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It is possible to visualise the data as points in a high-dimensional
space as was earlier argued. By placing all images in that space,
it is expected that some cloud of points will be present at a specific,
though somewhat confined, region. The breadth of this region or the
size of that cloud will depend on the variation amongst the images
(or more generally data) that is being visualised. PCA relies on Eigen
analysis to obtain the Eigen-vectors and Eigen-values of that cloud
of points. The highest Eigen-value will correspond to the most significant
Eigen-vector (see the single-headed arrow in Figure ![[*]](/IMG/latex/crossref.png)
).
It indicates the direction which best distinguishes the image data
and is expected to be the longest one too - that is - the one whose
magnitude is the greatest2.19. This is in fact what is considered to be the principal component
which describes that data.
In a recursive manner, at each stage of the process, the current principal component is virtually saved and put aside until only negligible components remain present. The recursion will therefore deal with simpler, more uniform data. More and more principal components are set aside and leave a data of lower dimensionality that occupies a relatively low volume in space. A smaller number of components can then be used to express the variation up to a comparatively high level of fidelity. The process is lossy, but so are some other stages in model construction including the choice of a finite number of landmarks. That loss is controlled in the sense that one can choose the minimal amount of variation that must be accounted for2.20. PCA is used to gain speed while retaining the best descriptors of variation or difference in shape and intensity. What this all comes down to is the acquisition of a model that is smaller in size and is easier to deal with. It is easier to deal with because: (1) it is smaller; (2) it is quicker to use and (3) some of its attributes are decomposed.
![\includegraphics[%%
scale=0.7]{./Graphics/pca.eps}](img27.png){kind=small}