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A direct method for stereo correspondence based on singular value decomposition

6
Brandyn Webb
10-02-2001
12:36 PM ET (US)
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
5
Markus Herrgard
10-01-2001
08:41 PM ET (US)
I think the SVD-based method itself appears to be pretty interesting mainly due to its simplicity and because there is a straightforward linear algebra solution. However, I read the Pilu paper first and was amazed that he just describes the steps of the method and does not even try to explain why the method works (i.e. the pairing matrix P obtained is the orthogonal matrix that maximizes the inner product of P and G). He also doesn't even bother trying to explain the method in his own words, but cites full sentences from the original Scott & Longuet-Higgins paper.

OK. So much for complaining about the papers. I actually have a real question on what replacing the matrix D (which has singular values on the diagonal) with the matrix E (which has ones on the diagonal) in SVD will do in general? Especially if you think of SVD from the PCA viewpoint, you are amplifying the components with low singular values compared to those with high singular values. I'm not quite sure how to interpret this.
4
Dave KauchakPerson was signed in when posted
10-01-2001
08:11 PM ET (US)
First, I'd like to say that I was a bit disappointed with the experimental setup, presentation and analysis of both papers, particularly the Pilu paper. I felt the papers should have done a better job of collecting an interesting set of cases/problems and presenting the results of those. In particular, the Pilu paper suggests that the algorithm presented works fairly well (though he does mention that maybe the method should only be used for bootstrapping), but only on a limited set of examples was tested and the actual analysis of the results from these examples is also minimal.

I was also wondering if anyone else had any better ideas for constructing G than is done in the Pilu paper. I think one of the key things that this paper presents is that we should try and include similarity information between the features when trying to do the matching. What I don't think the paper discussed enough, however, were the options that we have for doing this and the effectiveness of these options. The paper presents the normalized correlation as a one method for representing the similarity, but this is not the only way that this similarity could be presented. Also, the way in which the proximity and similarity portions are combined is fairly simplistic. One could easily consider at a minimum weighting these two things in some manner to get better results. The choices that they made should have been better justified either theoretically or experimentally.
3
Wu Junwen
09-29-2001
08:51 PM ET (US)
About the first paper, I'm not quite understand how the proximity matrix G reflect the proximity principle. I know the magnitude of G's element can reflect the proximity of two features, but can it garantee that the maximum element in P is corresponding to the one in G?(Change matrix E to D has no effect on the element's relative magnitude?)
2
Serge BelongiePerson was signed in when posted
09-28-2001
07:39 PM ET (US)
FYI, there's a nice interactive demo of the Ternus effect at http://vision.psy.mq.edu.au/~peterw/demos.html
1
andrew cosandPerson was signed in when posted
09-28-2001
04:40 AM ET (US)
For anyone who's interested, here's a link to Pilu's page on this project. Wish I'd known he already had slides made up...
http://www-uk.hpl.hp.com/people/mp/research/stersvd.htm

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