Decoding a Key into an Image

The encoded faces can be synthesized (or mapped back into image space) from their 60-coefficient key (c_j) by summing the mean face and the 60 weighted eigenvectors linearly. This forms an approximation n-tuple, $\vec{w}_x$, of each original face, $\vec{v}_x$, as given by Equation . The remapped approximations for Figure  are shown in Figure . Note the low degree of ``lossiness'' of the compression despite the large compression ratio of 7000 pixels to 60 coefficients (i.e., 117:1 compression).

 

$$\vec{w}_x=\bar{v} + \sum_{i=0}^{M-1} c_{x_i} \hat{e}_i$$ (4.36)

We can quantify the lossiness in the KL-based decomposition by measuring the variance between the original vector $\vec{v}_x$ and the optimal linear approximation vector $\vec{w}_x$. This residual variance, residue_x accounts for the total variance in our vector that cannot be spanned by the eigenvectors. The total residual variance in the dataset that cannot be spanned by the eigenvectors is denoted by $\lambda_{res} = \sigma_{res}^2$ and computed using Equation :

$$residue_x = \Vert \vec{v}_x - \vec{w}_x \Vert$$ (4.37)

$$\lambda_{res} = \sigma_{res}^2 = \sum_{x=0}^{N-1} \Vert \vec{v}_x - \vec{w}_x \Vert$$ (4.38)

We have thus described a method for converting a mug-shot image into a 60-scalar code which describes it in the KL domain. Furthermore, we have shown how this transformation can be inverted so that the 60-element code can be used to regenerate an approximation to the original image. Thus, the KL transform and ``inverse'' KL transform are added to our palette of tools.