McEwenIllumination.numeric_derivative

giant.ray_tracer.illumination:

McEwenIllumination.numeric_derivative(observations, rotation_to_inertial, delta=1e-06, max_inc=1.2217304763960306, max_emi=1.2217304763960306, max_phase=2.443460952792061)

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 use numeric finite differencing.

Mathematically the jacobian is

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

Here we compute this using numeric central finite differencing.

This is primarily used for verifying the analytic jacobian given by compute_photoclinometry_jacobian() which should be preferred in most cases due to its speed/efficiency.

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)

• delta (float) – The size of the perturbation to use in the finite differencing

• 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

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]