Onion Peeling (Dasch)
=====================


Introduction
------------

The "Dasch onion peeling" deconvolution algorithm is one of several described
in the Dasch paper [1]_. See also the :doc:`“two-point” <two_point>` and
:doc:`“three-point” <three_point>` descriptions.

How it works
------------

In the onion-peeling method the projection is approximated by rings
of constant property between
:math:`r_j - \Delta r/2` and :math:`r_j + \Delta r/2` for each data
point :math:`r_j`.

The projection data is given by
:math:`P(r_i) = \Delta r \sum_{j=i}^\infty W_{ij} F(r_j)`, where

.. math::
    W_{ij} =
    \begin{cases}
        0, & j < i, \\
        \sqrt{(2j + 1)^2 - 4i^2}, & j = i, \\
        \sqrt{(2j + 1)^2 - 4i^2} - \sqrt{(2j - 1)^2 - 4i^2}, & j > i.
    \end{cases}


The onion-peeling deconvolution function is :math:`D_{ij} = (W^{-1})_{ij}`.


When to use it
--------------

This method is simple and computationally very efficient. The article
states that it has less smoothing that other methods (discussed in Dasch).


How to use it
-------------

To complete the inverse transform of a full image with the ``onion_dasch`` method, simply use the :class:`abel.Transform <abel.transform.Transform>` class::

    abel.Transform(myImage, method='onion_peeling').transform

If you would like to access the ``onion_peeling`` algorithm directly (to transform a right-side half-image), you can use :func:`abel.dasch.onion_peeling_transform`.


Example
-------

.. plot:: ../examples/example_dasch_methods.py
    :include-source:

For more information on the PyAbel implementation of the ``onion_peeling`` algorithm, please see `PR #155 <https://github.com/PyAbel/PyAbel/pull/155>`_.


Citation
--------

.. |ref1| replace:: \ C. J. Dasch, "One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods", `Appl. Opt. 31, 1146–1152 (1992) <https://doi.org/10.1364/AO.31.001146>`__.

.. [1] |ref1|

.. only:: latex

    * |ref1|