It was mentioned in Example 11.13 that a credible job could be done of reconstructing appr...

It was mentioned in Example 11.13 that a credible job could be done of reconstructing approximations to the six original images by using only the two principal-component images associated with the largest eigenvalues. What would be the mean square error incurred in doing so? Express your answer as a percentage of the maximum possible error.

EXAMPLE: Moment invariants.

The objective of this example is to compute and compare the preceding moment invariants using the image in Fig. The black (0) border was added to make all images in this example be of the same size; the zeros do not affect computation of the moment invariants. Figures through (f) show the original image translated, scaled by 0.5 in both spatial dimensions, mirrored, rotated by 45° and rotated by 90°, respectively. Table summarizes the values of the seven moment invariants for these six images. To reduce dynamic range and thus simplify interpretation, the values shown are sgn(ϕi) log10(|ϕi | ) The absolute value is needed because many of the values are fractional and/or negative; the sgn function preserves the sign (interest here is on the invariance and relative signs of the moments, not on their actual values). The two key points in Table are (1) the closeness of the values of the moments, independent of translation, scale change, mirroring and rotation; and (2) the fact that the sign of ϕ7 is different for the mirrored image (a property used in practice to detect whether an image has been mirrored).

FIGURE (a) Original image. (b)–(f) Images translated, scaled by one-half, mirrored, rotated by 45° and rotated by 90°, respectively.

TABLE Moment invariants for the images in Fig.