Crossing Fibers Detection with an Analytical High Order Tensor Decomposition

Algorithm 2

Data: An homogenous polynomial of degree .

Result: with minimal.

(1) Calculate coefficients of from those of

.

(2) Construct the Hankel matrix from the

coefficients of .

(3) if  all the    minors of    are zero  then

   (tensors rank)

 else

  .

 Repeat

  (4) Compute from a square sub-matrix of

  dimension corresponding to a monomials

  basis of degree connected one

  of size . and its extension of dimension

   corresponding to

  the monomials basis of size ,

  which is the extension of .

    (5) Compute the matrix corresponding to

   the monomials basis multiplied by for

    and the multiplication matrix

  (6) Find the parameters such that

  and the matrix commute.

  if  solutions exist then

    Calculate the rank of and

   the rank of .

   if    then

    else

     ; Repeat Step 4.

   else

    ; Repeat Step 4.

  Until  the eigenvalues of    are simples

 with arbitrary real ;

(7) Calculate the eigenvalues of the common

eigenvectors of the multiplication matrix such

that , .

(8) Then solving the linear system in :

where are the eigenvalues found in Step 7.