[Insight-users] HessianRecursiveGaussianImageFilter

[Iván Macía]

Hi Won-Ki,

Regarding Q1 you are right, the filter computes in the 3D case the six
possible gaussian second order derivatives. Note that the filter uses
itk::RecursiveGaussianImageFilter internally.

Regarding Q2 I think you are also right. When computing second orderivative
(G*I)xx the filter applies recursive gaussian smoothing with first order
derivative twice in the x direction, but the number of smoothing filter is
fixed in the static member NumberOfSmoothingFilters = ImageDimension-2, so
in the 3D case there is only a single smoothing filter.

I made a small test with a 3D MRI image, applied HessianRecursiveGaussian
filter with sigma = 5.0 and extracted the six components of the hessian
tensor. The attached image shows an slice of the original image and the
sixth component (G*I)zz of the hessian matrix of the same slice. It seems
like there is smoothing in x but no smoothing in y (see the narrow
horizontal lines).

I made a Bug entry in the bug tracker with ID 5292
http://www.itk.org/Bug/bug.php?op=show&bugid=5292

Best regards

Ivan


-----Mensaje original-----
De: insight-users-bounces+imacia=vicomtech.es at itk.org
[mailto:insight-users-bounces+imacia=vicomtech.es at itk.org] En nombre de
Won-Ki Jeong
Enviado el: lunes, 02 de julio de 2007 21:26
Para: Insight-users at itk.org
Asunto: [Insight-users] HessianRecursiveGaussianImageFilter

Hi all

I have some questions about the HessianRecursiveGaussianImageFilter.

Q1. Is this filter (theoretically) same as applying Gaussian smoothing
and compute derivatives to the input image? For example, let G be a
Gaussian kernel and I be the input image, then is the filter compute
(G*I)xx, (G*I)xy, (G*I)xz, (G*I)yy, (G*I)yz, (G*I)zz (subscript means
partial derivative along that axis)?

Q2. If so, the implementation is not correct because, for example
(G*I)xx, the filter applies 1D recursive Gaussian order-1 derivative
filter twice along x and applies 1D recursive Gaussian filter along y,
that means along x the Gaussian is applied twice but there is no
Gaussian for z direction. Can somebody clarify this?

The implementation is in void GenerateData(void) in
itkHessianRecursiveGaussianImageFilter.txx.

Thanks a lot!

Won-Ki
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