Module jw_mf

Functions

calc_jw(data, params)
Spectral density function.

calc_S2_jw(data, params)
Spectral density function.

calc_S2_te_jw(data, params)
Spectral density function.

calc_S2f_S2_ts_jw(data, params)
Spectral density function.

calc_S2f_tf_S2_ts_jw(data, params)
Spectral density function.

calc_S2f_S2s_ts_jw(data, params)
Spectral density function.

calc_S2f_tf_S2s_ts_jw(data, params)
Spectral density function.

calc_diff_djw_dDj(data, params, j)
Spectral density gradient.

calc_aniso_djw_dDj(data, params, j)
Spectral density gradient.

calc_diff_S2_djw_dDj(data, params, j)
Spectral density gradient.

calc_aniso_S2_djw_dDj(data, params, j)
Spectral density gradient.

calc_diff_S2_te_djw_dDj(data, params, j)
Spectral density gradient.

calc_aniso_S2_te_djw_dDj(data, params, j)
Spectral density gradient.

calc_diff_djw_dPsij(data, params, j)
Spectral density gradient.

calc_diff_S2_djw_dPsij(data, params, j)
Spectral density gradient.

calc_diff_S2_te_djw_dPsij(data, params, j)
Spectral density gradient.

calc_S2_djw_dS2(data, params, j)
Spectral density gradient.

calc_S2_te_djw_dS2(data, params, j)
Spectral density gradient.

calc_S2_te_djw_dte(data, params, j)
Spectral density gradient.

calc_diff_S2f_S2_ts_djw_dDj(data, params, j)
Spectral density gradient.

calc_aniso_S2f_S2_ts_djw_dDj(data, params, j)
Spectral density gradient.

calc_diff_S2f_tf_S2_ts_djw_dDj(data, params, j)
Spectral density gradient.

calc_aniso_S2f_tf_S2_ts_djw_dDj(data, params, j)
Spectral density gradient.

calc_diff_S2f_S2_ts_djw_dPsij(data, params, j)
Spectral density gradient.

calc_diff_S2f_tf_S2_ts_djw_dPsij(data, params, j)
Spectral density gradient.

calc_S2f_S2_ts_djw_dS2(data, params, j)
Spectral density gradient.

calc_S2f_S2_ts_djw_dS2f(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2_ts_djw_dS2f(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2_ts_djw_dtf(data, params, j)
Spectral density gradient.

calc_S2f_S2_ts_djw_dts(data, params, j)
Spectral density gradient.

calc_diff_S2f_S2s_ts_djw_dDj(data, params, j)
Spectral density gradient.

calc_aniso_S2f_S2s_ts_djw_dDj(data, params, j)
Spectral density gradient.

calc_diff_S2f_tf_S2s_ts_djw_dDj(data, params, j)
Spectral density gradient.

calc_aniso_S2f_tf_S2s_ts_djw_dDj(data, params, j)
Spectral density gradient.

calc_diff_S2f_S2s_ts_djw_dPsij(data, params, j)
Spectral density gradient.

calc_diff_S2f_tf_S2s_ts_djw_dPsij(data, params, j)
Spectral density gradient.

calc_S2f_S2s_ts_djw_dS2f(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2s_ts_djw_dS2f(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2s_ts_djw_dS2s(data, params, j)
Spectral density gradient.

calc_S2f_tf_S2s_ts_djw_dtf(data, params, j)
Spectral density gradient.

calc_S2f_S2s_ts_djw_dts(data, params, j)
Spectral density gradient.

calc_diff_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_aniso_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2_te_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_diff_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_aniso_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2_te_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dDjdS2(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2_d2jw_dDjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dDjdS2(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2_te_d2jw_dDjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dDjdte(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2_te_d2jw_dDjdte(data, params, j, k)
Spectral density Hessian.

calc_diff_d2jw_dPsijdPsik(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dPsijdPsik(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dPsijdPsik(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_d2jw_dPsijdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dPsijdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2_te_d2jw_dPsijdte(data, params, j, k)
Spectral density Hessian.

calc_S2_te_d2jw_dS2dte(data, params, j, k)
Spectral density Hessian.

calc_S2_te_d2jw_dte2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2_ts_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_tf_S2_ts_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2_ts_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_tf_S2_ts_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dDjdS2(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2_ts_d2jw_dDjdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dDjdS2f(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2_ts_d2jw_dDjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dDjdS2f(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_tf_S2_ts_d2jw_dDjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dDjdtf(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_tf_S2_ts_d2jw_dDjdtf(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dDjdts(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2_ts_d2jw_dDjdts(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dPsijdPsik(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dPsijdPsik(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dPsijdS2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dPsijdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dPsijdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2_ts_d2jw_dPsijdtf(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2_ts_d2jw_dPsijdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2_ts_d2jw_dS2dts(data, params, j, k)
Spectral density Hessian.

calc_S2f_tf_S2_ts_d2jw_dS2fdtf(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2_ts_d2jw_dS2fdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_tf_S2_ts_d2jw_dtf2(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2_ts_d2jw_dts2(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2s_ts_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_tf_S2s_ts_d2jw_dDjdDk(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2s_ts_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_tf_S2s_ts_d2jw_dDjdPsij(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dDjdS2f(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2s_ts_d2jw_dDjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dDjdS2f(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_tf_S2s_ts_d2jw_dDjdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dDjdS2s(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2s_ts_d2jw_dDjdS2s(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dDjdtf(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_tf_S2s_ts_d2jw_dDjdtf(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dDjdts(data, params, j, k)
Spectral density Hessian.

calc_aniso_S2f_S2s_ts_d2jw_dDjdts(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dPsijdPsik(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dPsijdPsik(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dPsijdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dPsijdS2f(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dPsijdS2s(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_tf_S2s_ts_d2jw_dPsijdtf(data, params, j, k)
Spectral density Hessian.

calc_diff_S2f_S2s_ts_d2jw_dPsijdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2s_ts_d2jw_dS2fdS2s(data, params, j, k)
Spectral density Hessian.

calc_S2f_tf_S2s_ts_d2jw_dS2fdtf(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2s_ts_d2jw_dS2fdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2s_ts_d2jw_dS2sdts(data, params, j, k)
Spectral density Hessian.

calc_S2f_tf_S2s_ts_d2jw_dtf2(data, params, j, k)
Spectral density Hessian.

calc_S2f_S2s_ts_d2jw_dts2(data, params, j, k)
Spectral density Hessian.

Imports: sum

Function Details

calc_jw(data, params)

Spectral density function.

Calculate the spectral density values for the original model-free formula with no parameters {}
with or without diffusion tensor parameters.

The formula is:

                _n_
             2  \                1
    J(w)  =  -   >  ci . ti ------------
             5  /__         1 + (w.ti)^2
                i=m

calc_S2_jw(data, params)

Spectral density function.

Calculate the spectral density values for the original model-free formula with the single
parameter {S2} with or without diffusion tensor parameters.

The formula is:

                  _n_
             2    \                1
    J(w)  =  - S2  >  ci . ti ------------
             5    /__         1 + (w.ti)^2
                  i=m

calc_S2_te_jw(data, params)

Spectral density function.

Calculate the spectral density values for the original model-free formula with the parameters
{S2, te} with or without diffusion tensor parameters.

The model-free formula is:

                _n_
             2  \           /      S2             (1 - S2)(te + ti)te    \ 
    J(w)  =  -   >  ci . ti | ------------  +  ------------------------- |
             5  /__         \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                i=m

calc_S2f_S2_ts_jw(data, params)

Spectral density function.

Calculate the spectral density values for the extended model-free formula with the parameters
{S2f, S2, ts} with or without diffusion tensor parameters.

The model-free formula is:

                _n_
             2  \           /      S2            (S2f - S2)(ts + ti)ts   \ 
    J(w)  =  -   >  ci . ti | ------------  +  ------------------------- |
             5  /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=m

calc_S2f_tf_S2_ts_jw(data, params)

Spectral density function.

Calculate the spectral density values for the extended model-free formula with the parameters
{S2f, tf, S2, ts} with or without diffusion tensor parameters.

The model-free formula is:

                _n_
             2  \           /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ 
    J(w)  =  -   >  ci . ti | ------------  +  -------------------------  +  ------------------------- |
             5  /__         \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=m

calc_S2f_S2s_ts_jw(data, params)

Spectral density function.

Calculate the spectral density values for the extended model-free formula with the parameters
{S2f, S2s, ts} with or without diffusion tensor parameters.

The model-free formula is:

                   _n_
             2     \           /      S2s           (1 - S2s)(ts + ti)ts    \ 
    J(w)  =  - S2f  >  ci . ti | ------------  +  ------------------------- |
             5     /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                   i=m

calc_S2f_tf_S2s_ts_jw(data, params)

Spectral density function.

Calculate the spectral density values for the extended model-free formula with the parameters
{S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The model-free formula is:

                _n_
             2  \           /   S2f . S2s        (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ 
    J(w)  =  -   >  ci . ti | ------------  +  -------------------------  +  ------------------------- |
             5  /__         \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                i=m

calc_diff_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the original model-free
formula with no parameters {} together with diffusion tensor parameters.

The model-free gradient is:

                 _n_
    dJ(w)     2  \        dti    1 - (w.ti)^2
    -----  =  -   >  ci . ---  ----------------
     dDj      5  /__      dDj  (1 + (w.ti)^2)^2
                 i=m

calc_aniso_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the original model-free
formula with no parameters {} together with diffusion tensor parameters.

The model-free gradient is:

                 _n_
    dJ(w)     2  \   /      dti    1 - (w.ti)^2       dci           1       \ 
    -----  =  -   >  | ci . ---  ----------------  +  --- . ti ------------ |
     dDj      5  /__ \      dDj  (1 + (w.ti)^2)^2     dDj      1 + (w.ti)^2 /
                 i=m

calc_diff_S2_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the original model-free
formula with the parameter {S2} together with diffusion tensor parameters.

The model-free gradient is:

                   _n_
    dJ(w)     2    \        dti    1 - (w.ti)^2
    -----  =  - S2  >  ci . ---  ----------------
     dDj      5    /__      dDj  (1 + (w.ti)^2)^2
                   i=m

calc_aniso_S2_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the original model-free
formula with the parameter {S2} together with diffusion tensor parameters.

The model-free gradient is:

                   _n_
    dJ(w)     2    \   /      dti    1 - (w.ti)^2       dci           1       \ 
    -----  =  - S2  >  | ci . ---  ----------------  +  --- . ti ------------ |
     dDj      5    /__ \      dDj  (1 + (w.ti)^2)^2     dDj      1 + (w.ti)^2 /
                   i=m

calc_diff_S2_te_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the original model-free
formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free gradient is:

                 _n_
    dJ(w)     2  \        dti  /        1 - (w.ti)^2                         (te + ti)^2 - (w.te.ti)^2   \ 
    -----  =  -   >  ci . ---  | S2 . ----------------  +  (1 - S2) . te^2 ----------------------------- |
     dDj      5  /__      dDj  \      (1 + (w.ti)^2)^2                     ((te + ti)^2 + (w.te.ti)^2)^2 /
                 i=m

calc_aniso_S2_te_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the original model-free
formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free gradient is:

                 _n_
    dJ(w)     2  \   /      dti  /        1 - (w.ti)^2                         (te + ti)^2 - (w.te.ti)^2   \ 
    -----  =  -   >  | ci . ---  | S2 . ----------------  +  (1 - S2) . te^2 ----------------------------- |
     dDj      5  /__ \      dDj  \      (1 + (w.ti)^2)^2                     ((te + ti)^2 + (w.te.ti)^2)^2 /
                 i=m

                          dci      /      S2             (1 - S2)(te + ti)te    \ \ 
                       +  --- . ti | ------------  +  ------------------------- | |
                          dDj      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 / /

calc_diff_djw_dPsij(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Psi partial derivative of the original model-free
formula with no parameters {} together with diffusion tensor parameters.

The model-free gradient is:

                _n_
    dJ(w)     2 \    dci            1
    -----  =  -  >  ----- . ti ------------
    dPsij     5 /__ dPsij      1 + (w.ti)^2
                i=m

calc_diff_S2_djw_dPsij(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Psi partial derivative of the original model-free
formula with the parameter {S2} together with diffusion tensor parameters.

The model-free gradient is:

                   _n_
    dJ(w)     2    \    dci            1
    -----  =  - S2  >  ----- . ti ------------
    dPsij     5    /__ dPsij      1 + (w.ti)^2
                   i=m

calc_diff_S2_te_djw_dPsij(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Psi partial derivative of the original model-free
formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free gradient is:

                 _n_
    dJ(w)     2  \    dci       /      S2             (1 - S2)(te + ti)te    \ 
    -----  =  -   >  ----- . ti | ------------  +  ------------------------- |
    dPsij     5  /__ dPsij      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                 i=m

calc_S2_djw_dS2(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2 partial derivative of the original model-free
formula with the single parameter {S2} with or without diffusion tensor parameters.

The model-free gradient is:

                 _n_
    dJ(w)     2  \                1
    -----  =  -   >  ci . ti ------------
     dS2      5  /__         1 + (w.ti)^2
                 i=m

calc_S2_te_djw_dS2(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2 partial derivative of the original model-free
formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free gradient is:

                 _n_
    dJ(w)     2  \           /      1                 (te + ti)te         \ 
    -----  =  -   >  ci . ti | ------------  -  ------------------------- |
     dS2      5  /__         \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                 i=m

calc_S2_te_djw_dte(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the te partial derivative of the original model-free
formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free gradient is:

                         _n_
    dJ(w)     2          \                 (te + ti)^2 - (w.te.ti)^2
    -----  =  - (1 - S2)  >  ci . ti^2 . -----------------------------
     dte      5          /__             ((te + ti)^2 + (w.te.ti)^2)^2
                         i=m

calc_diff_S2f_S2_ts_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the extended model-free
formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \        dti  /        1 - (w.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
    -----  =  -   >  ci . ---  | S2 . ----------------  +  (S2f - S2) . ts^2 ----------------------------- |
     dDj      5  /__      dDj  \      (1 + (w.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                 i=m

calc_aniso_S2f_S2_ts_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the extended model-free
formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \   /      dti  /        1 - (w.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
    -----  =  -   >  | ci . ---  | S2 . ----------------  +  (S2f - S2) . ts^2 ----------------------------- |
     dDj      5  /__ \      dDj  \      (1 + (w.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                 i=m

                          dci      /      S2            (S2f - S2)(ts + ti)ts   \ \ 
                       +  --- . ti | ------------  +  ------------------------- | |
                          dDj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2_ts_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \        dti  /        1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
    -----  =  -   >  ci . ---  | S2 . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  (S2f - S2) . ts^2 ----------------------------- |
     dDj      5  /__      dDj  \      (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                 i=m

calc_aniso_S2f_tf_S2_ts_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \   /      dti  /        1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
    -----  =  -   >  | ci . ---  | S2 . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  (S2f - S2) . ts^2 ----------------------------- |
     dDj      5  /__ \      dDj  \      (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                 i=m

                          dci      /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ \ 
                       +  --- . ti | ------------  +  -------------------------  +  ------------------------- | |
                          dDj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2_ts_djw_dPsij(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Psi partial derivative of the extended model-free
formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \    dci       /      S2            (S2f - S2)(ts + ti)ts   \ 
    -----  =  -   >  ----- . ti | ------------  +  ------------------------- |
    dPsij     5  /__ dPsij      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                 i=m

calc_diff_S2f_tf_S2_ts_djw_dPsij(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Psi partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \    dci       /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ 
    -----  =  -   >  ----- . ti | ------------  +  -------------------------  +  ------------------------- |
    dPsij     5  /__ dPsij      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                 i=m

calc_S2f_S2_ts_djw_dS2(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2 partial derivative of the extended model-free
formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor
parameters.

The formula is:

                 _n_
    dJ(w)     2  \           /      1                 (ts + ti).ts        \ 
    -----  =  -   >  ci . ti | ------------  -  ------------------------- |
     dS2      5  /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                 i=m

calc_S2f_S2_ts_djw_dS2f(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2f partial derivative of the extended model-free
formula with the parameters {S2f, S2, ts} with or without diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \                 (ts + ti).ts
    -----  =  -   >  ci . ti -------------------------
    dS2f      5  /__         (ts + ti)^2 + (w.ts.ti)^2
                 i=m

calc_S2f_tf_S2_ts_djw_dS2f(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2f partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The formula is:

                   _n_
    dJ(w)       2  \           /       (tf + ti).tf                  (ts + ti).ts        \ 
    -----  =  - -   >  ci . ti | -------------------------  -  ------------------------- |
    dS2f        5  /__         \ (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                   i=m

calc_S2f_tf_S2_ts_djw_dtf(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the tf partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor parameters.

The formula is:

                          _n_
    dJ(w)     2           \               (tf + ti)^2 - (w.tf.ti)^2
    -----  =  - (1 - S2f)  >  ci . ti^2 -----------------------------
     dtf      5           /__           ((tf + ti)^2 + (w.tf.ti)^2)^2
                          i=m

calc_S2f_S2_ts_djw_dts(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the ts partial derivative of the extended model-free
formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without diffusion tensor
parameters.

The formula is:

                           _n_
    dJ(w)     2            \               (ts + ti)^2 - (w.ts.ti)^2
    -----  =  - (S2f - S2)  >  ci . ti^2 -----------------------------
     dts      5            /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                           i=m

calc_diff_S2f_S2s_ts_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the extended model-free
formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The formula is:

                    _n_
    dJ(w)     2     \        dti  /         1 - (w.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
    -----  =  - S2f  >  ci . ---  | S2s . ----------------  +  (1 - S2s) . ts^2 ----------------------------- |
     dDj      5     /__      dDj  \       (1 + (w.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                    i=m

calc_aniso_S2f_S2s_ts_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the extended model-free
formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The formula is:

                    _n_
    dJ(w)     2     \   /      dti  /         1 - (w.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
    -----  =  - S2f  >  | ci . ---  | S2s . ----------------  +  (1 - S2s) . ts^2 ----------------------------- |
     dDj      5     /__ \      dDj  \       (1 + (w.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                    i=m

                             dci      /     S2s             (1 - S2s)(ts + ti)ts   \ \ 
                          +  --- . ti | ------------  +  ------------------------- | |
                             dDj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \        dti  /               1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ 
    -----  =  -   >  ci . ---  | S2f . S2s . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  S2f(1 - S2s) . ts^2 ----------------------------- |
     dDj      5  /__      dDj  \             (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                 i=m

calc_aniso_S2f_tf_S2s_ts_djw_dDj(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Dj partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \   /      dti  /               1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ 
    -----  =  -   >  | ci . ---  | S2f . S2s . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  S2f(1 - S2s) . ts^2 ----------------------------- |
     dDj      5  /__ \      dDj  \             (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                 i=m

                          dci      /  S2f . S2s         (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ \ 
                       +  --- . ti | ------------  +  -------------------------  +  ------------------------- | |
                          dDj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2s_ts_djw_dPsij(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Psi partial derivative of the extended model-free
formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The formula is:

                    _n_
    dJ(w)     2     \    dci       /     S2s             (1 - S2s)(ts + ti)ts   \ 
    -----  =  - S2f  >  ----- . ti | ------------  +  ------------------------- |
    dPsij     5     /__ dPsij      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                    i=m

calc_diff_S2f_tf_S2s_ts_djw_dPsij(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the Psi partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \    dci       /  S2f . S2s         (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ 
    -----  =  -   >  ----- . ti | ------------  +  -------------------------  +  ------------------------- |
    dPsij     5  /__ dPsij      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                 i=m

calc_S2f_S2s_ts_djw_dS2f(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2f partial derivative of the extended model-free
formula with the parameters {S2f, S2s, ts} with or without diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \           /     S2s            (1 - S2s)(ts + ti).ts   \ 
    -----  =  -   >  ci . ti | ------------  +  ------------------------- |
    dS2f      5  /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                 i=m

calc_S2f_tf_S2s_ts_djw_dS2f(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2f partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The formula is:

                 _n_
    dJ(w)     2  \           /     S2s                (tf + ti).tf              (1 - S2s)(ts + ti).ts   \ 
    -----  =  -   >  ci . ti | ------------  -  -------------------------  +  ------------------------- |
    dS2f      5  /__         \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                 i=m

calc_S2f_tf_S2s_ts_djw_dS2s(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the S2s partial derivative of the extended model-free
formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion
tensor parameters.

The formula is:

                    _n_
    dJ(w)     2     \           /      1                 (ts + ti).ts        \ 
    -----  =  - S2f  >  ci . ti | ------------  -  ------------------------- |
    dS2s      5     /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                    i=m

calc_S2f_tf_S2s_ts_djw_dtf(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the tf partial derivative of the extended model-free
formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor parameters.

The formula is:

                          _n_
    dJ(w)     2           \               (tf + ti)^2 - (w.tf.ti)^2
    -----  =  - (1 - S2f)  >  ci . ti^2 -----------------------------
     dtf      5           /__           ((tf + ti)^2 + (w.tf.ti)^2)^2
                          i=m

calc_S2f_S2s_ts_djw_dts(data, params, j)

Spectral density gradient.

Calculate the spectral desity values for the ts partial derivative of the extended model-free
formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without diffusion
tensor parameters.

The formula is:

                             _n_
    dJ(w)     2              \               (ts + ti)^2 - (w.ts.ti)^2
    -----  =  - S2f(1 - S2s)  >  ci . ti^2 -----------------------------
     dts      5              /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                             i=m

calc_diff_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \      /    dti   dti           3 - (w.ti)^2        d2ti     1 - (w.ti)^2   \ 
    -------  =  -   >  ci | -2 --- . ---  w^2.ti ----------------  +  ------- ---------------- |
    dDj.dDk     5  /__    \    dDj   dDk         (1 + (w.ti)^2)^3     dDj.dDk (1 + (w.ti)^2)^2 /
                   i=m

calc_aniso_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti   dti           3 - (w.ti)^2       / dti   dci     dti   dci         d2ti   \   1 - (w.ti)^2        d2ci           1       \ 
    -------  =  -   >  | -2ci --- . ---  w^2.ti ----------------  +  | --- . ---  +  --- . ---  +  ci ------- | ----------------  +  ------- ti ------------ |
    dDj.dDk     5  /__ \      dDj   dDk         (1 + (w.ti)^2)^3     \ dDj   dDk     dDk   dDj        dDj.dDk / (1 + (w.ti)^2)^2     dDj.dDk    1 + (w.ti)^2 /
                   i=m

calc_diff_S2_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
     d2J(w)     2    \      /    dti   dti           3 - (w.ti)^2        d2ti     1 - (w.ti)^2   \ 
    -------  =  - S2  >  ci | -2 --- . ---  w^2.ti ----------------  +  ------- ---------------- |
    dDj.dDk     5    /__    \    dDj   dDk         (1 + (w.ti)^2)^3     dDj.dDk (1 + (w.ti)^2)^2 /
                     i=m

calc_aniso_S2_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
     d2J(w)     2    \   /      dti   dti           3 - (w.ti)^2       / dti   dci     dti   dci         d2ti   \   1 - (w.ti)^2        d2ci           1       \ 
    -------  =  - S2  >  | -2ci --- . ---  w^2.ti ----------------  +  | --- . ---  +  --- . ---  +  ci ------- | ----------------  +  ------- ti ------------ |
    dDj.dDk     5    /__ \      dDj   dDk         (1 + (w.ti)^2)^3     \ dDj   dDk     dDk   dDj        dDj.dDk / (1 + (w.ti)^2)^2     dDj.dDk    1 + (w.ti)^2 /
                     i=m

calc_diff_S2_te_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \      /    dti   dti  /             3 - (w.ti)^2                       (te + ti)^3 + 3.w^2.te^3.ti(te + ti) - (w.te)^4.ti^3 \ 
    -------  =  -   >  ci | -2 --- . ---  | S2.w^2.ti ----------------  +  (1 - S2) . te^2 ---------------------------------------------------- |
    dDj.dDk     5  /__    \    dDj   dDk  \           (1 + (w.ti)^2)^3                               ((te + ti)^2 + (w.te.ti)^2)^3              /
                   i=m

                             d2ti   /        1 - (w.ti)^2                         (te + ti)^2 - (w.te.ti)^2   \ \ 
                         +  ------- | S2 . ----------------  +  (1 - S2) . te^2 ----------------------------- | |
                            dDj.dDk \      (1 + (w.ti)^2)^2                     ((te + ti)^2 + (w.te.ti)^2)^2 / /

calc_aniso_S2_te_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti   dti  /             3 - (w.ti)^2                       (te + ti)^3 + 3.w^2.te^3.ti(te + ti) - (w.te)^4.ti^3 \ 
    -------  =  -   >  | -2ci --- . ---  | S2.w^2.ti ----------------  +  (1 - S2) . te^2 ---------------------------------------------------- |
    dDj.dDk     5  /__ \      dDj   dDk  \           (1 + (w.ti)^2)^3                               ((te + ti)^2 + (w.te.ti)^2)^3              /
                   i=m

                            / dti   dci     dti   dci         d2ti   \ /        1 - (w.ti)^2                         (te + ti)^2 - (w.te.ti)^2   \ 
                         +  | --- . ---  +  --- . ---  +  ci ------- | | S2 . ----------------  +  (1 - S2) . te^2 ----------------------------- |
                            \ dDj   dDk     dDk   dDj        dDj.dDk / \      (1 + (w.ti)^2)^2                     ((te + ti)^2 + (w.te.ti)^2)^2 /


                             d2ci      /      S2             (1 - S2)(te + ti)te    \ \ 
                         +  ------- ti | ------------  +  ------------------------- | |
                            dDj.dDk    \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 / /

calc_diff_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the original
model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci    dti     1 - (w.ti)^2
    ---------  =  -   >  ----- . --- . ----------------
    dDj.dPsij     5  /__ dPsij   dDj   (1 + (w.ti)^2)^2
                     i=m

calc_aniso_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the original
model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \   /  dci    dti     1 - (w.ti)^2         d2ci            1       \ 
    ---------  =  -   >  | ----- . --- . ----------------  +  --------- ti ------------ |
    dDj.dPsij     5  /__ \ dPsij   dDj   (1 + (w.ti)^2)^2     dDj.dPsij    1 + (w.ti)^2 /
                     i=m

calc_diff_S2_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the original
model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                       _n_
      d2J(w)      2    \    dci    dti     1 - (w.ti)^2
    ---------  =  - S2  >  ----- . --- . ----------------
    dDj.dPsij     5    /__ dPsij   dDj   (1 + (w.ti)^2)^2
                       i=m

calc_aniso_S2_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the original
model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                       _n_
      d2J(w)      2    \   /  dci    dti     1 - (w.ti)^2         d2ci            1       \ 
    ---------  =  - S2  >  | ----- . --- . ----------------  +  --------- ti ------------ |
    dDj.dPsij     5    /__ \ dPsij   dDj   (1 + (w.ti)^2)^2     dDj.dPsij    1 + (w.ti)^2 /
                       i=m

calc_diff_S2_te_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci    dti  /        1 - (w.ti)^2                         (te + ti)^2 - (w.te.ti)^2   \ 
    ---------  =  -   >  ----- . ---  | S2 . ----------------  +  (1 - S2) . te^2 ----------------------------- |
    dDj.dPsij     5  /__ dPsij   dDj  \      (1 + (w.ti)^2)^2                     ((te + ti)^2 + (w.te.ti)^2)^2 /
                     i=m

calc_aniso_S2_te_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \   /  dci    dti  /        1 - (w.ti)^2                         (te + ti)^2 - (w.te.ti)^2   \ 
    ---------  =  -   >  | ----- . ---  | S2 . ----------------  +  (1 - S2) . te^2 ----------------------------- |
    dDj.dPsij     5  /__ \ dPsij   dDj  \      (1 + (w.ti)^2)^2                     ((te + ti)^2 + (w.te.ti)^2)^2 /
                     i=m

                                d2ci       /      S2             (1 - S2)(te + ti)te    \ \ 
                           +  --------- ti | ------------  +  ------------------------- | |
                              dDj.dPsij    \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 / /

calc_diff_S2_d2jw_dDjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2 double partial derivative of the original
model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \        dti    1 - (w.ti)^2
    -------  =  -   >  ci . ---  ----------------
    dDj.dS2     5  /__      dDj  (1 + (w.ti)^2)^2
                   i=m

calc_aniso_S2_d2jw_dDjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2 double partial derivative of the original
model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti    1 - (w.ti)^2       dci           1       \ 
    -------  =  -   >  | ci . ---  ----------------  +  --- . ti ------------ |
    dDj.dS2     5  /__ \      dDj  (1 + (w.ti)^2)^2     dDj      1 + (w.ti)^2 /
                   i=m

calc_diff_S2_te_d2jw_dDjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2 double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \        dti  /   1 - (w.ti)^2              (te + ti)^2 - (w.te.ti)^2   \ 
    -------  =  -   >  ci . ---  | ----------------  -  te^2 ----------------------------- |
    dDj.dS2     5  /__      dDj  \ (1 + (w.ti)^2)^2          ((te + ti)^2 + (w.te.ti)^2)^2 /
                   i=m

calc_aniso_S2_te_d2jw_dDjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2 double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti  /   1 - (w.ti)^2              (te + ti)^2 - (w.te.ti)^2   \ 
    -------  =  -   >  | ci . ---  | ----------------  -  te^2 ----------------------------- |
    dDj.dS2     5  /__ \      dDj  \ (1 + (w.ti)^2)^2          ((te + ti)^2 + (w.te.ti)^2)^2 /
                   i=m

                            dci      /      1                 (te + ti)te         \ \ 
                         +  --- . ti | ------------  -  ------------------------- | |
                            dDj      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 / /

calc_diff_S2_te_d2jw_dDjdte(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - te double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                                _n_
     d2J(w)     4               \        dti                   (te + ti)^2 - 3(w.te.ti)^2
    -------  =  - (1 - S2) . te  >  ci . --- . ti . (te + ti) -----------------------------
    dDj.dte     5               /__      dDj                  ((te + ti)^2 + (w.te.ti)^2)^3
                                i=m

calc_aniso_S2_te_d2jw_dDjdte(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - te double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                          _n_
     d2J(w)     2         \   /       dti                        (te + ti)^2 - 3(w.te.ti)^2       dci            (te + ti)^2 - (w.te.ti)^2   \ 
    -------  =  - (1 - S2) >  | 2ci . --- . te . ti . (te + ti) -----------------------------  +  --- . ti^2 . ----------------------------- |
    dDj.dte     5         /__ \       dDj                       ((te + ti)^2 + (w.te.ti)^2)^3     dDj          ((te + ti)^2 + (w.te.ti)^2)^2 /
                          i=m

calc_diff_d2jw_dPsijdPsik(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - Psik double partial derivative of the
original model-free formula with no parameters {} together with diffusion tensor parameters.

The model-free Hessian is:

                       _n_
       d2J(w)       2  \       d2ci           ti
    -----------  =  -   >  ----------- . ------------
    dPsij.dPsik     5  /__ dPsij.dPsik   1 + (w.ti)^2
                       i=m

calc_diff_S2_d2jw_dPsijdPsik(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - Psik double partial derivative of the
original model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                         _n_
       d2J(w)       2    \       d2ci           ti
    -----------  =  - S2  >  ----------- . ------------
    dPsij.dPsik     5    /__ dPsij.dPsik   1 + (w.ti)^2
                         i=m

calc_diff_S2_te_d2jw_dPsijdPsik(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - Psik double partial derivative of the
original model-free formula with the parameters {S2, te} together with diffusion tensor
parameters.

The model-free Hessian is:

                       _n_
       d2J(w)       2  \       d2ci         /      S2             (1 - S2)(te + ti)te    \ 
    -----------  =  -   >  ----------- . ti | ------------  +  ------------------------- |
    dPsij.dPsik     5  /__ dPsij.dPsik      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                       i=m

calc_diff_S2_d2jw_dPsijdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - S2 double partial derivative of the original
model-free formula with the parameter {S2} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci            1
    ---------  =  -   >  ----- . ti ------------
    dPsij.dS2     5  /__ dPsij      1 + (w.ti)^2
                     i=m

calc_diff_S2_te_d2jw_dPsijdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - S2 double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci       /      1                 (te + ti)te         \ 
    ---------  =  -   >  ----- . ti | ------------  -  ------------------------- |
    dPsij.dS2     5  /__ dPsij      \ 1 + (w.ti)^2     (te + ti)^2 + (w.te.ti)^2 /
                     i=m

calc_diff_S2_te_d2jw_dPsijdte(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - te double partial derivative of the original
model-free formula with the parameters {S2, te} together with diffusion tensor parameters.

The model-free Hessian is:

                             _n_
      d2J(w)      2          \    dci             (te + ti)^2 - (w.te.ti)^2
    ---------  =  - (1 - S2)  >  ----- . ti^2 . -----------------------------
    dPsij.dte     5          /__ dPsij          ((te + ti)^2 + (w.te.ti)^2)^2
                             i=m

calc_S2_te_d2jw_dS2dte(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2 - te double partial derivative of the original
model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free Hessian is:

                    _n_
     d2J(w)       2 \               (te + ti)^2 - (w.te.ti)^2
    -------  =  - -  >  ci . ti^2 -----------------------------
    dS2.dte       5 /__           ((te + ti)^2 + (w.te.ti)^2)^2
                    i=m

calc_S2_te_d2jw_dte2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the te - te double partial derivative of the original
model-free formula with the parameters {S2, te} with or without diffusion tensor parameters.

The model-free Hessian is:

                            _n_
    d2J(w)       4          \             (te + ti)^3 + 3.w^2.ti^3.te.(te + ti) - (w.ti)^4.te^3
    ------  =  - - (1 - S2)  >  ci . ti^2 -----------------------------------------------------
    dte**2       5          /__                        ((te + ti)^2 + (w.te.ti)^2)^3
                            i=m

calc_diff_S2f_S2_ts_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \      /    dti   dti  /             3 - (w.ti)^2                         (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
    -------  =  -   >  ci | -2 --- . ---  | S2.w^2.ti ----------------  +  (S2f - S2) . ts^2 ---------------------------------------------------- |
    dDj.dDk     5  /__    \    dDj   dDk  \           (1 + (w.ti)^2)^3                                 ((ts + ti)^2 + (w.ts.ti)^2)^3              /
                   i=m

                             d2ti   /        1 - (w.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ \ 
                         +  ------- | S2 . ----------------  +  (S2f - S2) . ts^2 ----------------------------- | |
                            dDj.dDk \      (1 + (w.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 / /

calc_aniso_S2f_S2_ts_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti   dti  /             3 - (w.ti)^2                         (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
    -------  =  -   >  | -2ci --- . ---  | S2.w^2.ti ----------------  +  (S2f - S2) . ts^2 ---------------------------------------------------- |
    dDj.dDk     5  /__ \      dDj   dDk  \           (1 + (w.ti)^2)^3                                 ((ts + ti)^2 + (w.ts.ti)^2)^3              /
                   i=m

                            / dti   dci     dti   dci         d2ti   \ /        1 - (w.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
                         +  | --- . ---  +  --- . ---  +  ci ------- | | S2 . ----------------  +  (S2f - S2) . ts^2 ----------------------------- |
                            \ dDj   dDk     dDk   dDj        dDj.dDk / \      (1 + (w.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /


                             d2ci        /      S2            (S2f - S2)(ts + ti)ts   \ \ 
                         +  ------- . ti | ------------  +  ------------------------- | |
                            dDj.dDk      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2_ts_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \      /    dti   dti  /             3 - (w.ti)^2                        (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
    -------  =  -   >  ci | -2 --- . ---  | S2.w^2.ti ----------------  +  (1 - S2f) . tf^2 ----------------------------------------------------
    dDj.dDk     5  /__    \    dDj   dDk  \           (1 + (w.ti)^2)^3                                ((tf + ti)^2 + (w.tf.ti)^2)^3
                   i=m

                                                                (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
                                           +  (S2f - S2) . ts^2 ---------------------------------------------------- |
                                                                          ((ts + ti)^2 + (w.ts.ti)^2)^3              /


                             d2ti   /        1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
                         +  ------- | S2 . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  (S2f - S2) . ts^2 ----------------------------- |
                            dDj.dDk \      (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /

calc_aniso_S2f_tf_S2_ts_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti   dti  /             3 - (w.ti)^2                        (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
    -------  =  -   >  | -2ci --- . ---  | S2.w^2.ti ----------------  +  (1 - S2f) . tf^2 ----------------------------------------------------
    dDj.dDk     5  /__ \      dDj   dDk  \           (1 + (w.ti)^2)^3                                ((tf + ti)^2 + (w.tf.ti)^2)^3
                   i=m

                                                                (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
                                           +  (S2f - S2) . ts^2 ---------------------------------------------------- |
                                                                          ((ts + ti)^2 + (w.ts.ti)^2)^3              /


                            / dti   dci     dti   dci         d2ti   \ /        1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2
                         +  | --- . ---  +  --- . ---  +  ci ------- | | S2 . ----------------  +  (1 - S2f) . tf^2 -----------------------------
                            \ dDj   dDk     dDk   dDj        dDj.dDk / \      (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2


                                                                                                (ts + ti)^2 - (w.ts.ti)^2   \ 
                                                                         +  (S2f - S2) . ts^2 ----------------------------- |
                                                                                              ((ts + ti)^2 + (w.ts.ti)^2)^2 /


                             d2ci        /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ \ 
                         +  ------- . ti | ------------  +  -------------------------  +  ------------------------- | |
                            dDj.dDk      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2_ts_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci    dti  /        1 - (w.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
    ---------  =  -   >  ----- . ---  | S2 . ----------------  +  (S2f - S2) . ts^2 ----------------------------- |
    dDj.dPsij     5  /__ dPsij   dDj  \      (1 + (w.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=m

calc_aniso_S2f_S2_ts_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \   /  dci    dti  /        1 - (w.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
    ---------  =  -   >  | ----- . ---  | S2 . ----------------  +  (S2f - S2) . ts^2 ----------------------------- |
    dDj.dPsij     5  /__ \ dPsij   dDj  \      (1 + (w.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=m

                                d2ci         /      S2            (S2f - S2)(ts + ti)ts   \ \ 
                           +  --------- . ti | ------------  +  ------------------------- | |
                              dDj.dPsij      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2_ts_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci    dti  /        1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
    ---------  =  -   >  ----- . ---  | S2 . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  (S2f - S2) . ts^2 ----------------------------- |
    dDj.dPsij     5  /__ dPsij   dDj  \      (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=m

calc_aniso_S2f_tf_S2_ts_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \   /  dci    dti  /        1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                           (ts + ti)^2 - (w.ts.ti)^2   \ 
    ---------  =  -   >  | ----- . ---  | S2 . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  (S2f - S2) . ts^2 ----------------------------- |
    dDj.dPsij     5  /__ \ dPsij   dDj  \      (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                       ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=m

                                d2ci         /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ \ 
                           +  --------- . ti | ------------  +  -------------------------  +  ------------------------- | |
                              dDj.dPsij      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2_ts_d2jw_dDjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2 double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \        dti  /   1 - (w.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
    -------  =  -   >  ci . ---  | ----------------  -  ts^2 ----------------------------- |
    dDj.dS2     5  /__      dDj  \ (1 + (w.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                   i=m

calc_aniso_S2f_S2_ts_d2jw_dDjdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2 double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti  /   1 - (w.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
    -------  =  -   >  | ci . ---  | ----------------  -  ts^2 ----------------------------- |
    dDj.dS2     5  /__ \      dDj  \ (1 + (w.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                   i=m

                            dci      /      1                 (ts + ti)ts         \ \ 
                         +  --- . ti | ------------  -  ------------------------- | |
                            dDj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2_ts_d2jw_dDjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                    _n_
     d2J(w)      2  \        dti         (ts + ti)^2 - (w.ts.ti)^2
    --------  =  -   >  ci . ---  ts^2 -----------------------------
    dDj.dS2f     5  /__      dDj       ((ts + ti)^2 + (w.ts.ti)^2)^2
                    i=m

calc_aniso_S2f_S2_ts_d2jw_dDjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                    _n_
     d2J(w)      2  \   /      dti         (ts + ti)^2 - (w.ts.ti)^2       dci            (ts + ti)ts         \ 
    --------  =  -   >  | ci . ---  ts^2 -----------------------------  +  --- . ti ------------------------- |
    dDj.dS2f     5  /__ \      dDj       ((ts + ti)^2 + (w.ts.ti)^2)^2     dDj      (ts + ti)^2 + (w.ts.ti)^2 /
                    i=m

calc_diff_S2f_tf_S2_ts_d2jw_dDjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                      _n_
     d2J(w)        2  \        dti  /        (tf + ti)^2 - (w.tf.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
    --------  =  - -   >  ci . ---  | tf^2 -----------------------------  -  ts^2 ----------------------------- |
    dDj.dS2f       5  /__      dDj  \      ((tf + ti)^2 + (w.tf.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                      i=m

calc_aniso_S2f_tf_S2_ts_d2jw_dDjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                      _n_
     d2J(w)        2  \   /      dti  /        (tf + ti)^2 - (w.tf.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
    --------  =  - -   >  | ci . ---  | tf^2 -----------------------------  -  ts^2 ----------------------------- |
    dDj.dS2f       5  /__ \      dDj  \      ((tf + ti)^2 + (w.tf.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                      i=m

                               dci      /       (tf + ti)tf                   (ts + ti)ts         \ \ 
                            +  --- . ti | -------------------------  -  ------------------------- | |
                               dDj      \ (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2_ts_d2jw_dDjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                                 _n_
     d2J(w)     4                \        dti                   (tf + ti)^2 - 3(w.tf.ti)^2
    -------  =  - (1 - S2f) . tf  >  ci . --- . ti . (tf + ti) -----------------------------
    dDj.dtf     5                /__      dDj                  ((tf + ti)^2 + (w.tf.ti)^2)^3
                                 i=m

calc_aniso_S2f_tf_S2_ts_d2jw_dDjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                            _n_
     d2J(w)     2           \   /       dti                        (tf + ti)^2 - 3(w.tf.ti)^2       dci          (tf + ti)^2 - (w.tf.ti)^2   \ 
    -------  =  - (1 - S2f)  >  | 2ci . --- . tf . ti . (tf + ti) -----------------------------  +  --- . ti^2 ----------------------------- |
    dDj.dtf     5           /__ \       dDj                       ((tf + ti)^2 + (w.tf.ti)^2)^3     dDj        ((tf + ti)^2 + (w.tf.ti)^2)^2 /
                            i=m

calc_diff_S2f_S2_ts_d2jw_dDjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                                  _n_
     d2J(w)     4                 \        dti                   (ts + ti)^2 - 3(w.ts.ti)^2
    -------  =  - (S2f - S2) . ts  >  ci . --- . ti . (ts + ti) -----------------------------
    dDj.dts     5                 /__      dDj                  ((ts + ti)^2 + (w.ts.ti)^2)^3
                                  i=m

calc_aniso_S2f_S2_ts_d2jw_dDjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                             _n_
     d2J(w)     2            \   /       dti                        (ts + ti)^2 - 3(w.ts.ti)^2       dci          (ts + ti)^2 - (w.ts.ti)^2   \ 
    -------  =  - (S2f - S2)  >  | 2ci . --- . ts . ti . (ts + ti) -----------------------------  +  --- . ti^2 ----------------------------- |
    dDj.dts     5            /__ \       dDj                       ((ts + ti)^2 + (w.ts.ti)^2)^3     dDj        ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                             i=m

calc_diff_S2f_S2_ts_d2jw_dPsijdPsik(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - Psik double partial derivative of the
extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                       _n_
       d2J(w)       2  \       d2ci         /      S2            (S2f - S2)(ts + ti)ts   \ 
    -----------  =  -   >  ----------- . ti | ------------  +  ------------------------- |
    dPsij.dPsik     5  /__ dPsij.dPsik      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                       i=m

calc_diff_S2f_tf_S2_ts_d2jw_dPsijdPsik(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - Psik double partial derivative of the
extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                       _n_
       d2J(w)       2  \       d2ci         /      S2            (1 - S2f)(tf + ti)tf          (S2f - S2)(ts + ti)ts   \ 
    -----------  =  -   >  ----------- . ti | ------------  +  -------------------------  +  ------------------------- |
    dPsij.dPsik     5  /__ dPsij.dPsik      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                       i=m

calc_diff_S2f_S2_ts_d2jw_dPsijdS2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - S2 double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} and {S2f, tf, S2, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci       /      1                 (ts + ti)ts         \ 
    ---------  =  -   >  ----- . ti | ------------  -  ------------------------- |
    dPsij.dS2     5  /__ dPsij      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                     i=m

calc_diff_S2f_S2_ts_d2jw_dPsijdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - S2f double partial derivative of the
extended model-free formula with the parameters {S2f, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                      _n_
      d2J(w)       2  \    dci             (ts + ti)ts
    ----------  =  -   >  ----- . ti -------------------------
    dPsij.dS2f     5  /__ dPsij      (ts + ti)^2 + (w.ts.ti)^2
                      i=m

calc_diff_S2f_tf_S2_ts_d2jw_dPsijdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - S2f double partial derivative of the
extended model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                        _n_
      d2J(w)         2  \    dci       /       (tf + ti)tf                   (ts + ti)ts         \ 
    ----------  =  - -   >  ----- . ti | -------------------------  -  ------------------------- |
    dPsij.dS2f       5  /__ dPsij      \ (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                        i=m

calc_diff_S2f_tf_S2_ts_d2jw_dPsijdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                              _n_
      d2J(w)      2           \    dci           (tf + ti)^2 - (w.tf.ti)^2
    ---------  =  - (1 - S2f)  >  ----- . ti^2 -----------------------------
    dPsij.dtf     5           /__ dPsij        ((tf + ti)^2 + (w.tf.ti)^2)^2
                              i=m

calc_diff_S2f_S2_ts_d2jw_dPsijdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                               _n_
      d2J(w)      2            \    dci           (ts + ti)^2 - (w.ts.ti)^2
    ---------  =  - (S2f - S2)  >  ----- . ti^2 -----------------------------
    dPsij.dts     5            /__ dPsij        ((ts + ti)^2 + (w.ts.ti)^2)^2
                               i=m

calc_S2f_S2_ts_d2jw_dS2dts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2 - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
diffusion tensor parameters.

The model-free Hessian is:

                     _n_
     d2J(w)       2  \               (ts + ti)^2 - (w.ts.ti)^2
    -------  =  - -   >  ci . ti^2 -----------------------------
    dS2.dts       5  /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                     i=m

calc_S2f_tf_S2_ts_d2jw_dS2fdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor
parameters.

The model-free Hessian is:

                      _n_
     d2J(w)        2  \               (tf + ti)^2 - (w.tf.ti)^2
    --------  =  - -   >  ci . ti^2 -----------------------------
    dS2f.dtf       5  /__           ((tf + ti)^2 + (w.tf.ti)^2)^2
                      i=m

calc_S2f_S2_ts_d2jw_dS2fdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
diffusion tensor parameters.

The model-free Hessian is:

                    _n_
     d2J(w)      2  \               (ts + ti)^2 - (w.ts.ti)^2
    --------  =  -   >  ci . ti^2 -----------------------------
    dS2f.dts     5  /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                    i=m

calc_S2f_tf_S2_ts_d2jw_dtf2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the tf - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2, ts} with or without diffusion tensor
parameters.

The model-free Hessian is:

                             _n_
    d2J(w)       4           \             (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
    ------  =  - - (1 - S2f)  >  ci . ti^2 -----------------------------------------------------
    dtf**2       5           /__                        ((tf + ti)^2 + (w.tf.ti)^2)^3
                             i=m

calc_S2f_S2_ts_d2jw_dts2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the ts - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2, ts} or {S2f, tf, S2, ts} with or without
diffusion tensor parameters.

The model-free Hessian is:

                              _n_
    d2J(w)       4            \             (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
    ------  =  - - (S2f - S2)  >  ci . ti^2 -----------------------------------------------------
    dts**2       5            /__                        ((ts + ti)^2 + (w.ts.ti)^2)^3
                              i=m

calc_diff_S2f_S2s_ts_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \      /    dti   dti  /                  3 - (w.ti)^2                           (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
    -------  =  -   >  ci | -2 --- . ---  | S2f.S2s.w^2.ti ----------------  +  S2f(1 - S2s) . ts^2 ---------------------------------------------------- |
    dDj.dDk     5  /__    \    dDj   dDk  \                (1 + (w.ti)^2)^3                                   ((ts + ti)^2 + (w.ts.ti)^2)^3              /
                   i=m

                             d2ti   /               1 - (w.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ \ 
                         +  ------- | S2f . S2s . ----------------  +  S2f(1 - S2s) . ts^2 ----------------------------- | |
                            dDj.dDk \             (1 + (w.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 / /

calc_aniso_S2f_S2s_ts_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti   dti  /                  3 - (w.ti)^2                           (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
    -------  =  -   >  | -2ci --- . ---  | S2f.S2s.w^2.ti ----------------  +  S2f(1 - S2s) . ts^2 ---------------------------------------------------- |
    dDj.dDk     5  /__ \      dDj   dDk  \                (1 + (w.ti)^2)^3                                   ((ts + ti)^2 + (w.ts.ti)^2)^3              /
                   i=m

                            / dti   dci     dti   dci         d2ti   \ /               1 - (w.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ 
                         +  | --- . ---  +  --- . ---  +  ci ------- | | S2f . S2s . ----------------  +  S2f(1 - S2s) . ts^2 ----------------------------- |
                            \ dDj   dDk     dDk   dDj        dDj.dDk / \             (1 + (w.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 /


                             d2ci        /   S2f.S2s         S2f(1 - S2s)(ts + ti)ts  \ \ 
                         +  ------- . ti | ------------  +  ------------------------- | |
                            dDj.dDk      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \      /    dti   dti  /                  3 - (w.ti)^2                        (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
    -------  =  -   >  ci | -2 --- . ---  | S2f.S2s.w^2.ti ----------------  +  (1 - S2f) . tf^2 ----------------------------------------------------
    dDj.dDk     5  /__    \    dDj   dDk  \                (1 + (w.ti)^2)^3                                ((tf + ti)^2 + (w.tf.ti)^2)^3
                   i=m

                                                                  (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
                                           +  S2f(1 - S2s) . ts^2 ---------------------------------------------------- |
                                                                            ((ts + ti)^2 + (w.ts.ti)^2)^3              /


                             d2ti   /               1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ 
                         +  ------- | S2f . S2s . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  S2f(1 - S2s) . ts^2 ----------------------------- |
                            dDj.dDk \             (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 /

calc_aniso_S2f_tf_S2s_ts_d2jw_dDjdDk(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Dk double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                   _n_
     d2J(w)     2  \   /      dti   dti  /                  3 - (w.ti)^2                        (tf + ti)^3 + 3.w^2.tf^3.ti(tf + ti) - (w.tf)^4.ti^3
    -------  =  -   >  | -2ci --- . ---  | S2f.S2s.w^2.ti ----------------  +  (1 - S2f) . tf^2 ----------------------------------------------------
    dDj.dDk     5  /__ \      dDj   dDk  \                (1 + (w.ti)^2)^3                                ((tf + ti)^2 + (w.tf.ti)^2)^3
                   i=m

                                                                  (ts + ti)^3 + 3.w^2.ts^3.ti(ts + ti) - (w.ts)^4.ti^3 \ 
                                           +  S2f(1 - S2s) . ts^2 ---------------------------------------------------- |
                                                                            ((ts + ti)^2 + (w.ts.ti)^2)^3              /


                            / dti   dci     dti   dci         d2ti   \ /               1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2
                         +  | --- . ---  +  --- . ---  +  ci ------- | | S2f . S2s . ----------------  +  (1 - S2f) . tf^2 -----------------------------
                            \ dDj   dDk     dDk   dDj        dDj.dDk / \             (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2


                                                                                                  (ts + ti)^2 - (w.ts.ti)^2   \ 
                                                                         +  S2f(1 - S2s) . ts^2 ----------------------------- |
                                                                                                ((ts + ti)^2 + (w.ts.ti)^2)^2 /


                             d2ci        /   S2f.S2s          (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ \ 
                         +  ------- . ti | ------------  +  -------------------------  +  ------------------------- | |
                            dDj.dDk      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2s_ts_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci    dti  /               1 - (w.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ 
    ---------  =  -   >  ----- . ---  | S2f . S2s . ----------------  +  S2f(1 - S2s) . ts^2 ----------------------------- |
    dDj.dPsij     5  /__ dPsij   dDj  \             (1 + (w.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=m

calc_aniso_S2f_S2s_ts_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \   /  dci    dti  /               1 - (w.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ 
    ---------  =  -   >  | ----- . ---  | S2f . S2s . ----------------  +  S2f(1 - S2s) . ts^2 ----------------------------- |
    dDj.dPsij     5  /__ \ dPsij   dDj  \             (1 + (w.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=m

                                d2ci         /   S2f.S2s         S2f(1 - S2s)(ts + ti)ts  \ \ 
                           +  --------- . ti | ------------  +  ------------------------- | |
                              dDj.dPsij      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \    dci    dti  /               1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ 
    ---------  =  -   >  ----- . ---  | S2f . S2s . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  S2f(1 - S2s) . ts^2 ----------------------------- |
    dDj.dPsij     5  /__ dPsij   dDj  \             (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=m

calc_aniso_S2f_tf_S2s_ts_d2jw_dDjdPsij(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - Psij double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \   /  dci    dti  /               1 - (w.ti)^2                          (tf + ti)^2 - (w.tf.ti)^2                             (ts + ti)^2 - (w.ts.ti)^2   \ 
    ---------  =  -   >  | ----- . ---  | S2f . S2s . ----------------  +  (1 - S2f) . tf^2 -----------------------------  +  S2f(1 - S2s) . ts^2 ----------------------------- |
    dDj.dPsij     5  /__ \ dPsij   dDj  \             (1 + (w.ti)^2)^2                      ((tf + ti)^2 + (w.tf.ti)^2)^2                         ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                     i=m

                                d2ci         /   S2f.S2s          (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ \ 
                           +  --------- . ti | ------------  +  -------------------------  +  ------------------------- | |
                              dDj.dPsij      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2s_ts_d2jw_dDjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                    _n_
     d2J(w)      2  \        dti  /         1 - (w.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
    --------  =  -   >  ci . ---  | S2s . ----------------  +  (1 - S2s) . ts^2 ----------------------------- |
    dDj.dS2f     5  /__      dDj  \       (1 + (w.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                    i=m

calc_aniso_S2f_S2s_ts_d2jw_dDjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor parameters.

The model-free Hessian is:

                    _n_
     d2J(w)      2  \   /      dti  /         1 - (w.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
    --------  =  -   >  | ci . ---  | S2s . ----------------  +  (1 - S2s) . ts^2 ----------------------------- |
    dDj.dS2f     5  /__ \      dDj  \       (1 + (w.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                    i=m

                            dci      /     S2s            (1 - S2s)(ts + ti)ts    \ \ 
                         +  --- . ti | ------------  +  ------------------------- | |
                            dDj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_d2jw_dDjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                    _n_
     d2J(w)      2  \        dti  /         1 - (w.ti)^2              (tf + ti)^2 - (w.tf.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
    --------  =  -   >  ci . ---  | S2s . ----------------  -  tf^2 -----------------------------  +  (1 - S2s) . ts^2 ----------------------------- |
    dDj.dS2f     5  /__      dDj  \       (1 + (w.ti)^2)^2          ((tf + ti)^2 + (w.tf.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                    i=m

calc_aniso_S2f_tf_S2s_ts_d2jw_dDjdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2f double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                    _n_
     d2J(w)      2  \   /      dti  /         1 - (w.ti)^2              (tf + ti)^2 - (w.tf.ti)^2                          (ts + ti)^2 - (w.ts.ti)^2   \ 
    --------  =  -   >  | ci . ---  | S2s . ----------------  -  tf^2 -----------------------------  +  (1 - S2s) . ts^2 ----------------------------- |
    dDj.dS2f     5  /__ \      dDj  \       (1 + (w.ti)^2)^2          ((tf + ti)^2 + (w.tf.ti)^2)^2                      ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                    i=m

                             dci      /     S2s                (tf + ti)tf               (1 - S2s)(ts + ti)ts    \ \ 
                          +  --- . ti | ------------  -  -------------------------  +  ------------------------- | |
                             dDj      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_S2s_ts_d2jw_dDjdS2s(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2s double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                       _n_
     d2J(w)      2     \        dti  /   1 - (w.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
    --------  =  - S2f  >  ci . ---  | ----------------  -  ts^2 ----------------------------- |
    dDj.dS2s     5     /__      dDj  \ (1 + (w.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                       i=m

calc_aniso_S2f_S2s_ts_d2jw_dDjdS2s(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - S2s double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                       _n_
     d2J(w)      2     \   /      dti  /   1 - (w.ti)^2              (ts + ti)^2 - (w.ts.ti)^2   \ 
    --------  =  - S2f  >  | ci . ---  | ----------------  -  ts^2 ----------------------------- |
    dDj.dS2s     5     /__ \      dDj  \ (1 + (w.ti)^2)^2          ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                       i=m

                            dci      /      1                 (ts + ti)ts         \ \ 
                         +  --- . ti | ------------  -  ------------------------- | |
                            dDj      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 / /

calc_diff_S2f_tf_S2s_ts_d2jw_dDjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                                 _n_
     d2J(w)     4                \        dti                   (tf + ti)^2 - 3(w.tf.ti)^2
    -------  =  - (1 - S2f) . tf  >  ci . --- . ti . (tf + ti) -----------------------------
    dDj.dtf     5                /__      dDj                  ((tf + ti)^2 + (w.tf.ti)^2)^3
                                 i=m

calc_aniso_S2f_tf_S2s_ts_d2jw_dDjdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                            _n_
     d2J(w)     2           \   /       dti                        (tf + ti)^2 - 3(w.tf.ti)^2       dci          (tf + ti)^2 - (w.tf.ti)^2   \ 
    -------  =  - (1 - S2f)  >  | 2ci . --- . tf . ti . (tf + ti) -----------------------------  +  --- . ti^2 ----------------------------- |
    dDj.dtf     5           /__ \       dDj                       ((tf + ti)^2 + (w.tf.ti)^2)^3     dDj        ((tf + ti)^2 + (w.tf.ti)^2)^2 /
                            i=m

calc_diff_S2f_S2s_ts_d2jw_dDjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                                    _n_
     d2J(w)     4                   \        dti                   (ts + ti)^2 - 3(w.ts.ti)^2
    -------  =  - S2f(1 - S2s) . ts  >  ci . --- . ti . (ts + ti) -----------------------------
    dDj.dts     5                   /__      dDj                  ((ts + ti)^2 + (w.ts.ti)^2)^3
                                    i=m

calc_aniso_S2f_S2s_ts_d2jw_dDjdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Dj - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                               _n_
     d2J(w)     2              \   /       dti                        (ts + ti)^2 - 3(w.ts.ti)^2       dci          (ts + ti)^2 - (w.ts.ti)^2   \ 
    -------  =  - S2f(1 - S2s)  >  | 2ci . --- . ts . ti . (ts + ti) -----------------------------  +  --- . ti^2 ----------------------------- |
    dDj.dts     5              /__ \       dDj                       ((ts + ti)^2 + (w.ts.ti)^2)^3     dDj        ((ts + ti)^2 + (w.ts.ti)^2)^2 /
                               i=m

calc_diff_S2f_S2s_ts_d2jw_dPsijdPsik(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - Psik double partial derivative of the
extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                       _n_
       d2J(w)       2  \       d2ci         /  S2f . S2s        S2f(1 - S2s)(ts + ti)ts  \ 
    -----------  =  -   >  ----------- . ti | ------------  +  ------------------------- |
    dPsij.dPsik     5  /__ dPsij.dPsik      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                       i=m

calc_diff_S2f_tf_S2s_ts_d2jw_dPsijdPsik(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - Psik double partial derivative of the
extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                       _n_
       d2J(w)       2  \       d2ci         /  S2f . S2s         (1 - S2f)(tf + ti)tf         S2f(1 - S2s)(ts + ti)ts  \ 
    -----------  =  -   >  ----------- . ti | ------------  +  -------------------------  +  ------------------------- |
    dPsij.dPsik     5  /__ dPsij.dPsik      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                       i=m

calc_diff_S2f_S2s_ts_d2jw_dPsijdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - S2f double partial derivative of the
extended model-free formula with the parameters {S2f, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                      _n_
      d2J(w)       2  \    dci       /      S2s           (1 - S2s)(ts + ti)ts    \ 
    ----------  =  -   >  ----- . ti | ------------  +  ------------------------- |
    dPsij.dS2f     5  /__ dPsij      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                      i=m

calc_diff_S2f_tf_S2s_ts_d2jw_dPsijdS2f(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - S2f double partial derivative of the
extended model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                      _n_
      d2J(w)       2  \    dci       /      S2s               (tf + ti)tf               (1 - S2s)(ts + ti)ts    \ 
    ----------  =  -   >  ----- . ti | ------------  -  -------------------------  +  ------------------------- |
    dPsij.dS2f     5  /__ dPsij      \ 1 + (w.ti)^2     (tf + ti)^2 + (w.tf.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                      i=m

calc_diff_S2f_tf_S2s_ts_d2jw_dPsijdS2s(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - S2 double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} and {S2f, tf, S2s, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                         _n_
      d2J(w)       2     \    dci       /      1                 (ts + ti)ts         \ 
    ----------  =  - S2f  >  ----- . ti | ------------  -  ------------------------- |
    dPsij.dS2s     5     /__ dPsij      \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                         i=m

calc_diff_S2f_tf_S2s_ts_d2jw_dPsijdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} together with diffusion tensor
parameters.

The model-free Hessian is:

                              _n_
      d2J(w)      2           \    dci           (tf + ti)^2 - (w.tf.ti)^2
    ---------  =  - (1 - S2f)  >  ----- . ti^2 -----------------------------
    dPsij.dtf     5           /__ dPsij        ((tf + ti)^2 + (w.tf.ti)^2)^2
                              i=m

calc_diff_S2f_S2s_ts_d2jw_dPsijdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the Psij - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} together with
diffusion tensor parameters.

The model-free Hessian is:

                                 _n_
      d2J(w)      2              \    dci           (ts + ti)^2 - (w.ts.ti)^2
    ---------  =  - S2f(1 - S2s)  >  ----- . ti^2 -----------------------------
    dPsij.dts     5              /__ dPsij        ((ts + ti)^2 + (w.ts.ti)^2)^2
                                 i=m

calc_S2f_S2s_ts_d2jw_dS2fdS2s(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - S2s double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
diffusion tensor parameters.

The model-free Hessian is:

                     _n_
      d2J(w)      2  \           /      1                 (ts + ti).ts        \ 
    ---------  =  -   >  ci . ti | ------------  -  ------------------------- |
    dS2f.dS2s     5  /__         \ 1 + (w.ti)^2     (ts + ti)^2 + (w.ts.ti)^2 /
                     i=m

calc_S2f_tf_S2s_ts_d2jw_dS2fdtf(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor
parameters.

The model-free Hessian is:

                      _n_
     d2J(w)        2  \               (tf + ti)^2 - (w.tf.ti)^2
    --------  =  - -   >  ci . ti^2 -----------------------------
    dS2f.dtf       5  /__           ((tf + ti)^2 + (w.tf.ti)^2)^2
                      i=m

calc_S2f_S2s_ts_d2jw_dS2fdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2f - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
diffusion tensor parameters.

The model-free Hessian is:

                             _n_
     d2J(w)      2           \               (ts + ti)^2 - (w.ts.ti)^2
    --------  =  - (1 - S2s)  >  ci . ti^2 -----------------------------
    dS2f.dts     5           /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                             i=m

calc_S2f_S2s_ts_d2jw_dS2sdts(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the S2s - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
diffusion tensor parameters.

The model-free Hessian is:

                         _n_
     d2J(w)        2     \               (ts + ti)^2 - (w.ts.ti)^2
    --------  =  - - S2f  >  ci . ti^2 -----------------------------
    dS2s.dts       5     /__           ((ts + ti)^2 + (w.ts.ti)^2)^2
                         i=m

calc_S2f_tf_S2s_ts_d2jw_dtf2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the tf - tf double partial derivative of the extended
model-free formula with the parameters {S2f, tf, S2s, ts} with or without diffusion tensor
parameters.

The model-free Hessian is:

                             _n_
    d2J(w)       4           \             (tf + ti)^3 + 3.w^2.ti^3.tf.(tf + ti) - (w.ti)^4.tf^3
    ------  =  - - (1 - S2f)  >  ci . ti^2 -----------------------------------------------------
    dtf**2       5           /__                        ((tf + ti)^2 + (w.tf.ti)^2)^3
                             i=m

calc_S2f_S2s_ts_d2jw_dts2(data, params, j, k)

Spectral density Hessian.

Calculate the spectral desity values for the ts - ts double partial derivative of the extended
model-free formula with the parameters {S2f, S2s, ts} or {S2f, tf, S2s, ts} with or without
diffusion tensor parameters.

The model-free Hessian is:

                                _n_
    d2J(w)       4              \             (ts + ti)^3 + 3.w^2.ti^3.ts.(ts + ti) - (w.ti)^4.ts^3
    ------  =  - - S2f(1 - S2s)  >  ci . ti^2 -----------------------------------------------------
    dts**2       5              /__                        ((ts + ti)^2 + (w.ts.ti)^2)^3
                                i=m