Talk:Structure tensor: Difference between revisions

Revision as of 23:59, 19 August 2010

Revision as of 06:26, 20 August 2010

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* 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-->

== BogusFixed theory?incomplete definition ==

The definition of the structure tensor in [http://en.wikipedia.org/w/index.php?title=Structure_tensor&oldid=368781876 this version of the article] was incomplete and misleading. The eigenvalues of the matrix ''S'', as defined in that version, are simply <math>\lambda_1 = I_x^2 + I_y^2</math> (the square of the gradient modulus) and <math>\lambda_2 = 0</math>; the associated eigenvectors are the direction of the gradient and the same rotated 90 degrees. Thus that "structure tensor" is sumply a complicated way to express the gradient (minus its direction), and the coherence index is simply "gradient != (0,0)".<br/> The structure tensor makes sense only when that matrix is integrated over some neighborhood; and then it summarizes the distribution of gradient directions within that neighborhood.<br/>I have fixed that definition, hopefuly it is correct now. I also did some general cleanup of the article; I hope I did not lose anything important.<br/>--[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 06:26, 20 August 2010 (UTC)

==Removed passage on coordinate invariance==

<strike>TheI discussion about the eigenvalues and eigenvectors ofremoved this "structure tensor" seems to be nonsense.<br/> The eigenvectors of the matrix S are the direction of the gradient and the same rotated 90 degrees. The eigenvalues are simply <math>\lambda_1 = I_x^2 + I_y^2</math> (the square of the gradient modulus) and <math>\lambda_2 = 0</math>sentence, assince oneit candoes checknot byseem theunderstandable definitions.to Thusreaders thewho "coherencedo index"not isalready simply "gradient != (0,0)". Soknow what isit the point of all this mathematical mumbo-jumbo (other than to publish a few more papers)?<br/>This phrase seems to be meaningless,toomeans: "A significant difference between a tensor and a matrix, which is also an array, is that a tensor represents a physical quantity the measurement of which is no more influenced by the coordinates with which one observes it than one can account for it." The matrix S obviously depends on the coordinate system.<br/>--[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 1506:4526, 1920 August 2010 (UTC)

==Removed passage on tensor addition==

PS. The same holds for the three-dimensional case. The eigenvectors are the direction of the gradient and any two unit orthogonal vectors perpendicular to it. The eigenvalues are <math>\lambda_1 = I_x^2 + I_y^2 + I_z^2</math> and <math>\lambda_2 = \lambda_3 = 0</math>. <br/> If no one disagrees, I will try to fix the article.<br/>--[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 15:55, 19 August 2010 (UTC)</strike>

I removed this paragraph and picture, since they do not seem to be understandable to readers who do not already know what they mean: "<nowiki>[[Image:TensorAddition.png|thumb|Tensor addition of sphere and step-edge case]]</nowiki>Another desirable property of the structure tensor form is that the tensor addition equates itself to the adding of the elliptical forms. For example, if the structure tensors for the sphere case and step-edge case are added, the resulting structure tensor is an elongated ellipsoid along the direction of the step-edge case.<br/>--[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 06:26, 20 August 2010 (UTC)

Presumably what the author writes as <math>I_x^2</math> is not the square of something, but rather the integral of the derivative Ix^2 within a window; and ditto for the other three elements of ''S''. That seems to be the case in many applications. Yet the ''S'' matrix seems to be used in some cases as a surrogate of the Hessian of <math>I^2</math>, which includes ''S'' but second derivatives too. --[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 17:41, 19 August 2010 (UTC)

I have fixed the definition as above (added the missing integrals). I will soon be restoring the deleted content, with the proper fixes. --[[User:Jorge Stolfi|Jorge Stolfi]] ([[User talk:Jorge Stolfi|talk]]) 23:59, 19 August 2010 (UTC)