> Hi Alex,
>
> If I understand correctly, you are attempting to compare
>
> Gxx versus Gx( Gx )
>
> If so, part of the problem is that when we do Gxx, the Gaussian
> smoothing is applied only once, while, when using Gx(Gx) the
> Gaussian Smooting is applied twice.
>
> And... depending on how you are doing it,... it may not be the
> equivalent to Gxx, since the Gxx that we do requires:
>
> Dxx( G )
>
> which means that along X we do the equivalent of convolving
> with the second derivative of the Gaussian, while at the same
> time we still do the equivalent of convolve with a Gaussian along
> Y and along Z.
>
>
> The Scale normalization by the Gaussian will also apply
> twice on the Gx(Gx) version.
>
> All that said,
> It will be very healthy to run this on a synthetic image for
> which the theoretical values can be computed, so we have
> a reference value.
>
> It may well be that you might have uncovered a bug...
>
>
> Regards,
>
>
> Luis
>
> ---------------------------------------------------------------------------
> On Mon, Dec 14, 2009 at 7:30 PM, Oleksandr Dzyubak <
adzyubak@...> wrote:
>
>> Hi Luis,
>>
>> Looks like I am terribly missing something here...
>>
>> OK. For my testings I built objects with Gaussian profiles.
>> These are simple: Gaussian lines with various sigmas extruded
>> into Z-direction thus I got "Gaussian planes".
>> Objects: 3D Gaussian planes with Sigma=1,2,3,4,5,10 and intensities I=100.
>>
>> Now I use RecursiveGaussians and HessianRecursiveGaussian from the ITK stock
>> library
>> and calculate the second order derivative across the Gaussian profiles:
>> Hxx -- Hessian x-components, Gxx - Gaussian second order derivative,
>> and GxGx - Gaussian second order derivative taken sequentially as Gx(Gx)
>> using Gaussian first order derivative.
>>
>> Of course, all that was done with "NormalizeAcrossScale" ON and OFF.
>> Afterwards the maximum of absolute values were measured.
>> The results I summarized in the table below.
>>
>> Sigma ITK_Hxx_OFF ITK_Gxx_OFF ITK_Hxx_ON ITK_Gxx_ON
>> ITK_GxGx_OFF ITK_GxGx_ON
>>
>> 1 35.77 35.77 35.77
>> 35.77 19.47 19.47
>> 2 8.92 8.92 71.33
>> 71.33 4.87 19.47
>> 3 3.96 3.96 106.95
>> 106.95 2.16 19.47
>> 4 2.23 2.23 142.56
>> 142.56 1.22 19.47
>> 5 1.43 1.43 178.18
>> 178.18 0.78 19.47
>> 10 0.36 0.36 356.29
>> 356.29 0.19 19.47
>>
>>
>> As you can see from the table, Hxx is always equal to Gxx for both
>> "NormalizeAcrossScale" ON and OFF.
>> Ok. No surprises here. So far so good.
>> Should not be the values Scale indipendent though?
>>
>> Well, now lets take the second order derivative sequentially using Gx(Gx).
>> Now there is a dramatic difference here.
>> a) GxGx values do not match Gxx and Hxx anymore.
>> b) "ITK_GxGx_ON" version apparently demonstrates Scale Space invariance as
>> it should be.
>>
>>
>> Luis, I would highly appreciate any comments on that.
>>
>> Best regards,
>>
>> Alex
>>
>>
>> On Fri, Dec 11, 2009 at 7:39 PM, Luis Ibanez <
luis.ibanez@...>
>> wrote:
>>
>>> Hi Alex,
>>>
>>>
>>> The order for the Hessian follows the same
>>> pattern of the class
>>>
>>> Insight/Code/Common/
>>> itkSymmetricSecondRankTensor.h
>>>
>>> The elements store only the upper right triangle
>>> of the actual matrix.
>>>
>>> For example, in 3D you get
>>> the following pattern
>>>
>>> 0 1 2
>>> 3 4
>>> 5
>>>
>>> All that to say:
>>>
>>> Yes,
>>> the order that you are using is correct:
>>>
>>> Dxx Dxy Dxz
>>> Dyy Dyz
>>> Dzz
>>>
>>>
>>> BTW:
>>> You may want to use (or at least look) the filter:
>>>
>>> Insight/Code/BasicFilters/
>>> itkHessianRecursiveGaussianImageFilter.h
>>>
>>>
>>>
>>> Regards,
>>>
>>>
>>> Luis
>>>
>>>
>>> ---------------------------------------------------------------
>>> On Fri, Dec 11, 2009 at 12:33 PM, Oleksandr Dzyubak <
adzyubak@...>
>>> wrote:
>>>
>>>> Dear users,
>>>>
>>>> After I built the individual Hessian components,
>>>> in what order should I fill them up to further feed
>>>> into the SymmetricEigenAnalysisImageFilter?
>>>>
>>>> At the moment I am using the order below.
>>>>
>>>> H[0] = Dxx
>>>> H[1] = Dxy
>>>> H[2] = Dxz
>>>> H[3] = Dyy
>>>> H[4] = Dyz
>>>> H[5] = Dzz
>>>>
>>>> Is this correct order?
>>>>
>>>> Thanks,
>>>>
>>>> Alex
>>>>
>>>>
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>>
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