# Two Point (Dasch)¶

## Introduction¶

The “Dasch two-point” deconvolution algorithm is one of several described in the Dasch paper . See also the three_point and onion_peeling descriptions.

## How it works¶

The Abel integral is broken into intervals between the $$r_j$$ points, and $$P^\prime(r)$$ is assumed constant between $$r_j$$ and $$r_{j+1}$$.

## When to use it¶

This method is simple and computationally very efficient. The method incorporates no smoothing.

## How to use it¶

To complete the inverse transform of a full image with the two_point method, simply use the abel.Transform class:

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


If you would like to access the two_point algorithm directly (to transform a right-side half-image), you can use abel.dasch.two_point_transform().

## Example¶

# -*- coding: utf-8 -*-
from __future__ import division
from __future__ import print_function
from __future__ import unicode_literals

"""example_dasch_methods.py.
"""

import numpy as np
import abel
import matplotlib.pyplot as plt

# Dribinski sample image size 501x501
n = 501
IM = abel.tools.analytical.SampleImage(n).image

# speed distribution of original image
orig_speed = abel.tools.vmi.angular_integration_3D(origQ, origin=(-1, 0))
scale_factor = orig_speed.max()

plt.plot(orig_speed, orig_speed/scale_factor, linestyle='dashed',
label="Dribinski sample")

# forward Abel projection
fIM = abel.Transform(IM, direction="forward", method="hansenlaw").transform

# split projected image into quadrants

dasch_transform = {\
"two_point": abel.dasch.two_point_transform,
"three_point": abel.dasch.three_point_transform,
"onion_peeling": abel.dasch.onion_peeling_transform}

for method in dasch_transform.keys():
Q0 = Q.copy()
# method inverse Abel transform
AQ0 = dasch_transform[method](Q0, basis_dir='bases')
# speed distribution
speed = abel.tools.vmi.angular_integration_3D(AQ0, origin=(-1, 0))

plt.plot(speed, speed*orig_speed/speed/scale_factor,
label=method)

plt.title("Dasch methods for Dribinski sample image $n={:d}$".format(n))
plt.axis(xmax=250, ymin=-0.1)
plt.legend(loc=0, frameon=False, labelspacing=0.1, fontsize='small')
plt.savefig("plot_example_dasch_methods.png",dpi=100)
plt.show() For more information on the PyAbel implementation of the two_point algorithm, please see PR #155.

## Citation¶

  C. J. Dasch, “One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods”, Appl. Opt. 31, 1146–1152 (1992).