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Jorge Stolfi (talk | contribs) →The multi-scale structure tensor: Still think that shrink+filter+tensor+integrate+expand is better than two-parameter formulation |
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When editing articles in Wikipedia it is good manners to keep important material from other authors and not to delete material from others without a very good understanding of the contents. [[User:Tpl|Tpl]] ([[User talk:Tpl|talk]]) 08:15, 21 August 2010 (UTC)
*Sorry for that, but the original text was rather hard to understand.<br/> One problem with the original description is that its notation differed from that used in the rest of the article. It also seemed unnecessarily complicated, and failed to give the intuition behind the math.<br/>From any operator one can define a "multi-scale" version in an infinte number of ways. As I understand it, the "multiscale structure tensor" has three steps: (1) filter the image with some kernel ''h''<sub>''s''</sub> (2) compute the pointwise tensor matrix <math>\nabla'\nabla</math>, and (3) filter this tensor field with some other kernel ''w''<sub>''r''</sub>. The original text left the two radii ''r'',''s'' independent. However, if the parameter ''s'' is merely the radius of ''h''<sub>''s''</sub>, then shrink+filter+expand with a fixed-radius kernel ''h'' is equivalent to filtering with an ''s''-scaled ''h''<sub>''s''</sub>. Moreover, Gaussian is theoretically a good choice, but in practice one must use approximate discrete kernels, and compute the multiscale decomposition recursively by filtering with a fixed kernel ''h'' and then downsampling by a fixed ratio at each stage. That is, the first scale parameter ''s'' is beter understood as simply the resolution of the digital image, or the level in an image pyramid, rather than a parameter of the filter ''h''. This formulation has the advantage of forcing ''s'' to be truly a scale parameter, i.e. it excludes filters ''h''<sub>''s''</sub> that depend on ''s'' in a more complicated, non-scale-like way.<br/> It also seems more natural to specify the filtering scale ''s'' and the ratio ''r''/''s'', rather than ''r'' and ''s'' separately. (Note that if ''r'' << ''s'' the result is rather uninteresting.) But then, in the shrink+filter+expand formulation, the ratio ''r''/''s'' need not be mentioned explicitly, as it is already implicit in the choice of the mother (scale-inedpendent) kernels ''h'' and ''w''.<br/>In practice, in fact, one shoud omit the final 'expand' step unless strictly necessary, since it merely wastes a lot of space without performing any useful computation. That is another argument for handling the "multiscale" aspect by image scale reduction, rather than by parametrizing the structure operator. (And this observation holds for most other "multiscale operators".)<br/> Note also that ''h'' could be a band-pass filter rather than a low-pass one; that is, at each scale one analyzes detail with that scale 'only', and not any larger or smaller scales. (This is another common interpretation of the term "multiscale", e.g. in wavelet analysis.) Yet in that case one would still probably want to use a Gaussian window ''w'' for integration.<br/>So, I believe that my formulation in terms of shrink+filter+tensor+integrate+expand with scale-independent (but completely arbitrary) mother kernels ''h'' and ''w'' is mathematically equivalent to your formulation with two kernels depending on two parameters --- but is more parsimonious, and easier to understand.<br/>But I an not going to fight with you on this matter.<br/> All the best, --[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 22:58, 22 August 2010 (UTC)
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