PinholeModel.compute\_jacobian
==============================

.. currentmodule:: giant.camera_models.pinhole_model

:mod:`giant.camera_models.pinhole_model`\:

.. automethod:: PinholeModel.compute_jacobian

Example::

    >>> from giant.camera_models import PinholeModel
    >>> model = PinholeModel(kx=300, ky=400, px=500, py=500, focal_length=10, a1=1e-5, a2=1e-6,
    >>>                      misalignment = [[1e-12, -2e-14, 3e-10], [2e-15, 1e-13, 3e-10]],
    >>>                      estimation_parameters = ['multiple misalignments'])
    >>> model.compute_jacobian([[[0.5], [0], [1]], [[0.1, 0.2, 0.3], [-0.3, -0.4, -0.5], [4, 5, 6]]],
    >>>                        temperature=[10, -20])
    array([[  0.00000000e+00,  -3.75075000e+03,   0.00000000e+00,
              0.00000000e+00,   0.00000000e+00,   0.00000000e+00],
           [  4.00080000e+03,   2.98059600e-07,  -2.00040000e+03,
              0.00000000e+00,   0.00000000e+00,   0.00000000e+00],
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,
             -5.62612499e+00,  -3.00247537e+03,  -2.25045000e+02],
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,
              4.02330450e+03,   7.50150000e+00,  -1.00020000e+02],
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,
             -9.60191999e+00,  -3.00540096e+03,  -2.40048000e+02],
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,
              4.02640512e+03,   1.28025600e+01,  -1.60032000e+02],
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,
             -1.25025000e+01,  -3.00810150e+03,  -2.50050000e+02],
           [  0.00000000e+00,   0.00000000e+00,   0.00000000e+00,
              4.02858333e+03,   1.66700000e+01,  -2.00040000e+02]])

Mathematically the Jacobian matrix is defined to be

.. math::
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{c}} = \left[\begin{array}{cccc}
    \frac{\partial\mathbf{x}_P}{\partial f} & \frac{\partial\mathbf{x}_P}{\mathbf{k}} &
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{a}} &
    \frac{\partial\mathbf{x}_P}{\partial\boldsymbol{\delta\theta}}\end{array}\right]

where, using the chain rule,

.. math::
    :nowrap:

    \begin{gather}
    \frac{\partial\mathbf{x}_P}{\partial f} =
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{x}_I}
    \frac{\partial\mathbf{x}_I}{\partial f} \\
    \frac{\partial\mathbf{x}_P}{\partial\boldsymbol{\delta\theta}} =
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{x}_I}
    \frac{\partial\mathbf{x}_I}{\partial\mathbf{x}_C'}
    \frac{\partial\mathbf{x}_C'}{\partial\boldsymbol{\delta\theta}}
    \end{gather}

and

.. math::
    :nowrap:

    \begin{gather}
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{k}} = \left[\begin{array}{cccc} x_I & 0 & 1 & 0 \\
    0 & y_I & 0 & 1 \end{array}\right] \\
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{a}} = \left[\begin{array}{cc} k_x & 0 \\
    0 & k_y\end{array}\right] \mathbf{x}_I \left[\begin{array}{ccc} T & T^2 & T^3 \end{array}\right] \\
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{x}_I} = (1+a_1T+a_2T^2+a_3T^3)
    \left[\begin{array}{cc} k_x & 0 \\ 0 & k_y \end{array}\right] \\
    \frac{\partial\mathbf{x}_I}{\partial f} = \frac{1}{z_C'}
    \left[\begin{array}{c}x_C'\\y_C'\end{array}\right] \\
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{x}_I} = (1+a_1T+a_2T^2+a_3T^3)
    \left[\begin{array}{cc} k_x & 0 \\ 0 & k_y \end{array}\right] \\
    \frac{\partial\mathbf{x}_I}{\partial\mathbf{x}_C'} = \frac{f}{z_C'}\left[
    \begin{array}{ccc}1 & 0 & \frac{-x_C'}{z_C'} \\ 0 & 1 & \frac{-y_C'}{z_C'} \end{array}\right] \\
    \frac{\partial\mathbf{x}_C'}{\partial\boldsymbol{\delta\theta}} = \left[\mathbf{x}_C\times\right]
    \end{gather}

where :math:`\mathbf{k}=[k_x \quad k_y \quad p_x \quad p_y]` is a vector of the intrinsic camera parameters,
:math:`\mathbf{a}=[a_1 \quad a_2 \quad a_3]` is a vector of the temperature dependence coefficients,
:math:`\mathbf{x}_C'` is the camera frame point after applying the misalignment,
:math:`\boldsymbol{\delta\theta}` is the misalignment vector, :math:`\mathbf{x}_C` is the camera frame point
before misalignment is applied, :math:`\left[\bullet\times\right]` is the skew-symmetric cross product
matrix formed from :math:`\bullet`,
:math:`\mathbf{x}_P` is the pixel location, :math:`\mathbf{x}_I` is the gnomic location,
:math:`a_{1-3}` are the temperature coefficients, :math:`T` is the temperature, :math:`k_x` is the inverse of
the pixel pitch in the x direction, and :math:`k_y` is the inverse of the pixel pitch in the y direction.