5.3. Reduction of the first principal component by periodicity

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

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 accept-

Application of Principal Component Analysis to Image Compression

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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 appli-

This work was supported by the Universidad de Las Americas, Ecuador, and the Universidad

[1] Jackson JE. A User's Guide to Principal Components. John Wiley & Sons; 1991

[2] Diamantaras KI, Kung SY. Principal Component Neural Networks: Theory and Applica-

[3] Elsner JB, Tsonis AA. Singular Spectrum Analysis: A New Tool in Time Series Analysis.

able reconstructions of the image can be used.

cation of principal component analysis to image compression.

\* and Alfredo Mendez<sup>2</sup>

\*Address all correspondence to: wilmar.hernandez@udla.edu.ec

1 Universidad de Las Americas, Quito, Ecuador

tions. John Wiley & Sons; 1996

Plenum Press; 1996

2 Universidad Politecnica de Madrid, Madrid, Spain

Acknowledgements

Politecnica de Madrid, Spain.

Author details

Wilmar Hernandez<sup>1</sup>

References

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 principal components, the results shown in Figure 16.

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 original ones are obtained with componentsper (green).
