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<font size=3>I sent an email but I didn't see it appear on the mailing
list, so I send it again. Excuses if you got it twice:<br><br>
To: "Minjie Wu" <minjiewu@gmail.com>,
insight-users@itk.org<br>
Subject: Re: [Insight-users] Jacobian determinant image of the
deformation field<br><br>
Hi Minjie, <br><br>
You're right I think. I think the
itkDeformationFieldJacobianDeterminantFilter manual page is wrong.
<br><br>
<br>
T(x) = transformation<br>
u(x) = deformation field<br>
I = Identity matrix<br><br>
T(x) = x + u(x)<br>
jacobian of T = dT/dx = I + du/dx<br><br>
the manual page claims that det[ dT/dx ] = 1 + det[ du/dx ]<br><br>
but:<br><br>
det[ dT/dx ] = det[ I + du/dx ] <br><br>
does not in general equal det[ I ] + det[ du/dx ]
(which does equal 1 + det[ du/dx ] )<br><br>
you can try it for a 2d matrix:<br>
a b<br>
c d <br>
determinant = ad - bc<br><br>
add identity matrix I:<br>
a+1 b<br>
c d+1<br>
determinant = (a+1)(d+1) - bc = ad +1 + d+ a - bc which obviously
does not equal ad-bc+1<br><br>
Regards,<br>
Stefan<br><br>
<br><br>
<br>
At 21:37 24/08/06, Minjie Wu wrote:<br>
<blockquote type=cite class=cite cite>Hello All,<br><br>
<br>
I am trying to get the Jacobian determinant image of a deformation<br>
field. I tested two ways of computing the Jacobian determinant
image,<br>
the resulted images are similar in pattern, but different in the<br>
actual image intensity (range or histogram or voxelwise intensity<br>
diffrence).<br><br>
<br>
1. The first way I tried is by using the<br>
itkDeformationFieldJacobianDeterminantFilter filter with the<br>
deformationation field calcuated from 2, which gives me a
jacobian<br>
determinant image of intensity around 0, (-0.2 +0.2).<br><br>
In the manual page, it says:<br><br>
"Note that the determinant of a zero vector field is also
zero,<br>
whereas the Jacobian determinant of the corresponding identity warp<br>
transformation should be 1.0. In general, in order to obtain the<br>
effective deformation Jacobian determinant, 1.0 must be added to
each<br>
Jacobian value in the output."<br><br>
So I added 1.0 to the resulted image, the final intensity range for<br>
slice 78 is (0.8 1.2).<br><br>
<br>
2. The second way I tried is to using the example code,<br>
DeformableRegistration1.cxx. The jacobian calculation during<br>
itkFEMRegistrationFilter.h registration using ComputeJacobian().
The<br>
result image is similar in pattern as the first one, but for every<br>
voxel, the actual intensity value is different. The final intensity<br>
range for the same slice is (0.02 2.862).<br><br>
<br>
I am very confused by the results. I wonder if the results from 1
and<br>
2 are both jacobian determinant image of the deformation field and<br>
Which method I should use to calculate the precomputed deformation<br>
field?<br><br>
Any suggestion or help is appreciated. Thank you very much for your
help.<br>
-Minjie<br><br>
<br>
-- <br>
-Minjie<br>
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