| Data: Diffusion signal ; acquisition parameters |
| ; vectors sampling the unit sphere. |
| Result: The unique coefficients of the symmetric |
| positive definite tensor CT-FOD. |
| (1) Modeling the FOD function by symmetric |
| cartesian tensor of order and dimension 3. |
| (i) |
| with the coefficients of the tensor, and |
| the components of the gradient |
| vector . |
| To impose the positivity constraint, the homogeneous |
| polynomial of order in 3 variables, is |
| re-parameterized by a sum of squares of polynomials |
| of order according to the Ternary quartics |
| theorem [8], we notice that in our work : |
| (ii) |
| with real and positive weights; vectors |
| containing the coefficients of the order rank-1 |
| polynomials constructed from the unit vectors |
| sampling the unit sphere. |
| (2) By substituting given by (ii) in |
| (2), the signal can be approximated by |
| as following: |
| (iii) |
| is constructed for each vector or rank-1 |
| tensor , , sampling the unit sphere, and |
| contains the coefficients of these symmetric fourth |
| order tensors. The unknowns are then the weights |
| ; the values are simply obtained by minimizing |
| the following functional equation E: |
| (iv) |
| with , and normalized. |
| To ensure the positivity of the values, |
| the problem (iv) is solved using the efficient |
| constrained optimization algorithm Non Negative |
| Least Squares (NNLS) [8, 15]. |
| (3) The coefficients of the FOD tensor are |
| then estimated simply by multiplying the matrix , |
| of size containing the monomials of the rank-1 |
| symmetric fourth order tensor formed from vectors |
| , by the resultant vector of length . |
