# How can I represent the following mathematically? (Eigenfaces reconstrcution)

I got the following from a web site:

The feature vectors represent each image as a linear combination of the eigenfaces defined by the coefficients in the feature vector; if we multiply each eigenface by its corresponding coefficient and then sum these weighted eigenfaces together, we can roughly reconstruct the input image.

Zulfi.

• What more precisely is your question? – Juho Nov 15 '18 at 15:16
Why was "roughly" used in that article then? It was used only because we only have "20 eigenfaces that my training set generated". However, those are the most significant eigenfaces in the sense that their corresponding eigenvalues are the biggest 20. With only 20 eigenfaces, we can still reconstruct the input image but only "roughly". Another connotation of "roughly" comes from the fact that almost all numerical computations are approximations since, for example, we cannot expressed $$\sqrt 2$$ or even $$1/3$$ exactly in standard floating binary numbers.