[Insight-developers] Suggestion: Tensors and Tensor Image Filters

[Carl-Fredrik Westin]

Torsten and Gunnar,
Thanks for your inputs on a new tensor class in itk. We need this and 
there is an ongoing discussion how to go about this. I agree with 
Gunnar that we should be  general. The structure tensor and the 
diffusion tensor have similar constrains (symmetric, real eigenvalues 
in the ideal case, etc). However, my experience is that we should not 
try to be as efficient as possible, that is to use the fact that the 
tensor is symmetric, positive semi-definite  etc. With a wider tensor 
class, we can host other tensors as well,  such as the stress, the 
strain, and the deformation gradient tensor (common in nonlinear 
registration). Further, when we start differentiating these tensors, 
and get higher order tensors, it would be great to have structure in 
ITK to accommodate for this.

In summary, let's be as general defining a tensor class that we can 
afford to be, without complicating the use of them in applications such 
as DTMRI or local structure estimation. My suggestion is to not exploit 
the symmetry to begin with.

C-F

--
Dr. Carl-Fredrik Westin
Director, Laboratory of Mathematics in Imaging (LMI)
Assistant Professor of Radiology, Harvard Medical School
Brigham and Women's Hospital, 75 Francis St., Boston, MA 02115

Phone: (+1) 617-278-0639, Fax: (+1) 617-582-6033
Email:  westin at bwh.harvard.edu, Office: Thorn Building 323
http://lmi.bwh.harvard.edu/~westin


On Jan 31, 2005, at 19:25, Torsten Rohlfing wrote:

>
> Gunnar:
>
> Are those tensors you mentioned also symmetric? It seems that for 
> diffusion imaging, one can conveniently represent the tensors as 
> compact coefficient vectors by linearizing the upper right half of the 
> tensor (including the diagonal). Is the same true for the tensors you 
> mentioned? Can you think of applications where the symmetry constraint 
> would be a problem?
>
> Thanks!
>  Torsten
>
> Gunnar Farneback wrote:
>
>> Torsten wrote:
>>
>>> Also, can anyone think of a reason why the tensor dimension (i.e., 
>>> the tensor matrix size) should be a template parameter (as opposed 
>>> to fixed 3x3)? My current state of mind is that tensor image 
>>> dimensions should definitely be templated to allow processing of 2D 
>>> slices, whereas the tensors themselves really don't seem to make 
>>> much sense in anything other than 3D.
>>>
>>
>> If your aim is strictly Diffusion Tensor MRI, I don't think anything
>> but 3x3 is meaningful. However, it's also possible to represent the
>> orientation and structure of local features in an intensity image with
>> tensors, usually called orientation tensors or structure tensors.
>> These are in general NxN for N-dimensional images and at least 2x2,
>> 3x3, and 4x4 tensors are in common use.
>> The properties of these tensors are very similar to diffusion tensors
>> and many of the same operations on the tensors are relevant, so it
>> would be useful with a more general tensor class.
>>
>> I can provide further explanation and references on request.
>>
>> /Gunnar
>>
>
>
> -- 
> --
> Torsten Rohlfing, PhD           SRI International, Neuroscience Program
> Research Scientist              333 Ravenswood Ave
>  torsten at synapse.sri.com         Menlo Park, CA 94025
>   Phone: ++1 (650) 859-3379       Fax: ++1 (650) 859-2743
>
>     "Though this be madness, yet there is a method in't"
>
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