Anisotropy plugin output

[Waldir Leite Roque]

Dear Michael Doube and users

I have used the Anisotropy plugin (MIL) for a trabecular microCT sample (size 268 x 268 x 268) with the default settings of the plugin and the result can be seen in the attached file. I have some doubts:

1) From the Fabric Tensor Vectors the eigenvalues are given in the Fabric tensor values, however computing the eigenvalues for the current Fabric Tensor does not provide the eigenvalues given in the Fabric Tensor Values output. What is happening or am I misleading?

2) The Original Rotation Matrix is the Fabric Tensor and corresponds to the matrix that transforms the unit vectors (i, j, k) (source) to the fabric eigenvectors (say, e1, e2, e3) (target), is that correct?

Many thanks,

Waldir

[Michael Doube]

Dear Waldir,

1. The eigenvalues ("fabric tensor values") cannot be calculated from the eigenvectors ("Fabric tensor vectors"). The eigenvectors are 3×3 orthogonal unit vectors, with no scale/size - just orientation - which relate to the orientation of the texture in the sample. The eigenvalues (and vectors) are calculated by applying an eigendecomposition to an ellipsoid fitted to the points created by multiplying each line probe direction by the MIL in that direction. The eigenvalues relate to the spacing of the intercepts in the three orthogonal directions of the fabric tensor, but please note that the precision of the output provided here is inadequate to do anything useful. For details of the implementation, have a look at the code, particularly here:

https://github.com/mdoube/BoneJ/blob/master/src/org/doube/bonej/Anisotropy.java#L726

And Harrigan and Mann's paper:

Harrigan TP, Mann RW (1984) Characterization of microstructural

anisotropy in orthotropic materials using a second rank tensor. J Mater

Sci 19: 761-767.

http://dx.doi.org/10.1007/BF00540446

2. I think you have it right - the fabric tensor is effectively a rotation matrix that tells you the orientation of the texture relative to the coordinate frame of the input image. The inverse rotation matrix is used to rotate the input image into the coordinate frame of the image of the rotated specimen: this is because you work backwards, filling each pixel of the rotated image with a pixel looked up in the input image mapped by the inverse rotation matrix.

Michael