xopto.pf.util.helpers module

delta2g3(delta: float, g1: float) → float[source]

Computes the third Legendre moment from delta and the first Legendre moment of the scattering phase function.

delta = (1 - g3)/(1 - g1)

g3 = 1 - delta*(1 - g1)

Parameters

  • g1 (float) – First Legendre moment of the scattering phase function.

  • gamma (float) – Parameter gamma.

Returns

g3 – Third Legendre moment of the scattering phase function.

Return type

float

g2delta(g1: float, g3: float) → float[source]

Computes and delta from the first and third Legendre moment.

delta = (1 - g3)/(1 - g1)

Parameters

  • g1 (float) – First Legendre moment of the scattering phase function.

  • g3 (float) – Third Legendre moment of the scattering phase function.

Returns

delta – Similarity parameter delta.

Return type

float

g2gamma(g1: float, g2: float) → float[source]

Computes and return gamma from the first and second Legendre moment.

gamma = (1 - g2)/(1 - g1)

Parameters

  • g1 (float) – First Legendre moment of the scattering phase function.

  • g2 (float) – Second Legendre moment of the scattering phase function.

Returns

gamma – Similarity parameter gamma.

Return type

float

g2gammadelta(g1: float, g2: float, g3: float) → Tuple[float, float, float][source]

Computes the gamma and delta from the first three Legendre moments of the scattering phase function. Returns (g1, gamma, delta).

gamma = (1 - g2)/(1 - g1)

delta = (1 - g3)/(1 - g1)

Parameters

  • g1 (float) – First Legendre moment of the scattering phase function.

  • gamma (float) – Parameter gamma.

  • delta (float) – Parameter delta.

Returns

g – The first three Legendre moments (g1, gamma, delta).

Return type

tuple of 3 elements

gamma2g2(gamma: float, g1: float) → float[source]

Computes the second Legendre moment from gamma and the first Legendre moment of the scattering phase function.

gamma = (1 - g2)/(1 - g1)

g2 = 1 - gamma*(1 - g1)

Parameters

  • g1 (float) – First Legendre moment of the scattering phase function.

  • gamma (float) – Parameter gamma.

Returns

g2 – Second Legendre moment of the scattering phase function.

Return type

float

gammadelta2g(g1: float, gamma: float, delta: float) → Tuple[float, float, float][source]

Computes the second and third Legendre moment from gamma and delta parameters. Returns (g1, g2, g3).

gamma = (1 - g2)/(1 - g1)

delta = (1 - g3)/(1 - g1)

g2 = 1 - gamma*(1 - g1)

g3 = 1 - delta*(1 - g1)

Parameters

  • g1 (float) – First Legendre moment of the scattering phase function.

  • gamma (float) – Parameter gamma.

  • delta (float) – Parameter delta.

Returns

g – The first three Legendre moments (g1, g2, g3).

Return type

tuple of 3 elements

mhg_inverse_g1gamma(g1: float, gamma: float) → Tuple[float, float, bool][source]

Estimates the parameters g and b of the modified Henyey-Greenstein (MHG) scattering phase function that best fit the given values of g1 and gamma, where g1 is the first Legendre moment of the MHG phase function and gamma:

gamma = (1 - g2) / (1 - g1).

Parameters g and b of the scattering phase function MHG can be computed analytically.

Forward analytical model (Calabro et al. JBO 19(7) 2014):

g1 = b * g

g2 = b * g^2 + 2/5 (1 - b)

g3 = b * g^3

g_k = b * g^k where k > 2

…

Parameters g and b can be derived from a quadratic equation. The obtained values are valid only if:

b < 1 (if b == 1 … MHG simplifies to HG) or gamma < 1 + g1.

Parameters

  • g1 (float) – Target value of the first Legendre moment.

  • gamma (float) – Target value of parameter gamma = (1 - g2)/(1 - g1).

Returns

  • g (float) – Estimated values of parameter g.

  • b (float) – Estimated values of parameter b.

  • valid (bool) – A binary mask indicating valid values of g and b.

mhg_inverse_gammadelta(gamma: float, delta: float) → Tuple[float, float, bool][source]

Estimates the parameters g and b of the modified Henyey-Greenstein (MHG) scattering phase function that best fit the given values of gamma and delta, where gamma is

gamma = (1 - g2) / (1 - g1),

and delta is:

delta = (1 - g3) / (1 - g1),

Parameters g and b of the scattering phase function MHG can be computed analytically.

Forward analytical model (Calabro et al. JBO 19(7) 2014):

g1 = b * g

g2 = b * g^2 + 2/5 (1 - b)

g3 = b * g^3

g_k = b * g^k where k > 2

…

Parameters g and b can be derived from a third-order equation. The obtained values are valid only if 0 <= b <= 1.

Parameters

  • gamma (float or np.ndarray) – Target value of parameter gamma = (1 - g2)/(1 - g1).

  • delta (float or np.ndarray) – Target value of parameter delta = (1 - g3)/(1 - g1).

Returns

  • g (float or np.ndarray) – Estimated values of parameter g.

  • b (float or np.ndarray) – Estimated values of parameter b.

  • valid (bool or np.ndarray) – A binary mask indicating valid values of g and b.

sigma(gs: list) → float[source]

Computes parameter sigma from the given Legendre moments.

sigma = sum( (-0.5)^(i - 2) (1 - gi)/(1 - g1), for i = 2, 3, …

where gi is the i-th Legendre moment.

References:

Bodenschatz et al, JBO, 21(3) 2016.

Parameters

gs (tuple, list, ndarray) – First n Legendre moments as (g0, g1, g2, g3, …, gn)

Returns

sigma – Parameter sigma

Return type

float