McEwenIllumination.compute_photoclinometry_jacobian

giant.ray_tracer.illumination:

McEwenIllumination.compute_photoclinometry_jacobian(observations, rotation_to_inertial, max_inc=1.2217304763960306, max_emi=1.2217304763960306, max_phase=2.443460952792061, update=False, update_weight=0.005)

This computes the Jacobian matrix of the change in the illumination values given a change in the surface normal (represented by a change in the surface slope) and a change in the local albedo values.

Mathematically the jacobian is

$$\mathbf{J}=\frac{\partial I}{\partial \left[\begin{array}{ccc}{h}_x& h_y& \alpha \end{array}\right]}=\left[\begin{array}{ccc}\frac{\partial I}{\partial h_x}& \frac{\partial I}{\partial h_y}& \frac{\partial I}{\partial \alpha }\end{array}\right]$$

where

$$\begin{array}{r}\frac{\partial I}{\partial h_x}=\frac{\partial I}{\partial \hat{\mathbf{n}}}\frac{\partial \hat{\mathbf{n}}}{\partial \mathbf{n}}\frac{\partial \mathbf{n}}{\partial h_x}\\ \frac{\partial I}{\partial h_y}=\frac{\partial I}{\partial \hat{\mathbf{n}}}\frac{\partial \hat{\mathbf{n}}}{\partial \mathbf{n}}\frac{\partial \mathbf{n}}{\partial h_y}\\ \frac{\partial I}{\partial \alpha }={\alpha }_0(-(1-\beta ){\mathbf{n}}^T\mathbf{i}+2\beta \frac{-{\mathbf{n}}^T\mathbf{i}}{-{\mathbf{n}}^T\mathbf{i}+{\mathbf{n}}^T\mathbf{e}})\\ \frac{\partial I}{\partial \hat{\mathbf{n}}}=\frac{\partial I}{\partial \cos (i)}\frac{\partial \cos (i)}{\partial \hat{\mathbf{n}}}+\frac{\partial I}{\partial \cos (e)}\frac{\partial \cos (e)}{\partial \hat{\mathbf{n}}}\\ \frac{\partial I}{\partial \cos (i)}={\alpha }_0\alpha (-2\beta \frac{\cos i}{(\cos i+\cos e{)}^2})\\ \frac{\partial I}{\partial \cos (e)}={\alpha }_0\alpha (-2\beta \frac{\cos i}{(\cos i+\cos e{)}^2})\\ \frac{\partial \cos (i)}{\partial \hat{\mathbf{n}}}=-{\mathbf{i}}^T\\ \frac{\partial \cos (e)}{\partial \hat{\mathbf{n}}}={\mathbf{e}}^T\\ \frac{\partial \hat{\mathbf{n}}}{\partial \mathbf{n}}=\frac{\mathbf{n}{\mathbf{n}}^T-{\mathbf{n}}^T\mathbf{n}\mathbf{I}}{({\mathbf{n}}^T\mathbf{n}{)}^{1.5}}\end{array}$$

$\cos (i)$ is the cosine of the incidence angle, $\cos (e)$ is the cosine of the exidence angle, and all else is as defined before.

Parameters:

• observations (ndarray) – The observations as a numpy array with type ILLUM_DTYPE.

• rotation_to_inertial (ndarray) – The rotation that takes the frame the observations are expressed in into the the local frame for the surface (usually the local east north up frame)

• max_inc (float) – the maximum incidence angle to consider valid in radians

• max_emi (float) – the maximum emission angle to consider valid in radians

• max_phase (float) – the maximum phase angle to consider valid in radians

• update (bool) – A flag specifying whether to form an update Jacobian

• update_weight (float) – The weight of the prior values if doing an update

Returns:

the jacobian matrix as a n(+3)x3 array (where n is the number of observations) and a boolean array of length n specifying which rows of the jacobian matrix are valid

Return type:

Tuple[ndarray, ndarray]