onion_peeling.rst

Onion Peeling (Dasch)

Introduction

The "Dasch onion peeling" deconvolution algorithm is one of several described in the Dasch [1] paper. See also the two_point and three_point descriptions.

How it works

In the onion-peeling method the projection is approximated by rings of constant property between r_j − Δr/2 and r_j + Δr/2 for each data point r_j.

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

where

W_ij = 0  (j < i)

$$\sqrt{(2j+1)^2 - 4i^2} \, \, (j=i)$$

$$\sqrt{(2j+1)^2 - 4i^2} - \sqrt{(2j-1)^2 - 4i^2} \, \, (j &gt; i)$$

The onion-peeling deconvolution function is: 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 abel.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 abel.dasch.onion_peeling_transform.

Example

../examples/example_dasch_methods.py

or more information on the PyAbel implementation of the onion_peeling algorithm, please see Pull Request #155.

Citation

[1] Dasch, Applied Optics, Vol 31, No 8, March 1992, Pg 1146-1152.