Doc: Added regularization subsection in model-fitting section.

The "model" variable :math:`q_j` is bounded by the lower limit :math:`a_j` and the upper limit :math:`b_j` by construction.

The "engine" variable :math:`Q_j` can take any value, and is actually the parameter that is being adjusted in the fit, although it does not have any physical meaning.

In |SExtractor| three different types of changes of variables :math:`f_j()` are applied, depending on the parameter (:numref:`change_of_variable_table`).

In |SExtractor| three different types of transforms :math:`f_j()` are applied, depending on the parameter (:numref:`change_of_variable_table`).

.. _change_of_variable_table:

~~~~~~~~~~~~~~

Although minimizing the (modified) weighted sum of least squares gives a solution that fits best the data, it does not necessarily correspond to the most probable solution given what we know about celestial objects.

The discrepancy is particularly significant in very faint (|SNR| :math:`\le 20`) and barely resolved galaxies, for which there is, for example, a tendency to overestimate the elongation, known as the "noise bias" in the weak-lensing community :cite:`2004MNRAS_353_529H,2012MNRAS_424_2757M,2012MNRAS_425_1951R,2012MNRAS_427_2711K`.

To mitigate this issue, |SExtractor| implements a simple `Tikhonov regularization <https://en.wikipedia.org/wiki/Tikhonov_regularization>`_ scheme on some engine parameters, in the form of an additional penalty term in :eq:`loss_func`.

This term acts as a Gaussian prior on the selected *engine* parameters. However for the associated *model* parameters, the change of variable can make the (improper) prior far from Gaussian.

The discrepancy is particularly significant in very faint (|SNR| :math:`\le 20`) and barely resolved galaxies, for which there is a tendency to overestimate the elongation, known as the "noise bias" in the weak-lensing community :cite:`2004MNRAS_353_529H,2012MNRAS_424_2757M,2012MNRAS_425_1951R,2012MNRAS_427_2711K`.

To mitigate this issue, |SExtractor| implements a simple `Tikhonov regularization <https://en.wikipedia.org/wiki/Tikhonov_regularization>`_ scheme on selected engine parameters, in the form of an additional penalty term in :eq:`loss_func`.

This term acts as a Gaussian prior on the selected *engine* parameters. However for the associated *model* parameters, the change of variables can make the (improper) prior far from Gaussian.

Currently the only regularized parameter is :param:`SPHEROID_ASPECT_IMAGE` (and its derivatives :param:`SPHEROID_ASPECT_WORLD`, :param:`ELLIP1MODEL_IMAGE`, etc.), for which :math:`\mu_{SPHEROID\_ASPECT} = 0` and :math:`s_{SPHEROID\_ASPECT} = 1`

(:numref:`fig_aspectprior`).

.. _fig_aspectprior:

.. figure:: figures/aspectprior.*

   :figwidth: 100%

   :align: center

   Effect of the Gaussian prior on the :param:`SPHEROID_ASPECT_IMAGE` model parameter. *Left:* change of variables between the model (in abscissa) and the engine (in ordinate) parameters. *Right*: equivalent (improper) prior applied to :param:`SPHEROID_ASPECT_IMAGE` for :math:`\mu_{SPHEROID\_ASPECT} = 0` and :math:`s_{SPHEROID\_ASPECT} = 1` in equation :eq:`loss_func`.

.. _model_minimization_def: