# Example: Hansen–Law¶

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

import numpy as np
import abel
import matplotlib.pylab as plt
import bz2

# Hansen and Law inverse Abel transform of velocity-map imaged electrons
# from O2- photodetachement at 454 nm. The spectrum was recorded in 2010
# at the Australian National University (ANU)
# J. Chem. Phys. 133, 174311 (2010) DOI: 10.1063/1.3493349

# load image as a numpy array
imagefile = bz2.BZ2File('data/O2-ANU1024.txt.bz2')

rows, cols = IM.shape    # image size

# center image returning odd size
IMc = abel.tools.center.center_image(IM, method='com')

# dr=0.5 may help reduce pixel grid coarseness
# NB remember to also pass as an option to angular_integration
AIM = abel.Transform(IMc, method='hansenlaw',
symmetry_axis=None,
transform_options=dict(dr=0.5, align_grid=False),
angular_integration=True,
angular_integration_options=dict(dr=0.5),
verbose=True)

# convert to photoelectron spectrum vs binding energy
# conversion factors depend on measurement parameters
eBE, PES = abel.tools.vmi.toPES(*AIM.angular_integration,
energy_cal_factor=1.204e-5,
photon_energy=1.0e7/454.5, Vrep=-2200,
zoom=IM.shape[-1]/2048)

# Set up some axes
fig = plt.figure(figsize=(15, 4.5))
ax1 = plt.subplot2grid((1, 3), (0, 0))
ax2 = plt.subplot2grid((1, 3), (0, 1))
ax3 = plt.subplot2grid((1, 3), (0, 2))

# raw image
im1 = ax1.imshow(IM, extent=[-512, 512, -512, 512])
ax1.set_xlabel('x (pixels)')
ax1.set_ylabel('y (pixels)')
ax1.set_title('velocity map image: size {:d}x{:d}'.format(rows, cols))

# 2D transform
c2 = cols//2   # half-image width
im2 = ax2.imshow(AIM.transform, vmin=0,
vmax=AIM.transform[:c2-50, :c2-50].max(),
extent=[-512, 512, -512, 512])
ax2.set_xlabel('x (pixels)')
ax2.set_ylabel('y (pixels)')
ax2.set_title('Hansen Law inverse Abel')

# 1D speed distribution
#ax3.axis(xmax=500, ymin=-0.05, ymax=1.1)
#ax3.set_xlabel('speed (pixel)')
#ax3.set_ylabel('intensity')
#ax3.set_title('speed distribution')

# PES
ax3.plot(eBE, PES/PES[eBE < 5000].max())
ax3.axis(xmin=0)
ax3.set_xlabel(r'elecron binding energy (cm\$^{-1}\$)')
ax3.set_ylabel('intensity')
ax3.set_title(r'O\${_2}{^-}\$ 454 nm photoelectron spectrum')

# Prettify the plot a little bit:
plt.tight_layout()

# save copy of the plot
# plt.savefig('plot_example_hansenlaw.png', dpi=100)

plt.show()
```