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Jorge Stolfi (talk | contribs) →Bogus theory?: It seems that the definition was incorrect, that is all. |
Jorge Stolfi (talk | contribs) →Bogus theory?: Note that the definition is fixed |
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== Bogus theory? ==
<strike>The discussion about the eigenvalues and eigenvectors of 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>, as one can check by the definitions. Thus the "coherence index" is simply "gradient != (0,0)". So what is the point of all this mathematical mumbo-jumbo (other than to publish a few more papers)?<br/>This phrase seems to be meaningless,too: "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]]) 15:45, 19 August 2010 (UTC)
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>
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)
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