[Insight-users] Jacobian determinant image of the deformation field

[Stefan Klein]

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:

To: "Minjie Wu" <minjiewu at gmail.com>, insight-users at itk.org
Subject: Re: [Insight-users] Jacobian determinant image of the deformation 
field

Hi Minjie,

You're right I think. I think the 
itkDeformationFieldJacobianDeterminantFilter manual page is wrong.


T(x) = transformation
u(x) = deformation field
I = Identity matrix

T(x) = x + u(x)
jacobian of T = dT/dx = I + du/dx

the manual page claims that det[ dT/dx ] = 1 + det[ du/dx ]

but:

det[ dT/dx ] = det[ I + du/dx ]

does not in general equal det[ I ] + det[ du/dx ]    (which does equal 1 + 
det[ du/dx ] )

you can try it for a 2d matrix:
a b
c d
determinant = ad - bc

add identity matrix I:
a+1 b
c     d+1
determinant = (a+1)(d+1) - bc = ad +1 + d+ a - bc  which obviously does not 
equal ad-bc+1

Regards,
Stefan




At 21:37 24/08/06, Minjie Wu wrote:
>Hello All,
>
>
>I am trying to get the Jacobian determinant image of a deformation
>field. I tested two ways of computing the Jacobian determinant image,
>the resulted images are similar in pattern, but different in the
>actual image intensity (range or histogram or voxelwise intensity
>diffrence).
>
>
>1. The first way I tried is by using the
>itkDeformationFieldJacobianDeterminantFilter filter with the
>deformationation field calcuated from 2,  which gives me a jacobian
>determinant image of intensity around 0, (-0.2 +0.2).
>
>In the manual page, it says:
>
>"Note that the determinant of a zero vector field is also zero,
>whereas the Jacobian determinant of the corresponding identity warp
>transformation should be 1.0. In general, in order to obtain the
>effective deformation Jacobian determinant, 1.0 must be added to each
>Jacobian value in the output."
>
>So I added 1.0 to the resulted image, the final intensity range for
>slice 78 is (0.8 1.2).
>
>
>2. The second way I tried is to using the example code,
>DeformableRegistration1.cxx. The jacobian calculation during
>itkFEMRegistrationFilter.h registration using ComputeJacobian(). The
>result image is similar in pattern as the first one, but for every
>voxel, the actual intensity value is different. The final intensity
>range for the same slice is (0.02 2.862).
>
>
>I am very confused by the results. I wonder if the results from 1 and
>2 are both jacobian determinant image of the deformation field and
>Which method I should use to calculate the precomputed deformation
>field?
>
>Any suggestion or help is appreciated. Thank you very much for your help.
>-Minjie
>
>
>--
>-Minjie
>_______________________________________________
>Insight-users mailing list
>Insight-users at itk.org
>http://www.itk.org/mailman/listinfo/insight-users

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