OwenModel.compute\_jacobian
===========================

.. currentmodule:: giant.camera_models.owen_model

:mod:`giant.camera_models.owen_model`\:

.. automethod:: OwenModel.compute_jacobian

Example::

    >>> from giant.camera_models import OwenModel
    >>> model = OwenModel(kx=3000, ky=4000, px=500, py=500, 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{d}} &
    \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\mathbf{x}_I}
    \frac{\partial\mathbf{x}_I}{\partial f} \\
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{d}} =
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{x}_I'}
    \frac{\partial\mathbf{x}_I'}{\partial\mathbf{d}} \\
    \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}_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}{cccccc} x_i & y_i & 0 & 0 & 1 & 0 \\
    0 & 0 & x_i & y_i & 0 & 1 \end{array}\right] \\
    \frac{\partial\mathbf{x}_P}{\partial\mathbf{a}} = \left[\begin{array}{cc} k_x & k_{xy} \\
    k_{yx} & 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 & k_{xy} \\ k_{yx} & k_y \end{array}\right] \\
    \begin{split}
    \frac{\partial\mathbf{x}_I'}{\partial\mathbf{x}_I} = &
    \left(1 + \epsilon_2r^2+\epsilon_4r^4+\epsilon_5y_I+\epsilon_6x_I\right)\mathbf{I}_{2\times 2}+
    \left(\epsilon_1r+\epsilon_3r^3\right)
    \left[\begin{array}{cc}0 & -1 \\ 1 & 0 \end{array}\right]+\\
    &\left\{\left(2\epsilon_2r+4\epsilon_4r^3\right)
    \mathbf{x}_I+\left(\epsilon_1+3\epsilon_3r^2\right)
    \left[\begin{array}{c} -y_I \\ x_I\end{array}\right]\right\}
    \frac{\mathbf{x}_I^T}{r} +
    \mathbf{x}_I\left[\begin{array}{cc} \epsilon_5 & \epsilon_6\end{array}\right]
    \end{split}\\
    \frac{\partial\mathbf{x}_I}{\partial f} = \frac{1}{z_C'} \\
    \frac{\partial\mathbf{x}_I'}{\partial\mathbf{d}} = \left[
    \begin{array}{cccccc} r^2\mathbf{x}_I & r^4\mathbf{x}_I & y_I\mathbf{x}_I & x_I\mathbf{x}_I &
    r\mathbf{x}_{Ii} & r^3\mathbf{x}_{Ii} \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] \\
    \mathbf{x}_{Ii}=\left[\begin{array}{c}-y_I \\ x_I\end{array}\right]\\
    r=\sqrt{\mathbf{x}_I^T\mathbf{x}_I}
    \end{gather}

where :math:`\mathbf{k}=[k_x \quad k_{xy} \quad k_y \quad k_{yx} \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:`\mathbf{x}_I'` is the distorted gnomic location,
:math:`a_{1-3}` are the temperature coefficients, :math:`T` is the temperature, :math:`f` is the focal
length in units of distance, and
:math:`\mathbf{d}=[\epsilon_2 \quad \epsilon_4 \quad \epsilon_5 \quad \epsilon_6 \quad \epsilon_1 \quad \epsilon_3]` is a vector of the distortion coefficients.