Markus,

The SVD decomposition in this case can be viewed as breaking up the mapping (G) into two parts (plus a scaling step, D), the first of which rotates the space of features in the first image into the space of "matched" features, and the second which rotates from that to the feature space of the second image. The scaling matrix, D, is representative of how well matched the n'th feature pairing is, so by scaling that middle space (the matched feature space) accordingly, each pairing is only allowed to contribute according to how good of a pairing it is -- e.g., the last set of features (a pairing of one or more features from the first image with one or more from the second image), corresponding to the smallest eigenvalue (if not zero) may be a very poor match in terms of being a large displacement on the two images, and hence its corresponding entries in G will be small, and hence its contribution to the SVD breakdown of G should be small. Consider, though, how this "last" entry gets made: all of the better feature matches get discovered by the earlier eigenvectors, and those are removed from consideration by the orthogonality constraint. So the last eigenvector pair is going to "mop up" whatever features are left over in the two images, even if they happen to be pretty poor matches. The fact that they're poor matches is reflected in their small eigenvalue, in D. But note that even though they're poor matches, they are still a "best match" when you take into account that all of the other features have found mates already.

So when you replace D with E, you are saying now that you don't *care* how bad of a match it was, you want the features in the first image mapped 1:1 (in terms of weight if not literally) to the features in the second. So effectively you are discarding the distance information (in D) while preserving the logical feature pairings which were decided upon based on that distance information. Make sense?

-Brandyn