Research Article
Crossing Fibers Detection with an Analytical High Order Tensor Decomposition
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. |
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