure 16.

original ones are obtained with componentsper (green).

6. Conclusions

the structure of the covariance.

Using the almost periodicity of the first principal component, we can use less information to obtain acceptable reconstructions of the image. If in the first principal component of dimension 64 we repeat the first eight values periodically and use k principal components to reconstruct the image, we go from a reduction of k=64 to another of ½ � ð Þþ k � 1 8=64 =64. Figure 15 shows both the reconstruction of the image with 2 and 8 original principal components and the reconstruction of the image with 2 and 8 principal components, but with the first component

The first componentsper component is not the true one. Therefore, reconstructions from this set cannot be made with total precision. If we use to compare the 1-norm, 2-norm, and ∞-norm of the image and the corresponding reconstruction, with the original principal components and the principal components using their periodicity, we obtain, by varying the number of used

With the original principal components (blue), the original image can be completely reconstructed, while if we use only a few components, in this case 10 or less, approximations similar to the

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

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

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

brief introduction was made to a distance that depends on variances and covariances.

replaced by a vector whose coordinates have period 8, we call this componentsper.

This work was supported by the Universidad de Las Americas, Ecuador, and the Universidad Politecnica de Madrid, Spain.
