caps.diffusion_estimation.tensor_estimation.LSTensorEstimation

LSTensorEstimation

class caps.diffusion_estimation.tensor_estimation.LSTensorEstimation

Ordinary least square tensor fiting

Fits a diffusion tensor given diffusion-weighted signals and gradient info using a least square strategy [R1].

[+show/hide ols fit details]

Ordinary Least Sqaure Tensor Estimation: Diffusivity Modelisation

Notations:

• is the diffsuion signal in the direction and is denoted .
• is the signal without diffusion weighted and is denoted .

The signal attenuation can be modeled using the Stejskal-Tanner equation. Taking the logarithm of this equation:

For a second order tensor (a 3 by 3 symmetric positive-definite matrix),

solving the Stejskal-Tanner for more than 6 gradient directions results in an overdetermined linear system:

Where N is the number of gradient measures, and is the jth component of the ithe gradient direction .

Similarly, for a fouth order tensor, the linear system to solved can be written:

Finally,the linear system to solve has the form

To estimate the 6 parameters of a second order tensor, directions are required. To estimate the 15 parameters of the fourth order tensor, directions are required. Solving this system using a least square strategy leads us to find (the independant elements of ) minimizing the estimation error , where each row of contains the diffusion tensor monoms. The solution is then given by:

Such estimation minimize the quadratic error between the measure and the estimate (prediction). It has been shown that the least square solution is the best maximum likelihood estimator for an additive gaussian noise on the observations (in our case a gaussian noise on the ADC).

Warning

This noise model do not correspond to the acquisition noise which is Rician. On top of that, the positive nature of a diffusion tensor is not taking into account during the estimation: negative eigenvalues are often set to an arbitrary small positive value. To overcome this limitation, complex (often non linear) sytem has to be soved.

References

[R1]	(1, 2) Mori, S., 2007. Introduction to Diffusion Tensor Imaging. Elsevier.

Inputs

[Mandatory]

bvals_file: a file name (mandatory) the the diffusion b-values

bvecs_file: a file name (mandatory) the the diffusion b-vectors

dwi_file: a file name (mandatory) an existing diffusion weighted image

mask_file: a file name (mandatory) a mask image

model_order: an integer (mandatory) the estimated model order (even)

[Optional]

estimate_odf: a boolean (optional) estimate the odf

model_name: a string (optional) the name of the output tensor model file

output_directory: a directory name (optional) the output directory where the tensor model will be written

Outputs

tensor_file: a file name the result file containing the tensor model coefficients