<br>Hi Luis,<br><br>Looks like I am terribly missing something here...<br><br>OK. For my testings I built objects with Gaussian profiles.<br>These are simple: Gaussian lines with various sigmas extruded <br>into Z-direction thus I got "Gaussian planes".<br>
Objects: 3D Gaussian planes with Sigma=1,2,3,4,5,10 and intensities I=100.<br><br>Now I use RecursiveGaussians and HessianRecursiveGaussian from the ITK stock library<br>and calculate the second order derivative across the Gaussian profiles:<br>
Hxx -- Hessian x-components, Gxx - Gaussian second order derivative,<br>and GxGx - Gaussian second order derivative taken sequentially as Gx(Gx) <br>using Gaussian first order derivative.<br><br>Of course, all that was done with "NormalizeAcrossScale" ON and OFF.<br>
Afterwards the maximum of absolute values were measured.<br>The results I summarized in the table below.<br><br>Sigma ITK_Hxx_OFF ITK_Gxx_OFF ITK_Hxx_ON ITK_Gxx_ON ITK_GxGx_OFF ITK_GxGx_ON<br><br>1 35.77 35.77 35.77 35.77 19.47 19.47<br>
2 8.92 8.92 71.33 71.33 4.87 19.47<br>3 3.96 3.96 106.95 106.95 2.16 19.47<br>
4 2.23 2.23 142.56 142.56 1.22 19.47<br>5 1.43 1.43 178.18 178.18 0.78 19.47<br>
10 0.36 0.36 356.29 356.29 0.19 19.47<br><br><br>As you can see from the table, Hxx is always equal to Gxx for both "NormalizeAcrossScale" ON and OFF.<br>
Ok. No surprises here. So far so good. <br>Should not be the values Scale indipendent though?<br><br>Well, now lets take the second order derivative sequentially using Gx(Gx).<br>Now there is a dramatic difference here.<br>
a) GxGx values do not match Gxx and Hxx anymore.<br>b) "ITK_GxGx_ON" version apparently demonstrates Scale Space invariance as it should be.<br><br><br>Luis, I would highly appreciate any comments on that.<br><br>
Best regards,<br><br>Alex<br><br><br><div class="gmail_quote">On Fri, Dec 11, 2009 at 7:39 PM, Luis Ibanez <span dir="ltr"><<a href="mailto:luis.ibanez@kitware.com">luis.ibanez@kitware.com</a>></span> wrote:<br><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;">
Hi Alex,<br>
<br>
<br>
The order for the Hessian follows the same<br>
pattern of the class<br>
<br>
Insight/Code/Common/<br>
itkSymmetricSecondRankTensor.h<br>
<br>
The elements store only the upper right triangle<br>
of the actual matrix.<br>
<br>
For example, in 3D you get<br>
the following pattern<br>
<br>
0 1 2<br>
3 4<br>
5<br>
<br>
All that to say:<br>
<br>
Yes,<br>
the order that you are using is correct:<br>
<br>
Dxx Dxy Dxz<br>
Dyy Dyz<br>
Dzz<br>
<br>
<br>
BTW:<br>
You may want to use (or at least look) the filter:<br>
<br>
Insight/Code/BasicFilters/<br>
itkHessianRecursiveGaussianImageFilter.h<br>
<br>
<br>
<br>
Regards,<br>
<br>
<br>
Luis<br>
<br>
<br>
---------------------------------------------------------------<br>
<div><div></div><div class="h5">On Fri, Dec 11, 2009 at 12:33 PM, Oleksandr Dzyubak <<a href="mailto:adzyubak@gmail.com">adzyubak@gmail.com</a>> wrote:<br>
><br>
> Dear users,<br>
><br>
> After I built the individual Hessian components,<br>
> in what order should I fill them up to further feed<br>
> into the SymmetricEigenAnalysisImageFilter?<br>
><br>
> At the moment I am using the order below.<br>
><br>
> H[0] = Dxx<br>
> H[1] = Dxy<br>
> H[2] = Dxz<br>
> H[3] = Dyy<br>
> H[4] = Dyz<br>
> H[5] = Dzz<br>
><br>
> Is this correct order?<br>
><br>
> Thanks,<br>
><br>
> Alex<br>
><br>
><br>
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