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{{WikiProject Robotics|importance=mid|attention=no}}
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==Error on 3D structure tensor image==
It seems that there is a little confusion between the images for the 3D structure tensor. The description for the surfel image seems wrong, with the egein value relation being the one for the line, and conversly for the line below. The ellipsoid images of the right seems also messed up, or am I missing something ? <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/185.116.129.142|185.116.129.142]] ([[User talk:185.116.129.142#top|talk]]) 14:59, 24 March 2023 (UTC)</small> <!--Autosigned by SineBot-->
==Conceptual Explanation Request==
IMO this article (like many other in it's class) is in desperate need of a more in depth conceptual summary. It's wonderful that we have these exact mathematical descriptions, but the concepts for understanding how some of these things work do not require a degree in math. However *reading* about those concepts in these articles *does*.
--[[User:Catskul|Andy]] ([[User talk:Catskul|talk]]) 21:26, 24 June 2011 (UTC)
==paper?==
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:Not fully wikified but (arguably) looking better and good enough until edited? [[User:Rich257|Rich257]] 20:19, 25 September 2006 (UTC)
==Copied from net doc?==
* This article appears to have been taken from this page, almost verbatim: http://www.cs.cmu.edu/~sarsen/structureTensorTutorial/ [[Special:Contributions/147.4.36.7|147.4.36.7]] ([[User talk:147.4.36.7|talk]]) <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|undated]] comment added 18:27, 13 July 2010 (UTC).</span><!--Template:Undated--> <!--Autosigned by SineBot-->
** The article has now been subtantially rewritten, so this is not a problem anymore.--[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 00:12, 21 August 2010 (UTC)
== Fixed incomplete definition ==
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The coherence index was defined [http://en.wikipedia.org/w/index.php?title=Structure_tensor&oldid=368781876 in this version of the article] as 0 when the two eigenvalues were zero, that is, when the gradient was uniformly zero within the window. However, the formula for the general case does not have a definite limit when λ<sub>1</sub> and λ<sub>2</sub> both tend to 0, so any definition is equally wrong. Essentially, such a region can be regarded as totally isotropic or totally coherent, or anything in between, depending on what value one chooses to assign to 0/0.<br/> That article also stated that "[the coherence index] is capable of distinguishing between the isotropic and uniform cases." However, when λ<sub>1</sub> = λ<sub>2</sub> > 0, the first case of the definition yields 0, the same as the second case.<br/>pending clarification, I have removed this claim and merely noted that "some authors" define the index as 0 in the uniform case.<br/>--[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 06:40, 20 August 2010 (UTC)
== Name "Second moment matrix" ambigous/improper? ==
How standard is the name "second moment matrix"? I ask because the name is used in other areas, such as statistics and mechanics, but the meaning does not seem to be the same. Or is it? --[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 00:19, 21 August 2010 (UTC)
*The term "second-moment matrix" is a frequently used terminology in computer vision, because of an interpretation of the second-moment matrix in terms of second-order spectral moments of the Fourier spectrum. Formal statements about this can be found in the book by Lindeberg (1994) and the papers by Lindeberg and Garding (1996, 1997) cited among the references. [[User:Tpl|Tpl]] ([[User talk:Tpl|talk]]) 08:05, 21 August 2010 (UTC)
== The multi-scale structure tensor ==
Yesterday, I complemented this article with a description about the multi-scale structure tensor/second-moment matrix. I was, however, somewhat surprised by the way this text has been edited, with almost nothing left from the original text. In the revised article, there were also several statements that are incorrect and appear to be based on misunderstandings concerning the properties of this descriptor. Thus, it appears as if the revisions were not based on an understanding of the technical contents in the cited references. In the current version, I have reformulated this section with specific emphasis on explaining aspects of this theory that may not have been fully explicit for the author of the revisions. Please, let me know if the current text is more self-contained.
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)
==References to specific pages in references==
When referencing material from a rather extensive book, I included specific page number to make it possible for others to find the specific statements that are relevant for this article. This explanatory text was, however, removed by a previous editor. Does anyone know about a better way of inserting explicit page and section references, e.g. on the form (Author 2010; section 9.5), when referencing a particular section or page in a book? [[User:Tpl|Tpl]] ([[User talk:Tpl|talk]]) 08:15, 21 August 2010 (UTC)
* Sorry about that,too. Page and section references can often be better obtained from the book index and table-of-contents, or (for online reading) with search tools; so the value for readers who may want to check them should be weighted against the cost of cluttering the reference list with extra entries.<br/> An alternative to creating a separate <nowiki><ref>...</ref></nowiki> is the [[:Template:rp|"rp" template]]: the call <nowiki>{{rp|ch.23}}</nowiki> after the <nowiki></ref></nowiki> generates a superscript annotation, as in <sup>[1]</sup>{{rp|ch.23}}. Hope it helps, --[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 23:17, 22 August 2010 (UTC)
==Anisotropy is too abstract==
The direction of gradient varies in the neighborhood of the pixel at the curved edge. Is it better to talk about curvature instead of anisotropy? The formula for curvature can be easily found from the distribution of gradient. See for example Documentation tab at [http://outliner.codeplex.com/documentation Outliner project] --[[User:Wladik Derevianko|Wladik Derevianko]] ([[User talk:Wladik Derevianko|talk]]) 21:32, 2 May 2011 (UTC)
:Curvature is only one aspect of anisotropy. If there are variations in the direction/orientation of the gradient it may also be related to, e.g., presens of noise or of two or more lines/edges in the neighborhood. --[[User:KYN|KYN]] ([[User talk:KYN|talk]]) 07:32, 3 May 2011 (UTC)
==Typo==
"If we keep the local scale parameter s fixed [...]" should be "If we keep the local scale parameter t fixed [...]" <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/92.230.48.68|92.230.48.68]] ([[User talk:92.230.48.68|talk]]) 23:07, 8 March 2012 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
== Is it a tensor? ==
This matrix seems to not be a proper tensor in the sense of obeying rotational transformation rules.
Anyone care to explain otherwise? <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/132.3.33.81|132.3.33.81]] ([[User talk:132.3.33.81|talk]]) 16:01, 1 October 2013 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
Following that thought, the article begins "in mathematics", yet is entirly focused on image processing applications -- is there any reference to a formal treatment of this topic outside of computer graphics? <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/132.3.33.80|132.3.33.80]] ([[User talk:132.3.33.80|talk]]) 16:19, 1 October 2013 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
:It does not strictly satisfy the expected transformation properties of a proper [[tensor]]. But note that it is, in principle, constructed as the outer product of the image gradient and, hence, forms a 2nd order covariant tensor. This is then modified by computing a local average, typically weighted by a Gaussian kernel. As a result the structure tensor no longer transforms as a proper tensor with respect to scaling of the coordinate system. However, it transforms like a tensor with respect to rotation transformations(!), and this is what counts for the applications where it is used. To be useful also for various image scales, the structure tensor can be applied to a [[scale space]], and this is done in some applications. Haven't seen it used in computer graphics though. --[[User:KYN|KYN]] ([[User talk:KYN|talk]]) 18:09, 1 October 2013 (UTC)
: Thanks -- I had trouble proving out the rotational transformation but I've got it now. Rotation invariance should be noted in the article, though I obviously lack the expertise to work on it. <span style="font-size: smaller;" class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/132.3.33.79|132.3.33.79]] ([[User talk:132.3.33.79|talk]]) 20:09, 1 October 2013 (UTC)</span><!-- Template:Unsigned IP --> <!--Autosigned by SineBot-->
: The rotation transformation relies on the Gaussian kernel (called w in the article) being circular symmetric, something that is not mentioned in the intial definition of the structure tensor in the aricle. --[[User:KYN|KYN]] ([[User talk:KYN|talk]]) 20:23, 1 October 2013 (UTC)
== Possible error in equation? ==
Is there an incorrect equation in the section Complex Version?
The expression of <math>\lambda_2</math> (in terms of <math>\kappa_{20}</math> and <math>\kappa_{11}</math>) seems incorrect to me:
In my opinion it should be:
<math display="block">(|\kappa_{20}|-\kappa_{11})/2=\lambda_2</math>
instead of
<math display="block">(\kappa_{20}-|\kappa_{11}|)/2=\lambda_2</math>
as is stated there. <!-- Template:Unsigned --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Cocus|Cocus]] ([[User talk:Cocus#top|talk]] • [[Special:Contributions/Cocus|contribs]]) 11:52, 26 February 2018 (UTC)</small> <!--Autosigned by SineBot-->
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