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 . |